Stochastic p-timed Petri nets and their use for traffic queue estimation

Abstract

Proposes a method which uses a type of stochastic Petri net for filtering problems and presents the application of this method to the estimation of traffic queues in urban networks. An extension of probabilistic transition Petri nets, which associates to each place a distribution of timings, is developed. A firing algorithm is obtained by applying to the distribution of markings a composition operator on densities of probability. Links of the traffic network and vehicles are respectively modeled by places between two transitions and tokens. For a controlled link, another place is connected to one transition which is enabled by a signal. Interconnections between links are modeled by probabilistic transitions and sets of timed output places. Those places are connected to places gathering the outputs of several links. The filtering principle on one hand uses the firing algorithm and on the other hand applies Bayes' rule on observed transitions, corresponding to the actuation of loop sensors, and on measured variables correlated to the marking, corresponding to the individual presence time of vehicles on loop sensors. At each sampling time the algorithm performs measurement, correction and prediction steps. The resulting distribution of markings is used as the initial distribution for the next sampling time. Experiments with a microscopic traffic simulator show that, for a link, the filtering proposed is slightly better than the one based on Markov chains. For a simple system with two links the improvement is between 50% and 75%. For an intersection, the average quadratic error is about 1 veh/sup 2/. Tests with the simple system and vehicles vanishing indicate an average quadratic error on a link of 74 veh/sup 2/. Two possibilities to improve the estimation effectiveness are investigated: the addition of either a transition or a place. It leads to errors of 4 veh/sup 2/ and 3 veh/sup 2/ respectively. When an incident nullifies the discharge rate the queue estimator has to detect it. Two methods are proposed; one based on parameter estimation and the other on the extension of the Petri net. This last approach allows the detection of all the incidents. The application of the filtering method to other fields of control in transportation systems is a promising prospect.