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

A polling model with an autonomous server

Polling models are used as an analytical performance tool in several application areas. In these models, the focus often is on controlling the operation of the server as to optimize some performance measure. For several applications, controlling the server is not an issue as the server moves independently in the system. We present the analysis for such a polling model with a so-called autonomous server. In this model, the server remains for an exogenous random time at a queue, which also implies that service is preemptive. Moreover, in contrast to most of the previous research on polling models, the server does not immediately switch to a next queue when the current queue becomes empty, but rather remains for an exponentially distributed time at a queue. The analysis is based on considering imbedded Markov chains at specific instants. A system of equations for the queue-length distributions at these instant is given and solved for. Besides, we study to which extent the queues in the polling model are independent and identify parameter settings for which this is indeed the case. These results may be used to approximate performance measures for complex multi-queue models by analyzing a simple single-queue model

Analysis and optimization of vacation and polling models with retrials

We study a vacation-type queueing model, and a single-server multi-queue polling model, with the special feature of retrials. Just before the server arrives at a station there is some deterministic glue period. Customers (both new arrivals and retrials) arriving at the station during this glue period will be served during the visit of the server. Customers arriving in any other period leave immediately and will retry after an exponentially distributed time. Our main focus is on queue length analysis, both at embedded time points (beginnings of glue periods, visit periods and switch- or vacation periods) and at arbitrary time points.Comment: Keywords: vacation queue, polling model, retrials Submitted for review to Performance evaluation journal, as an extended version of 'Vacation and polling models with retrials', by Onno Boxma and Jacques Resin

Analysis of a class of distributed queues with application

Recently we have developed a class of media access control algorithms for different types of Local Area Networks. A common feature of these LAN algorithms is that they represent various strategies by which the processors in the LAN can simulate the availability of a centralized packet transport facility, but whose service incorporates a particular type of change over time known as 'moving sever' overhead. First we describe the operation of moving server systems in general, for both First-Come - First-Served and Head-of-the-Line orders of service, together with an approach for their delay analysis in which we transform the moving server queueing system into a conventional queueing system having proportional waiting times. Then we describe how the various LAN algorithms may be obtained from the ideal moving server system, and how a significant component of their performance characteristics is determined by the performance characteristics of that ideal system. Finally, we evaluate the compatibility of such LAN algorithms with separable queueing network models of distributed systems by computing the interdeparture time distribution for M/M/1 in the presence of moving server overhead. Although it is not exponential, except in the limits of low server utilization or low overhead, the interdeparture time distribution is a weighted sum of exponential terms with a coefficient of variation not much smaller than unity. Thus, we conjecture that a service centre with moving server overhead could be used to represent one of these LAN algorithms in a product form queueing network model of a distributed system without introducing significant approximation errors

Some aspects of queueing and storage processes : a thesis in partial fulfilment of the requirements for the degree of Master of Science in Statistics at Massey University

In this study the nature of systems consisting of a single queue are first considered. Attention is then drawn to an analogy between such systems and storage systems. A development of the single queue viz queues with feedback is considered after first considering feedback processes in general. The behaviour of queues, some with feedback loops, combined into networks is then considered. Finally, the application of such networks to the analysis of interconnected reservoir systems is considered and the conclusion drawn that such analytic methods complement the more recently developed mathematical programming methods by providing analytic solutions for sub systems behaviour and thus guiding the development of a system model

Performance modelling of the Cambridge Fast Ring protocol

The Cambridge Fast Ring is high-speed slotted ring. The features that make it suitable for use at very large transmission rates are the synchronous transmission, the simplicity of the medium-access-control protocol, and the possibility of immediate retransmission of erroneous packets. A novel analytical model of the Cambridge Fast Ring with normal slots is presented. The model is shown to be accurate and usable over wide range of parameters. A performance analysis based on this model is presented

Dynamic Service Rate Control for a Single Server Queue with Markov Modulated Arrivals

We consider the problem of service rate control of a single server queueing system with a finite-state Markov-modulated Poisson arrival process. We show that the optimal service rate is non-decreasing in the number of customers in the system; higher congestion rates warrant higher service rates. On the contrary, however, we show that the optimal service rate is not necessarily monotone in the current arrival rate. If the modulating process satisfies a stochastic monotonicity property the monotonicity is recovered. We examine several heuristics and show where heuristics are reasonable substitutes for the optimal control. None of the heuristics perform well in all the regimes. Secondly, we discuss when the Markov-modulated Poisson process with service rate control can act as a heuristic itself to approximate the control of a system with a periodic non-homogeneous Poisson arrival process. Not only is the current model of interest in the control of Internet or mobile networks with bursty traffic, but it is also useful in providing a tractable alternative for the control of service centers with non-stationary arrival rates.Comment: 32 Pages, 7 Figure

Lattice path counting and the theory of queues

In this paper we will show how recent advances in the combinatorics of lattice paths can be applied to solve interesting and nontrivial problems in the theory of queues. The problems we discuss range from classical ones like M^a/M^b/1 systems to open tandem systems with and without global blocking and to queueing models that are related to random walks in a quarter plane like the Flatto-Hahn model or systems with preemptive priorities. (author´s abstract)Series: Research Report Series / Department of Statistics and Mathematic