Bounds and Approximations for the Fixed-Cycle Traffic-Light Queue

Abstract

This paper deals with the fixed-cycle traffic-light (FCTL) queue, where vehicles arrive at an intersection controlled by a traffic light and form a queue. The traffic light alternates between green and red periods, and delayed vehicles are assumed to depart during the green period at equal time intervals. The key performance characteristic in the FCTL queue is the so-called mean overflow, defined as the mean queue length at the end of a green period. An exact solution for the mean overflow is available, but it has been considered to be of little practical value because it requires some numerical procedures. Therefore, most of the literature on the FCTL queue is about deriving approximations for the mean overflow. In deriving these approximations, most authors first approximate the FCTL queue by a bulk-service queue, approximate the mean overflow in the bulk-service queue, and use this as an approximation for the mean overflow in the FCTL queue. So far no quantitative comparison of both models has been given. We compare both models and assess the quality of the approximation for various settings of the parameter values. In this comparison and throughout the paper we do not restrict ourselves to Poisson arrivals, but consider a more general arrival process instead. We discuss the numerical issues that need to be resolved to calculate the exact expression for the mean overflow in both queues and show that clear computational schemes are available. Next, we present several bounds and approximations of the mean overflow that do not require numerical procedures. In particular, we derive a new approximation based on the heavy traffic limit and a scaling argument. We compare the new bounds and approximation with the existing ones. We elaborate on the impact of several parameters, like the length of the green and red period and the variance of the arrival distribution. Each of these parameters turns out to be crucial.

An exact root-free method for the expected queue length for a class of discrete-time queueing systems

“…Suppose that D(z) = z g − A(z) (16) and the functions A(z), B(z) are pgfs. Consider the following assumption.…”

“…Moreover, the precision of the found zeros also influences the result. To avoid these problems, some authors suggest approximations of the queue length; see, for example, [11,16,20]. Another approach is to use the matrix-analytic method; see [13].…”

Pollaczek contour integrals for the fixed-cycle traffic-light queue

“…The bulk-service queue serves as a popular approximation of the FCTL queue [16]. In fact, for Bernoulli arrivals with per time slot one or no arrival (which is case (i) in the proofs of Lemmas 3 and 4), this approximation becomes exact.…”

Proportional fair rate allocation for private shared networks

“…In our setup the green cycle is of duration g = 1 slot and the red cycle is of duraton r = τ − 1 slots. The characteristics of a FCTL have been much studied, see for example the overview in [3], with estimates of the average queue length given in [14] and [15].…”

“…where c = g/t > p. Proof: When c > 2p (so E(q x ) = 0), the second derivatives of w with respect to p and c are (14) impose the requirement that flow delay deadlines are met, while (15) ensures that the flow rates (including both information and dummy transmissions) can be scheduled by the network.…”