Abstract
The fixed-cycle traffic-light (FCTL) queue is the standard model for intersections with static signaling, where vehicles arrive, form a queue and depart during cycles controlled by a traffic light. Classical analysis of the FCTL queue based on transform methods requires a computationally challenging step of finding the complex-valued roots of some characteristic equation. Building on the recent work of Oblakova et al. (Exact expected delay and distribution for the fixed-cycle traffic-light model and similar systems in explicit form, 2016), we obtain a contour-integral expression, reminiscent of Pollaczek integrals for bulk-service queues, for the probability generating function of the steady-state FCTL queue. We also show that similar contour integrals arise for generalizations of the FCTL queue introduced in Oblakova et al. (2016) that relax some of the classical assumptions. Our results allow us to compute the queue-length distribution and all its moments using algorithms that rely on contour integrals and avoid root-finding procedures.
Highlights
The fixed-cycle traffic-light (FCTL) queue is an intensively studied stochastic model in traffic engineering [3,5,6,11,12,17,19]
Vehicles that arrive during a green period and meet no other vehicles in the queue are treated according to the following assumption: Definition 1 (FCTL assumption) For those cycles in which the queue clears before the green period terminates, all vehicles that arrive during the residual green period pass through the system and experience no delay whatsoever
The present paper extends that methodology to the transform domain and results in a contour-integral expression for the probability generating function (PGF) of the FCTL queue and some of the generalizations considered in [13]
Summary
The fixed-cycle traffic-light (FCTL) queue is an intensively studied stochastic model in traffic engineering [3,5,6,11,12,17,19]. Using analytic techniques suitable for dealing with such Markov chains, Darroch [5] obtained the probability generating function (PGF) of the steady-state overflow queue (the number of vehicles waiting in front of the traffic light at the end of a green period), and the PGF of the steady-state delay was obtained in van Leeuwaarden [17]. The traditional way of determining these remaining unknowns consists of two steps: finding the g − 1 complex-valued roots and using these roots as input for a system of linear equations whose solution gives the boundary terms Both steps can present difficulties, but were somehow considered unavoidable in the mathematically rigorous treatment of the FCTL queue [5,11,12, 17,19] and of related bulk-service queues [9,14,15].
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