A polling system whose stability region depends on a whole distribution of service times

We present an example of a single-server polling system with two queues and an adaptive service policy where the stability region depends on the expected values of all the primitives and also on a certain exponential moment of the service-time distribution in one of the queues. The latter parameter can not be determined, in general, in terms of any finite number of power moments. It follows that the fluid approximation approach may not be an appropriate tool for the stability study of this model.Comment: 6 page

Waiting times in polling systems with various service disciplines

We consider a polling system of N queues Q1,..., QN, cyclically visited by a single server. Customers arrive at these queues according to independent Poisson processes, requiring generally distributed service times. When the server visits Qi, i = 1,..., N, it serves a number of customers according to a certain visit discipline. This discipline is assumed to belong to the class of branching-type disciplines, which includes gated and exhaustive service. The special feature of our study is that, within each queue, we do not restrict ourselves to service in order of arrival (FCFS); we are interested in the effect of different service disciplines, like Last-Come-First-Served, Processor Sharing, Random Order of Service, and Shortest Job First. After a discussion of the joint distribution of the numbers of customers at each queue at visit epochs of the server to a particular queue, we determine the Laplace-Stieltjes transform of the cycle-time distribution, viz., the time between two successive visits of the server to, say, Q1. This yields the transform of the joint distribution of past and residual cycle time, w.r.t. the arrival of a tagged customer at Q1. Subsequently concentrating on the case of gated service at Q1, we use that cycle-time result to determine the (Laplace-Stieltjes transform of the) waiting-time distribution at Q1. Next to locally gated visit disciplines, we also consider the globally gated discipline. Again, we consider various non-FCFS service disciplines at the queues, and we determine the (Laplace-Stieltjes transform of the) waiting-time distribution at an arbitrary queue.

Heavy-traffic limits for Polling Models with Exhaustive Service and non-FCFS Service Order Policies

We study cyclic polling models with exhaustive service at each queue under a variety of non-FCFS local service orders, namely Last-Come-First-Served (LCFS) with and without preemption, Random-Order-of-Service (ROS), Processor Sharing (PS), the multi-class priority scheduling with and without preemption, Shortest-Job-First (SJF) and the Shortest Remaining Processing Time (SRPT) policy. For each of these policies, we rst express the waiting-time distributions in terms of intervisit-time distributions. Next, we use these expressions to derive the asymptotic waiting-time distributions under heavy-trac assumptions, i.e., when the system tends to saturate. The results show that in all cases the asymptotic wait