Queueing models for urban traffic networks

Roots, Symmetry, and Contour Integrals in Queuing-Type Systems

“…Example 3.2 (general bulk-service queue). Examples of systems with a pgf of the form (1.1) are the bulk-service queue (see [3]), the multiserver queue (see [12]), and Downloaded 11/18/21 to 130.89.15.141 Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/page/terms OBLAKOVA, AL HANBALI, BOUCHERIE, VAN OMMEREN, ZIJM certain traffic-light queues (see [19] and [18]). Each of these examples can be represented as a more general discrete-time bulk-service queue with arrivals that depend on the queue length prior to service.…”

“…For this example, X(z) is the pgf of the queue length, n + 1 is the size of service bulk, x j is the probability of having j customers in the queue, f j (z) = A(z) \sum n k=j z k , and D(z) = z n+1 -A(z), where A(z) is the pgf of the number of the arrivals during a time slot. This type of pgf occurs in many other queuing systems, e.g., in traffic models (see [19], [18]), but also in more general stochastic systems; see section 5 below.…”

“…The systems with these properties include the bulk-service queue (see [3]); the multiserver M/D/s and Geo/D/s queues, which are in some sense equivalent to the bulk-service queue with Poisson and binomial arrivals (see [12] and [14]); and the fixed-cycle traffic-light queue (see [18]). The traffic-light queue for a lane with detectors (see [19]) is an example of a queuing system with a pgf of the form (1.1) without these properties. For this queue, we can use the results for the systems with a general matrix M (z, z 1 , .…”

“…For example, in traffic models, an intersection can serve a number of vehicles during a green period, but when there is no queue, the vehicles may proceed without stopping with a higher speed than those that need to wait. As a result, more vehicles may pass the intersection; see [19], [18]. For the multiserver queues M/D/s and Geo/D/s, the queue length at times md, where d is the deterministic length of service, m \in \BbbN , behaves as the classical bulk-service queue length just after service with Poisson or binomial arrivals, respectively; see [12], [14].…”