___,_.,--_..,_..‘ :4 PLATOON DISPERSION DYNAMICS ONARTERIAL isI‘IIIg-Erfs‘i ;» g ' : .f " .. . WITHENTERINGVEHICLES; . V ”.._ ; Dissertation for thei‘begree of Ph - : MICHIGAN STATE UNIVERSH'YV- .- 1 ~ , -. 5 , . ~ 9-, KINKEUIIIIIAI} ' : + .1977 I 1 . ‘ . ,. 44' r ., N. 4-- . Ia: ‘1 4: ‘ _ . "I‘FP‘Iu ._ I V ’ ' I . 4 ..,, z . .. C ‘ n .J‘,’ ..,.,_,, I- l r)- .. I H)”: 'V‘" -, . r, 3.]. 1 ; -.. rhvlv.» 3", fl. .,,,, q r" r»..,' ',"_ " ""‘f‘ w .4, w, "4 ”vv .. ., g.,—..,..'I,r Date This is to certify that the thesis entitled PLATOON DISPERSION DYNAMICS ON ARTERIAL STREETS WITH ENTERING VEHICLES presented by KIN KEUNG LAI has been accepted towards fulfillment of the requirements for Ph-D- degree in fimiLEngineering Wm cf, 72/% Major professé May 27, 1977 0-7639 LG~.R.I" .’: I‘m-L A— AA.“ J; .1; LIL; MiClli? ”3321‘: L. .-..:,.' Uni Jersi I. .ro”."""\""f" :' 'w—K-‘d. ("7 “- «9' %W\ 45m ABSTRACT PLATOON DISPERSION DYNAMICS 0N ARTERIAL STREETS WITH ENTERING VEHICLES By Kin K. Lai The literature related to platoon dispersion was reviewed. A preliminary study of vehicular time headway within a platoon on arterial streets was conducted, and a stochastic model for the headway was de- veloped, which was found to be a gamma distribution. The results of the field study indicated no significant dif- ference between the dispersion of platoons under light and medium flow mainstream traffic. The dispersion of a platoon leaving from a signal and approach- ing a signal with or without entering vehicles was modeled as poly— nomial regressions with a degree of two. The model of dispersion of a passing platoon with entering vehicles was analyzed and related to the model of a dispersion platoon leaving an intersection with a signal. Analytical models of the measures of effectiveness (that is, average merging delay, average delay before merging, and capacity) for entering vehicles under the influence of platoon dispersion were Kin K. Lai developed. The best location of the entrance point, the distance for recovering the shape of the headway distribution after the entrance of sidestreet vehicles, and the optimal division of the in- lput sources were determined by use of the models. Simulation models were developed, and the validity of the anal- ytical models were examined by using the simulation results. PLATOON DISPERSION DYNAMICS ON ARTERIAL STREETS WITH ENTERING VEHICLES By Kin K. Lai A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY -Department of Civil and Sanitary Engineering 1977 6107mm ‘2 To My MOther and Father ii ACKNOWLEDGMENTS The author wishes to express his sincere appreciation to his major professor, Dr. William Taylor, for his valued guidance, assis- tance, and encouragement during the period of research and graduate study. As their friendship and his respect for Dr. Taylor have been satisfactorily developed, he sincerely cherishes as part of his accomplishment in this school. He also wishes to thank Dr. John Kreer, Dr. Francis McKelvey, and Dr. Connie Shapiro, who, too, served on his guidance committee, for their helpful suggestions and interest in the research project. Finally, the author especially wishes to express his deepest and heartfelt thanks to his wife, Alice, for her continual encourage- ment and understanding throughout the years of his graduate study. iii TABLE OF CONTENTS Page List of Tables . . . . . . . . . . . . . . . . . . . . . iv List of Figures. . . . . . . . . . . . . . . . . . . . . vii List of Appendices . . . . . . . . . . . . . . . . . . . xi CHAPTER ONE INTRODUCTION . . . . . . . . . . . . . . . . . . . . 1 1.1 Background 1 1.2 Definition of the problem 2 1.3 The basic approach 2 CHAPTER TWO REVIEW OF LITERATURE . . . . . . . . . . . . . . . . 6 CHAPTER THREE THE DISTRIBUTION OF VEHICULAR HEADWAYS WITHIN PIATOONS O O O O O O O O O O O O O O O O O O O O O I 1 5 3.1 Introduction 15 3.2 The proposed distribution 15 3.3 Comparison with observed data 24 3.4 Goodness of fit test 26 3.5 Significance of results 28 CHAPTER FOUR FIELD STUDIES PROCEDURES . . . . . . . . . . . . . . 30 4.1 Introduction 30 4.2 Determination of the input variables of the model ' 30 4.3 Selection of study sites 33 iv Page 4.4 Sample size requirement 35 4.5 Field data collection 37 4.6 Results of field studies 37 CHAPTER FIVE MATHEMATICAL MODELING OF PLATOON DISPERSION . . . . 39 5.1 Introduction 39 5.2 Data analysis 39 5.3 Development of the mathematical models 43 5.4 Analysis of the mean headway for the passing platoon with entering vehicles 50 CHAPTER SIX SOME ANALYTIC CONSIDERATIONS OF QUEUEING . . . . . 56 6.1 Introduction 56 6.2 Queueing consideration at the merging position 58 6.3 Queueing consideration before merging 67 6.4 Capacity and the best location of the minor stream 70 6.5 The required distance for recovering the shape of the headway distribution 75 6.6 Splitting flow to reduce delay of the entering vehicles 76 CHAPTER SEVEN SIMULATION ANALYSIS OF THE PROBLEM . . . . . . . . . 82 7.1 Introduction 82 7.2 Development of the phase one model 83 7.3 Development of the phase two model 89 7.4 Development of the phase three model 7.5 Evaluation of the results CHAPTER EIGHT SUMMARY AND CONCLUSIONS . . . . . . . . . 8.1' Summary of accomplishments 8.2 Applicability of models APPENDICES REFERENCES ' Vi 92 97 102 102 106 107 139 Figure 1.1 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 5.1 5.3 5.4 5.5 LIST OF FIGURES Procedural flow chart The negative exponential probability density function The shifted exponential probability density function The composite exponential cumulative distri- bution function The normal probability density function The lognormal probability density function The gamma probability density function The weibull probability density function Mean headway at different locations in the preliminary study Standard deviation of headway at different locations in the preliminary study Mean headway at different locations of platoon leaving from a signal Mean headway at different locations of moving platoon with entering vehicles Mean headway at different locations of platoon approach a signal Mean headway at different locations of platoon leaving from a signal with the combination of light and medium flow Mean headway at different locations of platoon approach a signal with the combination of light and medium flow. vii Page l7 l8 19 20 21 23 24 26 26 40 40 41 45 46 Figure 6.1 7.1 7.2 7.3 7.4 7.5 Translation of INTC Physical characteristics of phase one model Flow diagram for phase one model Physical characteristics of phase two model Flow diagram of phase two model Flow diagram of phase three model viii Page 77 84 86 89 92 95 Table 3.1 3.2 4.1 4.2 4.3 5.1 5.2 5.3 5.4 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 7.1 7.2 LIST OF TABLES Degree of freedom for different distributions Results of the goodness of fit test Characteristics of the study sites Standard deviation of headway at different locations in preliminary study Sample size requirement at different locations Results of the multiple comparisons of means for case 1 Results of the multiple comparisons of means for case 2 2 Equation results for m Equation results for m 3 Percentage of the passing time at different locations Calculate u; for different T Calculate otz for different T Average merging delay E(Z) Variance of merging delay Var (Z) Calculate the expected queue size N Calculate E(W)=N/A Calculate capacity for different T Calculate 6(3for different 9 and A Calculate 6(1/3,l/3,l/3) for different Simulation results of phase one model Analytical results of phase one model ix Page 28 29 34 36 36 44 45 51 51 58 65 65 66 67 71 72 74 80 81 85 87 Table 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 Greenshield's starting delay Simulation results Simulation results Simulation results Analytical results Analytical results Analytical results of phase two model of capacity of phase three model of phase two model of capacity of phase three model U-statistic for the comparison of the analytical and the simulation results Page 90 94 95 97 99 100 101 101a Appendix Appendix I Table I-l. Table I-2. Table I-3. Table 1-4 0 Appendix II Table Table Fig Table Table Table Table Table Table 11.1 o II-2. II-l. II-3. II-4. II-5. II-6 a 11-7 0 II-8. LIST OF APPENDICES The frequency of the headway sample within platoon leaving from a signal. The frequency of the headway sample within platoons with entering vehicles. The frequency of the headway sample within platoons approaching a signal. Mean and standard deviation of headway within platoon for all cases. on Incomplete gamma function F=flff(t)dt. co Polynomial equations of ftf(x) = a0 + 2 3 4 alt + a2t + a3t + aat oo Plotting of f£f(x)dx Formulation of fo(t) = f: f(x)dx /f:tf(t)dt Integrals of the forward recurrence time function F. T. Calculate K t) = f tf(t)dt O _ 'I Calculate M O(t) - fotfO (t)dt Calculate Ntt) = fgt2 f(t)dt Calculate No (t) = I: tzfo (t)dt xi Page 107 108 109 110 111 112 113 114 115 116 117 118 119 Appendix III (A) Proof ofut:1 (B) Proof of E0 6-11. (C) Proof of EQ 6-15. Appendix IV (A) Simulation program for Phase One model (B) Simulation program for Phase Two model (C) Simulation program for Phase Three model xii 120 123 125 127 132 CHAPTER ONE INTRODUCTION Background: When the smooth flow of traffic is interrupted by some kind of traffic control device, such as signals, the downstream flow of traf- fic will usually take the form of a "platoon." As each platoon moves down the street it disperses, and its time length increases as the time-headways between successive vehicles increase. As a result, the whole platoon length increases when the platoon moves down from the signal. If there are entering sources downstream of the signal, the entering vehicles have two possibilities for entering traffic: while the platoon is passing the entrance point or after it has passed. In the first case, the entrance point.wou1d be expected to be a con- siderable distance from the signal so that there would be a large gap within the platoon and entering vehicles. In the second case, the entrance would be expected to be closer to the signal, platoon length would be shorter, and it would pass through earlier. Information a- bout the behavior of vehicular platoons and how it affects entering vehicles is important to reduce delay time for entering vehicles. The purpose of this dissertation is to investigate the dynamics of platoon dispersion. Specifically, to find a model for the disper- sion of platoons and to use that model to investigate the effect on entering vehicles. Definition of the problem: The four objectives of this dissertation are as follows: (1) Develop a stochastic model for describing the dispersion of platoons approaching or leaving a signalized intersection, with or without entering vehicles. (2) Determine the delay time of entering vehicles before and upon merging. (3) Investigate the relationship between platoon dispersion and the capacity to accommodate vehicles entering downstream from the issuing intersection. (4) Determine an optimal system strategy for a traffic stream with variable vehicle entry points. The basic approach: Before discussing the approach of this dissertation, some def- initions must be reviewed. Headway is the time between succesive vehicles; theoretically it may range from zero seconds to infinity. In this dissertation, vehicular time headway was selected as the parameter for the platoon dispersion investigation. This parameter was chosen because it describes the interaction between vehicles in the car-following process, and it also describes the interaction in the merging process. Platoons include those vehicles queued at the intersection and those which join the queue after the light turns green, but within two seconds after the last queued vehicles arrival at the stop line. Since the behavior of the entering vehicles is mainly affected by the headways of the mainstream vehicles, vehicular headways will be used as a basis for the investigation of platoon dispersion. The research effort is divided into six principal phases. Phase 1 consists of a review of the literature describing platoon be- havior. This review illustrates the differences between approach used here and those used in previous studies and also suggests av- enues of future research. Phase 2 is the development of a stochastic model of vehicle time headways within a platoon released from a signalized intersec- tion. Using the results of Nemeth and Vecellio's report (23), the principal variables affecting platoon movement through linear sig- nal systems are identified as signal spacing, signal offset, and platoon size. In order to study the variation of headway as the vehicles move down the signalized intersection, the next signal's coordination with the prior one is also considered. The next step is to study the relationship between headway and the distance from the signal, given various platoon sizes. In Phase 3, data are collected under various conditions, that is, for different distances from the signal, different platoon sizes, and the presence or absence of entering vehicles. In Phase 4, the data are analyzed and a regression model is de- veloped to describe the relationShip between the parameters of the headway model and the distance from the signal and platoon size. Phase 5 studies platoon dispersion in connection with vehicles entering downstream from the signal. The effects of vehicles merging into the mainstream from any side street will be analyzed. The delay time of entering vehicles on the merging position and before merging is determined, as are the optimal locations of the merging points. The relationships derived from Phase 4 are tested for the platoon af- ter some vehicles merge into it. The distance required for the head- way distribution to recover its original shape will be determined, and the possibility of splitting the entering flow into two or more entry points will also be investigated. A computer simulation model is developed in Phase 6 to vali- date the analytical models. The analytical and simulation solutions of the same situation are compared. A flow chart illustrating the analytical procedures of the dissertation is shown in Figure 1-1. Figure 1—1. Procedural Flow Chart Definition of Problem Review of Literature Preliminary Study of Vehicular Time Headway Development of Stochastic Model of Vehicular Time Headway Field Studies of Platoon Behavior Development of Platoon Dispersion Models l l L _1 Analytical Studies Simulation Studies of Platoon Dispersion of Platoon Dispersion with Entering Vehicles with Entering Vehicles J 7 Comparison of Results of the Analytical and Simulation Studies CHAPTER TWO REVIEW OF THE LITERATURE Numerous studies have been conducted on the behavior of traffic leaving a signalized intersection. Experimental work on the dis— persion of traffic platoons has been reported by Lewis (19) (1958) and Graham and Chenu (10) (1962). .Pacey (26) (1956) and Robertson (27) (1969) have proposed theoretical models describing this process, and Nemeth and Vecellio (23) (1973) have developed a simulation model for the dispersion of platoons. Grace and Potts (9) (1964) and Herman, Potts, and Rothery (15) (1964) have attempted to combine the theoretical and experimental aspects of platoon behavior. A review of these studies follows. One of the early experimental studies was conducted by Lewis (19) in 1958. Platoon movement of traffic from an isolated signalized intersection was examined. The test site was a four-lane highway in Richmond, California. Observations of space-time data were made at five different locations downstream from a signalized intersection at distances up to 0.65 miles. Arrival time frequency distributions of the platoon at each of these five points were obtained. Analysis of this frequency distribution showed that three characteristics of pla- toon movement were linearly related to the distance downstream from the traffic signal. Those characteristics were (1) the maximum or- dinates of the equivalent normal distribution of the arrival times of 6 the N£h_vehicle, (2) the mean arrival time of the N£h_vehic1e, and (3) the time for the Pth_percentile of vehicles in a platoon to pass a point. Lewis concluded that a progression diagram for distances up to the study limits could be plotted for all vehicles, which would allow greater success in timing the downstream signals than would occur using random selection. Another experimental study was performed by Graham and Chenu (10) in 1962. Vehicles arrival time data also were collected at five locations downstream from a signalized intersection, at a distance of 1/35 mile, l/4 mile, 1/2 mile, 3/4 mile, and one mile. The arrival time of each vehicle at each location was recorded. A graphic illus— tration of the dispersion of the platoon at various distances from the signal was provided by histograms plotting the frequency distri- bution of vehicle arrivals within each 5 percent increment of time — length of the platoon. The author concluded that at distances as great as one mile downstream from the signal, the vehicles still were "bunched," since 77 percent of the vehicles remained in the platoon at that location. Pacey, (26) in an unpublished 1956 report, proposed a kinematic model to describe the dispersion of traffic platoons. He assumed that vehicle speeds in a platoon are normally distributed, so that the spread of the platoon can be accounted for and measured by the dispersion of vehicle speeds. Another assumption was that the speed of any individual vehicle remains constant as it moves down the road, that is, no interaction is assumed between vehicles. Based on these assumptions, Pacey formulated a distribution of travel time applicable to traffic departing from a saturated signalized intersection. The derived travel time distribution was shown to be a function of the normality of vehicle speeds, the mean and variance of the speed dis- tribution, and the distance over which travel times are distributed. The travel time distribution is as follows: 8(T) = { —(l/t —-§/a)2/252} (2.1) where T = travel time (in seconds); speed (in feet/sec.); 4 II < H mean of the speed distribution; 8 = o/a ; Q ll standard deviation of the speed distribution; and a = distance over which the travel times are distributed (in feet). Pacey's model was compared with observations made at two sites on dual carriageways of the times after the green signal at which vehicles passed two points, one near the signals and one 600 yards beyond them. Especially for moderate traffic volumes, Pacey's model proved a good fit to the actual data. A detailed theoretical study of Pacey's model was made by Grace and Potts (19) in 1964. They analyzed in detail the dispersion of a platoon as it moves down a street and emphasized the importance of allowing for this phenomenon to obtain efficient coordination of two traffic lights. A traffic density function was derived by the use of Pacey's model: _1 .. _ _2 _ K(X,T) — {m g { m(x) Z‘ZV—f } exp ( z ) dz.. (2 2) (17— This is a well-known form of the solution of the one-dimensional diffusion equation, 2 %§= “2(3) (2.3) where g(x) = k(x,o) is a known function; k(x,t) = the traffic density, equal to the number of vehicles per unit road length at a given point x and at time t; X = the distance down a highway; m = average vehicle speed; x/m = offset time (in seconds); x = x/m - t preset time equal to the time added to the beginning of the green phase of the second signal to allow for the spreading of a platoon leaving a prior signal; T = % t2; and <1= diffusion constant. If vehicle speeds are normally distributed with mean u and variance 02, then the diffusion constant is defined as u/O. Grace and Potts applied their theoretical model to the design of progressive signal systems. For certain assumed initial conditions regarding the traffic density function, the diffusion equation was used to provide analytical solutions to the coordination of two suc- cessive signalized intersections, allowing for the dispersion of the platoon. In 1964, Herman, Potts, and Rothery (15) conducted an experiment designed to test the kinematic model of traffic platoon behavior, in 10 particular, to test the detailed theoretical results obtained by Grace and Potts. Speed and arrival time of vehicles leaving an iso- lated intersection were measured at two locations, 757 feet and 2142 feet downstream from the signal. The results confirmed that the kinematic model accurately described the dispersion of platoons in medium volume traffic without interference. This was especially true for the lead vehicle in the platoon, while the behavior of the last vehicle varied from one platoon to another. In 1969, Robertson (27) of Road Research Laboratory, England, deveIOped a method to determine optimum fixed time traffic signal settings for a network of signalized intersections. In his "TRANSYT" model, a method was developed for predicting the dispersal of an average platoOn of traffic. Observations were made at four locations in west London, where traffic leaving a signal travelled at least 1,000 feet before reaching another signal or major junction. Each location had four observation sites, one just beyond the traffic sig- nal and the others at approximately 300, 600, and 1,000 feet down- stream. The time of passage of every vehicle was recorded at each point. By further analysis of these observations, a recurrence re- lationship was established to predict flow at each site downstream from a traffic signal given the input flow, previously predicted flow, and a smoothing factor. The recurrence equation is as follows: q’(i + t) = F.q(l) + (l-F). q’(i+t—l), (2.4) where q(i) = the flow in the igh time interval of the initial platoon (in veh/hr.); 11 q’(j) = the flow in the j£h_time interval of the predicted platoon (in veh/hr.); t = 0.8 times the average journey time over the distance for which the platoon dispersal is being calculated; and F = a smoothing factor. The smoothing factor, F, required for the best fit between the actual and calculated platoon shapes was found to be related to the journey time by the expression F = 1 (2.5) l + 0.5F The author points out that it would be reasonable to expect that the smoothing factor required should also be a function of "site factors" such as road width, gradient, parking, and so forth, but these aspects have not yet been investigated. The dispersion of a platoon of vehicles released from a signal- ized intersection has also been studied by Nemeth and Vecellio (1973), who developed a simulation model. Their purpose was to incorporate those variables affecting platoon movement into a model and to simu- late the behavior of a group of vehicles as it passes through a series of signalized intersections. Nine parameters were selected for inclusion in the variation analysis of platoon behavior: (1) traffic density; (2) .traffic volume; (3) mean velocity; 12 (4) standard deviation of velocity; (5) coefficient of variation of velocity; (6) mean spacing; (7) standard deviation of spacing; (8) coefficient of variation of spacing; and (9) mean time headway. The time and distance variation of the nine parameters for the observed platoons were plotted. From this variation analysis, dis— tance patterns were interpreted by the authors in relation to the spacing between signalized intersections, and time patterns were interpreted with respect to the values of the signal offsets. The authors concluded that these two variable, signal spacing and signal offset, affect platoon behavior. Two other variables were also investigated to determine their influence on platoon behavior: platoon size and lane of travel. The F - test was used to test the mean velocity difference in different lanes, and velocity - time patterns for platoons of various sizes were plotted. The authors concluded that lane of travel has no significant effect on platoon behavior, but platoon size does. Thus, the principal variables affecting platoon movement through linear signal systems were identified by Nemeth and Vecellio as sig- nal spacing, signal offset, and platoon size. In their simulation model, the processing of traffic between intersections was achieved by relating traffic dispersion to site - related travel time parameters. The variables mentioned above were incorporated into a travel time distribution function. Using the data obtained at the study sites, 13 regression equations were derived for the travel time distribution parameters. All the platoon studies reviewed thus far deal with platoons released from an isolated signalized intersection and traveling with no interference. In the experimental studies, such as that by Lewis and Chenu, no attempt was made to develop a model to describe the dispersion process. The kinematic model developed by Pacey, investigated by Grace and Potts and validated through experiments carried out by Herman and others, is useful for more general applications. By collecting speed distribution data at actual locations, a value of the diffusion constant czcan be computed and applied to the prediction of the spreading of platoons. Speed data obtained at different traffic volumes gave different a for low, medium, and high traffic flow. Since the kinematic model assumes that all vehicles travel at con- stant speeds normally distributed about a mean speed, many passing maneuvers and lane changes obviously have to take place in the pla- toon. As the volume increases, these maneuvers become more and more restricted, causing actual platoon behavior to deviate from the theory. Besides, the assumption of constant Speed is not realistic near traf— fic signals. Herman stated that the best fit of actual data for the model is the front of the platoon at medium traffic volume. The purpose of Nemeth's and Vecellio's simulation model was to generate mean delay and queue statistics at signalized intersections rather than attempt to predict the precise behavior of traffic be- 'tween intersections. 14 The model developed by Robertson was a deterministic one, which only predicts average platoon behavior. Thus, flexibility was restricted. From the above literature, it was found that several factors in- fluence platoon dispersion, among them, distance from the signal, signal offset, and platoon size. As the platoon moves down from the signal, it will disperse, in other words, the gaps between vehicles will increase as the position of vehicles from the signal increases. The approach taken in this dissertation differs in three aspects from the previous studies of platoon behavior. First, a stochastic model is developed to describe the dispersion of a platoon departing from and approaching a signalized intersection. Second, the case of allowing vehicles to enter the platoon downstream of the signal will be investigated. Third, the effect of platoon dispersion on the entering vehicles will be analyzed. CHAPTER THREE THE DISTRIBUTION OF VEHICULAR HEADWAYS WITHIN PLATOONS This chapter seeks to develop an accurate and reliable model for describing the distribution of vehicular headways within pla- toons released from a signalized intersection. The distribution of headways can be found by observing the times of arrival of successive vehicles in a given lane at a point along the lane. The headway of the i£h_vehicle is defined as: Hi = ti_1 -ti , (3.1) where ti-l is the arrival time of the vehicle ahead, and ti is the arrival time of the 1gp vehicle. If many vehicles are observed, the distribution of Hi can be obtained, and one can investigate whether this variable can be described by some theoretical distribution function. Vehicle headways are usually measured in seconds and theoreti- cally may range from zero to infinity. A zero headway means a col- lision on the roadway, and an infinite headway represents an empty roadway. Thus, the theoretical distribution of headways must be con— tinuous and positive. There are several distributions with these characteristics, and these will be discussed and compared below. Thepproposed distribution: It was first suggested by Adams (1) that the vehicles in light traffic passing a point at equal time intervals follow a Poisson dis- tribution. If this is the case, the distribution of headways can be 15 16 shown to be described by the negative exponential distribution. The Poisson distribution gives the probability, or proportion of a number of equal time intervals, during which any number of vehicles, n, will arrive at a point: - t n e ‘1 (qt) , i n’ n=0a132939.o. = 0 otherwise, P(x=n) (3.2) where qt is the mean number of vehicles arriving during the time intervals of t seconds. We are interested only in the probability of a headway greater than t seconds. This would mean that no arrivals occured in t seconds: P( H > t) = P (x=0). (3.3) Here, H is the headway. Thus, [I P (Hit ) 1— P(x=0) This is the cumulative distribution function of the vehicular headway. Therefore, the probability density function of headway is: f(t) = aggpm _<_ t) _ :1. _ -qt — dt(1 e ) = qe-qt o_<_tt+ulH>u)=P(H>t). t.u>0- (3.6) This equality shows the property of no memory. It implies that a knowledge of any headway gives no information at all about the length of the next headway. It is completely random. Figure 3—1. The Negative Exponential Probabilipy Density Function. f(t) q4 ¢D I. 2 3 4 III 6* 7 8 I? ' In Figure 3-1, it may be noticed that as the headways approach zero, the probability becomes larger. Applying this to our situation would assign higher probabilities to very low headways, which is not satisfactory. In the real situation, we know that the probability of a zero headway is, of necessity, zero. To take account of the fact that the negative exponential dis— tribution exhibits too high a probability in the zone near zero head- way, Newell (25) suggested that the headway distribution might con— form to the negative exponential distribution if shifted to the right a distance of T to allow for the existence of a certain fixed minimum 18 headway. Used by many researchers, the shifted negative exponential distribution is of the form — t... Hw=qe“ 9 t>. = 0’ otherwise. (3-7) A diagram of the distribution is shown in Figure 3-2. Figure 3-2. The Shifted Exponential Probability Density Function. f(t) +T+ 1 2 3 4 5 6 7 8 9 10 Another approach is to consider a composite distribution(ll) of headways in which some vehicles are free-flowing and others are restrained in platoons behind another vehicles. If the proportion which is free-flowing is (1—p), then the composite cumulative dis- tribution function is the following: P(H>t)=(1-p)e-t/fil + ‘(tra)/(fi2“a) (3.8) where bl = mean headway of free-flowing vehicles; h2 = mean headway of constrained vehicles; and a. = minimum headway of constrained vehicles. The diagram for the cumulative distribution (with p= 0.5) is shown in Figure 3-3. This composite distribution has also been used 19 Figure 3-3. The Composite Exponential Cumulative Distribution Function. F(t) 1m CompOSIte Constrain r Free Flow r I 1T r— I t T- 10 20 30 40 in recent studies of traffic flow on two-lane urban streets (17). A distribution which is most commonly used in phenomenal anal- ysis is the normal distribution. It has several characteristics which recommends it to traffic engineers: (1) It is symmetrical, (2) it assigns a finite probability to every finite deviation, and (3) the mode and median are equal to mean. The probability density function is given by 8XP{-(t-u)2/202}, (3.9) 0 2U —m:t:m f(t)= The distribution is completely specified by the location parameter 11 and the scale parameter 0'. A typical example of a normal prob- ability density function is given in Figure 3-4. 20 Figure 3—4. The Normal Probability DenSity Function. f(t) 1 23456 7‘89101112 A distribution of particular interest which has been found to fit headway data quite well is the Log-normal distribution. Consider now the family of p.d.f's:‘ 2 f(t) = ——l———-exp{— ilE%El—d t>0 tO/EU 20 _ (3.10) = 0, otherwise. The corresponding cumulative distribution function is t 2 F(t) = 319332—1 g;- Io exp{ - 2 , O/E; 20 (3.11) which, on writing {1n(Pv)}//O = u, becomes 1n t + 1n P G( o )’ (3.12) where (3.13) ‘21 is the Standardized normal probability integral. The results show that the random variable ln X is normally distributed, hence the name log-normal distribution given to EQ (3.11). The mean of the distribution is 1"1 exp (a«:2), and the variance is P-zeXP(02) (exp( 02)-l). For small values of 02 the distribution is nearly normal. For large values of 02, the distribution has large positive skewness. A typical example of log-normal probability density func- tion is shown in Figure 3-5. f(t) Figure 3-5. The Logenormal Probability Density Function. 0.8 0.6 0.4 0.2 T, 1 2 3 4 Tolle (26) proposed a more generalized version of a log-normal distribution by shifting the frequency curve by T. The addition of a new parameter requires no new theory since in this case T'will be predetermined, and the probability density function is t>T>O 2 f(t) = 1 exp {lE£$£%£l—9 20 (poo/2? (3.14) -= 0, otherwise. 22 A distribution which has‘the correct general form of the nega— tive exponential distribution but does not have the disadvantage of a sharp cutoff is the Erlang distribution: 1 ta-lexp(-t/b) t>0 f(t) ---—-- _ (a-1)-zba (3.15) = 0, otherwise. where a is a positive integer. For a = 1, this is a negative ex- ponential distribution. As a is increased, the density distribution becomes more peaked, indicating more regularity. As a approaches infinity, f(t) becomes constant. That is, vehicles arrive at equal. time intervals apart. Thus, a can be used as a measure of congestion. A slightly more general distribution is obtained if a is per- mitted to be a non-integer to form the well-known Pearson type III distribution, or the gamma distribution. Its probability density func— tion is 1 a f(t) t ”lexp(-t/b) tzo r(a)ba (3.16) = 0, otherwise, where P (a) is called the gamma function and is defined by the formu- a-l -Z e la P(a) = £0 2 dz. When 0<=a‘<1, the probability density function has an infinite ordi— nate at the origin: when a>1, the p.d.f. is zero at the origin and has a single maximum at b0 (a-l). A typical example of a gamma prob- ability density function is shown in Figure 3-6. If a random vari- able, X, is described by a gamma distribution, then the mean is ab, and the variance is abz. 23 The final distribution to be discussed is the Weibull distri- bution. This generates a family of cures as its parameters are given different values. When a random variable, X, has a Weibull distribution, its probability density function is: Figure 3-6. The Gamma Probabilipy Density Function. f(t) =%,b-l a=2,b=1 ctcm1 c f(t)= exp {-t/b } tzo b (3.17) = 0, otherwise. The distribution depends on the two parameters b and c. The expected value and variance of the Weibull distribution are given by E(H)= b'l/C -2/c 1 To; +1), VAR(H)= b {F(2/c + l) - (III/C + 1) )2} (3-18) A typical example of the probability density function of a Weibull distribution is shown in Figure 3-7. The weibull variate H with shape parameter c=l is the exponen- tial variate. It means that this distribution includes the case of random and nonrandom headway phenomena. 24 Figure 3—7. The weibull Probability Density Function. f(t) c=1,b=l Comparison with observed data: All the functions discussed previously have the characteristics of being the distribution of vehicular headways within platoons. This section describes the preliminary study of headway that was con- ducted by this author. In the study, field data were collected and compared with different theoretical distributions. The results were to provide the variance and approximate chang- ing pattern of headway along the road downstream from the signal, which information could be used to design the field study of a head- way dispersion model. This information was necessary to determine the sample size with acceptable accuracy and the best location for the field study of the platoon dispersion model. Seven of the eight distributions described above were compared with observed data: (1) negative exponential distribution; (2) shifted negative exponential distribution; (3) normal distribution; 25 (4) shifted log-normal distribution; (5) gamma distribution; (6) Erlang distribution; and (7) Weibull distribution. The composite negative exponential distribution was not chosen because in the case of platoon dispersion, the percentage of free-flow vehicles is almost zero. Headway data were collected in the center lane of a three—lane arterial in East Lansing, Michigan. The total length of the block is 1,200 yards. These data were collected at four sites; 150, 300, 700, and 1,000 yeards downstream from a signalized intersection. Obser— vations were made in the morning from 8:00 a.m. to 11:00 a.m., and consisted of about 100 platoons with a total of 400 vehicles. The site was selected because of the long distance between sig- nals and because there were no interruptions from side streets. These factors are important in the formulation of a theoretical model of how a platoon is dispersed. The headway data were collected only from those "within platoon" in the center lane of the road. Here, "within platoon" includes all vehicles upstream of the traffic signal waiting for the signal to turn green and those vehicles which join the queue (within two seconds) prior to the last vehicle reaching the stop line. Turning vehicles and those merging from other lanes are excluded. One hundred data sets were collected at each of the four sites, and goodness of fit tests were performed. The diffusion that took place throughout the section downstream from the intersection is shown in Figure 3-8, and the variance of the headways at different locations is shown in Figure 3—9. 26 Figure 3—8. Mean Headway at Different Locations in the Preliminary Study. .l..\ J U3 .1 \ Mean headway in sec. 5‘"? r 1 u 1— r . . w . distance 100 200 300 400 500 600 700 800 900 1000 (in yds.) r Figure 3-9. Standard Deviation of Headway at Different Locations in the Preliminary Study. §L_ Standard deviation of headway r I r r r r r r r 1’ distance 100 200 300 400 500 600 700 800 900 1000 (in yds.) Goodness of fit test: The degree of agreement between a theoretical probability distri— bution and the distribution of a set of sample observations constitutes a "goodness of fit" test. The assumption that our observations can be adequately described by a given theoretical probability distribution is 27 referred to as the null hypothesis. This test enables us to test this hypothesis for any probability distribution. "Goodness of fit" testing is accomplished by comparing the distribution of the ob— served data with the theoretical probability distribution specified in the null hypotheses. The most common goodness of fit tests in engineering statis- tics are the Chi-square tests (5), and the Kolmogorov-Smirnov test (18). The former is very powerful for large samples on the order of n z 100. The latter is very powerful when each observation is treated as an individual cell, if grouping is unnecessary, and small- er samples can be effectively analyzed. Since the sample size here was large, the Chi-square tests was applied, based on the test statistic g (01 - Ei)2 i=1 E1 (3.19) Q: where n number of cells; observed data points in cell i; and Oi Ei expected data points in cell 1. When the null hypothesis is true, that is, E1 is the expected value of 01 , one would feel intuitively that experimental values of Q should not be too large. If the null hypothesis is true, the test statistic is a random variable asymptotically following a-xz distribu— tion with (n-p-l) degrees of freedom, where p is the number of un— known parameters. Using the table for)(2 distribution, with (n—p-l) degrees of freedom, find c so that P (Q :c) =d . (3.20) 28 where¢x is the desired significance level of the test. The null hy- pothesis is rejected when the observed value of Q is at least as great as c. In our Chi-square test, there are different degrees of freedom for different distributions, as shown in Table 3-1. Table 3-1. Degree of Freedom for Different Distributions. Distribution Degree of Freedom Exponential n - 2 Shifted Expontial n - 2 Normal n - 3 Shifted Log-normal n — 3 Erlang I n - 3 Gamma n - 3 Weibull n - 3 Significance of results: The result of testing the hypothesis of fitting headways with the above proposed distributed by use of the Chi-square test is shown in Table 3—2. 7 Comparing the results of this testing, it can be seen that the gamma distribution is the best fit distribution for the observed pla— toon headways. It should be used as the basis for developing the platoon dispersion model. Due to Greenshield's constant starting de- lay, gamma distributed headway is true only starting close to the 29 signal, not at the signal Table 3-2. Results of the Goodness of Fit Test. Distribution Location 1 Location 2 Location 3 Location 4 Exponential 165.8802 92.8429 117.6418 104.6973 Shifted Ex- ponential 74.7796 52.3978 68.7166 47.0664 Normal 24.5386 5.6036 7.0281 10.9322 Shifted Log- normal 16.8066 15.1989 9.4884 10.3852 Erlang 12.0684 7.8025 7.2645 9.2501 Gamma 6.0743 3.6088 3.8878 3.5941 Wéibull 13.7735 3.5490 4.5246 9.9915 CHAPTER FOUR FIELD STUDIES PROCEDURES Introduction: The field studies of platoon dispersion are described and dis— cussed in this chapter. The dispersion characteristics of platoons leaving a signalized intersection are separated into several sub- models for investigation. Specifically, three situations were inves- tigated: (l) The behavior of a platoon leaving a signalized inter- section; (2) the behavior of a moving platoon with entering vehicles from side streets downstream from the signal. (We assumed the loca- tion of the entrance at a distance where the average headway was in steady state.); and (3) the behavior of a platoon approaching a traf— fic signal. To evaluate the dispersion behavior of a platoon down an arte— rial street, the changing pattern of the interaction between vehicles in the platoon was investigated, and headway between vehicles in the platoon was recorded. Determination of the input variables of the models: From the literature review, it was determined that several variables affect the behavior of platoon movement on arterial streets. Among them are the distance downstream from the traffic signal, sig— nal offset, and platoon size. These three variables were investigated 30 31 and their influence on platoon behavior determined by the research staff of the Transportation Research Center, The Ohio State Univer- sity (23) (1973). Lewis (19) (1958) found that distance downstream from a traffic signal was linearly related with the.arrival time of vehicles at a point. Since the time between successive arrivals is defined as head- way, we can conclude from his study that distance downstream from the signal and headway have the same relation. In Nemeth and Vecellio's experiments, they determined that im- proper signal offsets were one of the important factors causing traf— fic disturbances. This results in vehicle slowdowns or stoppages in platoons approaching a traffic signal. They also found that the initial acceleration characteristics of the smaller platoons (4 to 6 vehicles) are substantially different from those of the larger pla- toons (10 to 13 vehicles). Thus, it was reasonable to assume that headway would behave differently if the platoons varied in size. In the preliminary study of headway reported in chapter 3, it was noted that the average headway of vehicles in a platoon increases in the initial state, reaches a steady state, and then decreases in the final 300 yards preceding the next signal. The influence of sig- nal offset on average headways was also noted in our preliminary study. When the platoons were approaching the next signal, the aver- age headway decreased due to vehicles slowing down when they observed a red signal. We would imagine that if the signal offset was well coordinated with the prior signal, then every vehicle might pass smoothly. 32 Bleyl (2) (1972) reported that 300 yards prior to a signal, the average speed profiles for all signal patterns are the same. He classifies the signal display into six categories: (1) a green signal indication from the moment the lead vehicle reaches a point approximately 300 yards in advance of the signal, at which point a red signal indication is given; (2) a red signal indication until the signal is reached; (3) a red signal indication from the moment the lead vehicle reaches a point approximately 300 yards in advance of the signal, at which point a green signal is given; (4) a green signal indication during the entire approach; (5) a flashing yellow signal; (6) no signal at all. Bleyl found that the average Speed profiles of the vehicles approach- ing the signal have no influence on the last four signal displays. From steady state car-following theory, it is reasonable to assume the average headway is independent of the signal timing until the lead vehicle in the platoon reacts to the signal displays of the first two categories. Therefore, we concluded the following: (1) In the study of pla- toon dispersion leaving a signalized intersection (with or without en- tering vehicles), two independent variables to be considered are dis- tance from the signal and platoon size; and (2) in the study of pla- toon approaching the next signal, only the reaction of the platoon's lead vehicle to the signal display of the first two categories noted above is to be considered. 33 Selection of study sites: Having concluded that three factors (distance from the signal, platoon size, and signal offset) have a significant influence on pla- toon dispersion, it was decided that sites selected for data collec- tion should eliminate other variables that might influence the dis- persion. This meant that factors such as curb parking, grade, uncon- trolled access from side streets, and Opposing left turning traffic should be avoided. Additional site requirements were that the volume vary so that the data would include different levels of platoon size. Also, the downstream signal spacing should be large enough for the study pla- toons to reach steady state flOw. Since the objective also was to study the dispersion of the platoon leaving the signalized intersection with the possibility of a downstream entrance, two sites were chosen. One was a section of an arterial street with a cross—street downstream from the signalized intersection, and the other was an arterial with no nearby cross— street. Both sites are on two-way signalized urban arterials lo- cated in Lansing, Michigan, the first on North Grand River Avenue be- tween Logan Street and Airport Road, the second on Michigan Avenue between US-127 and Harrison Road. Platoon behavior was studied from the platoon formation point at the intersection of North Grand River and Airport Road and the intersection of Michigan and US-127. Some characteristics of the two sites are presented in Table 4-1. In the last chapter, the results of the preliminary study of headways at various locations indicated the average headway of vehicles 34 Table 4-1. Characteristics of the Study Sites 41 Distance of the lst entrance from the signal (yds.) O 880 Characteristics ‘Michigan Avenue Site N.Grand River Avenue Site' Total length (yds.) 1,200 2,024 Lane 3 2 'Max. speed (MPH) 35 45 Entrance point 0 2 increases up to a distance of around 350 yeards. Thus, four locations for data collection were established for Case One, 100, 200, 350, and 500 yards from the signal. It was believed that from these locations the behavior of the average headway cOuld be traced through the tran- sient state to the steady state. Case Two involved the platoon approaching the next signal. Bleyl's paper assumed that 300 yards prior to the next signal, the average headway is in a steady state. Thus, three locations were se- lected for data collection, 100, 200, and 300 yards prior to the next signal. Finally, Case Three concerned entering vehicles. It was assumed that the distance required to reach steady state after merging would be shorter than for platoons starting from the signal. Thus, the lo- cation for data collection were zero, 100, 200, and 300 yards from the entrance point. 35 Sample size requirement In any experiment, estimating the statistical measures of a given phenomenon through a sampling technique requires determination of a sufficient and economical sample size. The sample for this study had to be large enough to produce results with acceptable ac- curacy, but cost and time were also considerations. Therefore, the minimum number of observations required for the estimation of pla- toons' average headway dispersion behavior was determined. The headway of the vehicles within a platoon has been shown to be gamma distributed. If the number of data points from this distribution is sufficiently large, then the average will have a normal distribution according to the Central Limit Theorem. If H is the average of the observations from the sample, then P(|h-E(h)|: d)=a , (4-1) where d is the chosen margin of error, and a is a small probability. If we let d be 0.3 seconds and a be 0.05 seconds, we have the follow- ing equation to solve for n: 1.96 _ER__ ("3 0.3, (4-2) where 00 is the sample standard deviation. From Eq (4.2), we obtain 2 (1.96)2 00 n = ______________ (0.3)2 (4_3) Using the preliminary study of headways discussed in Chapter 3, the sanmle standard deviation of headways at various locations downstream L 36 from the signal are shown in Table 4-2. Table 4-2. Standard Deviation of Headway at Different Locations. Distance 150 300 700 1000 00 0.98 1.22 1.63 1.64 The sample size required for the four locations of Case 1 are approx- imately as shown in Table 4-3. Table 4-3. Sample Size Requirement at Different Locations. Distance 100 200 350 500 n 40 70 120 120 The sample standard deviation in the steady state was the maximum and is approximately equal to 1.64. Using this sample standard deviation to calculate the required sample size for the second and third cases would give average headways for each locations within f 0.3 seconds from the true mean with a probability of at least 0.95. Data on the value of sample standard deviations for the head- ways of the vehicles approaching a signal or with entering vehicles were not available at this stage of the study. Therefore, the sample size for the second and the third cases was derived by using the same ple standard deviation of the steady state. 37 Field data collection: Field data were collected between 7:00 a.m. and 8:00 a.m. and 3:30 p.m. and 5:30 p.m. at the two sites on a number of days during fall and winter of 1976. These hours were selected because they are morning and evening peak traffic periods. On all occa- sions weather conditions were favorable, with good visibility and dry-pavement. Measuring headways of an entire platoon by stopwatch was im- possible because of the speed with which it passed, especially large platoons. A photographic method was found to be more useful. A movie camera recorded the platoons as they passed each observation point along the street, and the headways between were vehicles re- Corded. The camera used was an ARRIFLEX l6S/B camera equipped with a power operation unit. Film capacity was 100 feet, which yields approximately 5,120 exposures per roll. Eastman Plus - X negative black and white 16 mm film was used. The camera had variable speed control, and a speed of 8 frames per second was used during the data collection. A total of ten lOO-foot rolls were used. The camera was mounted on a tripod at the selected locations to maintain a con— stant reference point. Results of field studies: Nemeth's and Vecellio's (22) (1971) field study tested the effect of platoon size on vehicle headways. Five categories were es- tablished: 9-10 vehicles, 12 vehicles, 16-17 vehicles, 20 vehicles, and over 20 vehicles. It was found that the effect of these platoon sizes on mean headway was not significant. 38 Therefore, for this study platoon size was grouped into three categories: under 5 vehicles; 6-9 vehicles, and over 9 vehicles. In general, platoons of under 5 vehicles are referred to as light flow, those with 6-9 vehicles as medium flow, and those with over 9 vehicles as heavy flow. For the second case, moving platoons with entering vehicles from side streets downstream from the signal, data were only avail- able under conditions of heavy flow on main and side streets. The result of the field studies are tabulated in Appendix I. CHAPTER FIVE MATHEMATICAL MODELING OF PLATOON DISPERSION Introduction: From the results of the field study, statistical relationships were obtained for headway patterns downstream from the signal. The first part of this chapter describes tests performed to determine the effect of traffic volume on headway dispersion; the second part for- mulates mathematical models for platoon dispersion. Data Analysis: This section analyzes the data collected for platoons of dif- ferent volume levels for different cases. The technique of multiple comparison of means (24) was used to determine whether or not the level of platoon volume had a statistically significant effect on the headway at different locations from the signal. The effect of level of platoon volume on platoon dispersion for the three cases previously described is shown in Figures 5-1, 5—2, and 5-3. Figure 5-1 was plotted by calculating the average headway for platoons under three different levels of platoon volume at a number of downstream locations from the signal. Figure 5-2 plots the aver- age headway for the moving platoon at a number of locations from the entrance with vehicles entering downstream. Figure 5-3 gives the average headway for platoons under different levels of platoon volume 39 40 Figure 5-1. Mean Headway at Different Locations of Platoon Leaving from a Signal. Sec 3.51 Light Flow 3.0, 5‘ ‘ Medium Flow -% m o m 2'. 5 T g . 3 Heavy Flow 2". 2.0% distance f I U r I . 100 200 300 400 500 (1“ yds°) Figure 5-2. Mean Headwayiat Different Locations of MbvingpPlatoon with Entering Vehicles. Sec 3.51 3.04 z z 'u g d ,3 2.5 5 0 Heavy Flow :1 2.0~ distance I I I T o 100 200 300 (1“ yds') Figure 5-3. Mean Headway at Different Locations of Platoon Approach a Signal. Sec 3.5? Light Flow 3. 0‘ Medium Flow >. 9 «g 2 o 5‘ .2 Heavy Flow a m 9’ 2.04 distance from 300 £00 160 next signal at a number of locations approaching the next signal. Multiple comparison of means requires the assumptions that the sample is from a normally distributed population having a mean Mi under different treatment (for example, three levels of platoon vol- ume) with different variance. In mathematical notations, this can be stated as . 2 Xij N (M19 01 )3 where j=l. . . . .J is the number of treatments. The null hypothesis is H : Mi = Mj, (5—1) 42 where i,j=1, . . . .J, and i > j. In the previous section it was shown that the sample was not drawn from a normal distribution. But with fairly large sample sizes, where the Central Limit Theorem is applied, the mean is normally distributed. In this study, data were collected under different treatments (platoon volume levels) at several locations from the signals. Thus, using a vector instead of a single variable, the new null hypothesis is defined as . 1 , Mi1 Mjl 1 Ho . M12 M32 M13 = M. , (54) 33 LMikJ LMjk .1 where i,j=l. . . .J, and i > j; J= total number of treatments; and K= total number Of locations. Eq (5-2) could be written as E; =‘EB , and the test statistic may be written as ij -+ C =Zd-S(_ _ dk Xi—xj), (5'3) where Zd standardized normal statistic under level of significance a ; _p S(§i _ i3) = a Kecomponents vector, and its m£h_component is equal to /_— 2 33nd Simz Ejm U1 n2 43 Sim = the sample standard deviation of headway at location m under i£h_treatment. + -)- i5 Then the vector I Mi -Mj I is compared with (.gk for all i and j, i >j. We reject Ho if + +4 ij 1M1 — Mjl :C (5-4) (3k and do not reject it otherwise. With the level of significance 0 of each component of the vector, the level of significance for the whole vector is Ok. I The results of the tests of the effect of platoon volume on headway at different locations are shown in Table 5-1, for the pla- toon leaving the signalized intersection, and Table 5-2, for the platoon approaching the next signal. In both cases, the effect of light flow and medium flow on headway was of no significant difference. Therefore, these two cate- gories were combined and labeled nonheavy flow. The results in mean headway after combining them are shown in Figures 5-4 and 5-5. The mean and standard deviation of headways for all cases are found in Table 1-4 in Appendix I. Develgpment of the mathematical models: The data analyzed in the previous section indicated that the relationships between distance and average headway spacing are non- linear. In other words, if we wanted to formulate this relationship in a mathematical form, it would not fit the form F(x) = Bo+Bl x +E. (5-5) 44 Table 5—1. Results of the Multiple Comparisons of Means for Case 1. m = l m = 2 m = 3 m = 4 | M3 - M1 | 0.2043 0.3386 0.5687 0.5464 C31 0.4653 0.3290 0.2986 0.3365 dk | M3 - M2| 0.1188 0.2459 0.4853 0.4697 C32 0.5076 0.3802 0.3425 0.2944 ak | M2 - M1 | 0.0855 0.0927 0.0834 0.0767 21 0.4843 0.3728 0.3597 0.3480 «R m: mth location from the signal; 1: light flow; 2: medium flow; 3: heavy flow; 0.01. 45 Table 5-2. Results of the Multiple Comparisons of Means for Case 3. m = m = 2 m = 3 I M3 - M1 I 0.4210 0.4300 0.5478 31 0.2546 0.2761 0.2688 Cu. | M3 - M2 l 0.2393 0.3586 0.4790 32 0.2730 0.2823 0.2742 Cuk | M2 - Ml I 0.1717 0.0714 0.0688 21 Cmk 0.2628 0.2936 0.2752 Figure 5-4. Mean headway at different locations of platoon leaving from a signal with the combination of light and medium flow. Sec Non Heavy Flow 3.0 h m g 2.5 __‘ 3 Heavy Flow .2 a 8 E 2.0 . ' distance 100 200 300 ' 400 ' 500 46 Figure 5-5. Mean Headway at Different Locations of Platoon Approach. Sec. 1 Non Heavy Flow 3.0 ‘ >. m 3 'o 8 Heavy Flow m 2.5 « :3 m m 21 200-1 1. distance T r ' from next 300 200 100 signal A nonlinear relationship can sometimes be approximated by an algebraic polynomial, which is intrinsically linear. Mathematically speaking, a polynomial in the base function{¢)i(x) } in any function of the form P(x) = no + on ¢l(x) + a2 (1)2 (x) + . . +0LIn ¢m(x), (5‘6) where the <fl.are constants. When the base functions are l,x,x2,x3. . . . . x“, then a polynomial 2 n (5-7) P(x)=ao+a x+ogx+.. .+ ax 1 is called an algebraic polynomial in one variable, and the polynomial is said to be of degrees n. According to the Weierstrass Theorem (13), if f(x) is a func— tion continuous on[ a,b] , and e > 0 is a given, when there exists 47 some integer n and an algebraic polynomial p(x) of degree n, then I f(x) — P(x) I < e for all x e [nab]. Basically, the theorem says that it is reasonable to seek an al- gebraic polynomial approximation to any continuous function in a finite interval. Thus, the problem now turns on the determination of the par- ameters of the polynomials. The simplest method used to formulate the functional relation- ship of two variables is the polynomial interpolation method, in which the interpolating function passes through the given points exactly. But this method has two disadvantages: (1) It is accurate only for the points within the study interval, that is, not for extrapolation; and (2) it is accurate only for deterministic functions. In our models, the average headway is a random variable which is normally distributed by the Central Limit Theorem. Furthermore, our data are "approximate", obtained experimentally. Therefore, in formulating the functional relationship, what is required is a method which allows the approximating function to differ from certain data points more than it does from others, since all data points are known only in what might be called a statistical sense. Least squares is such a method. A quite general statement of the least squares theorem is as follows: The functional value f(pj) (j=l,2. . . .n) is given at a point pj. A family of approximating functions, Qm(Pj, 309 319 32: - -9 am), (5‘8) 48 is chosen, where a0, a .am are parameters. In the method 1, 32, O O of least squares, a0, a1, a . . .am are determined such that 2’ {f (pj) - Qm (Pj’ a0, a1, . . . . a ) } (5-9) is a minimum. For the determined values of a0, a1, a2, . . .am, 1, . . . .am, the function Qm (Pj, £0, 31, . . .am) is the least squares approximation for the given data and the given approx— A say do, a imating functions. -1 Since the approximating function is taken as an algebric poly- nomial, _ 2 m Qm (x,ao, a1, a2, . . . .am) - a0 + alx + azx +. . .+amx .(5-10) In order to minimize n E { f (xj) - a - a xj — a xzj - . . .a X.m}2, (5-11) . o 1 2 m j J=1 the "normal" equation become: n a + a n n n m n 12 x +a22 x + amE x. = 2 f (xj); j=l j=l J _ j=l J 3:1 n n 2 n 3 n m+1 n aOX x + alz x + a 2 xj + . + amZ x. a, Z x.f(xj); j=1 J j=1 3:1 j=l J j=1 J and o . o s o o o o 49 and must be solved. A program for solving this set of equations is contained in a computer package available in the Computer Center, Michigan State University. Five curves from Figures 5-2, 5-4, and 5-5 were formulated within the range of collected data, and M (degree of the equation) was assumed to be 2 and 3. From the figures, it was found that after the defined distance, steady state was reached. For convenience, specific notations were given to each of these five curves. They were 2 NHLi = i§h_location of the curve of mean headway for non- heavy flow of Case 1; Hhi = 13h location of the curve of mean headway for heavy flow of Case 1; INTi = i£h_location of the curve of mean headway of Case 2; NHAPi = i£h_location of the curve of mean headway for non— heavy flow of Case 3; HAPi = igh location of the curve of mean headway for heavy flow of Case 3. If i is replaced by c, this represents the whole curve. After the equations were formulated, validity of the approximated 50 equations were examined and determined the degree of the poly- nomials. The coefficient of determination ( 7) which measures the proportion of total variation explained by the equation was used for this test. The coefficient of determination can be written as: n " __ z ( f(xi) - f(xi) )2 R = i=1 ' , (5-13) (f (xi) - :6 (xi) >2 i=1 where f(xi) = the value of f(xi) estimated by the regression e- quation; and.f—(xi) = the average of f(xi). The results of the least squares approximation of the models and their coefficient of determination are shown in Table 5-3 for M=2 and Table 5-4 for Me3. From the results, it was found that M=2 gave the desired accurate approximation. From the two tables, it was found that the 3rd term of the degree three model did not give any additional accuracy to the model. Therefore, Ms2 should be se— lected as the degree of the polynomials least squares approximation for the models. Analysis of the mean headway for the passing platoon with entering vehicles: Headway data for the passing platoon with entering vehicles were only available for heavy flow of the mainstream. In this section, the relationship between the mean headway for the passing platoon leaving a signaliZed intersection are analyzed. The results will be used as the basis for estimating the mean headway for the passing platoon with entering vehicles under non heavy flow. 51 2 Table 5-3. Equation Results for m=2: f(x) = a0 + a1 y +a2y where y = be +le. b b R2 NHLc 2.6981 0.1910 -0.0630 0.1000 -3 0.986 HLc 2.3148 0.1015 -0.0264 0.1000 —3 0.995 INTc 2.3639 0.0733 -0.0145 0.0133 -2 0.997 INHAPC 2.8125 0.1199 -0.0275 0.0200 -4 0.996 NAPC 2.4015 0.0720 —0.0337 0.0200 -4 0.994 Table 5—4. Equation results for m=3: f(x)= a o +a1y + azy2 + a3y3 where y = bo + blx. 30 al 32 33 b0 bl R HLc 2.3148 0.1015 -0.2640 -0.0160 0.0100 -3 0.995 INTC 2.3639 0.0733 -0.0145 40.0054 0.0133 —2 0.997 NHAPc 2.8125 0.1199 -0.0275 -0.0031 0.0200 -4 0.995 HAPC 2.4015 0.0720 —0.0337 -0.0025 0.0200 -4 0.995 52 From Figure 5—4, it can be observed that the curve INTC is similar to a portion of curve HLC . In other words, if INTc were shifted a constant distance, the two curves might be identical. The purpose of this section is to test this hypothesis. The two curves can be written as 2 = + + — HLc (x) BO le B2y 100_: x: 500; (5 14) _ 2 INT (x) = a +cx x (I x - 0 < x < 300. ‘(5-15) C O I 2 - - Therefore, we want to test the null hypothesis Ho : INTc (X) = HLC (x + 0), (5-16) or 2 Ho . ab — 80 + 0 81+' 6 82 (5-17) = B + B 0‘1 l 262 0‘2_82 To simplify the analysis, this nonlinear hypothesis is changed to a linear one. A value of 9 = 150 yards is assumed, since the first data set of INTC is approximately the midpoint of HLl and HL2. To test the equality of this reduced model and the full model, an F-test (21) must be performed. The full model is defined as the original one, that is, EQ (5—14) and EQ (5—15). The reduced model might be written as: 2 HLC (x) = B + 81 x + 82x . 100 5 X 5 500; (5-18) 0 53 2 2 INT (x) = (B +1508 + 150 B ) + (B +3008 )x +8 x ’ (5-19) C O 1 2 1 2 2 0 §_x :_300. We expect to findBO,3 and £2 to minimize 1 2 Q = %1( - HL (xi) )2+ I2 ( - INT ( ‘) ) yli C Y2; C X1 ’ (5-20) i=1 i=1 where yii = the mean headway of HLcat i£h_location; y21 = the mean headway of INTcat i£h_location; n1 = the total number of data points in HLC; and n2 = the total number of data points in INTC. The test statistic then leads to F = {SSE(R) - SSE(F) } /m1 -m2 (5-21) 9 . SSE(F) /m2 where SSE(R) = error sum fo squares of the reduced model; SSE(F) error sum of squares of the full model; ml = nl + n2 - pl; m2 n1 + n2 - p2; p1 = total number of parameters in the reduced model; and p2 = total number of parameters in the full model. The test statistic then is compared with an F-distributed statistic F a ( (m1 - m2), m2) under the level of significance a. The null hypothesis will be rejected if F 3 Fa,( ( ml - m2). m2) . (5-22) Solving forpo, 81 and ’52 to minimize Q, partial derivatives with respect topo’ )81’ and '52 are taken to obtain the normal equations. The resulting normal equations become: 54 B {n + n } +B-{gl x +150n +22 x } +8 {211x2 +22 x2 +30022 x +1502 n2 } o l 2 1 i=1 i= ‘1 2 1'1 21 2 i=1 11 i=1 21 1‘1 21 n1 {2 y +2 y .} 1_ _1 11 i— —1 21 n2 2 n2 n2 80 {21 xli + 150n2+2 x2 1} +81 {21 x11 +150 2n2+3002 x21+2 x: i} + 01=1 i-1211=1 i=1 i=12 n Bz{21x3 + 1503n +6750022 x + 45022 x2 +22 x3 } = i=1 “1 2 i=1 21 1= 1 21 i=1 21 n1 n2 n2 {21 ylix 11 + 150 L yZi + 1L YZiXZi} (5—23) i= —1 i= —1 1—1 n n n2 3 {21 x2 , 22 (x21+150) 2} +31{21x 3 + 15022 (x21+150) 2+ o 11 + . li i=1 i=1 i=1 i=1 n n1x4 2 n2 n2 x .(x +150) 2} +82 {2 + (150) 2(x21+150) 2+3002 x . 21 21 1121 1=l i=1x i=1 i=1 2 n2 2 _ n1 2 n 2 (x2i+ 150) +2 x 2i(x21+ 150) } — {E_ ylix 1 .— y21(x21+150) }. 1—1 1—1 1—1 Solving Eq (5-23) for 80,81, and 82, we obtain 80 = 1.84708 81 = 0.00272 (5-24) 32 =-0.00021. Substituting 80 ,81,and 82 into Eq(5-20), SSE(R) is obtained. The test statistic, F, is equal to 3.78. Under cx= 0.05, Fa.(3°2)=9'55° Thus, the null hypothesis was not rejected. The accepted hypothesis states that the INTc curve is identical with the portion of HLC starting from the location where the mean 55 headway is equal to the mean headway at the entrance point after the entering vehicles enter. This is reasonable because it implies that if both headway distributions have the same shape, they will have the same dispersion pattern. Therefore, for the case of non- heavy flow (where data were not available), the curve for platoon dispersion leaving the signal under non heavy flow is assumed to be valid for the case of entering vehicles. CHAPTER SIX SOME ANALYTIC CONSIDERATIONS OF QUEUEING Introduction: A number of theoretical papers (20) (28) have dealt with the queueing problem at unsignalized intersections where the minor street traffic is waiting for a sufficient gap to enter the main stream. Most papers have assumed that the arrival of both mainstream and minorstream traffic takes the Rflssondistribution. However, in chapter 3 it was shown that the headway within a platoon is gamma distributed. We will also assume that the mainstream traffic is composed of a series of platoons with no vehicles between platoons. We will call the time between platoons "idle" time and the time when the platoon is passing a given point the "passing" time. Because of the dispersion of the platoon, idle time will decrease as the dis- tance from the prior signal increases. Thus, in our queueing con— sideration, we have to seperately consider these two conditions. Another assumption is that the critical time gap is a step function. That is, the minor stream vehicles will accept the gap if it is greater than the critical time gap and reject it if the gap is less than the critical one. This assumption is suggested by the study by Blumenfeld (14), which showed that this assumption leads to no serious discrepancies in problems of engineering interests. 56 57 Multiple entry is permitted if the gap is sufficiently large, that is, if there is a critical time gap plus some move-up time for the second vehicles, and so on. The notations used throughout this chapter are defined below. f(t) = main stream headway distribution; A = arrival rate of minor stream (veh/sec); a,b = parameters of the mainstream headway distribution within a platoon (gamma distribution constants); s = cycle length; wi = the headway of the i£h_vehicle within a platoon; v = moving up time (in seconds); n = average number of headway within a platoon; d = distance of the minor street from the nearest upstream signal (in yards); and t = critical gap time (in seconds). Using these notations, we know that if the distance from the nearest upstream signal to the minor street and the average platoon size are given, chapter 4 can be used to calculate the parameters of the mainstream headway distribution within the platoon. Thus, in each cycle, platoon passing time will be equal to g wi, and the idle time will he s- ; wi. We further assume :hit 5 is always greater than g'wi; otheiwise, the platoon will merge. If we let p = g ‘wi/s, thin 100xp percent of the time at locations i will be i=1 passing time, and 100x(l-p) percent will be "idle" time. We also assume that the CYC13 length equals 60 seconds, which is the actual cycle length of the signal at which field data were obtained. The 58 percentage of passing time and idle time for different locations from the signal is shown in Table 6-1. Table 6-1. Percentage of thefpassing time at different locations. n 100 yds. 200 yds. 350 yds. 500 yds. Nonheavy flow 6 0.2251 0.2567 0.2990 0.3002 Heavy flow 11 0.3482 0.3791 0.4088 0.4135 Queueing considerations at the merging position: This section discusses the delay to a single vehicle waiting at a stop sign for a sufficiently large gap in the oncoming traffic. We first must define the terms "125?, which is different from gap. A lag is the interval of time between the arrival of a minor stream vehicle and the arrival of a major stream vehicle at the point where the streams cross. Assume that a vehicle arrives on the minor street at time t=0, and the driver waits for a gap greater than T seconds. There are two possibilities: The driver accepts the lag, or he re- jects the lag and subsequently accepts a gap. If we know the probability density function of the lag (fo(t)) and the probability density function of the gap (f(t)), then we will accept a lag with probability f: f0 (t) dt (6‘1) 59 and accept a gap with probability I: f (t) dt. (6-2) Using the terminology of Renewal Theory, fo(t) is known as the forward recurrence time, which is shown by Cox (6 ) (1962) as ” (6-3) £0 (t) = if f(x)“ p where u is the mean of the distribution f(t). Therefore, fo(t) is a function of t, and the parameters of the headway distribution f(t). Let Eq (6-1) be denoted by F0 and Eq (6-2) by F. We then further define a conditional probability, H(t), as follows: H(t) = Prob { A gap is accepted in the time interval (t,t dt) given that no gap was accepted in the interval (0, t)} With these definitions we can write the probability density of the merging delay as D(t) = 6(t) FO + H(t) F, (6—4) where (t) is a Dirac delta function, which accounts for the cases of zero delay, that is the acceptance of the lag. The first term of Eq (6-4) indicates that if a lag is accepted with a probability of F0, the waiting time is zero. The second term of Eq (6-4) indi- cates that if a gap is accepted with a probability of F, then the waiting time is t. By definition of a delta function, I: a (t) dt = 1, (6‘5) 60 and f: H (t) dt = 1, (6-6) where H(t), f(t), F0, and F are all positive functions. Thus, it is easily shown that D(t) is a probability density function, since I: D(t) dt = 1, (6-7) and D(t) 3_ 0 for all t. Let the Laplace Transform of f(x) be f*(s), by f *(s)=f: e Atf(t) dt. Taking the Laplace Transform with respect to D(t), the moment of the merging delay willbe obtained. The proof is given in Appendix III. From Eq (6-4), we have D*(S) = FH*(S), (6-8) In considering H(t) of Eq (6-4), the mainstream vehicle that arrives in time (t, t dt) must either have been the first vehicle to appear with density fo (t), and the lag was rejected, or a vehicle passed at (t-x), and its gap was rejected. From this, we can write the equation satisfied by H(t) as H (t) = fo(t) {l —«x(t)}+ f: H(x)f(t-x) (1- m(t—x) )dx, (6-9) where ll 0 a(t) a(t) if 0 §_t g T, and II I—‘ if t Z,T. Let (6-10) Go(t) = f 0(t) { l - a (t)} , 61 and c (t) = f (t) { .1 - (1(t) } . Eq (6-9) then can be written as H(t) = Go(t> +f: H(t) G (t-x)dt. (6‘11) Eq (6-11) becomes H*(S) = 60*(3) + H*(S) 3* (3), (6-12) H*(S) = 90*(3) , 1 -G*(S) (6_13) The moment of the merging delay can be found from Eq (6-4) to be dn u“ = <—1)n < —5;n9 D* (s) <6-14> t |s=o The proof Of Eq (6-14) and the first and second moment are pre- sented in Appendix III. The result is shown in Eq (6-15) and Eq (6-17) which is different from the result developed by Weiss and Maradudin (27) using the same model. _ l—_ m I - (6‘15) u t — F f0 t { F Go(t) + (1 FO)G(t) }dt. If we let Mo(t) = f0 t Go(t) dt, and M(t) = I: t G(t) dt, Eq (6-15) becomes ut1= Mo(t) + 1”F0 M(t), <6-16> F 62 and 2 _ w 2 2 ”t j§%_ { fot (F Go(t) + (1-Fo) F G(t) dt + (6-17) 2 f:t G(t) dt f:t (FGo(t)dt + (l-Fo) G(t))dt}. Let _ m 2 No(t) — fot Go(t) dt, and N(t) = f: t2 G(t) dt; Eq (6-17) then becomes u 2 = No(t) + 1-Fo N(t) + 2 uth(t). (6-18) 1: F F The variance of the merging delay is O 2 2 1 2 v (6‘19) In our headway model, F(t) has been shown to be gamma distri- buted; thus W , t _>_ 0 (6-20) where a and b are the parameters of distribution and are defined as real numbers. 63 In order to solve Eq (6-16) and Eq (6-18), F o (t), F(t), Mb(t), M(t), N G(t), and N(t) should be calculated first. By definition, F is the incomplete gamma function. Under different values of T, the value of F can be calculated by computer approx- imation for the different cases as defined in the last chapter. The result is presented in Table II-l of Appendix II. In calculating F 0a which is defined as the integral of f o, f o is defined as f o(t) = I: f(xl)dx , (6-21) u which is an indefinite integral of the headway distribution divided by its mean. There are several articles (18) which relate this indefinite integral to the cumulative Poisson distribution as b 1:0 but this is only true when "a" is an integer. Since "a" in gamma distribution is defined as a real number, a curve fitting an algorithm of fc, for eight different cases is presented. The integral of f(x) is approximated by computer, using nine different values of t ranging from 0 to 5. These nine points will plot all the curves of the resulting integral varying from one to O. 64 By using the Lagrange polynomial interpolation technique (13), the preposed polynomial will pass through five alternative selected points. The value at the selected points and at the midpoints between them where one would expect the error to be the largest are then examined, substituting those values into the polynomial equations and comparing the results with the original data. The proposed polynomial is shown in Table II-2 of Appendix II. The curves plotted by the polynomial equations are presented in Figure II-l of Appendix II. The selected points and their midpoints are then substituted into the polynomial equations, and one finds that the upper bound of the deviation of the equation from the integral is 0.05. Thus, the polynomial equations can be accepted as a good fit to the indefinite integral I: f(t)dt. fo is then equal to the proposed polynomial equation divided by the mean of the headway distribution; this result is shown in Table II-3, Appendix II. Fc, is defined as the integral of f0; by using computer approx- imation the result is shown in Table II-4 of Appendix II. Again using the computer approximation, M(t), Mb(t), N(t), and N(,(t) under different T for different cases is calculated, as shown in Table II-S through II-8 in Appendix II. With this information,“ i and otzare calculated under different T for different locations. These results are shown in Tables 6-2 and 6-3. 65 1 Table 6-2. Calculate,g; for Different T. T =.2 T = 3 T = 4 T = 5 NHL1 1.6307 9.6638 58.0139 404.4929 NHLZ 1.1548 6.3097 36.0276 242.1522 NHL3 0.8327 3.6102 15.9985 79.9109 NHL4 0.8141 3.5847 15.8166 78.0859 HLl 1.9675 11.4557 65.9667 421.5800 HL2 1.5794 10.4827 57.8873 853.5397 HL3 1.2981 7.3444 44.9245 318.1406 HL4 '1.2343 6.8330 39.2631 264.8093 2 Table 6-3. Calculate gA for Different T. '1:— T=2 T=3 T=4 T=5 NHLl 3.3679 90.2583 3423.4429 164033.5986 NHL2 1.7437 43.2742 1331.2554 58892.3497 NHL3 0.9688 14.8524 273.7818 6473.0088 NHL4 0.9130 14.5814 267.9273 6210.7805 HLl 4.8569 139.6327 4417.0574 178135.4064 HLZ 3.0980 115.0225 3560.0805 125251.8651 HL3 2.1631 60.1113 2062.0342 101547.4212 HL4 1.9678 50.4400 1577.5383 70388.9500 66 These results represent the merging delay for the minor stream vehicles while the mainstream platoon is passing, which occurs p percent of the time. The remaining (l-p) percent of the time there will be no merging delay. Thus, the overall expected merging delay will be E (Z) = P 0 (6-23) and the variance will be 2 Var (Z) = P0 t + p(l-p)( ut1)2 (6‘24) The proof of Eq (6—24) is given in Appendix III, and the results are shown in Tables 6-4 and 6-5. Table 6-4. Average Merging Delay E(Z). T = 2 T = 3 T = 4 T = 5 NHLl 0.3670 2.1753 13.0589 91.0513 NHL2 0.2964 1.6197 9.2482 62.1604 NHL3 0.2489 1.0794 4.7835 23.8933 NHL4 0.2444 1.0761 4.7481 23.4413 HLl 0.6850 3.9888 22.9696 146.7941 HLZ 0.5987 3.9739 21.9450 134.0269 HL3 0.5306 3.0024 19.3651 130.0558 HL4 0.5103 2.8254 16.2353 109.4986 67 Table 6-5. Variance of Mergigg Delay Var (Z). T = 2 T = 3 T = 4 T = 5 NHLl 1.2218 36.6041 1357.5797 65458.3323 NHL2 0.7020 18.7045 589.3893 26305.7369 NHL3 0.4349 7.1726 135.5082 3273.8831 NHL4 0.4131. 7.0758 132.9663 3144.9319 H11 2.5694 78.3968 2525.3985 102353.6166 HL2 1.7613 69.4613 2368.4217 76879.0861 HL3 1.2913 37.6053 1330.5592 65965.7520 HL4 1.1828 32.1745 1025.9936 46103.8799 Queueing Considerations before Merging; Using the previous results, the average delay for a vehicle before merging and the average queue length of the minor stream can be developed. In this section, Kendall's approach (16) for a queueing system with random inputs from the side and arbitrary service times is used to develop formulas for the average delay before merging and the average queue length. Let x (ti) denote the number in the system at time ti, where ti is the time Of completion of waiting of the ith vehicle. Let X (ti) be denoted by Xi. we can them.write for all m>0 xm+1 = Xm+ Am+1 - 6 , (6-25) 68 where 6 is introduced as a random variable such that =() x =Ov , 6 m (6-26) =1 ' xm>0 In Eq (6-25) A(m+l),is the number of vehicles which arrived during the waiting time of the (m+l)st vehicles arrival, say, Z m+1: independent random variable, A m+1 can in general be denoted by A, and Z m+1 by Z. Then conditionally A is a Poisson variable of mean Z, given that Z is the duration time. Also assuming that the steady state solution exists, we can take the expected value of Eq (6-25) and obtain E(Xm+1)= E(Xn) + E(A) -E (6). (6-27) Noting that E(anl)= E(Xflp = Nd in steady state, Eq (6-27) then becomes Nd = Nd + E(A) - E (6), (6-28) or E(5) = E(A). (6-29) The expected number of vehicles arriving during the merge wait- ing time of a vehicle is equal to the product of the arrival rate and the average merging waiting time, that is, E(A) = AE(Z). (6-30) Thus, we have . E(G) = AHZ) . (6-31) Squaring both sides of Eq (6—25) and taking the expected value as before yields E(X2m+l) = E(xm)2 + E(AZ) +86 2) + 2E(XmA) - 2E(6xm.) — 2E6A). (6-32) 69 Note thatd 2 = a ,xme = xm, , E(x;+1> = E. and A is independent of X. ands . Thus, we have E(AZ) + 8(8) —2E (xm) +2E (xm) E(A) -2 8(8) E(A) = 0, (6-33) or E(Az) + 18(2) -212(8(2))2 - 2Nd + 2188 8(2) = 0, (6-34) or 2 Nd = AE(Z) +>ECA ) ‘ AECZ) . . (6-35) 2(1 -2 e(Z) ) Now it is necessary to calculate E(Az), the second moment of the number of arrivals during the merge waiting time of any vehicle, and 2 2 E(A ) = VAR(A) + (E(A) ) 2 (6-36) =vmm>+xmu>f. where Var (A) = E(Var(AlZ)) + Var (E(AlZ) ). (6-37) The proof of Eq (6-37) is given in Appendix II. As mentioned earlier, AIZ is a Poisson variate and with mean 12 ; thus, Var (AIZ) :12 and E(AIZ) = AZ . Substituting into Eq (6-37), we obtain VAR (A) = E( 2') +VAR (22) =1 E( z)+z )2 VAR (z ) . (6—38) By using Eq (6—38) in Eq (6-35), we finally obtain 12 8(2)2 + 22 VAR (2) Nd = 1 E(Z) + . 2( 1— A E(Z) ) (6’39) 70 It has been shown that, (16) although Nd is the expected steady state system queue size at the time of a departure, it is equivalent to the expected steady system queue size at an arbitrary point in time. We denote this by N. Thus, N = 18(2) + 12 Var (2) +1 2 (8(2) )2 2( 1- 1 E(Z) ) ° (6-40) Eq (6-40) is often referred to as the Pollaczak—Khintchine formula. From it the expected waiting time in the system E(W) (which includes the merging delay) can be obtained through the well-known Little's formula: or E(W) = y_. 1 (6-42) If E(V) is the average waiting time for a vehicle before merging, then. E(V) = E - E(Z). (6‘43) A The result of N and E(W) under different T and different A for different cases is shown in Tables 6—6 and 6—7. Qapacity and the Best Location of the Minor Stream In calculating the capacity of the minor stream, it is assumed that a queue always exists on the minor street. For the installa- tion of an entrance downstream from the signal, such as the entrance to a parking lot or shopping center, capacity at the entrance point would be expected tobe the maximum. As before, the critical gap is T, and the‘move up time is v (which is the time required by the 71 nanm.m nmmH.o qnmo.o 8 «Nom.o owmo.o 8 wHHq.N quH.o cam aqoq.om cwHN.o mwmo.o 8 mmmo.o «Hoo.o 8 Homa.m NH¢H.O mam 8 onm.o omno.o 8 nqoa.a Hano.o 8 mmmo.m whoa.o Nam 8 nocm.o Nwmo.o 8 .mmmH.H w¢wo.o 8 noma.oa mnom.o Ham NNm¢.o memo.o wNHo.o Nmmm.H nmma.o memo.o mmmo.Ho ommq.o wwmo.o qqmz Nmmq.o mqoo.o Hmao.o momm.H mcma.o quo.o wqaa.qm qwmq.o mwoo.o mamz oamo.~ omoa.o meo.o maaw.mq wam.o nmmo.o 8 wqm¢.o oomo.o Namz wmma.c sooa.o Homo.o 8 Nam¢.o mmqo.o 8 Hmmw.H ouoa.o Hamz q n H m n H N u H q u H m n H N u H q n H m n H N n H ON\H n OH\H n .« mxa u .« .z muam moonu wmuoomxm wnu mamasoamo .olo anmH 72 oqmq.amfi oqma.m Nesm.o a osmo.m oomm.o 8 ommo.NH oaao.o «am ammo.wmm omam.q wean.o s ommm.o omao.o a moao.qfl oeoa.o mam a omno.e ommo.o a maeo.HH «Haa.o s meaq.ma oamm.o New 8 osmm.o amea.o a ommm.HH omqw.o 8 mmoo.om memo.H Hum. oqqm.m o~m~.a memN.o onm.aH samm.fi oweN.o mom~.mam omNH.~ oan.o egmz oqmm.a eaa~.H Nae~.o omoa.aa Noam.fi mqa~.o oeam.oam oNeH.N omqm.o mqmz omme.os soma.~ soam.o oaHH.mmq o~am.~ ohmm.o s os-.q oomm.o Ngmz oeHH.m~H qqmm.m «Hoe.o a omaw.s qamq.o a moaq.o omam.o quz_ a u H m u a N u a s u a m u a N u a a n a m u a N u H o~\a n K OH\H u a m\H u a <\z u have mumH=UHmo .nuo «Haas 73 second vehicle to move up to the first position after the first vehicle has left). Hence, with probability T (6-44) I f(t) dt 0 . a gap is less than the critical gap. In general, for a gap of size (T + (j-l)v, T +jv) with probability 'T + jv f . f(t) dt, T +(J‘1)V (6-45) j vehicles will enter the intersection. Thus, the capacity while the platoon is passing is C1 = q 2_ j J T+ JV f(t)dt (veh/sec). j T + (j+1)v (6‘46) If the move up time is v then the capacity during "idle" time is % veh/sec. Thus, the overall capacity becomes C = clp +1/v (l-p). (6-47) Since C1 and p both vary with distance from the nearest signal, a table can be constructed showing the capacity at different distances from the signal for different average platoon sizes. Assuming the move-up time to be two seconds, the result of C is tabulated in Table 6-8. For a specific value of critical gap T, the overall capacity is almost equivalent for different locations from the signal. Also, the capacity during the passing time is an in- creasing function and approaches steady state from 350 yards. The capacity during the idle time is a decreasing function and approaches II" 74 steady state from 350 yards. Table 6-8. Calculated Capacity for Different T. T = 2 T = 3 T = 4 T = 5 NHLl 0.4444 0.4053 0.3850 0.3879 NHL2 0.4437 0.3996 0.3780 0.3726 NHL3 0.4352 0.3957 0.3520 0.3540 NHL4 0.4362 0.3957 0.3655 0.3535 HLl 0.4067 0.3498 0.3307 0.3266 HL2 0.4098 0.3390 0.3168 0.3113 HL3 0.4076 0.3351 0.3039 0.2967 HL4 0.4087 0.3362 0.3029 0.2947 During the calculation of C, it can be seen that almost 80 percent of the capacity was from idle time. One of our assumptions is that no mainstream vehicles are allowed between platoons. If that assumption is relaxed and mainstream vehicles are allowed passing during the idle time, we would expect the capacity during idle time to be an increasing function, the same as the capacity for passing time. In that case overall capacity would reach a maximum at 350 yards. That is the best location for the entrance when the main- stream vehicles are allowed passing between platoons. 75 The Required Distance for Recovering the Shape of the Headway Distribution: After the minor stream vehicles enter the main stream, the shape and the parameters of the headway distribution change. In this section, we will investigate the distance required by the new pla- toon (with the entering vehicles) to recover the original headway distribution after dispersion. For the input vehicles, we consider only those vehicles waiting to enter the main traffic when a platoon is passing. If d and n are given, the passing time for the whole platoon is n _ 2 Wi =11 w , i=1 (6-48) _ n where w is the mean headway within the platoon. Thus,12 Wi i=1 vehicles will arrive to enter the mainstream during the passing time. n If the capacity C1£Wi (see the last section; here we only - i=1 n n consider the passing time) is greater than AZ Wi,then A X Wi i=1 i=1 vehicles will enter the mainstream during the platoon passing time. n n n If A 2 W1 is less than C 2 W1, then C 2 W1 vehicles will enter i=1 1i=l 11=1 the mainstream, and ( A —C1).B];Ni vehicles will wait until the 1:. platoon has passed. In the case where Aisyi vehicles enter the mainstream, platoon n . size becomes ((n+l) + l 2 W1 )Using the model developed in the last i=1 section of chapter 5, the required distance can be determined. In the case where demand exceeds the capacity during the pass- n ing time, only C12 Wican enter the traffic while the platoon is h=l 1 n, 2 Wi). 1i=1 Similarly, from chapter 5, we obtain the required distance. passing. The new platoon size then is ( (n + 1) + C An example will clarify how the models developed in chapter 5 can be used in this section. Supposel=l/10, the critical time gap is two seconds, and the entering point is 350 yards from the signal. Then n A 2 W1 = in w = 2.69 = 2 vehicles, C1 2 Wi = Can w= 7.4 = 7 vehicles, where the platoon is assumed to be in heavy flow. That means all two vehicles would be allowed to enter the mainstream while the pla— toon is passing. Platoon size after the minor street vehicles enter I]. then becomes ( (n + l) +A Z Wi ), which is equal to 14 vehicles. i=1 n Therefore, with the same passing time 2 Wi , the platoon now i=1 consists of 14 vehicles. The average headway of the new platoon n then became g lWi/ (number of vehicles - l), or 2.08 seconds. 1: Referring to Figure 6—1, the mean headway would change from A to B after the minor street vehicles entered. We could also say the platoon is in the situation of being located at d1, since at d1 the mean headway would be the same as B. According to the platoon dis- persion behavior, the platoon would need (350 - dl) yards to recover its mean headway of A. Spliting Flow to Reduce Delay of the Entering Vehicles: In this section we will consider the problem of spliting the queue into two or more queues to reduce the delay of a queue of 77 oocmumaw com com HZH ooq oezH «6 coauaflmamue com com .Hao muswam H ——-—-—.-—--.-------—--—-—o.------uq w .wsus cod P fin.o o.H rm.H cow KempeeH uean 78 vehicles waiting to enter the mainstream traffic. The analysis uses the results of section three of this chapter, that is, the average waiting time in the system E(W). If we split the queue into two streams, then we assume that the distance between the two entry points is sufficiently large so that the headway distribution returns to steady state after the first queue entry. While the technique de- veloped in the section is applicable to any number of successive entry points, our discussion is limited to the possibility of splitting the queue into three entry streams. Assuming 9 to be the rate of splitting, where 0: if 1, that is, if the rate of the minor street random arrival is A, than we split the arrival into two Poisson streams, with arrival rate lfifor the first entry and (l - 9) l for the second entry (if two streams are to be made). We then determine the optimal splitting, Gopt. For the case of two entry points, let us denote the distance of the first entry point from the signal as dl and the second as d2. If the average platoon size is n, then at d1 the average waiting time in the system E(wldl) is a function of d1, 61, and n: E(wldl) = f(dl, n, (=1 ). (6—49) The average waiting time in the system at d2 is E(wle) = g(d2-dl, n+h(6 1), (1-6 )1 ). (6-50) Eq (6-50) depends on (d2-dl), not d2 alone, because there are vehicles entering upstream of d2. The calculation of the parameters of the pla- toon reaches point d2 is different from the calculation of Eq (6-49). The term h(€)A) depends on the capacity at d1; if arrivals at dl exceed capacity C, then C vehicles will enter the traffic; otherwise 79 9 A will enter. Eq.(6-49) and (6—50) are used to calculate the total mean waiting time in the system as a function of the parameters: 1(0) = 0° E(wldl) + (l-G)E(w|d2). (6-51) Blumenfeld and Weiss suggested a parameter 6,(l3) which can measure the improvement effected by queue splitting, that is, 6 --—-——--° (6-52) -T (1) The smaller the value of 663, the greater the improvement effected by queue splitting. Using the data from field studies, and assuming 9 to be equal to 15, 1/3, 2/3, 14., and 3/4.5'%, 51/3, 52/3, 512, and (5)4 are calculated by using Eq (6-52). The critical gap T is assumed to be 3 seconds. The result is shown in Table 6-9. There it is found that equal split- ting, 9 = k, is the optimum splitting for two entry points. Similar calculations are performed for splitting minor traffic into three streams. Assuming that equal splitting was better than the other splittings in three entry points. Let d1, d2, and d3 be the distances of three entry points. We have T(9) = e E (wldl) + 92 E(wldz) + 93E (wld3), (6-53) where 91 + 92 + 93 = l, and 91, 92, and 93 are the splitting of the three entry points, respectively. The critical gap is assumed to be 3 seconds, and6(l/3 1/3 l/3)is calculated. The results are shown in Table 6-10.' It is found that5(l/3 1/3 l/3)is better than 6%. 80 HHwa.o Nqoa.o Nwmq.o HNHe.o «mom.~ wa8.N «Nam.a maem.o Aq\m.e\av Haw~.o oeam.o ammo.o wmeo.o omme.a AHmN.H omoH.H emow.o Ae\a.e\mV maea.o neao.o o~mq.o ow~q.o ooom.~ oww~.~ ONHM.H ammm.o Am\~.m\HV 0H\Hna maoa.o Haem.o oomm.o meom.o Naoa.a ~N©a.fi oemH.H «Hee.o Am\a.m\mv Hose.o cams.o aome.o mHHq.o Hmaa.a NHHN.H mo~o.a aaoe.o Aw.mv wowm.o wqu.o qwoa.o mNmH.o mumm.m Nmmm.m wNmm.H woum.o Aq\m.q\av mowm.o mmqm.o qum.o nomm.o oqwo.a emoa.H ammw.o meme.o Ae\a.e\mv mmmm.o mmoq.o Noaa.o mmma.o osmo.~ Heqe.~ coom.a oqae.o AM\~.m\HV m\Hu« mnmm.o omae.o amom.o aoe~.o one~.a owe~.a chow.o moam.o Am\H.M\~V meo¢.o mnoq.o mmefl.o mNmH.o ommo.a omae.fl oaom.o waem.o Ax.xv «as mg: «as Ham aqua mumz Ngmz Hgmz waauuaaam .« m uaouomwfio Hommwm oumaaoamo .mlo manna 81 The determination of splitting of the entry point not only depends upon the delay time,but also is a tradeoff between the costs and benefits from the splitting. clude that splitting into three is better than splitting into two, only that the delay time is decreased. Therefore, we cannot con- Table 6-10. Calculate 6 (l/3,l/3,l/3) for Different A. NHLl NHL2 NHL3 NHL4 HL1 HL2 HL3 HL4 A=1/3 0.2124 0.5427 1.4109 1.4249 * '* * 0.1547 A=1/5 0.4628 0.8811 1.6740 1.6688 0.1163+ 0.1262 0.3257 0.3894 A=1/10 0.6958 1.0515 1.7998 1.8095 0.3642 0.3843 0.5785 0.6337 * Approaching zero due to infinity delay occurred in TCl). CHAPTER SEVEN SIMULATION ANALYSIS OF THE PROBLEM Introduction: The analytic results of the behavior of vehicles entering into stream with a dispersing platoon were developed in chapter 6. To validate the model, an experiment was performed. Verification of the entire range of the model would require extensive data at var- ious volume levels, for the mainstream and entering traffic. It would be almost impossible to obtain sufficient data through field studies. Therefore, a computer simulation which could duplicate the real world situation was written. A simulation model consists of two basic phases: input data generation and bookkeeping data generation. The Monte Carlo tech— nique was used here to generate random events from some specified probability distribution. The models employed an event-oriented bookkeeping technique, updating the system status when events occur, recording relevant items, and calculating measures of the parameters of interest. The analysis conducted for this study was divided into three stages. In the first stage a model was developed of a single un- signalized intersection at which vehicles were arriving from two directions, and right-of—way was given to the mainstream. The second 82 83 stage expanded the model to include a signalized intersection with a single sidestreet downstream. With this model, the simulation of platoons leaving the intersection every signal cycle and arriving at the sidestreet was conducted. The final stage was the extension of the second stage to include multiple entrances downstream from the signalized intersection. The simulation models were designed such that the final results of the parameters of interest could be compared with the prior result to determine that steady state was reached. The programs were written in Fortran, and they were run on the CDC 6500 at Michigan State University. Development of the Phase One Model: The intersection used in the phase one simulation model was T-shaped, and the sidestreet was assumed to be a one-way street. Since it was established by Nemeth and Vecellio (1972) that the number of lanes exhibits no significant effect on platoon behavior, one lane of travel was assumed in the simulation model. To see the effect of entering vehicles on platoon dispersion, only right turns were permitted on the sidestreet. The physical characteristics of the phase one model are shown in Figure 7—1. Major street vehicles were given the right-of—way over entering vehicles whenever con— flicts in entering occured. Entering vehicles decelerated to a stop either at the intersection or in queue behind other stopped vehicles. It was assumed that an entering driver waiting to enter considers each time gap t in the mainstream traffic until he finds an acceptable 84 Figure 7-1. Physical Characteristics of Phase One Mbdel Intersection. @000. gap T, which he believes to be of sufficient length to permit his safe entry. Blumenfeld and Weiss (4) have shown that this assumption of a fixed critical gap is fairly realistic for estimating delays and capacities. Two input sources are involved in the present model: sidestreet arrivals and mainstreet arrivals. A common assumption made by many researchers (25) (14) is that arrivals from the sidestreet are random. In chapter 6 it was shown that the mainstreet headway conforms to a gamma distribution. Thus, for the simulation model, the interarrival time of the mainstreet vehicles was gamma distributed and that of side- street vehicles was random. The move—up time of the second vehicle in the sidestreet when the first vehicle entered traffic was assumed to be two seconds. Three parameters for the entering vehicles were recorded and measured in the simulation runs: average merging delay, average queue length and average total delay. Briefly, the simulation was processed as follows. First, a side- street vehicle was generated, and its arrival time was recorded. Mainstreet vehicles were generated until it was found that the arrival 85 time of the sidestreet vehicle lay between the arrival time of two successive mainstreet vehicles. The critical gap was then compared to the available lag. If the lag was rejected, successive main- street headway was generated and compared to the critical gap until an acceptable gap was found. 'The time spent in the merging position was recorded, and another sidestreet vehicle was generated and checked to determine whether its arrival time preceded the leaving time of the prior vehicle. If not, no delay was recorded as occurring before moving into the merging position; if so, this delay was re- corded. The process was repeated and new traffic was generated un- til the steady state solution was obtained. The logic flow diagram for the process is shown in Figure 7-2. The program was run under different conditions for comparison with the analytic solution from chapter 6. The results of the simulation program are shown in Table 7-1. Table 7-1. Simulation Results of Phase One Model. Average Merging Delay Average Delay Before Merging Q = 350 vph Q = 800 vph Q = 350 vph Q = 800 vph T = 2 0.7011 1.9362 0.2211 0.4225 T = 3 4.2420 7.2350 8.1240 738.3500 T = 4 16.2400 42.2631 1024.5000 2104.0000 Note: Q = mainstream volume. 86 Figure 7—2. Flow Diagram for Phase One Model. Initialization Generate gainstream arrival l 1 Update the "clock" & the arrival time i If the clock No exceeds time Yes E d limit n If there is No some sidestreet .4 ehicles waitin: dl Ye 0 enter r Generate minor stream_arrival No 4 the gap is Yes Record the J greater than merging delay Update the the critical arrival time the arrival time of the minor street less than instreet veh Record the delay before moving into the merging position. No 87 The results from Table 7-1 were run under a mainstreet volume 350 vph and 800 vph. On the average, this meant approximately 6 and 13 vehicles in the respective platoons. Minor street vehicles were assumed to arrive at the rate of 5 sec/veh., the entrance point was assumed to be 500 yards from the prior signal, and critical time gaps were assumed to be 2, 3, and 4 seconds. The simulation program is listed in Appendix IV. The comparable analytic solutions were those formulas in chapter 6 with p = 1, that is, under 100 percent passing time. The mainstreet vehicles were passing the intersections at the same volume levels and critical gaps used in Table 7-1, and the results are shown Table 7-2. Analytical Results of Phase One Model. Average Merging Delay Average Delay Before Merging Q = 350vph Q = 800vph Q = 350vph Q = 800vph T = 2 0.8141 1.2343 0.1882 0.4635 T = 3 3.5847 6.8330 9.6910 m T = 4 15.8166 39.2631 w w By the assumption of the analytical model, if the service rate is greater than the arrival rate, the average delay before merging will approach infinity. Therefore, three infinity delays appear in Table 7-2. To compare the results of the two models (analytical and simu- lation), a standardized statistical test was introduced to indicate 88 the reliability of the model forecasts. The U-statistic (16) measures the statistical correlation between two sets of data; it measures the agreement between the forecast (analytical) and ob- served éimulation) item frequencies. The accuracy of the forecast is judged by the magnitude of the U - value. If the discrepancy between the observed and predicted mean values of the level of activity in a cell is small, there still could be poor agreement between measured and predicted results in individual cells, and the U - statistic can measure.their significance. The U-statistic is calculated as 1 1/ {3‘2 6.0 0 0 0 2 0 0 2 0 0 0 0 l d1 : 100 yds. from the prior signal; d2 : 200 yds. from the prior signal; d3 : 300 yds. from the prior signal; d4 : 500 yds. from the prior signal. 108 Table I-2. The frequency of the headway sample within platoons with entering vehicles. Headways (in sec.) d1 d2 d3 d4 <1 0 1.0-1.5 16 1.5-2.0 29 26 27 21 2.0—2.5 39 46 44 45 2.5-3.0 25 24 32 31 3.0-3.5 7 13 8 9 3.5-4.0 2 3 5 7 4.0-4.5 2 2 l l 4.5-5.0 0 0 2 l 5.0-5.5 0 0 0 2 5.5-6.0 O 0 0 0 >6 0 0 0 0 d1 * neXt to the entrance point. d2 = 100 from the entrance point. d3 = 200 from the entrance point d4 = 300 from the entrance point 109 Table I—3. The frequency of the headway sample within platoons approaching a signal. Headways (in sec.) Light Flow Medium Flow Heavy Flow d1 d2 d3 d1 d2 d3 d1 d2 d3 < 1 0 0 0 l 0 0 2 1 0 l.0-1.5 7 5 2 15 8 4 19 10 8 1.5-2.0 18 13 13 33 13 10 35 19 23 2.0—2.5 34 18 21 26 20 25 26 33 27 2.5-3.0 30 32 28 16 28 30 18 23 28 3.0—3.5 18 25 33 12 26 27 12 18 20 3.5-4.0 9 16 15 11 16 12 4 l4 8 4.0—4.5 2 4 4 3 2 3 3 l 4 4.5—5.0 l 4 3 2 4 5 0 0 l 5.0-5.5 l 2 l 0 1 2 l l l 5.5-6.0 0 l 0 0 2 1 0 0 0 >6 0 0 1 0 o o 0 o 0 dl : 100 yds. from the next signal. d2 : 200 yds. from the next signal. d3 : 300 yds. from the next signal. 110 Table 1-4. Mean and standard deviation of headway within platoon 'for all cases. (A) Platoon leaving the signal 150 yds. 200 yds. 350 yds. 500 yds. P . 'Non heavy flow 2.2512 2.5670 2.9803 3.0024 0.855 0.8734 - 0.9878 0.9540 Heavy Flow 2.08975 2.2748 2.4533 2.4814 0.869 0.7921 0.8535 0.8633 (B) Platoon with entering vehicles. 0 yds. 100 yds. 200 yds. 300 yds. Heavy 2.1877 2.3523 2.4268 2.4886 flow 0.8256 0.7923 0.8756 0.9324 (C) Platoon approaching the signal. lOOyyds. 2004yds. 300 yds. Non heavy 2.5361 2.8857 3.0156 flow 0.8245 0.8973 0.8524 Heavy 2.2110 2.4914 2.5022 flow 0.8151 0.8122 0.8234 .J_ APPENDIX II on Table II-l. Incomplete Gamma Function F = 1;, f (t) dt. 111 T = 2 T =,3 T = 4 T = 5 NHLl 0.5703 0.1793 1 0.0364 0.0055 NHL2 0.7216 0.2799 0.0646 0.0104 NHL3 0.8479 0.4522 0.1508 0.0354 NHL4 0.8617 0.4586 0.1564 0.0367 HLl 0.4859 0.1445 0.0299 0.0049 8L2 0.5965 0.1718 0.0368 0.0064 HL3 0.6733 0.2379 0.0504 0.0076 'HLa 0.6936 0.2585 0.0583 0.0093 112 Table II-Z. Polynomial Equations of A: f(x)dx = ao+ a1t+ a2t2+a3t3+a4t4 a0 a1 . a2 a3 a4 NHLl 1.0 0.2749 -0.3936 0.0825 —0.0041 NHL2 1.0 0.0629 -0.0211 -0.0667 0.0134 NHL3 1.0 -0.0762 0.1821 -0.1280 0.0185 NHL4 1.0 -0.1036 0.2259 -0.1454 0.0205 HLl 1.0 0.3402 -0.5611 0.1586 -0.0137 HL2 1.0 0.2474 -0.3192 0.0455 0.0009 HL3 1.0 0.1334 -0.l352 -0.0238 0.0086 HL4 1.0 0.1056 -0.0914 -0.0395 0.0103 1.0. 0.8 0.6 0.4- 0.2~ Figure II—l. 113 Plotting of I: f(x)dx. 114 00 ft f(x)dx ‘ 2 3 4 Table II-3. Formulation of fo(t) = m = ao+a1t + a2t +a3t +a4t f0 t f(t)dt a0 31 8.2 33 a4 NHLl 0.4442 . 0.1221 —0.l728 0.0366 -0.0018 NHL2 0.3895 0.0240 -0.0082 -0.0259 0.0052 NHL3 0.3344 -0.0254 0.0609 -0.0428 0.0061 NHL4 0.3330 -0.0345 0.0752 -0.0484 0.0068 HLl 0.4785 0.1628 -0.2685 0.0759 -0.0065 HL2 0.4396 0.1087 -0.l403 0.0200 0.0004 HL3 0.4076 0.0543 -0.0551 -0.0097 0.0035 HL4 0.4029 0.0425 —0.0368 -0.0159 0.0041 115 Table II-4. Integrals of the Forward Recurrence Time Function E1 T = 2 T = 3 T i 4 T = 5 NHL1 0.2041 0.0432 0.0092 0.0012 NHL2 0.2638 0.0670 0.0145 0.0020 NHL3 0.3377 0.1192 0.0423 0.0059 NHL4 0.3427 0.1192 0.0348 0.0052 HL1 0.1830 0.0413 0.0082 0.0011 HL2 0.1988 0.0342 0.0070 0.0007 HL3 0.2366 0.0521 0.0117 0.0015 HL4 0.2557 0.0631 0.0135 0.0018 116 Table II—S. Calculate M(t) = I: tf(t)dt T = 2 T =.3 T = 4 T = 5 NHL1 0.6457 1.6016 . 2.0861 2.2206 NHL2 0.4450 1.5394 2.2728 2.5093 NHL3 0.2512 1.2477 2.2850 2.7927 NHL4 0.2310 1.2480 2.3220 2.8258 HLl 0.7343 1.5649 1.9530 2.0622 812 0.6249 1.6627 2.1461 2.5264 HL3 0.5143 1.5825 2.2247 2.4116 8L4 0.4844 1.5594 2.2407 2.4546 117 Table II-6. Calculate Mb(t) = f: t f o(t)dt T = 2 T = 3 T = 4 T = 5 NHLl 0.7297 1.1172 1.2309 1.2320 NHL2 0.7009 1.1785 1.3552 1.3560 NHL3 0.6365 1.1800 1.4860 1.4880 NHLA 0.6379 1.1878 1.4868 1.4890 811 0.7329 1.0733 1.1848 1.1870 HL2 0.7401 1.1357 1.2229 1.2240 HL3 0.7150 1.1621 1.3002 1.3010 HL 0.7145 1.1813 1.3480 1.3490 118 Table II-7. Calculate N(t) = I: t2f(t)dt T = 2 = 3 = 4 T = 5 NHLl 1.0197 .3879 .0410 5.6300 NHL2 0.7356 .4817 .9962 7.0336 NHL3 0.4259 .9666 .5606 8.8032 NHL4 0.3949 .9927 .7119 8.9348 HLl 1.1198 .1673 .4910 4.9696 HL 2 1.0078 .5775 . 2211 5. 7020 HL.3 0.8402 .5202 .6994 6.5178 HL 4 0.7940 .4844 .8184 6.7560 119 2 Table II-8. Calculate No(t) = I: t fo(t)dt = 2 T = 3 = 4 T = 5 NHLl .9116 1.8570 .2393 2.2400 NHL2 .9025 2.0766 .6752 2.6760 NHL3 .8361 2.1853 .2337 3.2350 NHL4 .8381 2.1735 .9933 2.9960 HLl .8985 1.7271 .1046 2.1060 HL2 .9298 1.8930 .1815 2.1820 HL3 .9124 2.0078 .4715 2.4730 HL4 .9152 2.0613 .6257 2.6300 APPENDIX III 120 n (A) Proof ofut D(t) is the probability density function of the merging delay. and D*(s) = 7: 0 "St D(t)dt. - n dI‘ * = m ‘1 = II Thus ( l) dsnD (s)lS=o lot: D(t)dt Ht Since F0 and F are constants, from EQ 6—3 0* (s) = F0 + H* (t) F by EQ 6-6 0* (s) = F0 + G0* (3) F l—G* (s) 1 _ d d ut — (-1)a§__D* (S)|S=O - E75; G: (S) l-G*(s) 8:0 d 60* (s) ._., <1—c* )_d_ 60* (s) - 050:) d (i-c*) ds -—-———-——- ds ds l-G*(s) Is=o ,2 (l - G* (3)) — I: c Go(t) dt f:G(.t)dt f: t G(t)dt = 00 - 0° . 2 (EQ-l) 1 — f0 G(t)dt (1- (O G (t)dt) Since I: G (t)dt = {1; f (t) dt = 1 - F and I: Go (t)dt = f2 f o(t)dt = l - E). _ ‘fTTf (t)dt 1—F (EQ-l)- 0 ° ’ Oth;f(t)dt F F 0 1 _ _ - _ _ pt - F ( Mo(t) 1 F0 (M (t)) ) F MO (t) + 1 - F0 M(t) F 121 2 d d2 0 *(s) And U = D* (s)ls=o = F———2 0 . t 662 ' d5 1-G*(s) |s=o 2 * ”' * * =d * * d * <1 Go (S) - d { d D (8)} _ d (l-G (9))3; Go (6)-Go (S)__(l-G (8)) ds2 _ ds ds ds ds l-G*(s) Is-o u-cuafi = { (1—G*)2._i_,[ < 1 -G*2d_Go*>1-(<1-g*3 2 z 2 - N" {18A (Xi ' M1+ M1 ‘M3) +ieA (Xi 'M1+M2 'M3) } = N— { N1 12A (X1 -M1)2 + N1(M1 “143)2 } + 1 _{NZ ’3 (Xi‘Mz)2 Nl N N2 isA ll 2 2 .51.. 0 .El. (M1 “M3) .E2_ 0 2 N2_(M2 -M3) N 1 + N + N 2 +N Noting that, in our case 2 2 3 = P 012 + 8 (M1 -PM1)2 + (l-P) (P M1) 124 2 2 = P 0 + P (l - P) M.l 1 Q. E. D. (C) Proof of EQ 6-15. Since for any random variable Z, X and any constant a, we have E (.z-a)2 = 8 <2 - 8(2))2 + (R (z) - :02 Using the relation 8 (z) = E (E(ZIX) ) Then VAR = E< (2 - E(Z) ).2IX) = M (2 -E (2 I102 Ix) + (E (zlx) - E(Z) )2 By taking expectation of both sides, we obtain, VAR (2) - E (VAR (zlx) ) + VAR (zlx). APPENDIX IV 0000.100 91 92 93 95 R) 14 15 13 125 SIMULATION PROGRAM FOR PHASE ORE NUDéL, PROGRAM MERGEItCUTLUT=05) ‘ DIMENSION ARRHP(1003).XLEAV(100 ) I=J:0 T=3. TNAIT=ABTT=TWATP=0 ARRHI=O TLMIT=3600. TMOVE=2. N:0 YK:9,9046 YA:3.3 EXPzfi. IF(YK.GE.1)GO T0 91 YK=1 ~ 00 TD 93 JK:YK YKC:JK Y0=YK~YKC R=RAHF(1) IF(R.LT.Y0)HO T0 92 YK=YKC GO TO 93 YKzYKC+1 KleK TR=1.U DO 96 11:1.K1 R=PANF(1) TR=TR*R Y=~ALOG(TR)/YA ABTTsABTTfY IF(ABTT.GE.TLHI*)GU T0 999 IF=ARRMI+IMOVE IF(I.E0.1)GO TU 14 . IFtARRMPcl).GE.1LEAV(1-1))00 TL 14 WAITB=XLEAV(I-17'AHRMP(I)+1HFVE THATB=TWATB¢WAITB M=M+1 ARRMP=XLEAV(I-l)+TMOVf CHECK=ARRMP(I) ‘ IF(CHECK.GT.ABTT)GU T0 1 GAPzABTT-CHECK IFGU 70 2-3 IF(FLOH.EQ.1033.;)00 TH 1:4 YK=1000 YA=3.33 GO TO 206 YK=160.0 YA=13033 GO TO 206 YK=4300 YA=6.66 GO TO 206 YK=2205 YA=5.00 GO TO 200 YK:14.40 YA=4.00 IF(YK.GE.1,) 00 T0 91 YKzl, GO TO 93 JKzYK YKC=JK Y0=YK-YKC R=RANF(1) IF(H.LT.Y0> GO *0 92 YK=YKC 00 T0 93 YK=YKC+1 K1=YK TR=1.0 DO 95 11:1:Kl R=RANF(1) TR=TR*R Y=—ALOG(TR)/YA ABTzABT+Y IF(ABT,0T.XLIHT) 00 70 909 IAUTzACT ICT=CT NOCYzlABT/ICT XNOCY=HOCY TT=CT*(ABT/CT~ALHCY) IFtTT.LE.RT) 00 T0 4 ‘ IF((NWG.EQ.0).AHD.(HwR.Eu.y))GC T3 41 GO TO 42 FAET=A8T+FARRT IF(NOCY-NNOCY) ?.7.6 IF((MNG.EQ.0).AhD.(H0R.FU.F))GC T0 43 GO TO 45 FAET:ABT+(RT-TT7+3.8+FAFHT IF(HOCY*NNOCY)5;5:6 1WR=INR+1 NWR:NWR*1 M8101=0 GO TO 1 NNOCY=NOCY GO TO 8 _ IF(LDIWG,EO.KDIHG) 00 T0 103 MSIGl=1 IWG=IHG+1 105 115 116 110 11 15 49 40 47 141 142 17 171 172 173' 174 176 277 GO T0 1 129 IF((TT-RT>.GE.YJ GU 70 115 YC:Y GO TO 116 YC=TT~RT IIMzNNG+NWR IFfYC.LE.(XHEAD(IIH)+TH0VE))C0 T0 113 LDIHG=KDING KDIHG=IHG GO TO 106 uwn=mwe+1 GO TO 106 ABT=ABT“Y IF(IM.E0.0) GO TU 9 JPS=JPS*IM INTT:IHR+ING JPS=IWTT-IPS IF(JPS.GT§M)GO T0 1T [Hza GO TO 11 JPS=N IMzN-JPS IPS=IPS+JP3 IPA38=JPS ABTT:FAET IFcABTT.LE.XLVLT) GO TO 15 GO TO 49 ' ABTT=XLVLT IFLABTT-ABAT)17:17.43 IF(IAS.EU.0)GU To 27 IF((ABTT-XABAT).GE.T) no Tn c7 1F(ABTT.GE.XABAT) nu Tn 147 GO TO 17 HAIT=1 XABTT=ABTT XABAT=ABTT GO TO 17 JL=JL+1 IF(WAIT.EO.1) UV T“ XLTGG TO 282 GO TO 283 YYK=5.7829 YYA=1/005613 GO TO 117 YYK=609326 r4 A '..‘ 179 178 281 117 81 82 83 19 27 145 46 YYA=1/0.3247 GO TO 117 YYK=806382 YYA=1/0.2971 130 GO TO 117 YYK=80247S YYA31/002758 GO TO 117 YYK=9.1641 YYA:1/0.3263 GO TO 117 YYK=80259 YYA=1/0.297 GO TO 117 YYK:9,9046 YYA=1/0v3031 GO To 117 YYK=8.2617 YYA=1/0,3003 IF(YYK,GE.1.)GU T0 81 YYK=10 GO TO 83 IKzYYK YYKC=IK YY0=YYK‘YYKC R=RANF(1) IF(R.LT.YYu) um T0 8? YYKzYYKC GO TO 83 YYK=YYKC+1 KK=YYK TR=1.0 DO 85 12:1.HK PzRANF<1> TR=TR*R YY=~ALOG=XHEAD(11-15+2o1 CONTINUE SP=14,5 ’ XDIST=800. NJA=0 QLVLT=0 IENT=0 QXLTI1)=0 NJL=0 QARTT=0 QTIH8=0 QHAIT=0 EXPP=10. EXP:1OI TMOVE=2. 0137:503. T=3. FLOW:1000¢ xLIMT=3600. N=0 XLVLT=0 NJA=0 HIAS=0 IPASS=0 TIMSs=0. TIMMS=00 JL=0 NNOCY=0 ABT=00 ABTT=00 JJ=0 GT=4OO RT=200 IAS=0 XFART=XDISTISP QABAT3-EXPP*ALOG(RANF(1)) A8AT:-EXP?AL00(HANF(1)) IPS=0 IM=0 JPS:0 XLVLT=0. LDIwc=o .).CXAT(1OO§) KDIHG=C INR=0 IHG=O r48161=0 133 CT=GT+RT NNG=0 NHR=0 FARRT=DIST/SP JJ=JJ+1 _ 152 [F(CT-XHEAD(JJ)I1:151:151 151 N=N+1 52 GO TO 1 1 IF(FLOW.EQ.300.’)GO T0 213 IFIFLOW-EQ.600.’)GO TO 2-2 IF(FLON.E0.800.‘)G0 T0 2I3 IFIFLow.EQ.1ooc-0)Go To 2:4 YK=1000 YA=3.33 GD 70 206 210 YK=160.0 YA=13.33 GO TC 206 202 YK=4000 YA=6,66 GO TO 206 203 YK=2205 YA=5.00 GO TO 200 204 YK=14040 YA=4.00 206 IF(YK.GE.1.) GO TO 91 YK=1. GO TO 93 91 JKzYK YKC=JK YQ=YK-YKC R=RAMF(1) IF(P.LT.YQ) GO TO 92 YK=YKC GO TO 93 92 YKzYKC+1 93 K1=YK TR=1.0 DO 95 11:1.K1 R=RANF(1) 95 TR=TQ*R Y=-ALOG(TR)/YA ’ = 8T+Y 2 1EIAST.GT.XLIMT: GU T0 999 IABTaABT ICT=CT NOCY=IABTIICT XNOCY=NOCY TT=CT*(ABT/CT-XN$8Y; ‘ . 5.8T) GO “ 1;:ILWE.EQ.0).AND.(NMR.ED.U)\GC T3 41 GO TO 42 41 FAET=A8T+FARRT - 7 6 0CY“NNOCY) 7: a w 4 4% 1;:?NNG.EQ.0).AWO.(NMR.EH.U))GU TJ 43 GO TO 45 3 FAET=ABT+(HT-TT)+3.8+FAPFT 45 11 15 49 40 147 47 141 142 17 171 IF(NOCY-NUOCY)5,5.6 INR=IWR*1 134 NHP=NNR+1. MSIGl=0 GO TO 1 NNOCY=NOCY GO TO 8 IFtLDIwG.Eo.KDIwG) Gm TO 135 MSIGl=1 ' ING=ING+1 GO T0 1 IF((TT-RT).GE.Y) GU T0 115 YC=Y GO TO 116 YC=TT~RT IIN=NNG+NNR , IFGO T0 346 IF(QABAT.LT.QXLT(M3LI)GO T0 445 GO TO 346 445 NAITS=0XLT(MJL)~0AHAT+NAITS QOXAB=QXLTIMJL>+TMUVF OXAT$OGXAB '80 To 340 346 OXATINJA);0ABAT QOXAR=OABAT GO To 340 999 D0 30 JTl=1.JL TIMM=XLT(JT1)-XAT(JT1) TIMHs=TIMMs+ TIMH 3a CONTINuE D0 333 MJ71=1aHJL QTIMM=OXLT(HJT1)-OXAT(HJT1) QTIMS=OTIMS+0TIMM 33o CONTINUE MSM=JL+HJL XMSNzMSM SUMM=T1MMS+QTIMS TIMSS=TIMH5+wAITs AVENM=SUMM/XMSH AVEINS=TIMss/XMSM PRINT 32. IPS PRINT 33. AVENI PRINT 34. AVEIHS PRINT 35. 3L PRINT 35. HJL PRINT 39,FLOH,EXP.T,UIST 39 FORMAT(* MAEN FLUH=*»F1*.4.vMIJOR FL3N=*aF1 .4.* T=*:F4.2}? C DISTANCE=*.F1 .4) 32 FORMAT(* VOL VFP 951 up = .LIjxy 33 FORMATI~ AVE wAITInG TING In THE w:a31me PLACE : .,pj,,4; 34 FORMAT(* AVE WAITING TIME IN TFE SYSTEM = *,F1‘.4) 35 EORMAT(* an In MIHUH STLEAH = «;I1 ) ND REFERENCES 10. ll. 12. 13. REFERENCES Adams, W.F. "Road Traffic Considered as a Random Series," Journal of the Institute of Civil Engineers. (November, 1936): 121. Bleyl, R.L. "Speed Profiles Approaching a Traffic Signal," Bureau of Highway Traffic Repprt, Pennsylvania State University, 1972. Blumenfeld, D.E., and Weiss, G.H. "0n Queue Splitting to Reduce waiting Times," Transportation on Research, 4. (l969): Blumenfeld, D.E., and Weiss, G.H. "On the Robustness of Certain Assumptions in the Merging Delay Problem," Transportation on Research, 4. (1970): 125—139. Cochran, W.G. "The X2 Test of Goodness of Fit," Annals of Mathematical Statistics, Vol.23 (1952): 245 & 315. Cox, D.R. Renewal Theory. London, Methuen, 1962. Draper, N.R., and Smith, H. Applied Regression Analysis. New York: Wiley, 1966. Drew, D.R. Traffic Flow Theory and Control. McGraw—Hill, New York: 1968. Grace, M.J. and Potts, R.B. "A Theory of the Diffusion of Traffic Platoons," Operation Research, Vol 12, (1964): 255-72. Graham, E.L., and Chenu, D.C. ”A Study of Unrestricted Platoon Movement of Traffic," Traffic Engineeripg_(April, 1962): ll-13. Grecco, W.L., and Sword, E.C. "Prediction of Parameters for Schahl's Headway Distribution," Traffic Engineering (February, 1968): 36-38. Greenshields, B.D., Shapiro, D., and Erickson, E.L. Traffic Performance at Urban Street Intersections. New Haven: Yale Bureau of Highway Traffic, (1947). Hamming, R.W. Numerical Methods for Scientists and Engineers, New York: McGraw—Hill, 1962. 138 14. 15. l6. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. .139 Herman, R., and Weiss, G. "Comments on the Highway-Crossing Problem," _gperation Research, Vol.9,No.6 (1961): 828-40. Herman, R.; Potts, R.B.; and Rothery, R.W. "Behavior of Traffic Leaving a Signalized Intersection," Traffic EngineeringAand Control. (January, 1964): 529-33. ITE Technical Committee 6P6. "Land Use and Demography Growth Versus Forecast," Traffic Engineeripg, Vol.47, (March, 1977): 42-44 0 Kell, J.H. "A Theory of Traffic Flow on Urban Streets," Proceedings of the 13th Annual Western Section Meeting, Institute of Traffic Engineers. (1969): 66-70. Kendall, D.G. "Some Problems in the Theory of Queues," Journal of the Royal Statistical Society, Series B, Vol.13 (1951): 151-185. Lewis, B.J. "Platoon M6vement of Traffic from an Isolated Signalized Intersection," Highwanyesearch Board Bulletin, 178, (1958): 1-11. Major, N.G., and Buckley, D.J. "Entry to a Traffic Stream," Proceedings of the Australian Road Research Board, Vol.1, Part 1 (1962): 206-228. Massey, F.J. "The Kolmogorov-Smirnov Test for Goodness of Fit," Journal of American Statistical Association, Vol.4 )1951): 68-78 0 Nemeth, E.A., and Vecellio,R.L. "Investigation of the Dynamics of Platoon Dispersion," Highway Research Record, No.334 (1970): 23-33. Nemeth, E.A., Vecellio, R.L., and Treiterer, J. "Effect of Signal Spacing on Platoon Dispersion," Engineering Experiment Station, Ohio State University, Final Report EES 311, (July, 1973). Neter, J., and Wasserman, W. (Applied Statistical MOdels. Homewood, 111.: IRWIN, (1974). Newell, G.F. "Statistical Analysis of Flow of Highway Traffic Through a Signalized Intersection," Applied Math, Vol.13 (1956): Pacey, G.M. "Progress of a Bunch of Vehicles Released from a Traffic Signal," Road Research Laboratory, Crowthorne, England, RRL Report RN/2665, (1956). ’5' 27. 28. 29. 30. '140 Robertson, D.I. ”TRANSYT: A Traffic Network Study Tool," Road Research Laboratory, Crowthorne, England, RRL Report LR/253, (1969). Tanner, J.C. "The Delay to Pedestrians Crossing a Road," Biometrika, Vol.38 (1951): 383-392. Tolle, J.E. "The Lognormal Headway Distribution Mode," Traffic Engineering and Control, Vol.13. (May, 1971): 22-24. Weiss, G.H., and Maradudin, A.A. "Some Problems in Traffic Delay," Qperation Research, Vol.10,No.l (1962): 74-104. ”'7Ifl'l‘m‘juflljjfllmHEHIQQHEIIMM‘ITF