Waiting times in queueing networks with a single shared server

Abstract

We study a queueing network with a single shared server that serves the queues in a cyclic order. External customers arrive at the queues according to independent Poisson processes. After completing service, a customer either leaves the system or is routed to another queue. This model is very generic and finds many applications in computer systems, communication networks, manufacturing systems, and robotics. Special cases of the introduced network include well-known polling models, tandem queues, systems with a waiting room, multi-stage models with parallel queues, and many others. A complicating factor of this model is that the internally rerouted customers do not arrive at the various queues according to a Poisson process, causing standard techniques to find waiting-time distributions to fail. In this paper we develop a new method to obtain exact expressions for the Laplace-Stieltjes transforms of the steady-state waiting-time distributions. This method can be applied to a wide variety of models which lacked an analysis of the waitingtime distribution until now.

Queue-length balance equations in multiclass multiserver queues and their generalizations

“…In combination with (2)-(4) it also gives a short proof of (1). In other words, one can obtain an expression for the PGF of the joint steady-state queue-length distribution in a large class of polling systems by just using the elementary balance equations (2) and (9), combined with the obvious relations (3) and (4). Short proof of (8).…”

“…Sidi et al obtain the joint queue-length distribution at arbitrary moments, as well as the joint queue-length distribution at departure epochs. The waiting-time distributions are obtained in a different paper [2]. For us it is slightly more convenient to refer to this latter paper in the analysis as described, because the authors in [2] use the same definition of V c i (z), the PGF of the joint queue-length at departure epochs, just after a departure from Q i and just before the arrival of the departing customer at the next queue.…”

“…For us it is slightly more convenient to refer to this latter paper in the analysis as described, because the authors in [2] use the same definition of V c i (z), the PGF of the joint queue-length at departure epochs, just after a departure from Q i and just before the arrival of the departing customer at the next queue. Take the formulas (3.2)-(3.6) of [2]. From (3.2), which is the counterpart of our (2), one can express (in the notation of the present paper) the differences of PGFs at visit beginning and visit completion epochs into those at service beginning and service completion epochs: [2], which is the counterpart of (1) above, in terms of differences V b i (z) − V c i (z), as was also done in [4].…”

“…Take the formulas (3.2)-(3.6) of [2]. From (3.2), which is the counterpart of our (2), one can express (in the notation of the present paper) the differences of PGFs at visit beginning and visit completion epochs into those at service beginning and service completion epochs: [2], which is the counterpart of (1) above, in terms of differences V b i (z) − V c i (z), as was also done in [4]. This gives…”

“…The first term on the right-hand side gives the fourth plus fifth term in (42). One could argue that some results in [2] and [17] could have been derived faster by starting from (44).…”