INTERACTION BETWEEN SIGNAL SETTINGS AND TRAFFIC FLOW PATTERNS ON ROAD NETWORKS Gaetano FUSCO1, Guido GENTILE1, Lorenzo MESCHINI1, Maurizio BIELLI2, Giovanni FELICI2, Ernesto CIPRIANI3, Stefano GORI3, Marialisa NIGRO3
Abstract. The paper illustrates methodology and preliminary results obtained in the project of basic research “Interaction between signal settings and traffic flow patterns on road networks”, funded by the Italian Ministry of University and Research. The object of the project is to develop a general procedure to study, model and solve the problem of the optimal road network signal settings, by taking into account the interaction between signal control systems and traffic flow patterns. Different macroscopic and microscopic modelling approaches are discussed and applied to test cases. Moreover, different signal control strategies are introduced and tested by numerical experiments on a wide area of the road network of Roma.
1.
Introduction
The paper describes the objectives, the methodology and the preliminary results of the research project “Interaction between signal settings and traffic flow patterns on road networks”, granted by the Italian Ministry of University and Research with the Fund for Investments on Basic Research (FIRB). The project joins three research units, belonging respectively to the University of Rome “La Sapienza”, the University “Roma Tre” and the Institute for Information System Analysis (IASI) of the Italian National Council of Research.
1
Università degli Studi di Roma La Sapienza, via Eudossiana, 18, Roma,
[email protected];
[email protected];
[email protected] 2
Istituto di Analisi dei Sistemi ed Informatica, CNR, viale Manzoni, 30, Roma,
[email protected];
[email protected] 3 Università degli Studi Roma Tre, Dipartimento di Scienze dell’Ingegneria Civile, Via Vito Volterra, 62,
[email protected];
[email protected];
[email protected]
The object of the research is to develop a general procedure to study, model and solve the problem of optimal road network signal settings, by taking into account the interaction between signal control systems and traffic flow patterns. This problem is interesting by both a theoretical and an application point of view, since several mathematical studies and experimental results have shown that usual signal setting policies, which simply adjust signal parameters according to the measured traffic, may lead to system unstable solutions and deteriorate network performances. At core of the problem is the difference between a user equilibrium flow pattern, where individuals choose their paths in order to minimize their own travel time, and a system optimizing flow that minimizes total delay of all users. The problem is object of an intensive research activity by the scientific community by many years (Cfr. [4] and [27]. Although the combined problem of signal settings and traffic assignment is a well known non convex problem, a systematic analysis of the objective function of the global optimization signal setting problem highlighted in several examples large quasi-convex intervals for even different levels and patterns of traffic demand [5]. Still open problems are recognizing which conditions give rise to non convexity of the objective functions as well as individuating possible real-time strategies to keep the state of the traffic network near system optimal conditions. The global optimal signal settings on a road network is a complex problem that involves dynamic traffic patterns, users’ route choice and real-time application of suitable control strategies. With several noticeable exceptions (Cfr. [13], [26], [1] and [2]), the problem is usually tackled by following an equilibrium approach, that is, by searching for a possibly optimal configuration of mutually consistent traffic flows and signal variables. The equilibrium approach assumes, but does not reproduce, the existence of a process where drivers correct their route choice day-to-day, according to the modified network performances, until no further improvement could be achieved. The traffic flow is modeled by a stationary relationship between link travel time and link flow. Although the delay at node approaches can be taken into account as it may be added to the link cost, neither a realistic modeling of traffic congestion on a road network or an explicit representation of real-time traffic control is possible. In order to investigate real-time applications, the research project aims at extending the static modeling framework usually adopted in the scientific literature to apply dynamic assignment models and deal with both coordinated arteries and traffic-flow responsive signal settings. The study is being developed focusing on the following topics: • Modelling of traffic flow along coordinated arteries and urban traffic networks; • Integration of signal control and dynamic assignment problems; • Advanced models and methods for signal settings and traffic control.
2.
Modeling framework
Many models with increasing levels of complexity have been analyzed in order to identify the most suitable for different possible applications to arteries or networks, in static or dynamic contexts.
2.1.
Analytical traffic model for synchronized arteries
A simple analytical traffic model that describes the average delay along a synchronized artery (see ref. [24] for further details) has been here extended to apply different hypotheses of drivers’ behavior. Such hypotheses concern the capability of drivers to accelerate and catch up the tail of a preceding platoon, if any. Specifically, the following assumptions have been introduced: a) all vehicles follow the same trajectory and, after having been stopped at a signal, accelerate at a given rate until they reach a given cruise speed; b) all vehicles can accelerate along a link and catch up the previous platoon; c) all vehicles can accelerate up to a given maximal acceleration rate, so that they catch up the previous platoon only if the link is long enough. This model has been developed specifically to assess synchronization strategies along signalized arteries. Delays at every approach of the artery are computed by checking, for each arriving platoon, which condition occurs among the following: a) the platoon arrives during the red (or -if a queue already exists at that approach- before the queue is cleared, so that the first vehicle of the platoon is stopped) and ends during the green; b) the platoon arrives during the green (or after the queue, if any, has been cleared) and ends during the red; c) the platoon arrives during the green after the queue, if any, has been cleared and ends during the green of the same cycle; d) the platoon arrives during the green and ends after the green of the next cycle. A different analytical expression for the delay holds for each condition. Since the existence and the length of a queue can not be determined before all platoons have been analyzed, the delay computation requires an iterative procedure that classifies the different platoons progressively. It is worth noting that such a procedure involves few iterations, because the platoons can both catch up each other along the links and recompose themselves at nodes, when more platoons arrive during the red phase.
2.2.
Dynamic traffic network loading
The cell transmission model [6] has been here extended to deal with coordinated arteries, by introducing the following changes: • in less than critical conditions, vehicles can move forwards by a generic quantity, dependent on the traffic density on the link; • capability of simulating even complex signalized intersections; • apply dynamic shortest paths and follow different OD pairs in the network loading process. The extended cell transmission model has been selected as the most suitable mathematical model to assess and calibrate real-time feedback signal control systems that use the relative link occupancy as control variable, like Traffic Urban Control does [7].
2.3.
Microscopic simulation of traffic flow
A microscopic simulation model aimed at reproducing the behavior and interaction of vehicles on road networks has been developed by the IASI [11]. This microscopic simulator is able to manage different aspects of vehicles’ dynamic, that is: acceleration and deceleration, based on the car-following principles [6] and on the state of downstream signals; lane changes and overtakes, based on gap-acceptance rules and vehicles’ paths. Vehicles are generated according to the negative exponential distribution, where the number of vehicles to be generated every minute can be time varying in order to simulate the variation of vehicular flows during the day. The system includes a graphic interface, developed in the UNIX environment in C language with the public domain graphic libraries X11. The micro-simulation is animated on a reproduction of the road network, where the signals and the vehicles are visualized.
2.4.
Dynamic traffic assignment model
Dynamic traffic assignment models are the most advanced modeling framework to simulate traffic patterns in congested urban networks, as they can both simulate the flow progression along links and reproduce the complex interaction between traffic flow and route choice behavior of users. Several software packages for solving dynamic traffic assignment are now available. Two of the most popular and advanced (Dynasmart [9] and Dynameq [8]) have been taken into consideration in this research and are object of a systematic comparison. Both have been applied to a selected area of the road network of Roma, having 51 centroids, 300 nodes, 870 links, 70 signalized junctions (18 of which are under control). The figure below shows the study network as displayed by the two software packages.
Figure 1. The study network in Dynameq (on the left) and Dynasmart (on the right) Both models are simulation based, but they differ as far as: the modeling approach of traffic flow (Dynameq assumes a fundamental triangular diagram, Dynasmart a modified Greenshield model [11]); the route choice (two phases for Dynameq -a path generation phase in which one new path at each iteration is added until the maximum predefined number of paths is reached, and a convergence phase in which no new paths are added- one
phase for Dynasmart, which does not introduce any distinction between the generation phase and the convergence phase); the simulation process (Dynameq uses an event based procedure, Dynasmart a macroparticle simulation model); convergence criterion (time based for Dynameq, flow based for Dynasmart). Thus, Dynameq aims especially at providing a detailed representation of traffic flow by following a microsimulation approach. It reproduces the traffic behavior at junction approaches by an explicit lane changing model and resolves conflicts at nodes by applying a gap acceptance rule. It simulates pre-timed signal traffic control and ramp metering. On the other hand, Dynasmart provides a simpler framework to model the traffic flow and aims rather at allowing simulating the impact of real-time strategies of traffic control and information systems (like actuated traffic signal, ramp metering, variable message signs and vehicle route guidance) on users’ behavior. Even if the tests of the models are still in progress, the following remarks can be made. Comparison between static and dynamic assignment models shows that they provide not so different patterns of traffic flows (R2=0.61), but, as expected, the static model overestimates the traffic flow, as it assumes that the whole trip demand is assigned within a steady-state simulation period. Dynasmart and Dynameq predict very similar trends of the overall outflow of the network. However, their results are quite different as far as the route choice and, consequently, the traffic flow patterns. Finally, simulation experiments highlight that a relevant issue of dynamic simulation assignment models is their capability to reproduce a temporary gridlock, which is actually experienced on the network in the rush hour. In fact, if the dynamic assignment model is forced to prevent the gridlock, the goodness-of-fit of simulated flow with respect to observed values decreases from R2=0.54 to R2=0.46.
3.
Signal Control strategies
The traditional approach to signal control assumes traffic pattern as given and signal parameters (cycle length, green splits and offsets) as design variables. However, the combined problem of signal settings and traffic assignment generalizes the signal settings problem to concern the interaction between the control action and the users’ reaction. Usually, it takes as design variables only green splits of signals at isolated junctions. In the Global Optimization of Signal Settings Problem (Gossp), the whole system of signals is set to minimize an objective function that describe the global network performances. In the Local Optimization of Signal Settings Problem (Lossp), flow-responsive signals are set independently each other either to minimize a local objective function or following a given criterion, like equisaturation [29] or Po policy [25]. In the research project the usual approach is being extended in two directions: taking into account signal synchronization and develop signal control strategies that can be suitably applied in real-time.
3.1.
Pre-timed synchronization
Signal synchronization of two-way arteries can be applied by following two different approaches -maximal bandwidth and minimum delay-, although a solution procedure that applies the former problem to search for the solution of the latter one has been developed [23]. Specifically, it is well known that, given the synchronization speed and the vector of distances between nodes, the offsets that maximize the green bandwidth are univocally determined by the cycle length of the artery. Such a property of the maximal bandwidth problem has been exploited to facilitate the search for a sub-optimal solution of the minimum delay problem. Thus, a linear search of the sub-optimal cycle length is first carried out starting from the minimum cycle length for the artery and then a local search is performed starting from the offset vector corresponding to that cycle length. The analytical model described in section 2.1 makes it possible a quick evaluation of the artery delay without involving any simulation. The performances of the solutions obtained by applying the analytical model are then assessed by dynamic traffic assignment. Moreover, two different strategies for one-way signal synchronization are also introduced in this paper. The simplest strategy consists of starting the red of each node just after the end of the primary platoon (that is, the platoon with the most number of vehicles). In the second strategy, the red is checked to start after the end of the every platoon and the position giving the least delay is then selected. Preliminary tests conducted by carrying out a systematic analysis of the optimal offsets on an artery of Roma underline that the latter strategy outperforms the former one, especially for large traffic flows. Such an occurrence highlights that the performances of the artery are heavily affected by the complex process of platoon re-combination at nodes, so that the strategy that favor the main platoon can be ineffective if it is stopped later, while secondary platoons can run without being delayed. Numerical experiments concerning the analytical model have been carried out on Viale Regina Margherita, a 6-node artery situated into the study network of Rome, in the peak-off period. In the first experiment, the common cycle length of the artery has been set equal to the maximal cycle length of the junctions. The green times have been adjusted consequently to keep the green splits unchanged. Thus, the offsets are the only control vector. In the second experiment, also the cycle length has been varied from the minimum cycle for the most critical junction of the artery (namely, 32s) until its current value (108.5s). As before, the objective functions of the two problems are the maximal bandwidth and the minimum delay for the artery. The corresponding values of maximum bandwidth and sub-optimal minimum delay offsets are reported in Table 1 and Table 2, respectively.
Node N. 1 2 3 4 5 6
Node N. 1 2 3 4 5 6
Offset of maximum bandwidth [s] Cycle [s] 32 42 52 62 72 82 92 102 0 0 0 0 0 0 0 0 14.9 19.5 24.2 28.8 33.5 38.1 42.8 47.4 14.9 40.5 50.2 59.8 33.5 38.1 42.8 47.4 30.9 19.5 50.2 59.8 69.5 79.1 88.8 47.4 15.5 20.4 9.2 61.1 34.9 39.8 44.6 100.5 17.7 23.3 2.9 3.4 40.0 45.5 51.1 5.6 Table 1. Offset of maximum bandwidth. Offset of minimum delay [s] Cycle [s] 32 42 52 62 72 82 92 102 0 0 0 0 0 0 0 0 31.9 39.5 45.2 10.8 12.5 64.1 76.8 94.4 15.9 5.5 45.2 59.8 4.5 38.1 42.8 47.4 28.9 19.5 26.2 37.8 12.5 59.1 64.8 47.4 18.5 41.4 27.2 61.1 35.0 39.8 44.6 84.5 19.7 32.3 4.9 3.4 43.0 54.5 63.1 7.6 Table 2. Sub-optimal offset of minimum delay
108.5 0 104.7 50.4 50.4 52.6 6.5
108.5 0 1.7 47.5 50.5 85.1 5.9
Figure 2 plots the value of delay at nodes along the artery for different values of the cycle length and the sub-optimal offsets. The analytical delay model described in section 2.1 has been applied, by assuming acceleration capabilities of vehicles and then allowing a platoon catching up the preceding one. As expected, the delay function is non convex. The lowest value is obtained for a cycle length of 42s, while other local minima are obtained for the cycle lengths of 72s and 102s.
120
Delay [s/veh]
100 80 60 40 20 0 32
42
52
62
72
82
92
102
Cycle length [s]
Figure 2. Estimation of node delay along the artery by the analytical traffic model for different values of the cycle length.
3.2.
Actuated signal control based on logic programming
The traffic actuated strategies are constructed on the basis of the current configuration of the traffic flows or of the current number of vehicles at the controlled junctions. Thus, both the measurement of data and the elaborations aimed at identifying the traffic signal setting at each instant are carried out in real time. The literature proposes several control methods that use mathematical programming; amongst others, successful examples of adaptive systems are the SCOOT system [15], SCATS [18], UTOPIA [19], COP [14]. New approaches have been recently proposed with the development of models based on fuzzy logic, such [3], [17]. Traffic control based on logic programming have also recently appeared. The method adopted in this paper, described in [10], [11] and [12], is one of the first models and applications of this type. It is an adaptive method actuated by vehicles that adopts logic programming to model and solve the decision problems associated with traffic control. Such a method can be applied with success to urban intersections with high levels of traffic where many different and unpredictable events contribute to large fluctuations in the number of vehicles that use the intersection. The logic programming methods based on vehicle counts make it possible to design the traffic control strategies with a high degree of simplicity and flexibility. The system makes use of a very efficient logic programming solver, the Leibniz System [28], that is capable of generating fast solution algorithms for the decision problems associated with traffic signal setting. This method has been applied to a system of two signalized and coordinated junctions in Roma, Italy. The junctions, which are 180 meters apart and lay on a urban arterial road used to access and egress the city, are characterized by high congestion peaks and flow conditions varying rapidly over time. The schema of the study area is reported in Figure 3.
In cooperation with the Mobility Agency of Rome, a preliminary analysis of technical aspects, signal plans, traffic flows and congestion levels has been performed. Some historic samples of traffic data have been therefore integrated with a data collection campaign with the use of temporary loop detectors. Collected data have been used to calibrate and validate the simulation of the system with the micro simulator described in section 2.3. To take into account the problems related to the quantity and quality of available traffic data, both incomplete and at an aggregate level, this application required to integrate and refine the detected data with analytic processes based on mathematical traffic flow models. To this end, a suitable macroscopic model based on the simplified theory of kinematic waves [21] was developed to estimate the queue length of each stream at a given intersection, as a function of vehicles counts taken at an upstream section. Via Affogalasino
Via Fanella
3 180 m
6
suburbs
city center
4
1
5
7 Via Portuense
2 Regulator
Via del Trullo
Loop detector connected both to the local regulator and to the central system Loop detector connected only to the local regulator
Figure 3. Scheme of the two junctions for the application of the signal control based on logic programming
3.3.
Real-time feedback network signal regulator
Real-time network traffic control strategies usually follow a hierarchical approach that optimizes cycle length, green splits and offsets by evaluating their impacts on the network performances at different level of time and space aggregation. The Traffic Urban Control (TUC) model formulates the real-time traffic urban control system as a Linear-Quadratic (LQ) optimal control problem based on a store-and-forward type of mathematical modeling [7]. The basic control objective is to minimize the risk of oversaturation and the spillback of link queues by suitably varying, in a coordinated manner, the green-phase durations of all stages at all network junctions around some nominal values.
We are now aiming at exploiting the results obtained in [5], where at least one local minimum of the objective function of the global optimization of signal settings was found out and then can be assumed as a desired state of the network. We so extend the approach followed in TUC and we introduce a Linear-Quadratic regulator designed to control the distance to the desired state of the network, and specifically the green split vector and traffic flow patterns that are solution of the global optimization of signal settings problem. The LQ optimal control can be then be formulated as follows: g(k ) = g d − L[x(k ) − x d ] (1) where the following notations are introduced: g = green time vector (at time k); gd = desired green time vector (solution of static Goss problem); x = link flow vector (at time k); xd = link flow vector (solution of static Goss problem); L = gain matrix. The generalized cell transmission model briefly described in section 2.3 has been recognized as the most suitable approach for simulating such a real-time control strategy in the short-term. However, a wider analysis involving the long-term rerouting behavior of drivers and the en-route diversion of drivers, which may occur when large increase of congestion is experienced on some links of the network, necessarily requires the application of a dynamic equilibrium traffic assignment model. It is worth noting that such diversions are taken by the control system as random disturbances of the process, unless a real-time route guidance system exists that provides the drivers with updated information on the current or, better, on the future traffic conditions of the network.
4.
Signal control and dynamic assignment problems
The integration of signal control and dynamic assignment has a twofold goal. On one hand, we aim at assessing the effectiveness of the synchronization algorithm introduced in section 3.1 and, on the other hand, we aim at appraising the effect of signal control on users’ route choice. In the first experiment the cycle length of 108.5 seconds, which better fits the current conditions, is used and the corresponding suboptimal offset vector (last column of Table 2) has been applied in simulation to the artery. The equilibrium dynamic traffic assignment of the total off peak hour trip demand has been so carried out with Dynameq. Results reported in Table 3 show a significant reduction of delay on the artery (about 13%), which is even more important if we consider that, due to its improved performances, the artery attracts 13.8% more traffic. The total travel time on the whole network decreases as about 2%, although the study area is much wider than the influence area of the artery and, more important, the objective function of the synchronization algorithm accounts only the travel time of the artery.
Current scenario Synchronization Difference Average arterial travel time 176.5 154.0 -12.7% [s] Average total arterial travel length 41.4 47.1 +13.8% [veh·km] Average total network 87.6 85.5 -2.4% travel time [veh·h] Table 3. Comparison of arterial and network results pre and post synchronization. A further investigation concerns the analytical traffic model used by the synchronization algorithm. Two issues are here dealt with. The first regards the capability of such a simple model to estimate the average travel time along the artery. The second issue concerns the impact of the interaction between traffic control and drivers’ route choice to the solution found. As signals have been set by assuming the traffic flows as independent variables and the dynamic traffic assignment provided the dynamic equilibrium traffic flows on the artery, the analytical delay model has been applied again by taking the average equilibrium traffic flows as input. The travel times on the artery computed by Dynameq and by the analytical model are compared in Figures 5 and 6. Although the analytical model is based on a much simpler theory that assumes stationary and homogeneous traffic flow, the two models provide very similar results. As far as the two assumptions considered in the analytical model, namely the possibility or not that platoons can always catch-up each other, the former provided a better correspondence with the microsimulation performed by Dynameq.
Arterial travel time direction 1 [s
240 Simulation
220 200
analytical model (with catchup)
180
analytical model (without catchup)
160 140 120 100 80 5
25
45
65 85 105 125 Simulation interval [min]
145
165
Figure 4. Comparison between simulation and analytical model (Arterial direction 1, Cycle length 108.5 s).
Arterial travel time direction 2 [s
240 220
Simulation
200
analytical model (with catchup)
180
analytical model (without catchup)
160 140 120 100 80 5
25
45
65 85 105 125 Simulation interval [min]
145
165
Figure 5. Comparison between simulation and analytical model (Arterial direction 2, Cycle length 108.5 s) However, the differences between the two models become larger when shorter values of cycle are used. In such cases, the green is long just enough to allow the traffic flow being served. Even a small random increase of traffic produces a temporary over-saturation and a relevant increase of delay, which can not be predicted by a stationary model. Indeed, much larger differences have been recorded between the microsimulation and the well-known Webster model, which has been applied to compute the delay at the lateral approaches. Thus, although the analytical model predicted that shorter cycle lengths would reduce delays on the artery, the results obtained by the dynamic traffic assignment model indicate that, even if the travel times along the artery improve, the overall performances of the network are slightly worse.
5.
Conclusions and further developments
The research project “Interaction between signal settings and traffic flow patterns on road networks” aims at extending the global signal settings and traffic assignment problem to a more realistic dynamic context and at developing effective signal control strategies that lead the network performances to a stable desired state of traffic. Different approaches to model traffic flow are introduced and compared: the analytical model of delay on a synchronized artery; a generalized cell transmission model; a microsimulation model. Two dynamic traffic assignment models, Dynasmart and Dynameq, are applied to a real-size network. the two-way synchronization algorithm that applies the analytical delay model to carry out a linear search of the minimum delay solution, which revealed to be effective in
reducing the travel times along a synchronized artery even if, moreover, the traffic flow on the artery increased as it became more convenient. Current research concerns a Linear-Quadratic regulator designed to keep signals settings and traffic flows stably near a desired state, namely the solution of the global optimization of signal settings problem.
References [1] Abdelfatah A.S., Mahmassani H. (1998). System optimal time-dependent path assignment and signal timing in traffic networks, paper presented at the 77th Annual Meeting of Transportation Research Board. [2] Abdelghany K., Valdes D., Abdelfatah A. and Mahmassani H. (1999) Real-time traffic assignment and path-based signal coordination: application to network traffic management, 78th Annual Meeting of Transportation Research Board, Washington, January 1999. [3] Bingham E. (2001) Reinforcement learning in neurofuzzy traffic signal control. European Journal of Operational Research, 131, 232–241. [4] Cantarella, G.E., Improta and G., Sforza, A. (1991). Road network signal setting: equilibrium conditions. In: Papageorgiou, M. (Ed.), Concise Encyclopedia of Traffic and Transportation Systems. Pergamon Press, pp. 366–371. [5] Cipriani E., Fusco G. (2004). Combined Signal Setting Design and Traffic Assignment Problem, European Journal of Operational Research, 155, 569–583. [6] Daganzo C.F. (1997). Fundamentals of transportation and traffic operations. Pergamon, Great Britain, 147-156. [7] Dinopoulou, V., C. Diakaki, and M. Papageorgiou (2002). Applications of the urban traffic control strategy TUC. Proceedings of the 9th Meeting of the EURO Working Group on Transportation, Bari, Italy, 2002, pp. 366-371. [8] Dynameq user’s manual – Release 1.1. November 2005 [9] Dynasmart-P Version 1.0 User’s Guide. September 2004 [10] Felici G., Rinaldi G., Sforza A., Truemper K. (1995). Traffic control: a logic programming approach and a real application. Ricerca Operativa, 30, 39-60. [11] Felici G., Rinaldi G., Sforza A., Truemper K. (2002). A methodology for traffic signal control based on logic programming. IASI-CNR Report n. 581, ISSN: 1128-3378. [12] Felici G., Sforza A., Rinaldi G., Truemper K. (2006). A logic programming based approach for online traffic control. To appear on Transportation Research C, available online at www.sciencedirect.com. [13] Ghali M.O. and Smith M.J. (1995). A model for the dyamic system optimum traffic assignment problem. Transp.Res.-B, Vol.28B, pp.155-17.
[14] Head K.L., Mirchandani P.B., and Sheppard D. (1992). Hierarchical framework for real time traffic control. Transportation Research Record, 1324, 82-88. [15] Hunt P.B., Robertson D.I., Bretherton R.D., and Winton R. (1981). SCOOT - A traffic responsive method of coordinating signals. Transportation and Road Research Laboratory Report, LR 1014, Crwothorne, Berkshire, England. [16] Jayakrishnan R., Hu T., Mahmassani H. S. An evaluation tool for advanced traffic information and management systems in urban networks. Transportation Research-C, 2(3):129-147, May 1994. [17] Kosonen I. (2003). Multi-agent fuzzy signal control based on real-time simulation. Transportation Research C, 11, 389-403. [18] Luk J.Y.K. (1984). Two Traffic Responsive Area Traffic Control Methods: SCAT and SCOOT. Traffic Engineering and Control, 30, 14-20. [19] Mauro V., Di Taranto D. (1990). UTOPIA. Proceedings of the 6th IFAC / IFIP / IFORS Symposium on Control Computers and Communication in Transportation, Paris, France, 1990. [20] Mirchandany P., Head L. (2001) A real-time traffic signal control system architecture, algorithms, analysis. Transportation Research C, 9, 415-432. [21] Newell G.F. (1993) A simplified theory of kinematic waves in highway traffic, part I: general theory; part II: queuing at freeway bottlenecks; part III: multi-destination flows, Transportation Research B, 27, 281-313. [22] Niittymaki J., Turunen E. (2003). Traffic signal control on similarity logic reasoning. Fuzzy Sets and Systems 133, 109-131. [23] Papola N. and Fusco G. (1998), Maximal Bandwidth Problems: a New Algorithm Based on the Properties of Periodicity of the System, Transportation Research-Part B, 32(4), pp.277-288. [24] Papola N. and Fusco G. (2000b), A new analytical model for traffic signal synchronization, Proceedings of 2nd ICTTS Conference, Beijing, China, 31 July-2 August, 2000. [25] Smith M.J. (1980). A local traffic control policy which automatically maximises the overall travel capacity of an urban road network. Traffic Eng. Control, pp.298-302. [26] Smith M.J. and Ghali M.O. (1990). Traffic assignment, traffic control and road pricing. In C.Daganzo (ed.), Proc. of the 12th International Symposium on Transportation and Traffic Theory, pp.147-170, New York, Elsevier. [27] Taale, H. and van Zuylen, H.J., (2001). The combined traffic assignment and control problem––an overview of 25 years of research. In: 9th World Conference on Transport Research, Seoul, Chorea, July, 22–27. [28] Truemper K. (1998). Effective Logic Computation, Wiley-Interscience Pub., New York.
[29] Webster (1958). Traffic signal settings. Road research technical paper no. 39, HMSO, London.
