Priority Queuing Systems with and without Feedback

Abstract

The problems discussed in this paper arose in the theory of telephone traffic, with a central processing system, such as a computer, performing numerous functions in as exchange. However, several of its more important results are applicable in many businesses offering multiple services, including ones working on a nonpriority basis. Such results as those determining the amount of storage to allocate to each service, and assessing whether the system can cope with the expected number of arriving customers are independent of the priority set, so that these results have very wide applicability.

Joint Queue-Length Distribution for the Non-Preemptive Multi-Server Multi-Level Markovian Priority Queue

“…The vast majority of effort concerning the joint distribution has focused on the two-level problem. The single source of previous work addressing the joint queue-length distribution for more than two priority levels comprises the papers of Wignall [12] and Wignall and Enns [13]. These consider single-server systems involving multiple queues with distinct arrival rates ranked by priority level, and with feedback -requiring deterministic or probabilistic transitions between queues before system exit is achieved.…”

“…The present problem can, in principle, be constructed as a special case of this scheme. In [12,13], equations that must be solved recursively are presented for the multivariate probability generating function (PGF); but no general solution is provided, and manual solution becomes increasingly cumbersome as the number of priority levels grows beyond a small number. No actual probability mass functions (PMFs) are computed.…”

“…By contrast, we present an explicit closed-form expression for the joint PGF given any number of priority levels, that lends itself to practical numerical evaluation of the joint PMF. In [12], explicit results are confined to the probability that the system is empty, and the probability that a given queue is being served at a random observation. In [13], the distribution of the maximum queue length during a busy period in the presence of probabilistic feedback is also given.…”

Joint Queue-Length Distribution for the Non-Preemptive Multi-Server Multi-Level Markovian Priority Queue

“…The vast majority of effort concerning the joint distribution has focused on the two-level problem. The single source of previous work addressing the joint queue-length distribution for more than two priority levels comprises the papers of Wignall [12] and Wignall and Enns [13]. These consider single-server systems involving multiple queues with distinct arrival rates ranked by priority level, and with feedback -requiring deterministic or probabilistic transitions between queues before system exit is achieved.…”

“…The present problem can, in principle, be constructed as a special case of this scheme. In [12,13], equations that must be solved recursively are presented for the multivariate probability generating function (PGF); but no general solution is provided, and manual solution becomes increasingly cumbersome as the number of priority levels grows beyond a small number. No actual probability mass functions (PMFs) are computed.…”