In this paper we generalize the martingale of Kella and Whitt to the setting of Lévy-type processes and show that under some quite minimal conditions the local martingales are actually L 2 martingales which upon dividing by the time index converge to zero a.s. and in L 2. We apply these results to generalize known decomposition results for Lévy queues with secondary jump inputs and queues with server vacations or service interruptions. Special cases are polling systems with either compound Poisson or more general Lévy inputs. Keywords: Lévy-type processes, Lévy storage systems, Kella-Whitt martingale, decomposition results, queues with server vacations

We introduce a new service discipline, called the synchronized gated discipline, for polling systems. It arises when there are precedence (or synchronization) constraints between the order that jobs in different queues should be served. These constraints are described as follows: There are N stations which are "fathers" of (zero or more) synchronized stations ("children"). Jobs that arrive at synchronized stations have to be processed only after jobs that arrived prior to them at their corresponding "father" station have been processed. We analyze the performance of the synchronized gated discipline and obtain expressions for the first two moments and the Laplace-Stieltjes transform (LST) of the waiting times in different stations, and expressions for the moments and LST of other quantities of interest, such as cycle duration and generalized station times. We also obtain a "pseudo" conservation law for the synchronized gated discipline, and determine the optimal network topology that minimizes the weighted sum of the mean waiting times, as defined in the "pseudo" conservation law. Numerical examples are given for illustrating the dependence of the performance of the synchronized gated discipline on different parameters of the network.

We consider a polling model of two M=G=1 queues, served by a single server. The service policy for this polling model is of threshold type. Service at queue 1 is exhaustive. Service at queue 2 is exhaustive unless the size of queue 1 reaches some level T during a service at queue 2; in the latter case the server switches to queue 1 at the end of that service. Both zero- and nonzero switchover times are considered. We derive exact expressions for the joint queue length distribution at customer departure epochs, and for the steady-state queue-length and sojourn time distributions. In addition, we supply a simple and very accurate approximation for the mean queue lengths, which is suitable for optimization purposes. AMS Subject Classification (1991): Primary: 60K25, Secondary: 90B22 Keywords & Phrases: Queueing, polling, ATM, threshold service, queue length distribution. 1 Introduction In this paper we consider a model of two M=G=1 queues, which are served by a single serve...

In this paper we present a new approach to derive heavy-traffic asymptotics for polling models. We consider the classical cyclic polling model with exhaustive service at each queue, and with general service-time and switch-over time distributions, and study its behavior when the load tends to one. For this model, we explore the recently proposed mean value analysis (MVA), which takes a new view on the dynamics of the system, and use this view to provide an alternative way to derive closed-from expressions for the expected asymptotic delay; the expressions were derived earlier in [32], but in a different way. Moreover, the MVA-based approach enables us to derive closed-form expressions for the heavy-traffic limits of the covariances between the successive visit periods, which are key performance metrics in many application areas. These results, which have not been obtained before, reveal a number of insensitivity properties of the covariances with respect to the system parameters under heavy-traffic assumptions, and moreover, lead to simple approximations for the covariances between the successive visit times for stable systems. Numerical examples demonstrate that the approximations are accurate when the load is close enough to one.

Sarkar and Zangwill (1991) showed by numerical examples that reduction in setup times can, surprisingly, actually increase work in process in some cyclic production systems (that is, reduction in switchover times can increase waiting times in some polling models). We present, for polling models with exhaustive and gated service disciplines, some explicit formulas that provide additional insight and characterization of this anomaly. More specifically, we show that, for both of these models, there exist simple formulas that define for each queue a critical value z * of the mean total setup time z per cycle such that, if z õ z*, then the expected waiting time at that queue will be minimized if the server is forced to idle for a constant length of time z * 0 z every cycle; also, for the symmetric polling model, we give a simple explicit formula for the expected waiting time and the critical value z * that minimizes it.

Previous work in the stability analysis of polling models concentrated mainly on stability of the whole system. This system stability analysis, however, fails to model many real-world systems for which some queues may continue to operate under an unstable system. In this paper we address this problem by considering queue stability problem that concerns stability of an individual queue in a polling model. We present a novel approach to the problem which is based on a new concept of queue stability orderings, dominant systems, and Loynes ’ theorem. The polling model under consideration employs an m-limited service policy, with or without prior service reservation; moreover, it admits state-dependent set-up time and walk time. Our stability results generalize many previous results of system stability. Furthermore, we show that stabilities of any two queues in the system can be compared solely based on their (λ/m)’s, where λ is the customer arrival rate to a queue. ©2000 Elsevier Science B.V. All rights reserved.

Abstract—We study the concept of Ferry based Wireless Local Area Network (FWLAN), in which a number of isolated nodes are scattered over some area and where communication between a node and the outer world, or communication between the nodes, are made possible via a message ferry. The Ferry has a predetermined cyclic path which collects packets from a node and delivers packets to it when it is in the vicinity of the node. We use the mathematical theory of polling systems to study the performance of the FWLAN. We consider three different architectures and each of them is mapped to a polling model. The polling disciplines that are needed for modeling the FWLAN involve non-standard variants of gating disciplines. Our goal is to design the routes of the Ferry as well as the points where it should stop to distribute and collect packets. This mathematical modeling brings another dimension to the classical related vehicle routing problem due to the radio channel: the cyclic path of the ferry need not touch every node. The distance between the node and the fairy at the point when communication occurs determines the transmission rate and hence the service time and thus the system’s capacity. I.

We consider a polling system where a group of an infinite number of servers visits sequentially a set of queues. When visited, each queue is attended for a random time. Arrivals at each queue follow a Poisson process, and service time of each individual customer is drawn from a general probability distribution function. Thus, each of the queues comprising the system is, in isolation, an M/G/∞-type queue. A job that is not completed during a visit will have a new service time requirement sampled from the service-time distribution of the corresponding queue. To the best of our knowledge, this paper is the first in which an M/G/∞-type polling system is analysed. For this polling model, we derive the probability generating function and expected value of the queue lengths, and the Laplace-Stieltjes transform and expected value of the sojourn time of a customer. Moreover, we identify the policy that maximises the throughput of the system per cycle and conclude that under the Hamiltonian-tour approach, the optimal visiting order is independent of the number of customers present at the various queues at the start of the cycle. 1

This paper considers a batch service polling system. We first study the case in which the server visits the queues cyclically, considering three different service regimes: gated, exhaustive, and globally gated. We subsequently analyze the case (the so called ‘Israeli Queue’) in which the server first visits the queue with the ‘oldest’ customer. In both cases, queue lengths and waiting times are the main performance measures under consideration. 1

Abstract—We study the concept of Ferry based Wireless Local Area Network (FWLAN), in which a number of isolated nodes are scattered over some area and where communication between a node and the outer world, or communication between the nodes, are made possible via a message ferry. The Ferry has a predetermined cyclic path which collects messages from a node and delivers messages to it when it is in the vicinity of the node. We use the mathematical theory of polling systems to study the performance of the FWLAN. We consider three different architectures and each one of them is mapped to an appropriate polling system. The polling disciplines that are needed for modeling the FWLAN involve non-standard variants of gating disciplines. Our goal is to design the routes of the Ferry as well as the points where it should stop to distribute and collect messages. This mathematical modeling brings another dimension to the classical related vehicle routing problem due to the radio channel: the cyclic path of the ferry need not touch every node. The distance between the node and the fairy at the point when communication occurs determines the transmission rate and hence the service time and thus the system’s capacity. I.