Markovian polling systems with an application to wireless random-access networks

Motivated by an application in wireless random-access networks, we study a class of polling systems with Markovian routing, in which the server visits the queues in an order governed by a discrete-time Markov chain. Assuming that the service disciplines at each of the queues fall in the class of branching-type service disciplines, we derive a functional equation for (the probability generating function of) the joint queue length distribution conditioned on a point in time when the server visits a certain queue. From this functional equation, expressions for the (cross-)moments of the queue lengths follow. We also derive a pseudo-conservation law for this class of polling systems. Using these results, we compute expressions for certain system parameters that minimise the total expected amount of work in systems that arise from the wireless random-access network setting. In addition, we derive approximations for those parameters that minimise a weighted sum of mean waiting times in these systems. Based on these expressions, we also present an adaptive control algorithm for finding the optimal parameter values in a distributed fashion, which is particularly relevant in the context of wireless random-access networks

Cyclic-type polling models with preparation times

We consider a system consisting of a server serving in sequence a ¿xed number of stations. At each station there is an in¿nite queue of customers that have to undergo a preparation phase before being served. This model is connected to layered queuing networks, to an extension of polling systems, and surprisingly to random graphs. We are interested in the waiting time of the server. The waiting time of the server satis¿es a Lindley-type equation of a non-standard form. We give a suf¿cient condition for the existence of a limiting waiting time distribution in the general case, and assuming preparation times are exponentially distributed, we describe in depth the resulting Markov chain. We provide detailed computations for a special case and extensive numerical results investigating the effect of the system’s parameters to the performance of the server

Polling systems with batch service

Motivated by applications in production and computer-communication systems, we study an N-queue polling system, consisting of an inner part and an outer part, and where products receive service in batches. Type-i products arrive at the outer system according to a renewal process and accumulate into a type-i batch. As soon as Di products have accumulated, the batch is forwarded to the inner system where the batch is processed. The service requirement of a type-i batch is independent of its size Di. For this model we study the problem of determining the combination of batch sizes ~D(opt) that minimizes a weighted sum of the mean waiting times. This model does not allow for an exact analysis. Therefore, we propose a simple closed-form approximation for ~D (opt), and present a numerical approach, based on the recently-proposed mean waiting-time approximation in [1]. Extensive numerical experimentation shows that the numerical approach is slightly more accurate than the closed-form solution, while the latter provides explicit insights into the dependence of the optimal batch sizes on the system parameters and into the behavior of the system. As a by-product, we observe near-insensitivity properties of ~D(opt), e.g. to higher moments of the interarrival and switch-over time distributions