A NON-ITERATIVE METHOD FOR URBAN TRAFFIC SIGNAL TIMING Dissertation for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY GAIL HAROLD GROVE 1977 m] LIE TEAL t Nlic'tigm State Uim c-rsrty (,7, . .w—m This is to certify that the thesis entitled A Non-Iterative Method for Urban Traffic Signal Timing presented by Gail Harold Grove has been accepted towards fulfillment of the requirements for Doctor of Systems Philosophy degree in Science Major professor Wfiflww 0-7639 ABSTRACT A NON-ITERATIVE METHOD FOR URBAN TRAFFIC SIGNAL TIMING BY Gail Harold Grove The primary objective of this research was to improve the basic TRANSYT traffic network signal timing method by replacing Robertson's time consuming hill- climbing optimization process with a much faster and thus less costly running non-iterative split and offset calculation technique. The TRANSYT/G splits were obtained by using differential calculus methods on the network objective function. This objective function was constructed of tractable approximations to the stops, uniform and random delay terms for each link within the network. The TRANSYT/G offsets were determined from the Morgan and Little maximal bandwidth offsets modified so as to include: an improved apportionment of unequal directional bandwidths, queue growth and decay allow- ances by means of partial excess green shifts, and an application to networks. Gail Harold Grove Based upon the results of testing two data sets from Ft. Wayne, Indiana and Washington, D.C., it was concluded that: 1. This author's split estimation method is superior to Robertson's in terms of lower objective function and higher system speed values. This is true for its use in both TRANSYT/G for on-line control applications and in TRANSYT for more accurate off-line signal timing optimization studies, and, 2. TRANSYT/G's vastly improved computer running time versus TRANSYT makes it a potential candidate for an on-line signal timing con- trol method. A secondary result of this research was the de- velopment of a finite traffic queue dispersion model for use in off-line simulation studies requiring a more accurate model than the infinite queue version presently being used in TRANSYT. A NON-ITERATIVE METHOD FOR URBAN TRAFFIC SIGNAL TIMING BY Gail Harold Grove A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering and Systems Science 1977 é;gcxtfifi I; ACKNOWLEDGEMENTS I wish to thank my major Professor, Dr. John B. Kreer for his many helpful suggestions and guidance throughout my research and my committee members for their timely review of this dissertation. A special thank you is to be extended to my wife Sandra for her patience and continual encourage- ment during the many months of this research effort. ii Chapter I II III IV TABLE OF CONTENTS INTRODUCTION 1.1 Benefits of Improved Urban Traffic Flow 1.2 Previous Signal Timing Optimization Procedures 1.3 Preview THE TRANSYT MODEL 2.1 Objective Function 2.2 TRANSYT Flow Model 2.2.1 Basic Assumptions 2.2.2 Flow Histograms 2.2.3 Traffic Queues 2.2.4 Platoon Dispersion 2.2.5 Stops, Uniform and Random Delays 2.2.6 Dummy Links 2.3 Optimization Procedure TRANSYT/G NEAR-OPTIMUM SPLIT CALCULATION 3.1 Split Bounds 3.2 Tractable Model of Stops and Uniform Delay 3.3 Tractable Model of Random Delay 3.4 Near—Optimum Split 3.5 Two-Phase Double Cycling 3.6 Non-Overlapping Multiphase Single Cycling 3.7 Overlapping Multiphase Single Cycling 3.8 Split Sensitivity ARTERIAL OFFSET CALCULATION 4.1 Time-Space Diagrams 4.2 Maximal Arterial Bandwidth iii Page 10 11 12 14 19 24 30 33 35 40 42 52 57 63 68 72 72 72 Chapter Page V TRANSYT/G OFFSET CALCULATION 87 5.1 Unequal Bandwidth Apportionment 88 5.2 Excess Green Adjustment 91 5.3 Network Considerations 94 VI NETWORK APPLICATIONS 99 6.1 Link Numbering System 99 6.2 Selection of Networks 100 6.3 Input Data 104 6.4 Comparison of TRANSYT/G versus TRANSYT 107 VII CONCLUDING REMARKS 112 7.1 Conclusions 112 7.2 Suggested Future Research 113 APPENDIX A: TRAFFIC QUEUES AND THEIR DISPERSION 114 APPENDIX B: TRANSYT AND TRANSYT/G FLOW CHARTS AND LISTINGS 128 BIBLIOGRAPHY 143 iv Table 2.1 3.1 6.2 A.l A.2 A.3 LIST OF TABLES Random Delay Values Random Delay Derivative with Respect to Degree of Saturation Random Delay Derivative with Respect to Split Split Sensitivity Comparison of TRANSYT/G to TRANSYT - Part I Comparison of TRANSYT/G to TRANSYT - Part II Expected Queue Length, E(n) Queue Length Variance, 02 Travel Time, t Page 27 46 48 70 108 110 119 121 126 Figure 2.1 2.2 2.3 2.8 3.1 LIST OF FIGURES A Typical TRANSYT Flow Histogram A Typical Uniform Flow Histogram Green, Amber, and Red Time Intervals in a Cycle Length Departures and Queue Length for Non- Uniform Arrivals on the jth Link Node-Link Diagram and Stoplines Platoon Dispersion as a Function of Travel Time Random Delay Versus Normalized Saturation Convex and Non-Convex Objective Functions Phase A and B Time Interval Rela- tionships Departures and Queue Length for Uniform Arrivals on the jth Link Normalized Split Relative to Arrivals Double Cycled Timing of a Two-Phase Signal Single Cycled Timing of a Non-Over- lapping Four-Phase Signal Four-Phase Overlapping Signal Model Single Cycled Timing of an Over- lapping Four-Phase Signal Typical Time-Space Diagram vi Page 13 13 16 17 22 23 28 32 38 43 53 54 58 64 65 73 Figure Page 4.2 Typical Constant Velocity Time- Space Diagram. 74 4.3 Two Group 1 Signals Limiting the Green Band 78 4.4 Two Group 2 Signals Limiting the Green Band 79 4.5 A Group 1 Signal and a Group 2 Signal Limiting the Green Band 30 4.6 Adjustment of the Green Bands 84 5.1 Total Bandwidth Apportionment as a Function of Cumulative Arrivals 90 5.2 Shifting of Excess Green Time for Queue Clearances 92 5.3 Interconnection of Two, Two-Way Veins 96 6.1 General Link Numbering System and a Specific Example 101 6.2 Node-Link Model of Modified Ft. Wayne, Indiana Area 102 6.3 Vein Interconnection Model of Figure 6.2 103 6.4 Node-Link Model of Modified Washington, D.C. Area 105 6.5 Vein Interconnection Model of Figure 6.4 . 106 A.1 Smoothing FactOr Versus Travel Time 127 8.1 TRANSYT and TRANSYT/G General Flow Diagrams 130 8.2 TINPUT Type 4 Card Input Section Listing 131 3.3 SPLIT Listing 132 3.4 OFFSET Listing 136 vii LIST OF SYMBOLS F = the network objective function, Fj = the objective function for the jth link, fi = the objective function for the ith node, Sj = the vehicle stops per second on the jth link, Dj = the vehicular uniform delay on the jth link, Rj = the vehicular random delay on the jth link, Wj = the delay weighting factor on the jth link, K = the link stop penalty factor, tgj = the start of the green inverval for the jth link, taj = the start of the amber interval for the jth link, trj = the start of the red interval for the jth link, tej = the time when the queue length first decays to zero for the jth link, ij(t) = the arrival flow rate for the jth link, oj(t) = the departure flow rate for the jth link, Gj = the green interval saturation flow rate for the jth link, Ij(t) = the cumulative number of arrivals at time t on the jth link, Ojtt) = the cumulative number of departures at time t on the jth link, viii the queue length at time t on the jth link, the cycle length, the platoon dispersion or smoothing factor, the travel time down a link, the average travel time, the expected travel time, the degree of saturation on the jth link, the oversaturation time allowed, the random delay slope in the oversaturation region for the jth link, the optimum cycle length, the total lost time per cycle, the lost time for the primary phase, the ratio of maximum directional flow to saturation flow for the jth phase, the minimum green time for the jth phase, Ithe jth phase, the normalized signal split for the ith node, th the integer valued signal split for the i node, the near-optimum normalized split for the ith node, the bounds on the integer value split, the sum of the amber and minimum green times h for the jt phase, ix IN. 3 IN. t.. 13 ol h the uniform arrival flow rate on the jt link, the uniform arrival flow rate on the jth link during the kth half cycle, the capacity efficiency of the jth link, h the capacity efficiency of the jt link h during the kt half cycle, the ratio of total arrivals to GjC on the jth link, the number of links terminating at the ith node th h the start of the i time interval for the jt phase, the start of the ith time interval for the jth h phase during the kt half cycle, th the jth phase split for the i node, the random delay approximation constants for the jth link, th the weighted sum of stops for the i node, the weighted sum of uniform delays for the ith node, the weighted sum of random delays for the ith node, the second partial derivatives of the fi with respect to the Sij' the phase split vector for the i the red time for the jth node, th node, X 13(5) '5'! the outbound (inbound) directional bandwidth, h the location of the jt node downstream from the reference point, the outbound (inbound) vehicle speed along the jth link, th h the travel time from the i node to the jt node in the outbound (inbound) direction, th the relative offset from the i node to the jth node measured between the center of the red intervals of the two nodes, the minimum green for the set of nodes along an arterial, the offset shift of the jth node for unequal directional bandwiths, the outbound (inbound) platoon length, the ratio of total arrivals to Gj on the jth link, the maximum equal bandwith, the queue clearance offset shift of the jth node relative to the critical node, the excess green to the right of the rear edge of the outbound (inbound) bandwidth for the jth node relative to the critical node, the outbound (inbound) average cumulative arrivals, xi CJ' zn(e) E(n) the TRANSYT/G offset for the jth node relative to the critical node, the steady state time-independent probability of n vehicles being in the system, the ratio of the queue arrival rate to the service rate, the finite maximum number of vehicles in the queue, the probability generating function for the discrete distribution of a finite maximum allowable queue length, the expected number of Vehicles in the queue, the variance of E(n). xii CHAPTER I INTRODUCTION 1.1 Benefits of Improved Urban Traffic Flow Modern society has become more time conscious in its day to day activities, especially when commuting be- tween home, work, and recreation areas. As a result, man is intolerant of unnecessary delay in his travel. A significant portion of our population must contend with driving through urban areas having considerable conges- tion of vehicles. Thus, one of the primary duties of a transportation engineer or planner is to improve traffic flow through our cities by decreasing vehicle travel time. In light of our energy shortage, it has become increasingly important to reduce the amount of vehicle stop and go movement which wastes so much of our dwindling fuel supply. Virginia engineers (CR1) claim a potential savings of 125,000 gallons per day for a computer con- trolled network of 400 signals in washington, D.C. Based upon sixty cents per gallon, this savings could be $75,000/day in this area alone. Increasing traffic densities in urban areas, limited space for additional roadways, and the escalating cost of roadway construction make it mandatory to l undertake research which contributes to the efficient utilization of our existing urban roadway facilities. The research described in this thesis is directed to- wards increasing the efficient use of our existing urban street networks, while simultaneously providing improved service to the drivers. This has been accomplished by the development of a real-time algorithm for calculating traffic signal synchronizations which could be used for on-line computer control of an urban network of signalized intersections. 1.2 Previous Signal Timing Optimization Procedures In Hillier's delay-difference method (HIl), he considered tree or ladder-type networks which could be simplified into a single link by combining links in parallel and in series. The objective was the minimiza- tion of delay in each link where flows were considered uniform. The delay on a link was assumed to be a func- tion of the offsets of the two signals at the ends of this link and is independent of all other network off- sets. This algorithm calculates the sum of the two directional delays by varying the integer valued offsets and selects the offset minimizing the sum of the two delays. This method appears time consuming for cal- culating the offset for a single link but has the ad- vantage that computations increase linearly with the number of links unlike most other models. Allsop extended Hillier's work to general non- simplifiable grid networks (ALl) and later used a dynamic programming concept which minimizes the total delay and gives a global minimum (AL2). However, this technique is too slow to be used for real-time control of medium sized networks of approximately 50 inter- sections. Inose, et a1. (INl) developed an algorithm applicable only to tree networks in which the common cycle time is selected as the maximum of the individual, ideal intersection cycle times assuming uniform arrivals at isolated intersections. Splits are then selected by minimization of delay while offsets are calculated as a function of the heavier of the directional vehicle volumes on the links connecting each pair of nodes. Inose, et a1., assume no turning movements and no platoon dispersion. Any link which would form a closed loop of links within the network is not used in the offset cal- culations, thus restricting application to tree networks. This method is simple and effective provided that there are very few turning movements. It is unlikely that the total delay is minimized. SIGOP (TRl) was the first systematic signal timing optimization program applicable to a general network. The length of each split phase for a signal is set pro- portional to the total or critical flow in each phase. The total flow was considered to be the sum over all lanes and approaches while the critical flow was the maximum noted per lane through the given node. The optimization procedure determines the offset differences for each link and optimal offsets for the entire net- work. The objective function is’a linear combination of stops and delays. An n by n matrix inversion is required, where n equals the number of nodes. Two deficiencies are apparent in SIGOP. First, the offset optimization procedure produces a local optimum rather than a global optimum. Second, since the stochastic effects on each link are ignored, the lower bound on cycle time is selected as the optimum especially when- ever the capacities of the signalized intersections are approached. Shortly, thereafter, Robertson (R01) developed a more sophisticated model and more effective optimiza- tion process called TRANSYT. A detailed description will be found in Chapter II. TRANSYT has been field tested in London and Glasgow with favorable results. Kaplan and Powers (KPl) presented a comparison of SIGOP and TRANSYT as applied in San Jose and Glasgow; each area having high signal density. They concluded that: 1. There is no measurable difference between SIGOP and TRANSYT for short cycle times, 2. Double cycled TRANSYT produced travel times which were 5 percent less than SIGOP and single cycled TRANSYT in the Glasgow network, and 3. The TRANSYT model appeared more accurate in predicting travel times. For more details on these earlier algorithms, Munjal and Hsu's comparative study (MHl) is an excellent summary of the state-of-the-art circa 1972. Since this time, several other signal timing methods have appeared in the literature. Messer et a1. (MWl) have applied a variable sequence, multiphase, progression optimization program to the real-time control of a Dallas, Texas arterial. Good progressions were obtained and no apparent problems due to variable phase sequencingwere experienced in this pilot study. Gartner (GAl) confirmed by direct field observa- tions on a major Toronto arterial that his microscopic flow pattern analysis reduced traffic delay. Gartner and Little (GL1) presented a dynamic programming approach to obtain the splits and offsets of a general network. They extended Hillier's combination method to general networks. Actual field evaluations were being planned at the time of their article. Lieberman and WOo's SIGOP II (LWl) is composed of a flow model and a dynamic programming methodology which minimizes an objective function composed of vehicle delay, stops, and excess queue length. Turning movements, lane channelization, multiphase control, signal split constraints, platoon disperson (from TRANSYT), and short term volume variations are included. SIGOP II is claimed to possess computational speed which varies linearly with the number of nodes and is somewhat slower than SIGOP but faster than TRANSYT. However, it appears that program tapes and the field validation are incomplete at this time. Even though this review of previous contributions was brief, it should be noted that a vast amount of time and effort has been expended by many researchers and that many volumes are devoted to their results. However, one should proceed onward to the preview of this author's work. 1.3 Preview Chapter II describes the objective function, the traffic flow model and the optimization procedure of the existing TRANSYT computer program. The variables utilized in the optimization of the objective function are the individual signal splits and the offsets between the signals comprising the network. Appendix A contains a derivation of the dispersion of traffic queues and an extension to finite queue lengths. The main body of this author's research comprises Chapters III and V. Chapter III is a detailed deriva- tion of the near-optimum split calculation employed in TRANSYT/G. Chapter IV is a brief description of arterial offsets, while Chapter V details the TRANSYT/G offset method. The offsets are determined from an extension of the Morgan-Little maximal bandwidth concept. Both splits and offsets are calculated in a non-iterative fashion, thus vastly reducing the computer run time and cost as compared to the existing TRANSYT model. Portions of the Ft. Wayne, Indiana and the Wash- ington, D.C. downtown areas were selected for a comparison of the improved TRANSYT/G program versus the TRANSYT pro- gram. Chapter VI contains the two network applications while Appendix B includes general computer program flow charts and listings. Conclusions and suggested future research ob- jectives are included in Chapter VII. CHAPTER II THE TRANSYT MODEL The mathematical model characterizing traffic flow is composed of a set of expressions or equations representing the traffic dynamics and its ultimate con- trol based upon the network's physical configuration and the behavior of the traffic operating within its boundaries. For control purposes, we desire a model capable of providing information of significant detail for the calculation of the selected traffic signal timing vari- ables. However, it must not be so detailed that it is incapable of real-time computer speeds encountered in on-line control situations. The Traffic Network Study Tool (TRANSYT) de- veloped by Robertson (R01, R02) is a method of optimizing traffic signal settings using an off-line computer model of the known network system conditions. It is composed of three main elements: 1. A network objective function for ranking different sets of signal settings, 2. A traffic flow model used solely for gen- erating avalue of the objective function for a given set of signal timings, vehicle flows and network geometry, and 3. A hill-climbing optimization process that alters the signal settings and determines if the ob- jective function has been reduced or not. Before proceeding further on this subject some terminology must be defined. In an urban traffic net- work which is being modeled, each signalized inter- section will be represented by a node and each signifi- cant directional traffic stream leading into an inter- section will be represented by a link. This concept of a link applies to special bus-only lanes, pedestrian crossings and heavy flow left turn lanes in addition to the normal through traffic lanes. In a network, one traffic signal is usually chosen as the reference sig- nal. Then the phasing.of any other network signal is de- termined by the time interval between the centers of the main street reds of the two signals under consideration. This interval ranges from zero up to the cycle length and is called the offset. The other parameter of in- terest is the split, defined as the ratio of the main street green and amber time intervals to the cycle length. The synchronization problem requires the specification 10 of a split and offset for each signal in the network such that some predetermined objective will be achieved. 2.1 Objective Function In TRANSYT, Robertson defined the overall net- work function, F, as F = jgl Fj = jgl [Ksj + Wj(Dj + Rj)J, (2.1) where 83. = the vehicle stops per second on the jth link, K = the stop penalty common to all links in the network, Dj = the vehicular uniform delay on the jth link, Rj = the vehicular random delay on the jth link, Wj = the delay weighting factor on the jth link, Fj = the objective function for the jth link, and 2. = the number of links comprising the network. A typical urban traffic stream is composed of many different types of vehicles, such as automobiles, various sized trucks, buses and pedestrians, each with their own average speeds. Thus, each link j is allowed to have its own delay weighting factor Wj' In general, most links are assigned unity weighting. However, it may be desirable to have non-unity delay weighting on certain links such as those containing bus-only traffic. By use of the objective function different sets of signal timings, in this case splits and offsets, can 11 now be compared. The set of timings having the smallest objective function is considered to be the best of those sets investigated. If a network having n signalized intersections and 2 links, the stop penalty K and the link delay weighting factors W., j = l,...,£ are given then the 3 stops, Sj' the uniform and random delays, Dj and Rj respectively, for all 2 links can be determined from a traffic flow model of the urban traffic network. 2.2 The TRANSYT Flow Model The major characteristics of vehicular flow in a signalized network are represented in the TRANSYT flow model. 2.2.1 Basic Assumptions The basic assumptions of the TRANSYT flow model are: 1. All major intersections of the network have traffic signals or are controlled by a priority rule, 2. All of the signals in the network have a common cycle time or one-half of it, commonly referred to as 'double cycling,‘ 3. Traffic enters the network at a constant Specified rate on each approach, and 4. The percentage of vehicle volumes turning at each signalized intersection remains constant through- out the entire cycle. 12 2.2.2 Flow Histograms All traffic behavior calculations required for the stops and delay terms are made by manipulation of three types of histograms and representation of indi- vidual vehicles is not required. These three flow histograms are: 1. Arrival - the traffic flow histogram that would arrive at the stop line at the end of the link if it were not impeded by a signal at the stop line, 2. Departure - the traffic flow histogram that departs from the stop line of a link, and 3. Saturation-—the traffic flow histogram that would leave the stop line if there-were enough flow to exceed the intersection's capacity. The actual flow histogram for any link during any one cycle will vary from the average histogram due to the random nature of individual vehicles and is accounted for in the objective function's random delay term des- cribed later in this chapter. The general histogram form is illustrated in Figure 2.1 and.is significantly more representative than uniform models such as Figure 2.2. 13 Flow Rate, 0.0 0.2 0.4 0.6 0.8 1.0 Normalized Cycle Time Figure 2.1 A Typical TRANSYT Flow Histogram Flow Rate 0.0 0.2 014 0.6 0.8 1.0 Normalized Cycle Time Figure 2.2 A Typical Uniform Flow Histogram 14 2.2.3 Traffic Queues Before proceeding further, several definitions are required for the development of the basic queueing process used in the TRANSYT flow model. These defini- h tions for the jt link are: ij(t) = the arrival flow rate from the arrival histogram (vehicles/hour), oj(t) = the departure flow rate from the depar- ture histogram (vehicles/hour), Gj = the green interval saturation flow rate from the saturation histogram (vehicles/ hour), Ij(t) = the cumulative number of arrivals at time t, Oj(t) = the cumulative number of departures at time t, Qj(t) = the queue length at time t, tgj = the start of the green interval, taj = the start of the amber interval, trj = the start of the red interval, tej = the time when the queue length decreases to zero, and C = the cycle length. The arrival flow rate for each link during each cycle length spans three different time intervals, namely; the green, the amber, and the red time intervals. Such 15 a representation is illustrated in Figure 2.3, covering two cycle lengths. The basic cumulative arrival and departure re- lations are defined as: I.(t) f i.(T)dT, (2.2) J rj and t Oj(t) ft O.(T)dT, (2.3) rj 3 where t . < t < t . + C. r3 — - r3 This traffic flow model assumes that the arrivals are periodic, that is; the same arrival histo- gram is repeated cycle after cycle, thus ij(t) = ij(t - DC), n = 1,2,... 0 (204) Several important requirements of this flow model are that the signalized intersection must not be saturated, namely; t .+C . o + _ O o + - o o . Ij(trj C) ftrj 13(r)dr < (tr) C th)GJ,(2 5) and that the arrival rate during the green interval does not equal or exceed the saturation flow, '. t < . f . < < . + . . iJ( ) G3 or tgj _ t ta] C (2 6) Based upon these assumptions, requirements and-Figure 2.4, it can be shown that the vehicle flow.entering a 16 camcoq maoxo m CH mam>umch wEHB mom can .H0QEd .cmmuw m.~ whamwm no no no no 0m 9+. u 0+. u o . u . u 0+.» 0+.» .u .u 17 G. ————————————————————————— 3 Arrival I Rate I I o I g, ___________ 3 Departure Rate 0 Qj max ——————————— Queue Length 0 r I l 1 . t . t . t . t .+ tr] 93 e) a] r3 C Figure 2.4 Departures and Queue Length for Non-Uniform Arrivals on the jth Link 18 link also exits that link during a cycle length. Thus, since the queue must clear prior to the end of the green interval, we have Oj(trj + C) = Ij(trj + C). (2.7) In a similar manner, all arrivals that have accumulated from the start of the red interval up until the time when the queue decays to zero must depart during the time from the start of the green to the same point when the queue clears. Thus, Oj(tej) = Ij(tej) . (2.8) During the remainder of the cycle when there is no queue, the departures equal the arrivals at each point in time, i.e., oj(t) = 1j(t) for tej : t < trj + C, (2.9) or r3 3 + C) - Ij(tej)(2.10) Over the total cycle length, the departure rate can be summarized as 0 for t . < t < t . r3 — 93 ' t = O I o O oJ( ) GJ for tgj i t < teJ (2 11) i 0 t f t o < t < O + O J( ) or e) _ tr] C 19 The model requires the treatment of each link and its associated flow histograms for two successive cycle lengths before proceeding to the next link. The first cycle allows for proper development of queues and flow histograms while the second cycle length is used for the calculation of the stops and delays for the ob- jective function. Due to the previously mentioned un- saturated flow assumptions and requirements, the queues will decay to zero and remain zero only once during any one green time interval. Thus the queue is always zero at ta" This is illustrated in Figure 2.4 and is re- 3 presented mathematically by Ij (t) trjgtgtgj Qj(t) = Ij(t) — oj(t) = Ij(t)-(t-tgj)Gj for tgjf-titej 0 tej_<_t 1 + 0.5t and E(T) = the expected travel time. The side constraint on Robertson's equation is required to ensure that the magnitude of flow will de- crease as the platoon disperses. If this constraint were violated the sequence {ij(-)} would be monotonically increasing and thus would not be an accurate representa- tion of the actual platoon dispersion phenomenon. 21 The smoothing factor A is bounded above and below by l and 0 respectively and decreases in a nega- tive exponential manner as the travel time, t increases from zero. Thus, this so-called exponential smoothing is a function of the travel time down a link. . A statistical derivation of traffic queues and their dispersion is presented in detail in Appendix A. As an example of platoon dispersion, consider a portion of an arterial represented by the node-link dia- gram of Figure 2.5. For the following case: il(t) = 1000 vehicles/hour, for all t G1 = 2000 vehicles/hour t = 0 seconds r1 t = 25 seconds 91 tel = 50 seconds C = 60 seconds t = 5, 10, and 20 seconds. The 01(t) departure pattern leaving stopline l and the i2(t) arrival histogram appearing at the downstream stopline 2 were calculated and plotted in Figure 2.6. It should be noted that as the travel time, t, in- creases, the maximum amplitude of 12(t) decreases and a longer time is required for i2(t) to decay to zero. This dispersion mechanism is a basic part of the TRANSYT flow model. 22 A”: No mmcflamoum one Emummwo xcflqiwooz m.m musmflm some 23 mafia Hm>mua mo coauocsm m mm cofimuommwa cooumHm m.~ whamwm cm on om om ov one a o p b p p . p / Fl L mocoomm om u u 3; ma mosoomm oH u u 31a I‘u fiv 2b ,n-v . noncomm m u u N Auv fl 3 He 24 2.2.5 Stops, Uniform and Random Delays Next, the general stops and delay relations can be written from the arrival queue length graphs illus- trated in Figure 2.4. h link of The vehicle stops per second on the jt the traffic network can be determined by considering the stops due to the vehicle queueing during a time interval dt to be ij(t)dt. Thus the total vehicle stops per h cycle for the jt link is the area under the arrivals graph of Figure 2.4, namely; t . — e] . — So - o d - o o o o J ftrj 13(t) t 13(te3) (2 17) By use of equations 2.8-2.11, the stops equation can be written as t . = = e] S. O(tej) ftrj oj(t)dt Gj(tej - tgj). (2.18) Similarly, the delay due to vehicle queueing during this same time interval dt is Qj(t)dt, thus giving the total uniform delay per cycle for the jth link as te. D- = f 3 Q-(tldt. (2.19, J trj 3 Then using the definitions of queue length and cumulative arrivals, 25 t . ' t . 93 e) _ _ ft . Ij(t)dt + It . [Ij(t) (t tgj)Gdet D. = 3 r3 9: = [tej [ft i (T)dTJdt - G (t - t )2/2 (2 20) trj trj j j ej 93 ° ° All TRANSYT flow calculations are made on the basis of average flow rates and queues that are expected to occur during each cycle unit. However, in real life situations, the random behavior of individual vehicles will tend to fluctuate about the average levels of the flow histograms. Robertson (R01) observed in his field studies that a cycle time which is long enough to clear a queue during one green time period for uniform arrivals may not be sufficient for complete queue decay for random arrivals during every cycle. Thus an extra delay term, called the random delay, Rj' solely dependent upon the h degree of saturation at the stop line on the jt link, was added to the link's objective function, namely: 1. Both the original TRANSYT random delay model and that used in its revision are illustrated in Table 2.1 and Figure 2.7 for comparison. 2.2.6 Dummy Links In a network comprised of links forming closed loops, it is necessary to estimate some departure Table 2.1 Random Delay Values 27 X. OriginaL Revised TRANSYT 3 TRANSYT R. Rj mj = 0.0001 mj = 0.001 ij = 0.01 mj = 0.1 0.0 0.000 0.000 0.000 0.000 0.000 0.1 0.003 0.003 0.003 0.003 0.003 0.2 0.013 0.012 0.012 0.012 0.012 0.3 0.032 0.032 0.032 0.032 0.031 0.4 0.067 0.067 0.067 0.066 0.064 0.5 0.125 0.125 0.125 0.124 0.117 .6 0.225 0.225 0.225 0.222 0.200 .7 0.408 0.408 0.407 0.398 0.333 .8 0.800 0.800 0.795 0.756 0.546 .9 2.025 2.020 1.977 1.671 0.878 .0 in 49.751 15.565 4.762 1.365 .1 -3.025 1002.493 102.429 12.001 2.007 .2 -l.800. 2001.249 201.239 21.155 2.769 .3 -l.408 3000.833 300.830 30.799 3.610 .4 -1.225 4000.625 400.623 40.608 4.501 .5 -1.125 5000.500 500.499 50.490 5.423 1.6 -l.067 6000.417 600.416 60.410 6.365 . -1.032 7000.357 700.357 70.353 7.320 1.8 -1.012 8000.312 800.312 80.309 8.285 . -1.003 9000.278 800.278 90.275 9.256 2.0 -1.000 10000.250 1000.250 100.248 10.233 10.04 6.04 J R. Random Delay, .5 O 0.0 28 Normal Operation Region (0 < xj < 1.0) Revised TRANSYT with mj = 0.1 Original TRANSYT +00 Oversaturation Region (Xj > 1.0) 1.0 2.0 Normalized Saturation, X j '00 Original TRANSYT Figure 2.7 Random Delay Versus Normalized Saturation 29 patterns since each genuine link is processed only once in calculating the objective function. A so-called "dummy link" must be introduced in parallel with a genuine link to break any set of links forming a closed loopa This inserted dummy link has the same total flow as the genuine link but its arrival histogram is assumed to be uniform over the cycle length. The arrival pattern for the next downstream link in the loop is calculated from the appropriate percentage of the dummy link's departure pattern. Then the flow histograms for the remaining links comprising the closed loop are calculated in the normal solution sequence. The rules for dummy link in- clusion are: 1. Every closed loop of links must be broken by a dummy link. 2. One dummy link can be used to break several closed loops provided that the loops have at least one common link. 3. The dummy link should be introduced at a node within a loop at which the degree of satura- tion of the corresponding link is high or the volume flow rate entering the downstream link is low. 4. Use as few dummy links as possible. 30 2.3 Optimization Procedure The optimization portion of TRANSYT makes use of a special search procedure developed by Robertson to accomplish a hill-climbing process. In Great Britain, where TRANSYT was conceived, most existing traffic signal equipment use 50 step control mechanisms. Thus Robertson's optimization increment sizes are based upon the 50 step cycles where each in- crement must be less than one half the number of steps in the cycle. The hill-climbing process takes the first incre- ment in the increment list and adjusts the offset of the first intersection or node in the node list for a local minimum of the network's objective functions. The offset of the second node is then adjusted in a similar manner and so on until the end of the node list is reached. At this point, the second increment is used and each node is reoptimized in turn. The process is considered complete when all the nodes have been optimized for all of the in- crements. Robertson suggests the following two increment lists for a 50 step cycle: a) 7, 20, 7, 20, 7, 1, l for offsets and b) 7, 20, -l, 7, 20, l, -l, l for splits and off- sets . 31 The 7 step increments are used to find an approx- imate local minimum while the 20 step increments avoid getting trapped in that minimum. The positive unity steps are for fine-tuning the approximate offsets while the -1 steps allow adjustments of the splits. If the number of steps in the cycle length differ from 50, then the 7 and 20 step increments should be ad- justed proportionately. Although Robertson's optimization procedure appears to give very good results, it has several drawbacks. First of all, the optimization process may produce a local Optimum instead of the desired global optimum. The prob- lem being that the objective function is not necessarily convex for the general case. A function F defined on an open interval (a,b) is said to be convex if for each x,y 6 (a,b) and each A, 0 g A i l, we have F(lx + (1 - l)y) : AF(x) + (1 - l)F(y) (2.26) This definition and Figure 2.8 illustrates the concept of convexity. The second drawback to Robertson's hill-climbing process is that the number of computations involved in- crease in the order of the square of the number of inter- sections and thus restricts its use to off-line applica- tions. 32 h .cowuocsm m>wuomflno xm>cou llll III l'l'i. h .cofluocsm w>fluomnno xm>soolcoz Figure 2.8 Convex and Non-Convex Objective Functions CHAPTER III TRANSYT/G NEAR-OPTIMUM SPLIT CALCULATION The most widely known method of selecting a reasonably good split for a signalized intersection is that attributed to Webster, first published in his classic paper in 1958 (WEl). A very good summary and comparative evaluation of this technique was presented by Gerlough and Wagner in 1967 for an isolated inter- section (GWl) and in 1969 for an arterial (WAl). Wagner et al. state that any systematic approach for determining the split must be at least as well defined as Webster's. Webster used a combination of Poisson arrival model theory and computer simulation in order to deduce his formula for average delay per vehicle at a signalized intersection. From this, he derived ex- pressions for optimum cycle length and split that min— th imized his total delay relation at the i intersection. These were: _ 1.5L + 5 C0 - n , (3.1) l - Z Y i=1 3 and 5 3i "' js ' (302) IA ' 1.5LA ‘ fi- 34 where C = the optimum cycle length, 0 rA = the effective red for the primary phase A, L = the total lost time per cycle, LA = the lost time for the primary phase A, Yj = the ratio of maximum directional flow to saturation flow for phase j, and n = the number of signal phases. It should be noted that the lost time includes the effect of the starting delay and reduced flow during the amber time. Webster's or some refinement of it is rapidly becoming the standard. Before proceeding, several observations concerning Webster's split for a two-phase signal can be made for various levels of traffic flow density (GWl). These are: 1. For light flow - the split can be varied greatly from the optimum without producing an oversatura- tion effect on either phase, and 2. For medium flow - the split can be varied somewhat from the optimum before approaching oversatura- tion, and 3. For heavy flow - only a small variation in split can cause oversaturation. The TRANSYT program contains an optional pro- cedure for determining allocation of green times for two or multiphase signals based upon equalization of satura- tion on all phases. This technique is a refinement of 35 Webster's and appears to give a good initial starting point for iterative optimization schemes. The philosophy and objective of this author's research was to develop a non-iterative signal timing algorithm. Thus, a near-optimum set of splits will be determined for our general traffic network on a single pass through the TRANSYT traffic flow model. The gen- eral method employed by this author was to write the ob- jective function in terms of the network signal splits, and then calculate the optimum split set by calculus methods. In developing these signal split relations, only two phase single cycle signals will be considered in the first portion of this chapter thus preventing the deriva- tions from becoming overly complex and obscuring the technique at hand. Later, this method will be extended to the double cycling and multiphase cases. 3.1 Split Bounds Of course, the loosest set of bounds on the sig- nal split normalized with respect to the cycle length, would be 0 < si< 1. System requirements however, dictate a somewhat tighter set; for example when consider- ing the minimum green time per phase, we have: 5A 5B o < ——-£_s i l - C— < 1, (3.3) 36 = minimum green time in cycle units for the primary phase (IPA) , gB = minimum green time in cycle units for the secondary phase 6p 3’ , C = cycle length in cycle units. A minimum green time is required to allow any queues to discharge and for pedestrians to cross the street. When considering the cumulative arrivals and departures at any given intersection, some additional quantities such as the ratio of arrivals to saturation flows can be intro- duced into the overall bounds on the split. These quantities may or may not be a tighter restriction, de- pending upon the given set of system conditions, thus use of minimum-maximum functions are required. As previously shown in Figure 2.4, the total de- partures during the non-red time must equal the total arrivals during the entire cycle in order for the queue to clear prior to the start of the next red time interval. For link j, we have trj+C trj+C I.(t . + C) =- f o.(t)dt = f o.(t)dt t . . 3 r3 r] J tgJ J trj-I-C i It dot . (3.4) 93' The split for a two-phase signal is the ratio (bf the primary phase A green and amber time intervals to 37 the cycle length (cross-hatched in Figure 3.1). From this we have: si = (trA + C - tgA)/C tgA = trB (3.5) trA = th Using the definition for the split from above, the phase cumulative arrival relations for Figure 3.1 become: I (t A +C)umucH mafia m mac 4 mmmsm H.m ousmam mu mm \\\\\\\\ 4m u «on «H 39 The unnormalized upper and lower bounds may not result in an integer value and thus must be restricted as such. Specifically, each real valued bound having a non-zero decimal part must be rounded up to the next highest in- teger. This may be accomplished by the use of two well defined mathematical functions, namely the 'greatest integer' and the 'signum' functions: [hj] E the greatest integer less than or equal to 13., J and +1 if hj > 0 sgn(hj) : 0 if hj = 0 (3.9) -1 1f hj < 0 . We then define E. e E. + s h. - h. 3.10 where _ I.(t.+c) h. = 3 .51 , j = A,B. (3.11) J Gj Since we are setting up integer bounds on an integer valued split parameter, Ei, where the hj are real, the rounding of the hj values must be done such that the kj values will be within the hj bounds. That is) hA 5_kA 5.31.: c - kB‘: 0 - hB . (3.12) From this set of split bounds, it is required that 40 z E. < c (3.13) i 3 “ to insure a valid split value. In the case of equality, kA = 51 = c - RE, the split is uniquely determined by its bounds. By denoting the sum of the minimum green plus amber time intervals for the jth phase as gj, we have the bounds on the integer split for the intersection of two one-way streets as: max(gA,k ) 3 si 1 C - max(gB,kB). (3.14) A For the intersection of two bidirectional streets, the summation of the ratios of cumulative arrivals to the saturation flow rates per phase will govern the bounds calculation, thus 2 max(g.,k.) < E. < C - Z max(g.,k.), (3.15) A 9’8 where the link subscript j = l,2,...,£. 3.2 Tractable Model of Stops and Uniform Delay The least complicated signalized intersection that may occur in a general network is the corner inter- section composed of two one-way streets, each having a uniform arrival rate as in Figure 3.2. After one com- plete cycle of simulation, the queues will have had suf- ficient time to develop properly. The general delay and stops equations from Section 2.2 can then be written 41 easily from the rectangular area under the arrival rate curve and the triangular area under the queue length curve of Figure 3.2. Thus, since the input is constant at INj over the whole cycle length, we have for link (D II t . e3 - ft . 1j(t)dt IN. t . - t . = G. t . - t . , 3.16 r] 3( 8 r3) 3( e (33) ( ) J 3 t . t . e] 93 _ ft . Qj(t)dt INjft . (t trj)dt r3 I] C II t . _ - e3 - (G. INj)ft . (t te.)dt J g] 3 l 2 l 2 = -IN. t . - t . + — G. ~ IN. t . - t . 3.17 2 3‘ 9] r3) 2 ( J 3" e) 93) ( ) where the time at which the queue clears is IN. . = t o + t - - t u o 3018 tea 9] ( 93 r3)(§;l=—Ifi;) ( ) Substitution back into the stop and delay equations re- sults in . = t . - t . . 3.19 S] ( 93 r3)°3 ( ) and 1 2 0. = —»t - t . , 3.20 J 2‘ 9) r3) C3 ( ) where 0. IN. c. = _J_____l . (3.21) 42 Most of the intersections comprising a general urban traffic network have many links with non-uniform arrival rates, thus the areas under the arrivals, ij(t), and queue length, Qj(t), curves (Figure 2.4) represent- ing the stops and delay are not rectangular and tri- angular respectively as in the uniform case (Figure 3.2). Thus, by using the stops and delay equations based upon uniform arrivals, some degree of error will be introduced into the solution of the general network. But, since the goal is to achieve only near-optimum signal settings in a trade-off for real time operation, these uniform arrival stops and delay equations will be accepted as an accurate enough model. 3.3 Tractable Model of Random Delay The revised TRANSYT random delay model described in Section 2.2.5 can be rewritten as: Rj mj(4_mj).2 1 (2 mj)Xj+Xj - 2 + (2 - mj)Xj}. (3.22) In order to write the random delay in terms of the normalized split, 81, for the ith intersection, we recall equation 2.23 from Chapter II for the degree of h saturation of the jt link. For a two phase signal this relation becomes: 43 G. ———————————————————————— 3 Arrival Rate IN. 3 0 G. __________ 3 Departure Rate I I 0 I l I l I I I | = - .. Qj max _ _________ I , slope (Gj INj) slope = I IN. ' I 3 : I Queue Length I | I 0 ' ' t . t . t . t . t .+C r3 9] e3 33 r3 Figure 3.2 Departures and Queue Length for Uniform Arrivals on the jth Link 44 Ij(tr. + C) E1. . G s.C = s. for 3 E vA .1 l X. = (3.23) I.(t . + C) h 63(1r3 s‘IC l-s for j 6 QB’ j i i I.(tr. + C) where h. s 1' Tl . (3.24) 3 jS Now the derivative with respect to the split can be derived as: dR. 3R. 3X. —1 = ——J-a O dsi Xj si 1 2X3 ' (Z'm-) 3X. 1- 2- . X.+X. where r-I (t + C) -h -x2 i r' _ ' = ' . 2; - h. for J 6 (FA G.C S. s. 3 ax. 3 1 1 8 =l (3.26) 31 2 I~(tr- + C) h. x. LGjC(1 - Si) (1-31) j It is observed that this form of the random delay and its derivative with respect to the normalized split leads to a very complicated form when attempting to derive the near-optimum split calculation equations. With the use of an appropriate approximation to the random delay derivative, a tractable answer will be shown to exist. 45 The random delay derivative with respect to the degree of saturation, Xj, was evaluated for several dif- ferent values of mj and is illustrated in Table 3.1. From this we can see that the first derivative, or slope, and mj are reciprocals of each other in the oversatura- tion region (Xj > 1.0). In order to decide upon an acceptable approxima- tion of the random delay with respect to the split, first one must determine the range over which the approximation must hold. We see that <0 for jeeA D.) w 1' (3.27) ds. V . 1 i 0 for j 6 QB . The limits of this derivative are: dR -w for j 6 TA 1im+(d—s-}) = (3.28) 31-20 J. N 0 for j 6 ‘PB: R,0 for j 6 IA dR. 11m (53%) = (3.29) Si"1 +oo for j 6 ‘PB . From Section 3.1, we found that the normalized split is constrained within the bounds: max(gA,kA) max(g§,kB) < s. < 1 - si(m1n) = C —’ i _, C = si(max).(3.30) 46 Table 3.1 Random Delay Derivative with Respect to Degree of Saturation TRANSYT 3Rj/3Xj 3 m. = 0:0001 m. = 0.001 m. = 0.01 ‘m. = 0;1 3 3 3 J 0.0 0.000 0.000 0.000 0.000 0.1 0.059 0.059 0.059 0.058 0.2 0.141 0.141 0.140 0.138 0.3 0.260 0.260 0.259 0.249 0.4 0.444 0.444 0.441 0.412 0.5 0.750 0.749 0.740 0.659 0.6 1.312 1.309 1.279 1.045 0.7 2.526 2.514 2.397 1.667 0.8 5.991 5.915 5.261 2.651 0.9 24.589 23.237 15.328 4.061 1.0 5024.876 507.783 52.381 5.683 1.1 9975.211 976.946 86.194 7.092 1.2 9993.765 993.895 94.937 8.077 1.3 9998.225 997.254 97.511 8.698 1.4 9998.439 998.449 98.542 9.085 1.5 9999.000 999.005 99.048 9.331 1.6 9999.306 999.308 99.331 9.494 1.7 9999.490 999.491 99.504 9.606 1.8 9999.609 999.610 99.619 9.685 1.9 9999.691 999.692 99.698 9.744 2.0 9999.750 999.750 99.754 9.787 47 For the usual minimum green and amber time in- tervals in a 60 unit cycle length, 13 5,max(gj,Ej) for all j. (3.31) Thus the bounds on the normalized split for a 60 unit cycle length become: 0.217 E 51.3 0.783 . (3.32) These bounds can be shown to remain reasonably constant for other cycle lengths. Thus our random delay approximation will be considered for split values be- tween 0.2 and 0.8. For a typical link having a 2000 vehicle/hour/ lane saturation flow rate, the m3. term will commonly be in the 0.0001 to 0.1 range and the hj term will be less than 1.0. Table 3.2 represents a tabulation of the de/dsi relations over the above m. and hj ranges. J The following random delay approximation was selected: 3:12. dsi = mj (Kljsi + KZj) . (3.33) From Table 3.2, the Klj and sz constants were calculated to give: h. 5% (41.6631 - 33.33) . j e In R If (3.34) D: m h. _l _ . mj (41.663i 8.33) . 3 6 e8 , 48 Table 3.2 Random Delay Derivative with Respect to Split Phase A: dRA/dsi hA 0.5 1.0 1n.A 0.01 0.001 0.0001 0.01 0.001 0.0001 X. 0.2 -l,248.6 -12,498.6 -124,998.6 -2,499.6 -24,999.6 -249,999.6 0.3 - 552.5 - 5,552.4 - 55,552.4 -1,110.6 -ll,110.6 -lll,110.6 0.4 - 301.7 - 3,112.7 - 31,237.5 - 624.3 - 6,249.3 - 62,499.3 0.5 - 104.8 - 1,015.6 - 10,049.8 - 399.0 - 3,999.0 - 39,999.0 0.6 - 10.0 - 11.9 - 12.1 — 276.3 - 2,776.2 - 27.776.2 0.7 - 2.7 - 2.9 - 2.9 - 201.5 - 2,038.1 - 20,405.4 0.8 - 1.2 - 1.2 - 1.2 - 150.9 - 1,556.4 - 15,618.8 Note: For the phase B derivative, dRB/dsi' replace the si heading with l - si table values. and change the sign of the resulting 49 for 0.2 :_si 3 0.8. At first, it may appear that this linear approx- imation to the random delay derivative is overly simpli- fied. However, Robertson points out that even when the degree of saturation, xj, is as large as 0.9 and the offset is the best possible, typically the random delay is no more than half the total delay (R01). Thus this author considers this linear approximation adequate for the objectives of this research. 3.4 Near-Optimum Split From Section 2.1, the objective function for each link j, was given as the weighted sum of the stops and delays or F. = KS. + w. D. + R. 3.35 3 3 3I 3 3I. ( ) and the total network objective function, F, as a summa- tion of the link Fj's. This can also be written in terms of the node fi's, namely 2 n F = 2 F. = 2 fi , (3.36) j=l 3 i=1 where 2 = the number of links in the network, and n = the number of nodes within the network. Assuming that the Fj's for the links terminat- ing; at any node, i, are dependent and those Fj's associated with any other node are independent of this 39t-. results in: 50 n £i 3:22 i=1 j=l [KSj + Wj(Dj + Rj)] . (3.37) where 21 = the number of links terminating at node i. The equations for stops and uniform delay from Section 3.2 for each link j can be written in terms of the split as: (l - Si)ch for j 6 QA Sj = (3.38) sich for j 6 m3, 1 2 2 . 4 f (1 Si) C cj for 3 C 0A Dj = (3.39) l 2 2 . :- SiC Cj for j E (93. where all terms have been previously defined. Thus, the total network objective function can be written as: n F = Z {KC[(1 - s.) 2 c. + s. E c.] i=1 1 jéqh J 1 jégh J 02 2 2 + 3—[(1 - s.) 2 W.c. + s. X W.c.] 1 .- i '6 j 3 thh 3 (PB . + Z W.R. . 3.40 j J J ( ) iflhere all the links are separated into two sets, namely those belonging to phase A, (a) and phase B, (03), 1‘ e spective 1y . 51 Now considering any node, i, the objective func- tion fi for all 1i links feeding that node can be minimized with respect to its split, Si’ by the First Derivative Test (OLl) as follows: - d a _ 0 - ds. (fi) — KCl .2 cj + .2 c.] 1 JEWA JEQB C2 + 5—[—2(l - si) 2 chj + 23i '2 W.cj] jeQA JECPB W.h.Kl. W.h.K2. +siX—JfiJ—l+X—lfiL—l. (3.41) 3' 3' 3' Then solving for the split si to provide a near min- imization of the objective function for node i, gives: 5. = 1 2 W.h.K2. W.h.K2. X (C W.c. + KCc. - _l_l__l) - Z (KCc. + jeeA 3 3 3 j . $643 3 i» .(3.42) W.h.K . 2(C2W.c. + —1—J—l1) . j j m. J J In order to be sure that 3i does indeed minimize the node objective function fi' the second derivative with respect to the split must be positive or indicate con- vexity in an a neighborhood about §i. Then by the Second Derivative Test (0L1), d2 2 W.h.K1. (f.) =czw.c. +z—J—J—l> 0, (3.43) 2 1 ~ . j 3 . m. 68- 3 3 3 J. 52 since all terms are positive. Thus the near-optimum split 81 can be used for any two phase traffic signal where each phase may have numerous links. The near-optimum normalized split has been plotted in Figure 3.3 for the case of a two phase signal located at the intersection of two one-way streets. The delay weighting factors, Wj’ were assumed equal to unity, the stop factor equal to 4 and saturation flows of 4500 vehicles per hour. From Figure 3.3, it is observed that the split equals 0.5 for any situation having equal arrivals, INj, and saturation flows, Gj' on both phases. A more general statement can be made: the normalized split will equal 0.5 for any situation having equal cj terms. These cj terms were previously defined as the ratio of the product of the saturation flow and the h arrival flow to their difference for the jt link. 3.5 Two-Phase Double Cycling Sections 3.1 through 3.4 of this chapter have covered the two-phase single cycling cases, however it may be advantageous to use double cycling in certain instances where the traffic flow rates are reasonably light or the cycle length is long. Figure 3.4 illus- trates the double cycling timing of a two-phase signal and will constitute a reference for the following deriva- ‘tion. As a direct result of this diagram the following related timing variables are: 53 1.0‘ 0.8: Secondary u Flow Rate w-I 0.6‘ '3. 250 U) 'U 0.4‘ 0 .3 500 '3 0.2. 750 g 1000 z 0.0 - . r . 0 250 500 750 1000 Primary Flow Rate Figure 3.3 Normalized Split Relative to Arrivals 54 Hmcmflm mmmamIOBB o no mowfiwa omaomo mansoa v.m shaman m m mm H NmHu AM u HmHu M Nm Hmmu Nflmu HoIcoz m «o mcflefia omaoao mamcwm m.m musmflm u on» _ as ._ __... a u a u on» _ oe ._ ._ o u o u mu» _ me _ ._ mm» m u an» _ me _ L <8 a 59 For an N phase non-overlapping traffic signal, there. are (N-l) split terms. Thus for our four phase example at the ith intersection, t . . = r3 9] sij C (3.56) S. = (t . - t .)cj (3.57) _ 1 2 . D. — 2(t . - t .) cj where 3 = A,B,C (3.58) The stops and uniform delay equations are then written as: sj = (l - Sij)ccj 3 = A,B,C (3.59) SD = (SiA + Sis + SiC)CCD _ 1 2 2 . _ Dj - 2(l Sij) C Cj J — A,B,C (3.60) _ 1 2 2 DD ’ 2(siA + Sis + Sic) C cD Let Si, Di, and R1 represent the weighted sums of the stops, uniform and random delay terms for the ith signal, respectively. Then take the first partial derivatives of Si, Di, and ii with respect to the phase splits, s.. to form: 13 3S. 81 = KC(cD - c ) 3 ij 351 2 3813 - C [(siA + siB + siC)W’DcD - (l - sij)chj] (3.61) 881 W.h 38. = m (Kljsij 4' K23.) , J = A,B,C. 60 Then to insure that a minimum does exist for the objec- tive function, fi, for the ith signal, from section 3.4, 2. 1 f. = 2 KS. + W. D. + R. 1 -=1[ 3 3‘ J 3)] J = Si + Di + Ri . (3.62) The second partial derivatives in turn appear as the forms: azfi 2 W.h.K1. ..= 2=C(Wc +w.c.)+—l—l—l>o 33 35.. D D J 3 m3 13 (3.63) azfi 2 . ij ’W‘CWDCD> 0' 3 =A'B'C° 1k 13 By considering L.. = U + V. > 0 (3.64) 33 J ij = U > 0, it is easily shown that L L L AA BA CA AB BB CB AC BC CC ==U(V'AVB + VAVC + VBVC) + VAVBVC > 0. (3.65) 61 Thus since the L.. > 0 and A > 0, the objective func- 1] tion for the ith intersection has a relative minimum of f1 (siA'siB'siC) at the p01nt (giA'giB'siC)' Then by setting the first derivatives of the intersection's objective function with respect to the phase splits to zero, the general equation becomes: esi 301 381 ° = '3‘;— + 5‘..— + 's— ij ij ij - _ 2 - _ - KC(cD cj) + C [(siA + siB + SiC)WDCD (l sij)chj] W.h. + —3—lmj (Kljsij + sz) 3 = A,B,C. (3.66) By rearranging terms, we have the following set of equa- tions: 2 h.K2. h K _ 2 2 , j 13 — sijIC WDcD + (C cj + mj )Wj] + (S + S )CZW c ik ii D D jpk.£ = A,B,C and j # k # z . (3.67) Let 2 hIKZ = KC - + . - _1_l w YJ (CJ CD) (C cJ ) 3 w = 02w c + (c2 + Eifli)w (3 68) 3 I) D C Inj 3 ° 2 . W = C W C J = AIBIC° 62 Then the set of equations can be written in matrix form as: yA WA w w SiA YB = V we w Sis h_y L”w w WC“ 31C) or g = P! - 31 (3.69) The set of splits for our multiphase situation can be determined by pre-matrix multiplication by W71 or, 31 = 3'1 : z (3.70) The matrix §i is a unique vector of phase splits for the ith intersection provided [WI # 0; a condition which can be shown true if wj # w # 0 for all 3. This matrix representation can be used for any number of phases (N :_2) constituting any signal's requirements. If there are N phases, then there will be (N-l) phase splits and the Si, W71, and 3. matrices will have dimensions equal to (N-l) by l, (N-l) by (N-l), and (N-l) by 1, respectively. To extend this result to apply to situations where there are multiple links per phase, simply replace the yj, and wj with 2 y. and X wj respectively. 3 j 63 3.7 Overlapping Multiphase Single Cycling The more general form of multiphase single cycling has some overlapping of green time intervals between the various signal phases. Consider Figure 3.6 of a specific example of an overlapping four phase signal. The respective red, green, and amber time intervals for this signal are illustrated in Figure 3.7. Note that there are only 2 independent phase Split terms required, namely for phases A and C since the phase B green is an overlap of the greens of phases A and C. Phase D is a dependent phase. Similar to Section 3.6, we can write: th = tgA = tr0 tgC = trA (3.71) t = t = t gD rB rC Thus for an N phase traffic signal with M overlapping green time intervals, there are (N - M - 1) split terms. These splits are - t t . . = r3 193 - = sij C where 3 A,C. (3.72) Then the stops and uniform delay terms can be written as: (1 - s..)ch 1) Sj = (l - siA - siC)CCB 3 = A,C (3.73) (SiA + siC)CCD 64 Fi ure . g 3 6 Four-Phase Overlapping Signal Model 65 Hmcmwm ommsquzom mcwmmmaum>o so no mcwfiwe omaomo mamcwm h.m ousmwm on o p Q 9 i emu can on» _ u e ._ L o u o a new m e ._ .1 m p m u and. fl 9 — 46 «m ’ = A,C. (3.74) In an analogous manner to the non-overlapping green's case of Section 3.6, it can be shown that: i = KC(C - C - c.) 35. 1 _ 2 _ _ _ - - asij — C [(siA + siC)WDcD (l siA SiC)WBcB (l sij)chj] 8R. W.h. 1 = _l_l. - = asij mj (Kljsij + szl . 3 A,C. (3.75) The minimum of the objective function, fi’ was obtained from examination of the second partial derivatives: azfi 2 W.h.Kl. ij = EETTY — C [W’DcD + WBcB + chj] + —Jfi%——l-> 0 13 (3.76) 32fi 2I J 0 c L. = -——————— = c w c + w c > 3 = A, . 3k asikasij D D B B Then using equation set 3.64, it can be shown that LAA LCA U + VA U A = = LAC LCC U U + VC = U(vA + VC) + VAVC > 0, (3.77)_ 67 insures that the objective function has a relative min- §. ) results from 1mum. This minimum (SiA' 1C 8f. _ 1 =2 .. .. 2 .. 0 - asij KC[cD cB cj] + C [(siA + SiC)WDCD th. 3 = A,C. (3.78) Rearrangement of terms provides: 2 K2.W.h. KC[cj + CB - cD] + C (chj + WBCB) - mj -s [C2(Wc +Wc)+(C2 +fljfi)W] "ij DD BB cj mj j 2 + Sikc (WDCD + WBcB) j 7‘ k . (3.79) The above equation can be set into matrix re— presentation by letting 2 K2.W.h. yj - KC(cj + cB - CD) + C (chj + CBWB) - -—%gJ—J- = C2( W + W ) + (C2 + Eifli)w (3 80) wj CDD BB Cj mj j ' - C2(c W + W ) w' DD CBB' Then 68 [WA w][:iA] I w WC ic g = g . s. . (3.81) I——I "< ‘4 ‘I_“’3 II or By comparing these results with those from Sec- tion 3.6, we can see that each overlapping green reduces the dimensions of the W. matrix by l. 3.8 Split Sensitivity It has been shown in Kreer (KRl) that signal timing histograms developed by TRANSYT are insenitive to large volume changes within the network unless the volume changes cause one or more intersections to be- come saturated. Knowledge of the sensitivity of the split equa- tion will be of value to the traffic engineer when plan- ning initial or changes to his traffic signal timing system. In order to determine the magnitude of the split sensitivity to cycle length, primary and secondary arrival flows, the split equation of Section 3.4 was used. For the ith node, the following sensitivity equations were derived from the split equation by ordinary calculus methods: Asi 331 = 2C[WAcA(l-si) - chBgi] + K(cA - cB) 75C 2’ 5C denom. ' Asi asi C[K + CWA(1 - 81)] cA 2 ——-e: = ( ),and AINA 5INA denom. INA 69 A3. 881 C[K + CWBéi] c 1 _ B 2 KIN ‘35IN denom. (IN ) ' B B B W h K W h K where denom.=C2(Wc +Wc)+i§——]-‘-§+—B—§—]-’-§. A A B B mA mB To illustrate these sensitivities, consider the intersection of two one-way streets controlled by a two phase signal with the following values: 1. Cycle length, C = 60 seconds, divided into 60 cycle units, 2. Delay weighting factors, WA = WB = l, 3. StOp penalty factor, K = 4, 4. Saturation flows, GA = GB = 4500 vehicles/ hour, 5. Random delay terms, hA/mA = hB/mB = 50. These sensitivity equations are then solved for the AC, AINA, and AINB out varying the split, Si’ by‘i 1 cycle units. For values which can be tolerated with- example, consider the situation where the primary and secondary flow rates are initially equal to 500 and 250 vehicles/hour respectively. The primary flow rate can increase by 6% to 530 vehicles/hour before the integer valued split estimate will increase by one cycle unit (0.0167 for the normalized split). A number of other variations in the cycle length, primary and secondary flow rates have been evaluated and summarized in Table 3.3. 70 e.meu e.meu a oom. oooa oooa ~.eeH o.emu ~.eou_ ooe. oooH one o.emH o.omH m.emu ohm. oooH oom o.ooH m.e~H m.-H has. oooH omw o.onu . .o.emu V m.emH «we. con oooH o.e~H m.mnn H.eow, Hmo. oom one H.o~H H.o~H a oom. oom oom o.omu o.mHH o.oeu oom. oom ohm m.e~H o.omu m.-H mew. one oooH e.oHH_ m.~mH o.emw moo. omm one m.mHH o.omH o.oeH eon. omm oom o.mHH o.m&H s oom. ohm om~ Anson\moaownm>v Amocooomv Anson\mmaowno>v mzHo azHo co an mzH mzH MHGD maowu H H_m0 mmcmnu umamm a How CQHHMMHMbF omuwmwmmoz mmumm 30HMIHM>HHH¢ mue>aoanoom peaon "m.m wanes 71 One can conclude that the sensitivity of the split equation is dependent upon the actual magnitude of the cross flow rates in addition to their relative values. CHAPTER IV ARTERIAL OFFSET CALCULATION 4.1 Time-Space Diagrams For years, traffic engineers have used time- space diagrams as a visual aid in studying the behavior of vehicle movement on an arterial. These diagrams illustrate the split for each signalized intersection along an arterial as well as the offsets between the signals. It should be noted that the vehicle speeds be- tween any two adjacent signals can be different from those for any other set of adjacent signals. This re- sults in the outbound and inbound green bands taking on a zig-zag slope (Figure 4.1) instead of the constant slope bands for constant velocity along the entire street (Figure 4.2). 4.2 Maximal Arterial Bandwidth Now consider an urban bidirectional roadway with a number of signalized intersections all with a common cycle time. The widths of the outbound and inbound progression green bands, shown in Figures 4.1 and 4.2, are the outbound and inbound maximal bandwidths, respectively. A progression is determined by the splits, the offsets, and the cycle length. 72 Distance 73 f r 1 Cf I [' Outbound :3 — I:___’_'J IIIIIIII IIIIIIII Inbound _ _ [:2 — II: Time f r Figure 4.1 Typical Time-Space Diagram Distance 74 f r x5 :3 :3 x4 (:2 E: [:3 Outbound x3 5: ’ t::I I: Inbound 1‘; _ _ l::] x1 — [2:] Time f r Figure 4.2 Typical Constant Velocity Time-Space Diagram 75 The objective function for an arterial is some- times chosen as the maximization of bandwidth. Morgan and Little (LMl) have developed an algorithm based upon this concept where they offer solutions to the following two problems: 1. Given a common cycle length, splits for each signal, the vehicle speeds between adjacent signals, de- termine the set of offsets which will produce bandwidths which are equal in each direction and as large as possible, and 2. Adjust the set of offsets to increase one of the two bandwidths, when possible, while giving the other direction the largest bandwidth then possible. Before proceeding, a set of terms need to be defined for the remainder of the maximal bandwith dis- cussion. These are: rj = red time for node j on the street being studied (cycles), b(E) = outbound (inbound) directional band- width (cycles), x. = location of node j downstream from the reference point (feet), vj(V;) = outbound (inbound) vehicle speed along link 3 (feet/sec.), Tij(Tij) = travel time from node i to node j in the outbound (inbound) direction (cycles), 9(5) 76 cycle length (sec./cycle), relative offset from node i to node 1 measured between the center of the red intervals of the two nodes (cycles), minimum green of all nodes along the arterial (cycles), offset shift of node j for unequal bandwidth (cycles), outbound (inbound) platoon length (seconds), number of nodes or signals along the arterial. By convention, 0 : eij < l and the set {8..Ij = 1,...,m} for any i is called a synchronization of the m 13 along the street in question. signals The travel times between the ith and jth signals are determined from and the Tij are obtained by replacing each I with I. Then for two sequential intersections, namely i (i + l), (j-l kii Tk,k+1 3 > 1 < 0 j = i (4.1) I i-l [1:23 Tk'kfl j < i and 77 T = xi+1 ' xi i,i+1 viC - (4.2) _ = x1 x1+1 Ti,i+1 §.C 1 By definition, a signal is called 'critical' if one side of its red touches the green band in one direction and the other side touches the green band in the other direc- tion. Morgan and Little's approach was to maximize the directional bandwidths b and b. It was determined that all critical signals must fall into at least one of the two groups defined as follows:.. a) Group 1: consist of signals with reds touching the front of the outbound and the rear of the inbound, and b) Group 2: where reds touch the front of the in- bound and the rear of the outbound. With the aid of Figures 4.3, 4.4, and 4.5 for the various combinations of Group 1 and 2 signals the offset rela- tions can be written as: _ _1_ - 1 . This leads to two possible solutions (0 and 1/2) and is 'half-integer synchronization' for the maximal equal bandwidth case. By defining man (-) = mantissa of (°); obtained by dropping the integer portion and adding unity if the result is negative; and dij = 0, 1/2, we have 78 ocmm comma on» mcfluflfieq mamcmflm H macho 039 m.v shaman N x+.. L )- I /. . ..Li7 it? I 79 comm cmmuw on» mcfluflfiflq mamcmfim m macaw 039 ¢.v musmfim w _/ 3o 23 Ila . . n A mm w_m .8. D H 2 _ x+ oI.II1 x+ Q on TI \\ / \\\ /// \\ / \ Tn // N n." I N \ All w |/.lv_mfimv 13.!an o Illw. fl 1# PI 80 ocmm cmmuw one OGHuHEHH HmcmHm m msouu m can HocmHm H macaw fl m.v musmHh Kulllwx + neon IIIIIv. KIIIIIIHx + Salli mI Lem-.. I... .4“... ..H .H l ZI. fl TH / / \ \ // / \ H / / \ L / / \ \ / , \ x , , \ \ 1 , \ \ , , \\ \ K r _ _ . \ . 81 _ 1 - eij — man[2('rij + Tij) + dij]° (4.4) If the ith signal's red touches the front of the outbound green band, it appears as shown in Figure 4.3a. By taking th the right side of the i '5 red as the origin, the tra- h. jectory (not shown) touching the right side of the jt 5 th red passes the i signal at a point in time: __ l _ 1 -' .. uij - l manI2(ri rj) + 2(Tij Tij) dij] .(4.5) h The trajectory touching the left of the jt red passes the ith signal at uij - rj. Thus since dij = 0 or 1/2 th and the i red must touch the front, we get the main re- sult of Morgan and Little's paper, namely: B = max{max min maqu..(d..) - r.], 0}, (4.6) i j dij 13 13 J and a maximal equal bandwidth synchronization, {6C1,...,ecm}. Now in an effort to summarize the Morgan-Little offset method for equal bandwidths, the next five steps are listed. 1. Calculate the set Y s (lej 6 [l,m]) from y1 = 0 ). (4.7) 82 2. Calculate the set 2 = (zjlj 6 I-1,m]) from zl = 0 z. = z. + (x.-x ).(4.8) 3 3-1 20 j-l) v. 3. Calculate the set U = (uijli,j e [:l,m]) from uij(d) = l - man(yj-yi-dij), dij = 0,1/2. (4.9) 4. After completing the previous three steps, we have the maximum equal directional bandwidths B = maxfmix min max (uij(dij) - rj), 0}. (4.10) 3 .. 1] 5. Then a two-way synchronization (ecl"'°'ecm) for maximal bandwidths is determined from ecj = man(zj - zC + dcj)' (4.11) In general, either the outbound or the inbound flow is greater; very seldom will they be exactly equal. This unequal flow condition is a function of the time of day and relative locations of residential, work, and recreational areas in any given geographical setting. The resulting morning and afternoon peak traffic flows may dictate different bandwidths as a function of time during the day or week. Morgan and Little suggested dividing the total available bandwidth, 2B, between the two directions on the basis of platoon lengths p and 5, where they 83 defined platoon length (in seconds) as the hourly volume times the vehicle headway divided by 3,600. The shifting procedure used to obtain the unequal directional bandwidths, b and b, was developed from Figure 4.6. To shift the front of the outbound band from f to f' in order to obtain the directional bandwidth b, we have aj = max[(ucj - 1) - (B - b), 0] (4.12) for max[0,B] £.b i g . To obtain the inbound bandwidth b, the shift of the rear of the inbound band from r to r' is achieved by oj = max[b - (ucj - rj), 0] (4.13) for max[0,B] i b i 9. An important point to remember is that only one of these bandwidth shifts can be made for any one arterial since the second is dependent on the other and the total avail- able bandwidth. Summarizing the unequal directional flow cases, we take the following steps. The total bandwidth can be divided between the two opposing directions by first calculating the minimum green time band. 84 oooonoH / /.iy / z / nooom noose on» no noosunsnoa 2fT2/Z/. «I /// / oxm /I./ / / // ’/ ,z / / ,/ / / / "I / / / w.¢ onsmHh n e n __ n n n n O n = = = E. o _ rt 8 0 8 1 _ . FL. I q «II Ihu \H \N\\ ITII- OX \ x \ AT: o .14 \ \ \ \\ x \ oosonuso \ \ \ \ \ \\ xx \ \ \ \\_\ . n hi .un g = min(l - r.) = l - max(r.). (4.14) i i i 1 Then for the inbound arrivals less than the outbound (5 < p). 1. Calculate the larger bandwidth from min(9.ZB p/(p + 5)) if p+§ : 2B = g p :_ZB (4.15) min(9:P) other Calculate the adjustment set (01,...31m) from aj = max(uCj - l + b - B, 0), (4.16) where the ucj's were a result of step 3 above. Calculate the smaller bandwidth from B = max(ZB - b, 0). (4.17) Then an adjusted two-way synchronization (001,...,0cm) is obtained from ecj = man(zj - zc + dcj - aj). (4.18) If the opposite case exists (5 > p), then Calculate the larger directional bandwidth from 'min(g. 23 13/(p + 5)) if p+§ _<_ 23 g _ p > ZB (4.19) III II min(g,p) other 86 Calculate the adjustment set (01,...,om) from oj = max(B + rj - ucj, 0). (4.20) Calculate the smaller directional bandwidth from b = max(ZB - B, 0). (4.21) Then an adjusted two-way synchronization (0C1,...,6Cm) 'is obtained from ecj = man(zj - zC + dcj - aj). (4.22) CHAPTER V TRANSYT/G OFFSET CALCULATION Now that the maximal arterial bandwidth concept of Morgan and Little has been described in Chapter IV, several modifications will be presented. Even though the basic maximal bandwidth procedure has appeal for non-iterative offset calculations, this author considers this algorithm to have several disadvantages, namely: 1. The criterion given for the apportionment of unequal bandwidths does not include turning movements, 2. Queue growth and decay are ignored, and 3. The arterial concept needs to be extended to networks Because of these disadvantages, the set of Morgan-Little offsets may or may not minimize the previously defined objective function of stops and delays for a network. A reasonably good set of signal offsets may be obtained provided that the effects of these above men- tioned detriments are sufficiently minimized. This author feels that an optimization procedure such as the hillclimbing method is not required. In its place, 87 88 Morgan and Little's method was used as a base from which the TRANSYT/G offsets were determined. In addition to the terms defined in Chapter IV, several terms pertaining to the TRANSYT/G offset adjustd ments are listed below. Bcj = the offset shift of node j relative to the critical node for queue clearance (cycles), ch(ch) = the excess green to the right of the rear edge of the outbound (inbound) bandwidth for node 3 relative to the critical node (cycles), u(fi) = the outbound (inbound) average cumulative arrivals, and ¢cj = the TRANSYT/G offset for node 3 relative to the critical node (cycles). 5.1 Unequal Bandwidth Apportionment In the opinion of this author, Morgan and Little's definition of platoon length does not reflect the de- pendence upon turning movements and thus is not responsive enough to the vehicle flow rates within the system. There- fore it is suggested that the total bandwidth should be apportioned between the two directions of flow on the basis of the cumulative number of arrivals, Ij(trj + C), for each direction of flow. The turning movements are taken into account in the definitions of the average arrival 89 demand, u. The cumulative arrivals averaged over all the links comprising the outbound direction of flow is l m = — z o o + 0 S. 1 u m =113(tr3 C) ( ) There exists a similar relation 0 for the inbound direc- tion. Since the cumulative arrival terms are reasonably estimated in the TRANSYT/G flow model, the turning move- ments can exert their influence upon the offset determina- tions. As an illustration of this total bandwidth apportionment, consider the two three-signal arterials of Figure 5.1. In both cases, (a) and (b), Morgan and Little would apportion the total bandwidth equally since 8 = p = 3600 = 0.277 . (5.2) This author would apportion the total bandwidth unequally for both cases. In case (a) where u < E, 1000 + 300 + 300 = 3 = 533., and (5.3) 5 = 1000 + lgOO + 1000 = 1000. For case (b), u > E and u = 1000 + 1300 + 1500 = 1333., (5.4) i = 1000 + 1%00 + 1000 = 1000. 90 mHm>HuH¢ 0>HHMHsfiso mo COHuocsm a mo ucmEcoHuuomm< auoHsocmm Hopes H.m musmHm 3V Hn\£0> oomH Hn\:0> OOOH no}? 83 P 3 N N H .329. oooHC no}? 83 (C 23 IN H H£\£0> oooH no}? oom >2}? oooH 7 Exam; oom >I m H A n M as? 83 C as? 83 C K H no}? 83C 91 With the aid of Figure 5.1, it is obvious that unequal bandwidths are called for, thus the use of u(fi) is pre- ferred to p(p) for apportionment. 5.2 Excess Green Adjustment Now the second shortcoming of Morgan and Little's offset method will be alleviated, namely the consideration of the requirements imposed on the offsets by the need to start the clearing of queues prior to the arrival of the next platoon. Panyan (PAl) describes the situation where the progression band occupies the total green time-of the sig- nal having the minimum green. He suggests shifting 311 the excess green of the remaining signals to the left, earlier in time, to allow the queues to start decaying prior to the arrival of the next platoon. This queue clearance shift of excess green time can be derived from Figure 5.2 by taking a reference point at the center of the Group 1 critical signal's red time interval. For signal 3, we have: 1 . man[0Cj 2(rc + rj) - b - Ic.sgn(3 c)] Y 3 (5.5) C3 - r.) + I .sgn(j - c)] l manlec. + E(rc 3 c] ch 3 where these excess green times lie within the following bounds: 92 C I H O I .2 _ _Ifl. ~.m onooan “r rH fA-mo .no N e+o+mI -r'\ 93 0‘: ch :_l - rj - b, (5.6) 0 i'ch £|1 - rj - b . Since the magnitude of ch and §cj are un- equal for the unequal bandwidth cases, one only wants to th shift the excess green of the 3 signal by the smaller of the ch and ch quantities so as not to restrict either directional bandwidth. Thus the shift of the 3th offset relative to the critical node, c, would be Bcj = min(ycj,ycj) Z 0. (5.7) This queue clearance shift would be applied to all sig- nals other than the critical signals resulting in the start of all red times touching the rear of the pro- gression green bands. There is one important drawback to Panyan's shift of all the available excess green time, namely: that none of the excess green is available to the right of the pro- gression band for stragglers at the end of the platoon. Thus Panyan's 100% excess green shift may not be the Optimum choice. This author proposes that the excess green shift should be some intermediate value between Panyan's 100% shift and the 0% shift inherent in the Morgan-Little method. This proposed shift for the jth signal is made relative to the Morgan-Little offsets, 6 ., described in Chapter IV. The resulting TRANSYT/G 0] offsets are: 94 ¢cj = maniacj - ijchQHIj - 0)] (5.8) where the multiplicative shift terms, kj, range between 0.0 and 1.0 and are determined during the initial check- out of each network application. Again referring to Figure 5.2, consider the possibility that the jth signal is also critical, j = c', along our selected street. Two possibilities exist: c' belongs to either Group 1 or to Group 2. 'It is noted that for c 6 Group 1, and c'e Groupl,y .560 and ; ,=0, or CC CC (5.9) c' 6 Group 2, ch' = 0 and yea, 75 0. In either situation, Bcc' = 0 and is the expected shift between two critical signals along the same street. 5.3 Network Considerations Now that the author's offsets have been developed for arterials, an extension to networks requires the pro- per interface of signal timings between the arterials and cross streets. The offsets for a general network are determined by applying the author's arterial offset concept to a subset of all the possible streets within the network according to the following constraints. Each signalized street selected will be referred to as a vein. When constructing a vein-interconnection model: 95 1. Each intersection must be included in one of the veins, 2. All veins after selection of the first must begin at an intersection belonging to a pre- viously selected vein to insure prOper relative timing, 3. No closed loops are to be formed when selecting these veins, 4. Attempt to select streets with the heavier traffic flows as veins first, and 5. Attempt to select as few veins as possible. The vein-interconnection model is simply an over- lay of the general network such that no set of veins form a closed loop. It forms the sequential order in which the offsets are estimated. The first three rules are inherent in this author's offset method while rules 4 and 5 generally help one construct a vein-interconnection model having a lower final objective function value. The actual equations for the start of the green and red time intervals can be visualized from Figures 5.2 and 5.3. By definition, let 6 = the relative shift due to the phasing at the interconnecting node between two veins, such as at node j in Figure 5.3. 96 loss N CH0> mcH0> m03lo39 .039 m0 coHuomccooumucH m.m ousoam Since node 1 is the first to be considered in our example, we can select 6 = 0, or any other convenient value. Thus for 6 = 0 on vein 1, the :pA -green for node 1 will start at zero. Next, the cpA green for node 2 starts at man[%-(r2 - r1) + ¢c2 - 001]. For vein 2, belonging to TB here, 6 equals the newly calculated :pA. node 3 starting green time minus its red time interval. In summary, the green and red time intervals on a two-way vein are shifted according to: 1 . I = _ _ - tgj man[2(rj r1) + ¢cj 001 + 6] (5.6) where 6 = 0 for vein 1 {t'. - r. for all other veins where g1 1 i is the cross phase at the interconnecting node and t'. = t'. + t . - t . f l < ' < 5.7 U 93 (r3 93) or -3-n ( ) The primed and unprimed symbols represent the new and previous values respectively. If the vein is a one-way, there is only one bandwidth and it equals the minimum green of all the nodes associated with that particular vein. In this case, equation 5.6 is replaced by t'. = manlr. - r 93 3 1+T lj + (1 - kj)(l - rj - g) + 5] (5.8) 98 In the next chapter, two applications to general urban traffic networks will be illustrated in detail. CHAPTER VI NETWORK APPLICATIONS Now that this author's signal timing method has been developed in the previous chapters, two applica- tions to actual urban areas will be illustrated. First, a 21 intersection portion of the Ft. Wayne, Indiana downtown area was selected for comparison of TRANSYT/G versus the standard hillclimbing TRANSYT. Certain non- critical modifications of the network configuration were made. Vehicle volumes, speeds, turning movements and physical dimensions were supplied by Ft. wayne officials. The second application consisted of a modified 38 intersection portion of washington, D.C. This loca- tion was selected due to availability of geometric and traffic data similar to the Ft. Wayne data. To evaluate the effectiveness of the new signal timing method proposed by this author, data from.the above mentioned example areas were used in two computer programs, namely; TRANSYT and TRANSYT/G. 6.1 Link Numberipg System The link numbering system is keyed to the node numbers such that node number 1. becomes the first part 99 100 of the link number. The last digit of the link number takes on a value from 0 to 7 depending upon the direc- tion of flow into the ith node. Links entering the node from the north, east, south, west, northeast, southeast, southwest, and northwest are assigned the values 0,...,7, respectively for their last digit (Figure 6.1a). Figure 6.1b illustrates this technique for an inter- section of two two-way streets at node number 9. 6.2 Selection of Networks In Ft. Wayne, an area bounded by Main Street on the north, Jefferson on the south, Fairfield on the west, and Lafayette on the east was selected. The node-link model selected for this region consists of 21 nodes, one for each signalized intersection, and 56 links, one for each directional flow including 6 dummy links required to break loops for the TRANSYT calculations. Main and Harrison Streets are two-way while the other eight streets are one-way, as indicated in Figure 6.2. Using the guidelines outlined in Chapter V, a vein interconnection model required for TRANSYT/G was selected and is illustrated as heavy black in Figure 6.3. Since this vein interconnection model is not unique, other suitable models could have been selected. In Washington, D.C., the node-link model selected consisted of 38_nodes and 134 links including 21 dummy 101 90 91 93 (b) 92 Figure 6.1 General Link Numbering System and a Specific Example 102 41 53 151 213 \43/ 4;; \3/ 150 F 31 43 141 203 “\or 32 ,c 142 192 F J 3"’<0 i/ 140 \14 19 21 33 131 193 Figure 6.2 Node-Link Model of Modified Ft. Wayne, Indiana Area 103 Vein 5 Vein 3 Vein l @ .. Figure 6.3 Vein Interconnection Model of Figure 6.2 104 links. The area includes nine two-way and six one-way streets (Figure 6.4). Similar to the Ft. wayne example, a vein-inter- connection model was selected for TRANSYT/G (Figure 6.5). 6.3 Input Data The input data required for TRANSYT is described in detail in Robertson's User's Manual (R02) and will not be reprinted here due to its length. The same data set used for TRANSYT can be used for TRANSYT/G with one ex- ception; the type 4 hillclimbing step size card has been replaced by the new type 4 vein list cards. One card or file line is required for each vein. The format con- sists of right justified quantities in the standard five column field widths used in TRANSYT. Field 1 contains a 4. Field 2 contains the direction of flow indicator per the following code: bbbbl for one-way flow going east, bbbb2 for one-way flow going north, bbbb3 for one-way flow going west, bbbb4 for one-way flow going south, bbbb5 for two-way flow going east and west, . bbbb6 for two-way flow going north and south, assuming north at the top of the vein interconnection model diagram. Fields 3 - 16 contain the node numbers comprising the vein and are in the order in which they are encountered in the vein model. 105 991 972 992 9—@ 970 990 I 973 1001 9931002 962 @—® N <——— 96o 100 963 891 I 1003 832 892 @—® 830 890 833 881 I 893 822 882 F l I 82! 82 880 88 910 823 871 883 810 870 900 813 611 873 86 [ 862 ,1 612 610 650 860 863 60i113fl3 85 852 A 602 840 590 630 77 772 73 732 773 721 I 733 O 7’ 760 720 700 763 723 6 753 743 Figure 6.4 Node—Link Model of Modified Washington, D.C. Area 106 e @9600 8 e n V .i e @963 .. . n .m 7 6 V 2 .i w 4 .. Geotooem 5 0006061660 1 eosefieeonoo. 72 6.4 ion Model of Figure 107 6.4 Comparison of TRANSYT/G versus TRANSYT To compare the new split and offset calculations of TRANSYT/G to the previous hillclimbing TRANSYT method, the following computer runs were made for both the Ft. Wayne and the Washington data on a Burroughs B7700 system: 1. Initial conditions with equal splits and zero offsets, 2. TRANSYT (STARl) splits without hillclimbing on offsets, 3. TRANSYT/G (SPLIT) splits without offset calcula- tions, 4. 7-Step hillclimbing TRANSYT with STARl splits, 5. 7-Step hillclimbing TRANSYT with SPLIT splits, 6. TRANSYT/G with no excess green shift, _ 7. TRANSYT/G with full excess green shift, and 8. TRANSYT/G with partial excess green shift. These results are summarized in Table 6.1. By comparing run numbers 3 to 2 and also 5 to 4 for each data set, one can conclude that this author's split calculation method (SPLIT) is superior to Robert- son's (STARl). This is evidenced by that fact that the use of SPLIT produced lower objective function and higher system speed values than those produced by STARl. An- other observation can be made concerning runs 4 and 5. This author's split estimation method produces a better set of initial conditions for the hillclimbing procedure. 108 .msnn HmHuuom How 0Hn0HHm>w #02 I t mm.mH no.mNN mh.HH mo.m Ho.ohH N.o mmN N m mw.mH Nh.va mm.NH oo.m we.mbH N.m moN N h cv.mH hm.mmN mm.HH mo.m mH.NmH N.m moN N m mH.mH mo.vHN oo.HH mo.m mo.HmH m.evv MHv.Nm Nmo m Ho.mH ww.mHN mH.HH mm.m No.5mH m.va eNm.mm one o mm.vH Hm.emN vh.HH mo.m mm.mmH « «MH H m oh.vH om.mmN MN.HH mm.w mm.¢0N s emH H N HN.¢H oh.th hm.HH oh.h hm.HNN « vMH H H . . Amoco: mmV .U.a .soummHzmmz ”N 0Hmeoxm .mmeoa Hm.moH mo.m oH.o oo.oe H.m «AH N o hm.mH mm.vHH Nm.m mH.o mm.mm H.m NHH N h ¢¢.mH Hm.m0H 0H.m mH.m mm.om H.m .NHH N o mm.mH hm.ooH mm.e mH.w mn.vn m.vHH Hmm.NH com m uh.mH mN.NOH mm.¢ mN.v mm.mh w.mHH Hmv.MH mom. v mw.mH mm.qu on.m mH.w mm.mHH 4 mm H m mm.mH ov.mvH mm.m mN.v mm.mHH « mm H N vm.oH mm.omH ow.m vm.v mm.NNH « mm H H Amoco: HNV MGMHocH .osNMK .um “H 0Hmemxm ommmmooum H.00m Aun\ Aun\ mstH moHHusm 2.3.5 TIL :53 nodes .3153 1.8.... so .oz no .oz h0Hoo monn oEHB ammbm omoam soHuoczh macaw Eoosom EHOMHGD cam mcHusou Ewummm 0>Huoonno Hobos Hobos Hobos Dmu Insm «0 mm: cam H been I ewmzame op o\ewmz¢me mo conauoosoo H.o oHone 109 In both examples, fewer total number of calculations were required and lower objective functions were obtained. Run numbers 6, 7 and 8 are similar in nature ex- cept for the amount of excess green shift adjustment to the offsets. The shift values for runs 6, 7 and 8 were zero (Morgan-Little), full (Panyan) and partial (Grove) respectively. Comparison of these runs indicate lower objective function and higher system speed values for this author's partial shift for both examples. Thus, the partial shift is preferred. It should be emphasized that the partial shift varies from one network to another and is determined from several trial runs during either the preliminary checkout of a new signal timing system or during on-line system operation. Finally, comparison of runs 8 to 4 points out the potential use of TRANSYT/G versus TRANSYT (Table 6.2). In the opinion of this author, the benefit of TRANSYT/G's extremely fast computer running time relative to TRANSYT's far outshadows its slightly higher objective function (about 3%) and lower system speed (about 1%) values. In fact, from runs 4 and 8, the improvement in running time can be estimated from the CPU times as follows: a) the Ft. Wayne data ran l§§i§~= 37.3 times faster, and b) the Washington data ran $2353 = 73.0 times faster 110 o.me.H H.oeei NH.HI oH.oI mm.m+ He.e+ om oouooasnoz m.em“H m.~HHI Ho.HI o~.oI mH.m+ -.m+ Hm memos .om AIIIV canoe. .oomo inc Aus\eso Ass AIIIV osae dam one swoon smunmm soauossm o>aooonno noooz pom . mo sumo noosouooueo is some ewmzame I is some o\esmz 0, 114 115 including all items in line and those being served. If the arrival rate, a, is less than the service rate, b, the queueing system is said to be stable and there exists a finite time independent probability of the queue being in any state n. On the other hand, if the ratio (%) > 1, the queue length continues to grow with time. When considering the idea of the queue length (number of vehicles in a queue) for an urban traffic net- work, first consider a directional traffic stream having Poisson arrivals and an exponential service rate. Let Pn(t + At), n > 0 represent the probability that the queueing system contains n vehicles at time (t + At). Let At be such that only one vehicle can arrive or depart during the At time interval. Then there are only three ways in which this queueing system can reach state n during the time t to t + At, i.e., l. the system remains in state n, or 2. the system changed from state (n - 1) to state n, or 3. the system changed from state (n + 1) to state n. If the probability of one_arrival in At is (a A t) and the probability of one departure in At is (b A t), the corresponding probabilities of no arrivals or departures are (1 - a A t) and (l - b A t). 116 Since traffic signals are spaced some finite dis— tance apart in an urban network, the queue lengths must be restricted to some finite value of N vehicles, where N is a function of signal separation distance, split, offset, and cycle length. Thus, if the maximum number of vehicles in the queueing system is limited to N such that vehicles arriving when n > N will not be able to join the queue, we have, neglecting higher order terms : Pn(t + At) = Pn(t)[1 - (a+b)At] + Pn_1(t)[a A t] + Pn+1(t)[b A t], 0 < n < N t+At) P0(t)[1-aAt] +P1(t)[bAt], n= 0(A.l) P (t + At) ll 2 PN(t)[l - b A t] + PN_1(t)Ia A t], n Passing to the limit with respect to At results in: Pn(t) = -(a+b)Pn(t) + aPn_1(t) + an+1(t) 0 < n < N, Po(t) = -aPo(t) + bP1(t) n = 0, IA.2) PN(t) = -bPN(t) + aPN_l(t) n = N. Setting the time derivatives equal to zero and eliminating time results in: (l-I-)I)Pn=Pn+1-I-APn_l 0mue msmum> Houomm mcflnuoofim H.¢ munmwh u .msau Ho>mua as on on OH c Ll - b b coo .~.o \ m. m coupom (I\\\\\ m“ A8 u zv o>ouo Am u zv m>ouo m. .v.o.n low u as m>ouo u conunwaom w 0 J .o.o v .m.o o.H APPENDIX B This appendix contains a general flow diagram of Robertson's TRANSYT signal timing program and this author's modified version, TRANSYT/G, illustrated in Figure 8.1. Due to the length of these two computer pro- grams, only listings of the following portions of TRANSYT/G will be included: 1) the Type 4 Vein Card Section of TINPUT (Figure 3.2), 2) the complete SPLIT subroutine (Figure 3.3), and 3) the complete OFFSET subroutine (Figure B.4). A general description of the function of the various other subroutines called by TRANSYT and TRANSYT/G is as follows: TINPUT - reads all the input data and checks for appropriate order and boundedness, STARl - an optional calculation of the splits based upon equal saturation of cross phases (TRANSYT only), HILLCL - performs Robertson's hillclimbing optimiza- V tion of splits (optional) and offsets (TRANSYT only), 128 129 SUBPT - calculates the delays, stops, and objec- tive function for a given set of signal settings, TOUTP - outputs the previously calculated informa- tion for each node and link along with their totals, and SPLOT - plots out the link flow histograms. 130 TRANSYT :) (TRANSYT/G D ‘B 0 Figure 3.1 TRANSYT and TRANSYT/G General Flow Diagrams c- c- c- C- 240 131 SUBROUUINE IINPUI(ICL40) MISSING SECTION (SEE ORIGINAL IRANSYI LISIING). YHE NEXT SECIION PROCtSSES A TYPE 0 CARD(A VEIN CARD) IF(IPCARDoLV.S.OR.IPCARD.GT.0)GU I0 110 IF(WR11.EU.3)GD 70 246 IF(1PCARD.EQ.0)GO ID 205 wRITE( b.5110) IF(NR1|.NE.U) NRIIE(1115110) 5110 FORMAT(“OCARD'90X9”CARD”p38Xp”NUDE/VEIN LIST”/ 5 c- c- 2&5 246 247 250 255 120 258 256 257 1” NO. TYPE”) WPITE( b.5000)lCRNU:(ICARD(I)oI=lclb) IF(lel.NE.0) NR11E(11:5040)ICRNO;(ICARD(I)oI=1,16) NAR7=NARI+I L0(NARI)=ICARD(2) HN(NART):0 ' DU 250 [33016 J=ICARU(I) lF(J.EU.0)Gn ID 250 IF(J.GI.0)60 T“ 2“? NFAUL1:1 IFAULI(I)=IMINUS IF(NN(NART).GE.IS)GO IU 255 NN(NARI)=NN(NARI)+I LN(NARVnI-2)=J CUNIINUt GU 10 258 WHIIE( b.5120) IF(lel.~E.0) NR11E(1|:5120) FURNAI('0100 MANY NODES PER VtIN IN LIST ' LIMII la'/) NFAULT=1 00 25b 131016 IF(IFAULI(I) .EO. IMINUS) 60 ID 257 CUNIINUE 60 10 100 leTtt 6:5050) IFAULI HRIYE( 6'2000) WRITE( 6:2003) lF(hR11.EQ.0) GO TO 100 WRITE(II,5050) [FAULT WRIYEUIpZUOO) wRIIt(I1p2003) 60 It) 100 MISSING SECIIUN (sac ORIGINALTRANSYT LISIING). REIURN END Figure 8.2 TINPUT Type 4 Card Input Section Listing c- c. c- No» INOU‘t-UN" OONO‘U‘bMN" 132 SUBROUTINE SPLIT REAL*U STP£N(?00)oRUNTlM(200)oVLHKMS(200)'HGHT(200) IRKA(200)0RKH(200)0RKC(200)0RKD(200I0RKE(200) INTEGER NLIST(SNIOLNUM(20070ICARD(16)IIFAULT(IO) pNURD(79II)pNUHAX(7)0KNBIAS(7)0N00MAX(7) OITLE(20) INTEGER NLINK(50)INBIAS(7rSUIINMIN(7950)oLOUTN(200) oLBSTRT(2:200)oLBFN5H(20200)ILDSTRT(21200) ILFSTRT(ZIZOOJILFFNSH(ZIZOO)OLVEHC(200) oLTOTF(200)oLUNIF(200)ILLN0(40200) oLJT(flo200),LNUMO(200),NNUMO(50)oISIZES(15) oNSTAGE(50)oKLIST(50)oLPL0T(240)oLFUDG(40200) 'IN(50)INK(SUIIIIrNDSAT(200)INTSAT(200) IIPTOUT(12000)01HARK(IT) ILDFN$H(20200)IL3ATF(200)OLENTF(“0200) COHMON SIPEN,RUNTIM.VEHKHS,HGHI.CONv1aCONVSoCOva ICONVTpCDNVBICONVA'CONVBOCUNVCIDSToDFNoSTOPP ITIMIOLDPIOPINDEXIPAOPBOPCOTOTRTITVKIBUSRT OBVKINLISToLNUMIICARDIIFAULTIITLEINPNoMFAULT ,NFAULT'IFNDIILOIPI[STEPSoINODESrILINKS IISTLSToINODCIIVoNUSfoNOSLpNPLTS'ICYCLE .NLINKpNBIAS:NMINoLOUTNoLBSTRTpLBFNSH pLDSTRTpLDFNSHpLFSTRToLFFNSHoLVEHCoLSATF pLTOTF,LUNIF:LLNOpLENTFOLJTILNUNOINNUHO rISIZESoIMARKoNSlAGE.KLISToLPLOTpLFUDGrHRIl COMMON/BLKA/NRI(200)'NARTpLU(20)pNN(20)pLN(ZOplb) IP=1 IL=l NOSEzo N0$L=0 IF(NRII.E0.0)GU TU 199 THE TRAFFIC PATTERNS, DELAYS, ETC FOR THE INITIAL SETTINGS ARE CALCULATED AND THE SIGNAL AND LINK DETAILS ARE UUTPUT(UPTIONAL). J=IMARK(6) IF(J.EQ.O)GO T0 5 HFAULT:0 IL:0 CALL SUBPT(IPTUUTaRKAoRKBpRKCoRKDoRKEI IFIJ.LU.I)GO T0 5 Iv=1 TF(NR11.E0.3)IV=3 NFAULT=0 CALL TUUTP(RKA9RKBoPKCoRKDoRKE) NOSE=0 NDSL=H MFAULT=0 CALL SUBPT(IPTOUTIRKAIRKBIRKCORKDORKE) Iv=l IF(NRIT.EQ.3)IV=3 Figure B.3 SPLIT Listing c- c- c- c- c- C- c- 199 200 21() 211 181 133 NFAULT=0 CALL TUUTP(RKA'RKBIHKCIRKDIRKE) THIS VERSION HILL ADJUST SPLITS FOR UNGROUPED SIGNALS HAVING 2 STAGES, l GREEN/STAGE ONLY; OTHERS LEFT AS IS. SET UP NK(A,*) LINK TERMINATION MATRIX. D” 200 I=1pINODES INTIJ=0 DO 210 J=I:ILTNKS I=LUUTN(J) IF(I.LT.0)GO T0 210 IN(I)=IN(I)+I KON=IN(I) NKCIoKONJ=J NK(1,11)=IN(T) anTINUE FOR EACH NUDE I. CALC NEw SPLIT AND ADJUST GREEN BAND FOR EACH LINK EXITTNG FROM NODE I. lx=ISTEPS XC=IX CL:1CYCLE 1:0 I=I+I IF NUDE I HAS MORE THAN 2 STAGES OR IS TO BE GROUPED NTTH A LATTER NUDE. LEAVE SPLIT AS IS. NNK= NK(l.Tl) IF(NSTAGE(I). GT. 2. 0R. NLIST(I) LT. 0)GO T0 230 CA: 0. 3C3: U. hKA: 0.;WK83 0. ucA=0.:wcH=u. WKKA=0.;NKK3=0, K=0 Ksk+l NA=NK(I,K) JB=L8$TRTTI.NA) NS=NbIAS(JB.I) JBH=LBFNSH(I.NA) NF=NBIASTJBB,I) IFINF.LT.NS)NF=NF+IX NGI=NF-NS IF(NGI.GT.IX)NGI=NGI-Ix NR1(NA)=IX-NGI IF(LNUH(NA).LT.0)GU T0 189 A:LTOTF(NA) 6=LSATF(NA) C=8*A/(B-A) D=O.S*TIM*A NNN:I.25*AAXC/B+0.5 ann(JB.T)=MAx0(NHIN(J8,I),NNN) Figure B.3 (cont'd.) ISO 187 189 184 168 171 I72 192 191 134 IF(J8.NE.))GU T0 186 CA=CA+C NAA=NKA+NGHT(NA)*41.66*D WKKA=NKKA-WGHT(NA)*33.53*D wcA=wcA+wGHTTNA)*C 60 T0 187 C5=CH+C WKH=WKB+wGfiT(NA)*QI.6b*D hKKb=WKKB-NGHT(NA)‘8.33*D «Cb=wc8+wGHT(NA)*C CONT INUT' IF(K.LI.NNK)GO TU 18! AIC:(CL*CL*NCA)-(WKKA+HKKB)+STOPP*CL*(CA'CB) IC=XCAAICI(CLACLt(NCA+WCB)+HKA0NK8)*.S NA=NK(I.1) JB=LB$TRT(IONA) JRB=LBFNSH(I.NA) Kx=LNUM(NA)/2 hx=ath KK=1 AK=KK+I Nb=NK(I,KK) KX8=LNUM(N8)/2 KXH=2*KXB KIJ=2 IFTKX8.EQ.LNUM(NH))KIJ:1 KIK=2 IF(Kx.Eu.LNUM(NA))KIK=1 IF(KIK.NE.KIJ)GO T0 188 IF(KK,LT,NNK)GO T0 180 KJB=LBSTRT(1pN8) KJ88=LBFNSH(laNB) IC3MINU(M‘X“(IC0NMIN(JUI1))oIX'NHTN(KJBrI)) Kxx=LNUM(NA) TF(JB.tU.l)Gn TO 171 Lx=NbIAS(KJ88,I)+IC-NR1(NA) NA=KJT5H GO TO 172 Lx=NRIAS(JBB,1)+lc-NR1(NB) MA=J88 NBIASIMArII=SCAL(LXoISIEP5) k=0 K=K+I NA=NKITvKI JB=LBSTRT(1,NA) NS:~BTAS(JB'I) JHB=L8FNSH(1,NA) NF=NBIAS(JBB.I) NGI=SCAL(NF-NS.ISTEPS) NRITNA)=IX-SCAL(NGTaISTEP8) Figure B.3 (cont'd.) 135 IF(K.LT.NNK)GO In 191 230 IF(I.LT.INUDES)GO T0 211 L- OUTPUT FOR NEW SPLITS. HFAULT=O IL=1 CALL SUHPT(IPTOUT:RKA.RKB.RKcoRKD,RKE) 1v=1 IF(leI.EU.3)IV=3 NFAULT=2 CALL TUUTPIRKAIRKBIRKC'RKUpRKE) QETUF‘N END Figure B.3 (cont'd.) c- C- C- c- c- c- I 2 I 2 NOWCUAN‘ C’INO‘U‘BMM" I 136 SURROUTINE UFFSET THIS SUNRDUTINE IS ENTERED FROM THE MAIN PROGRAM AND PERFORMS THE ESTIMATION SEUUENCE NECESSARY TO FIND THE OFFSET SETTINGS. TNU SUBROUTINES ARE ENTERED FROM "OFFSET” - ”SUHPT” T0 CALCULATE DELAYS ETC AFTER EACH SIGNAL CHANGE AND "TOUTP' T0 UUTPUT RESULTS To THE LINE PRINTER. REALAu STREN1200).RUNTIMTZOO).VEHKMSTZUO).RGHT(200) 'RKA(200)'RKB(200)'RKC(200)ORKD(ZOO)ORKE(200) OPINIISI INTEGER NLIST(SU).LNUM(200).ICARD(Ib)oIFAULT(16) .NUS(IS):ILL(15)oN00(15).NP(15)IKS(15) .TTLE(20) INTEGER NLINKISU)INBIASI7050)INMIN(7050)IL0UTN(200) nLBSTRT(2.2001oL8FNSHT2pzoo).LDSTRT(2:200) OLFSTRT(3:200)0LFFNSH(23200)OLVEHC(200) oLTOTF(200)oLUNIF(20”)oLLNO(“IZUOI .LJT(0,200),LNUMUTBUU).NNUM0(SO)oISIZESTTS) .NSTAGE(SH),KLIST(50).LPL0T(2&0).LFUDG(ao200) pIPTOUT(12000)pIMARK(11) oLDFNSH(2.200)oLSATP(200)pLENTF(H.200) CUMMON STPtN,RUNTIH.vEHKMS.NGHT.C0NVI.CONV5.C0NVS pCONVToCUNVBpCONVA,CONVBoCONVCoDSTaDFNoSTOPP ITIHoULDPIIPINDLXIPAproPCoTUTRToTVKoBUSRT IBVKINLISTILNUMI[CARDIIFAULTOITLLINPNaHFAULt INFAULTI[ENDIILIIPIISTEPSIINODESIILINKS rISTLST.INUDCpIVoNUSEpNOSLpNPLTSoICYCLE oNLINKpNBIASoNMINILOUTNILBSTRToLBFNSH .LDSTRT,LDFNSHpLFSTRTpLFFNSHaLVEHC.LSATF ,LTUTF,LUNIF,LLN0,LENTF.LJT,LNUM0.NNUM0 oISIZES:IMAHKoNS[AGEIKLISIILPLOIILFUDGoWRII CUMMON/BLKA/NRI(200),NART.L0(20),NN(20).LN(20,16) CUMMUN/BLKB/EGS DIWENSION Y(1S).Z(15).TT1ST.TB(TS).R(IST DIMENSION DLCT15).UCT15),TH(15),AL(15).EGS(15) NRITE(11.789) DO 777 J=IQNAPT EGSIJ)=0.0 READ(12./)EGS(J) 771 CUNTINUE 789 FORMAT1' INSERT 0.0-~1.0 EXCESS GREEN SHIFT FOR"! “EACH VEIN 1 LINE AT A TIME“) 1P=1 IL=1 N05E=u NDSL=0 Xc=ISTtPS CL=ICYCLF IF(IMARK(I).NE.10)GO T0 400 THE TRAFFIC PATTERNS. DELAYS: ETC FOR THE INITIAL Figure 3.4 OFFSET Listing C- C- “00 509 c- 520 521 522 523 526 524 137 SETTINGS ARE CALCULATED AND THE SIGNAL AND LINK DETAILS ARE DUTPUT. J=IMARK(6) IF(J.EU.0)GO T0 5 MFAULTzu IL=U CALL $UHPT(TPTUUTaRKAIRKBoRKCoRKDoRKE) IFIJ.EU.I)GU T0 5 TVs! IFTwR11.EU.3)IV=3 NFAULT=0 CALL TUUTP(HKA,RKB,RKC.RKD.RKE) NUSE=0 NUSL=0 MFAULTzu CALL SUBPT IIPTOUTIRKApaKBoRKCpRKDrRKL) IV=1 IF(WRII.EQ.S)IV=3 NFAULT=D CALL TUUTP (RKAoRKBoRKCIRKDoRKE) CONTINUE CALC NEW UFFSETS FOR EACH NUDE 8 ADJUST LINK GREENS. M=0 IND=0 M=M+I N=NN(M) TN=1800tN P30 PBR=0 IF(L0(M).GE.S)GO T0 550 SET UP FOR ONE-RAY STREET. 1:5-L01M) J=0 J=J+1 NLNzLNTM,J) NA=10ANLN+I'T DU 521 IA=T.ILINKS IF(NA.EU.LNUM(IA))GO T0 522 CONTINUE IF(J.EQ.N)GU T0 524 NA2=TO*LN(M,J+1)+I’T DU 523 IB=IoILTNKS IF(NA2.E0.LNUN(IB))GO T0 526 CONTINUE TIJ)=CONV1AAVE(LJToIB) PR:NRT(IA) R(J):RRACONVA IFTJ.LT.N)GU T0 520 G=1.-R(1) DO “I J=20N Figure 3.4 (cont'd.) 138 XG=I.-R(J) IF(G.GT.XG)G=xG uI CUNTINUE Na:N-I 00 Tn 916 c- SET UP FOR Two-NAY STREET. 550 1:7-LUTM) an 610 J=J¢1 NLN=LN(M,J) IFTJ.EQ.1.AND.LN(M.J).GT.LN(MoJ41))I=I+2 NAP=IU*NLN§I'T IFTJ.F0.1)KI=TSIGN(2.LN(MoJ+IJ-LNTMoJ)) 00 611 IA=1.ILINKS IF(NAP,EQ,LNUM(IA))G0 TU 612 011 CONTINUE 612 P8R=PBR+LTOTF(IA) NA=NAP+KI DU 613 18:1.ILINKS IF(NA.EU.LNHH(IH))GU TU 61a 615 CONTINUE old P:P+LTUTF(IH) mnaao IF(J.EU.N)GU T0 615 NA2=TU*LN(M.J+1)+KI+I-1 DO 616 IE=1.ILINKS IFINA2.EN.LNUM(IE))GO Tu 617 616 CONTINUE 617 T(J)=CONv1*AVE(LJT.It) T8(J)=CONv1*AvE(LJT,TAT 615 RR=NR1(18) R(J)=RR*CUNVA IF(J.LT.N)GD Tn 610 PBR=PBRITN 621 R=PITN N2=N-1 C' CALC Y(I) K ZTI) FUR tACH SIGNAL. Y(1)=". 2(1)=0. DO 10 1:20N IFILUTM).LE.4)Tb(I-I)=0. Dx=.5t(T(I-I)+T8(I'l))/CL Y(I)=Y(I-1)+Dx-.5*(R(I)-R(I-1)) uxx=.s*(1(1-1)-TB(I-1)T/CL 10 Z(I)=Z(T-1)+Dxx C' CALC MAX EQUAL BANDNIDTH. CB=-1.1 00 an 1:1,N 88:1.1 J=0 Figure 3.4 (cont'd.) 30 21 20 40 C- “5 So 55 60 65 139 J=J+l YY=Y(J)-Y(I) U‘)=1.'SCAL(YYIIO) YY=Y(J)-Y(1)-.S US=1.-$CAL(YY.1.) UUR=U0-R(J) USR=US-R1J) IF(UOR.LT.USR)GO T0 3 TH(J)=0. AL(J)=U0 D=UOR ‘60 10 4 FH(J):0.5 ALtJ)=HS D=U5R Ib(D.LT.SB)SH=D IF(J.LT.N)6U TU 30 IFISB.LF.CB)GD T0 20 CB=SB KC=T DO 21 J=lon DLC(J)=TH(J) UC(J)=AL(J) CUNTINUE IF(CB.LT.U)CB:0. CHZ=2.*CB BS=C6 BBS=CH CALC MAX EUUAL BANDWIUTH SYNCHRONIZATION. DU 35 .1:le ZZ=Z(J)-Z(KC)+DLC(J) THTJJ=SCAL(ZZ.1.) CALC MIN GREEN. G=IO-R(‘) DU “0 [=20N XG:lo-R(I) 1F(G.GT.XG)G=XG CONTINUE PPH:P+PBR lF(PBR-P)45o105o70 CALC ADJUSTMENTS FOR lNBOUND LT OUTBOUND. IF(PPH,GT.C82)GU T0 50 BS=ANIN1(69CBZ*PIPPB) GU TU 60 1F(P.GE.CBB)GO T0 55 BS:AMINI(G:P) GO TO 60 88:6 DO 65 J=TaN AL(J)=AMAX1(uc(J)-l.+BS-CB.0.) Figure 8.4 (cont'd.) 7o 75 60 85 90 95 100 105 884 883 882 879 832 878 880 916 140 335:AMAX1(C62-BS.0.) IF(LU(M).LE.0)BBs=0. GO TO 95 CALC ADJUSTMENTS FUR INBUUND GT OUTBOUND. IFTPPB.GT.C82)GO TU 75 st=AMIN1(G.CBZ*PBR/PPB) Go To 85 IF(PBR.GE.CBZ)GU T0 80 BBS=AMIN116oPBPJ GO TO 85 885:6 '00 Q0 J=loN AL(J)=AMAX1(HHS+R(J)-UC(J).0.) HS:AMAX1(CBB-BBS.0.) AL(KC)=U. DU ‘00 JS‘oN YY=TH(J)-AL(J) TH(J)=$cAL(YY,l.) CONTINUE MAKE EXCESS GREEN ADJUSTMENTS T0 OFFSETS. DU 880 J=IIN AJ=J-KC TC=0. J1=J J2:Kc-1 IF(J-KC)863,879,884 J1=KC J2=J-l DU 862 JJ=J1.J2 TC=TC§T(JJ) TC=TCICL GC:TH(J)-.S*(R(KC)+R(J))-8$-SIGN(TC'AJ) ch=1.-R(J)-us 1 ; GC=SCAL(GC:1.) TCB=0. IF(J.E0.KC)GO T0 878 DU 832 JJ=leJ2 TCB=TCH-TB(JJ) 1C8=TCBICL GCB=TH(J)+.S*(R(KC)'R(J))*$IGN(TCB04J) ccau=1.-R(J)-BBS GCB=SCAL(GCB.1.) Hc=AMINl(GCoGCB) TH(J)=TH(J)-EGS(MJ*SIGN(BCoAJ) IF(THTJ).LT.0.)TH(J)=TH(JJ+I. TH(J):TH(J)*XC+.S SET UP INTERFACE FROM UFFSETS TO NBIAS(*o*). TT=0. DO 907 J=IIN IF(H.E0.1.AND.J.E0.1)DEL=0. Figure 3.4 (cont'd.) 926 927 901 770 771 911 908 909 887 907 700 141 IF(J.GT.1)GO T0 926 IF(M.GT.1)GO T0 907 NS=DEL+.S GO TO 901 IF(L0(M).GE.S)GO Tn 927 T1=T1+T(J-1)/CL ES=(1.'EGS(H))*(1.'R(J)-G) us:(11+R(J)-R(1)+DEL+£S)*XC+.5 GO TO 901 YY=TH(J)-TH(1) N8=YY*(.S*(R(J)-R(l))+DEL)*XC*.5 -NS=SCAL(NS.I$T£PS) NLN=LN(M0J) DU 770 KKK=1,INODES IF(NLN.EQ.NLIST(KKK))GU T0 771 CONTINUE IF(L0(M).LE.GIGU T0 911 1:1 IF(LU(M).tQ.S)I=2 NA=10*NLN+I'I DU 908 IA=1.ILINKS IF(NA.tu.LNUM(TA))GO TU 909 CUNTINUE JB=LHSTRT(1.IA) JHB=LBFNSH(1'IA) NF:NBIAS(JHB'KKK)‘NBIAS‘JBpKKK) NBIAS(JH,KKK):NS =NS NF=NF+NS NF=SCALTNF.ISTEPS) NBIAS(JBB:KKK)=NF IT(N,EU,NART)GD T0 907 IF1NLN.E0.LN(M+1'1))XGzAtc0NVA-R(J) IF(M.E0.NANT-1)GO T0 907 00 887 L=2.NART-M IF(NLN.NE.LN(M+Lpl))GU T0 887 X62:A*CONVA-R(J) 1ND:M+L GO T0 907 CONTINUE CONTINUE 0EL=XG IF(IND.EQ.0)GO T0 700 IF(M.EU.IND-1)DEL=XGZ IFtM.LT.NART)GO T0 509 OUTPUT FOR NEW SIGNAL TIMINGS. MFAULT=0 IL=I CALL SUBPT(IPTOUTpRKAoRKBaRKC:RKD.RKE) IV=1 Figure 3.4 (cont'd.) 142 TFIWRIT.EQ.3)IV=5 ISTLST=2 NFAULT=ISTLST CALL TOUTP(RKAoRKH'RKCIRKDoRKE) C- FURTHER COPIES OF THE FINAL OUTPUT ARE PRINTED C- IF SPECIFIED BY THE CONTENTS OF IMARK(9). IF([MARK(9).LT.2.0R.IMARK(9).GT.b)GU T0 201 IY=IMAPK(9)-1 DU 509 IZ=T:IY CALL TOUTP (RK‘pRKUoRKCoRKUoRKE) 309 CONTINUE 201'1F(NPLTS.EQ.0)GO IU 202 CALL SPLOT (IPTDUT) 202 RETURN T.TTI) FUNCTION SCAL(AIX) c- THIS FUNCTION SCALES (A) BY IX) SUCH THAT t- (A) LIES BETWEEN 0.0 AND (X). SCALzA TI IF(SC‘L.GE.".)GU T0 12 SCAL=SCAL+X GO TO 11 12 IF(SCAL.LT.X)RETURN SCAL=SCAL'X 60 T() 12 RETURN END FUNCTION AVE(X,K) C- THIS FUNCTION AVERAGES THE XItpK) VECTOR COMPONENTS. DIMENSION x(d,200) COUNT=0. AVE:O. DO 1 1:1,4 [FIX(IoK).E0.0.)GU TO 1 COUNT=COUNT+1. ' AVE=AVE+X(I.K) 1 CONTINUE IFIAVE.E0.0.)RETURN AVE=AVEICOUNT RETURN END Figure 3.4 (cont'd.) BIBLIOGRAPHY (ALI) (ALZ) (CR1) (DRl) (GAl) (GL1) (GWl) (H11) (1N1) (KPl) BIBLIOGRAPHY Allsop, R.E., "Selection of Offsets to Minimize Delay to Traffic in a Network Controlled by Fixed-Time Signals,” Transportation Science, 1 (1967), 1-13. Allsop, R.E., 0 timization Techniques for Re- ducing Delay to Traffic—in Signalized Road Networks, Ph.D. Thesis, University College, London, 1970. Cronk, D.L., "Modern Highways News Report," Rural and Urban Roads, (February 1974), 9. Drew, D.R., Traffic Flow Theory and Control, McGraw-Hill, Inc., I968. Gartner, N., "Microscopic Analysis of Traffic Flow Patterns for Minimizing Delay on Signal- Controlled Links," Highway Research Board, Record No. 445, (1973), 12-23. Gartner, N., Little, J.D.C., "Generalized Combination Method for Area Traffic Control,“ Transportation Research Board, Record No. 531, (1975), 58-69. Gerlough, D.L., Wagner, F.A., "Improved Criteria for Traffic Signals at Individual Intersections," National Cooperative Research Program, 32(1967). Hillier, J.A., "Appendix to Glasgow's Experi- ment in Area Traffic Control," Traffic Engineering and Control, 7, 9, TI9635, 569-571. Inose, H., Fujisaki, H., Ramada, T., “A Road Traffic Control Theory Based on a Macroscopic Traffic Model,” J.I.E.E., 87, 8, (1967), 55-67. Kaplan, J.A., Powers, L.D., "Results of SIGOP- TRANSYT Comparison Studies," Traffic Engineering, 143* (KRl) (LMl) (m1) (MH1) (MW1) (0L1) (PAl) (R01) (R02) (SE1) (TRI) 144 Kreer, J.B., "Factors Affecting the Relative Performance of Traffic Responsive and Time-of- Day Traffic Signal Control,” Transportation Research, Vol. 10, (1976), 75-81. If Little, J.D.C., Martin, B.V., and Morgan, J.T., ”Synchronizing Traffic Signals for Maximal Bandwidth,“ Highway Research Board, Record No. 118, (1966), w21- 47. Lieberman, Edward 8., W00, James L., ”SIGOP II: A New Computer Program for Calculating Optimal Signal Timing Patterns," Transportation Re- search Record, Record No. 596, (1976), 16- 21. Munjal, P.K., Hsu, Y.S., ”Comparative Study of Traffic Control Concepts and Algorithms," Highway Research Board, Record No. 409, (1972), “-800 Messer, Carroll J., Whitson, Robert H., Dudek, Conrad L., and Romano, Elio J., "A Variable- Sequence Multiphase Progression Optimization Program,” Trans ortation Research Board, Record No. 445, (19735, 53-33. Olmstead, J.M.H., Advanced Calculus, New York: Appleton-Century-Crafts, Inc., I96I. Panyan, W. D., Urban Traffic System Simulation and Control, Ph. D. Thesis, Michigan Statefi Uni- versity, 1969. Robertson, D.I., ”TRANSYT: A Traffic Network Study Tool," Trans ort and Road Research Laboratory, TRRL Report LR-253, (19695. Robertson, D.I., "TRANSYT Method for Area Con- trol Program-Manual," Trans rt and Road Research Laboratory, NTIS PB- 4IU§4, (August I972). Seddon, P.A., "The Prediction of Platoon Dis- persion in the Combination Methods of Linking Traffic Signals,” Transportation Research, 6, (June 1972), 125-156. "SIGOP: Traffic Signal Optimization Program, A Computer Program to Calculate Optimum.Co- ordination in a Grid Network of Synchronized Traffic Signals," Traffic Research Corp., New York, (1966). (WAl) (WEI) 145 Wagner, F.A., et al., "Improved Criteria for Traffic Signal Systems on Urban Arterials," National Cooperative Research Program, 73, (19697} Webster, F.V., "Traffic Signal Settings," Road Research Laboratory, Technical Paper 39, London, (1958). MTITITTWNISIMIW WWIITITIHYII WWW“ 3 1293 0306‘! 8908