Despite traffic circles, four-way stop signs, lights regulated by timers or sensors, and other methods, the management of urban intersections remains problematic. Consider that transportation systems have all the features of so-called complex systems: the great number of state and control variables, the presence of uncertainty and indeterminism, the complex interactions between subsystems, the necessity to optimize several optimization criteria, and active behavior of the controlled process, to name just a few. Therefore, a mathematical approach to these systems can resolve their complex issues more elegantly than other methods.Addressing both efficiency and traffic safety issues, Optimal Traffic Control: Urban Intersections examines the traffic control optimization problem and presents a novel solution method. Using an approach based on control theory, graph theory, and combinatorial optimization, the authors derive a full mathematical description of the traffic control problem and enumerate all combinatorial aspects. The result is a set of algorithmic solutions to various problems along with computer implementation that you can incorporate into real traffic control systems for immediate results. The book concludes by evaluating how the choice of a complete set of signal groups influences intersection performance.Although modern cities throughout the world have a unique character influenced by culture, geography, and population, most of them share one main feature: busy intersections and the issue of controlling the traffic traveling through them. The development of information technologies, especially computer and telecommunications techniques, has changed the complexity of the problem and influenced the development of new solutions. Clearly stating the issues and presenting a possible solution, this book shows you how to take full advantage of all the capabilities of microprocessor-based traffic signal controllers.

Preface	p. xi
Introduction	p. 1
Mathematical Model of Traffic Process on a Signalized Intersection	p. 9
General mathematical description of the dynamic process on a signalized intersection	p. 11
Uncontrolled system inputs	p. 17
Input components-traffic streams	p. 17
Mathematical models of arrival flows-traffic streams	p. 18
Modeling arrival flow with the Poisson process	p. 19
Modeling arrival flows with more complex stochastic processes	p. 21
Traffic stream parameters	p. 22
Basic relations in the set of traffic streams	p. 24
Conflictness relation of traffic streams	p. 25
Nonconflictness relation of traffic streams	p. 26
Compatibility relation of traffic streams	p. 27
Signal group	p. 33
Signal group definition	p. 34
The relation of identical signal indications (Identity relation)	p. 36
The complete set of signal groups	p. 39
Collection of all complete sets of signal groups	p. 39
Number of signal groups in a complete set of signal groups	p. 44
Compatibility relation of signal groups	p. 48
Relation of partial ordering in collection D[subscript b]	p. 53
Traffic control	p. 57
Control variables	p. 58
The control-signal plan	p. 62
Control vectors-phases	p. 62
Signal plan structure	p. 64
Queues-state of the traffic process on isolated signalized intersections	p. 69
Definition of the state	p. 69
State transformation	p. 71
Deterministic model of state transformations (regular vehicle arrivals)	p. 71
Stochastic queuing models	p. 75
The output function	p. 81
Control Problem Statement	p. 85
General statement of traffic control problem (signal plan choice)	p. 87
The set of feasible controls (signal plans)	p. 91
The constraints that define the set of feasible controls	p. 92
Constraints on control variables	p. 95
The constraint of one interval of green indication during the cycle	p. 95
Constraints of minimal duration of green indication intervals	p. 97
The constraints of maximal red times	p. 99
The flow balance (capacity) constraints	p. 99
The set of feasible control vectors	p. 108
The intergreen time constraints	p. 114
Minimal intergreen times for pairs of traffic streams	p. 115
Minimal effective intergreen times for signal group pairs	p. 119
The extension of the set of feasible phases	p. 120
The relation of green indications succession	p. 124
The relation and graph of phase transitions	p. 127
The phase transitions relation	p. 127
The phase transition graph	p. 131
Some features of the phase transition graph	p. 137
Structural constraints on phase transition	p. 141
Minimal intergreen time constraints for phases	p. 145
The constraint on the sum of phase durations	p. 148
Mathematical expressions of signal plan constraints	p. 148
Optimization criteria	p. 161
The general form and features of optimization criteria	p. 161
Types of optimization criteria	p. 162
Optimization criteria related to capacity	p. 166
Theoretical capacity	p. 167
Practical capacity	p. 168
Capacity per cycle	p. 169
Saturation degree of a traffic stream	p. 170
Capacity factor	p. 171
The delay at an intersection	p. 172
Mathematical expectation of delay when cycle time is known	p. 176
Mathematical expectation of delay when cycle time is not given	p. 177
The number of vehicle stops	p. 178
Signal plan parameters	p. 180
The cycle time	p. 180
Green time of a signal group	p. 180
The total number of control vectors (phases)	p. 181
The Method of Optimal Signal Plan Determination	p. 183
The statement of the problem of finding the optimal closed path on graph G[subscript s]	p. 185
The method of finding the optimal closed path on graph G[subscript s]	p. 187
Elements of the method	p. 188
The relaxation and extension of J[subscript c](u(.)) function	p. 188
The elimination criterion	p. 194
Branching rules	p. 195
Bounding rules	p. 207
Branch-and-bound recursive operation	p. 208
Determination of Optimal Control (Signal Plan)	p. 215
Capacity optimization	p. 217
The capacity per cycle	p. 218
Maximization of the intersection capacity per cycle	p. 220
Equalizing saturation degrees of vehicle traffic streams	p. 235
Minimization of the sum of square differences between saturation degrees of traffic streams	p. 236
Minimization of the maximal saturation degree	p. 239
Capacity factor maximization	p. 240
Delay minimization	p. 253
Delay minimization in the case the cycle time is known	p. 253
Delay minimization when cycle time is not given	p. 257
Extreme values of signal plan parameters	p. 259
Maximization of effective green times	p. 259
Cycle time minimization	p. 260
Cycle time maximization	p. 262
Extreme values of the length of signal plan structure	p. 263
Determining the maximal length of signal plan structure	p. 264
Determining the minimal length of signal plan structure	p. 267
Effects of the Choice of the Complete Set of Signal Groups on Intersection Performance	p. 269
The relation of partial ordering (refinement) and the set of feasible controls	p. 271
The heuristics for the choice of the complete set of signal groups	p. 273
Appendices
Graphs, cliques	p. 289
Equivalence relation	p. 295
Pseudocodes of programs CLIQ and MINA	p. 297
Refinement relation, Hasse diagrams	p. 299
Effective values of green, red, and intergreen times	p. 301
Determination of the control vectors transition graph	p. 309
Description of STECSOT program (STructurE and Cycle Split Optimization Technique)	p. 321
The proof of delay function convexity	p. 329
References	p. 339
Index	p. 347
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Optimal Traffic Control: Urban Intersections

9781420062687

1420062689

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