Showing papers in "Queueing Systems in 1986"

Queueing systems with vacations—a survey

Abstract: Queueing systems in which the server works on primary and secondary (vacation) customers arise in many computer, communication, production and other stochastic systems. These systems can frequently be modeled as queueing systems with vacations. In this survey, we give an overview of some general decomposition results and the methodology used to obtain these results for two vacation models. We also show how other related models can be solved in terms of the results for these basic models. We attempt to provide a methodological overview with the objective of illustrating how the seemingly diverse mix of problems is closely related in structure and can be understood in a common framework.

On queueing network models of flexible manufacturing systems

Abstract: This paper outlines the state-of-the-art in studying flexible manufacturing systems (FMSs) using analytical queueing network models. These include Jackson networks, reversible networks and approximate models of non-product-form networks. The focus is on identifying the major features of the models as they relate to the operational characteristics of FMSs. Prescriptive models concerning the optimal design and/or operational control of FMS networks are also discussed.

Assembly-like queues with finite capacity: bounds, asymptotics and approximations

Abstract: In this paper we study a queueing model of assembly-like manufacturing operations. This study was motivated by an examination of a circuit pack testing procedure in an AT & T factory. However, the model may be representative of many manufacturing assembly operations. We assume that customers fromn classes arrive according to independent Poisson processes with the same arrival rate into a single-server queueing station where the service times are exponentially distributed. The service discipline requires that service be rendered simultaneously to a group of customers consisting of exactly one member from each class. The server is idle if there are not enough customers to form a group. There is a separate waiting area for customers belonging to the same class and the size of the waiting area is the same for all classes. Customers who arrive to find the waiting area for their class full, are lost. Performance measures of interest include blocking probability, throughput, mean queue length and mean sojourn time. Since the state space for this queueing system could be large, exact answers for even reasonable values of the parameters may not be easy to obtain. We have therefore focused on two approaches. First, we find upper and lower bounds for the mean sojourn time. From these bounds we obtain the asymptotic solutions as the arrival rate (waiting room, service rate) approaches zero (infinity). Second, for moderate values of these parameters we suggest an approximate solution method. We compare the results of our approximation against simulation results and report good correspondence.

Approximation with generalized hyperexponential distributions: weak convergence

Abstract: Generalized hyperexponential (GH) distributions are linear combinations of exponential CDFs with mixing parameters (positive and negative) that sum to unity. The denseness of the class GH with respect to the class of all CDFs defined on [0, ∞) is established by showing that a GH distribution can be found that is as close to a given CDF as desired, with respect to a suitably defined metric. The metric induces the usual topology of weak convergence so that, equivalently, there exists a sequence of GH CDFs that converges weakly to a given CDF. This result is established by using a similar result for weak convergence of Erlang mixtures. Various set inclusion relations are also obtained relating the GH distributions to other commonly used classes of approximating distributions, including generalized Erlang (GE), mixed generalized Erlang (MGE), those with reciprocal polynomial Laplace transforms (Kn), those with rational Laplace transforms (Rn), and phase-type (PH) distributions. A brief survey of the history and use of approximating distributions in queueing theory is also included.

Finite capacity assembly like queues

Abstract: In assembly lines, service involves assembling units coming from more than one source. In queue terminology, we may consider this situation as one in which service is rendered only to groups of customers — one from each class. In this paper we give procedures to determine response time characteristics of such a system under Markovian assumptions when a finite capacity restriction is imposed. This restriction is imposed to reflect reality as well as to make analysis tractable. In the course of this study, we also give a recursive technique to determine the distribution of the time taken for a specific number of departures in a Poisson queue from an arbitrary initial state. We demonstrate that this distribution is related to the response time distribution of the assembly-like queue. We believe that this procedure will also be of independent interest.

A central-limit-theorem version of L =λ W

Abstract: Underlying the fundamental queueing formulaL=λW is a relation between cumulative processes in continuous time (the integral of the queue length process) and in discrete time (the sum of the waiting times of successive customers). Except for remainder terms which usually are asymptotically negligible, each cumulative process is a random time-transformation of the other. As a consequence, in addition to the familiar relation between the with-prob ability-one limits of the averages, roughly speaking, the customer-average wait obeys a central limit theorem if and only if the time-average queue length obeys a central limit theorem, in which case both averages, properly normalized, converge in distribution jointly, and the individual limiting distributions are simply related. This relation between the central limit theorems is conveniently expressed in terms of functional central limit theorems, using the continuous mapping theorem and related arguments. The central limit theorems can be applied to compare the asymptotic efficiency of different estimators of queueing parameters. For example, when the arrival rateλ is known and the interarrivai times and waiting times are negatively correlated, it is more asymptotically efficient to estimate the long-run time-average queue lengthL indirectly by the sample-average of the waiting times, invokingL=λW, than it is to estimate it by the sample-average of the queue length. This variance-reduction principle extends a corresponding result for the standard GI/G/s model established by Carson and Law [2].

Queueing models of secondary storage devices

Abstract: Queueing theory has occupied an important role in the analysis of computer storage structures and algorithms. In this survey we focus on secondary or auxiliary storage devices, which often comprise the principal bottleneck in the overall performance of computer systems. We begin with descriptions of the more important devices, such as disks and drums, and a general discussion of related queueing models. Server motion and dependent successive services are salient features of these models. Widely used, generic results are presented and then applied to specific devices. The paper concludes with a discussion of open problems.

Almost sure comparison of birth and death processes with application to M/M/s queueing systems

Abstract: Given conditions, which concern the infinitesimal parameters of two birth and death processes, the processes are constructed on the same probability space such that one process is almost surely larger than the other. Application is made to M/M/s queueing systems. Stochastic comparisons of queue length and virtual waiting time in two M/M/s systems are obtained.