Simple Queueing Research Articles

Transient Behaviour of an M/M/1/N queue

For a simple queue with finite waiting space the difference equations satisfied by the Laplace transforms of the state probabilities at finite time are solved and the state probabilities have been obtained. The method economizes in algebra and the simple closed form of the state probabilities is used to obtain important parameters.

Explicit Steady-State Solutions to Some Elementary Queueing Models

Several simple queueing models, commonly used as textbook examples, have explicit steady-state solutions, which to date have escaped notice. These solutions are easily computable. In addition to their didactic interest, the results presented here are also useful in the analysis of more complex queueing models.

Information theoretic approximations for M/G/1 and G/G/1 queuing systems

This paper presents new results concerning the use of information theoretic inference techniques in system modeling and concerning the widespread applicability of certain simple queuing theory formulas. For the case when an M/G/1 queue provides a reasonable system model but when information about the service time probability density is limited to knowledge of a few moments, entropy maximization and cross-entropy minimization are used to derive information theoretic approximations for various performance distributions such as queue length, waiting time, residence time, busy period, etc. Some of these approximations are shown to reduce to exact M/M/1 results when G = M. For the case when a G/G/1 queue provides a reasonable system model, but when information about the arrival and service distributions is limited to the average arrival and service rates, it is shown that various well known M/M/1 formulas are information theoretic approximations. These results not only provide a new method for approximating the performance distributions, but they help to explain the widespread applicability of the M/M/1 formulas.

Flow Control in Packet-Switched Networks by Gradual Restrictions of Virtual Calls

A network flow control method in virtual circuit-oriented packet-switched networks, such as international packet-switched networks based on the CCITT X.75 protocol, is proposed. Lengths of individual queues to outgoing routes are observed constantly in a switch (or node), and the virtual calls connected to a congested outgoing route are restricted gradually depending upon the queue length. Being proportional to the number of virtual calls restricted simultaneously, the amount of processing load to carry out flow control can be decreased by the introduction of gradual restrictions. Priorities of calls can also be considered in flow control. The proposed flow control method is analyzed theoretically by a simple queueing model and basic traffic characteristies are shown. Optimum restriction factors and threshold values are obtained. Through several numerical examples, the effectiveness of the method is demonstrated. The amount of processing load is also analyzed and evaluated quantitatively.

Capacity analysis of the Mass Storage System

We have examined the procesos f staginddestaging of data with a view toward establishing certain performance characteristics of the MSS. The main issue we address is: What is the maximum throughput achievable on the MSS? The analysis is based on a simple queuing model, the validity of which has been established by comparing its predictions with the measurements gathered in our operational environment. The results have been found of value in monitoring and tuning MSS usage, in evaluating alternative design changes, and in workload planning.

Variance Reduction Techniques for the Simulation of Markov Processes, I: Multiple Estimates

A method for reducing the variance of simulation-generated estimates is proposed and discussed. The method may be applied to the estimation of steady state parameters of discrete and continuous time Markov chains, semi-Markov processes, and regenerative discrete time Markov processes on a general state space (such as the waiting time process in a multiple-server queue). The method is similar to the technique of control variables, but differs in that the means of the controls need not be explicitly known. Numerical results for a variety of simple queueing models are presented.

The Almost Regenerative Method for Stochastic System Simulations

The regenerative method for simulations of stochastic systems allows data collection at the entry times to a single recurrent state of the process of interest. Estimates of estimator variance are then easily computed since the generated observations have the desirable property of being independent and identically distributed. Relative to a fixed run length, however, the mean time between entries into this single state may be excessively long for complicated stochastic systems (e.g., a congested network of queues), thus providing few observations and poor variance estimates. The almost regenerative method, a relaxation of the regenerative method, can, under frequently encountered conditions, alleviate this problem by allowing data collection at the entry times to a set of states of the process of interest. Empirical evidence from simulations of simple queueing networks supports the intuitive notion that the almost regenerative method can provide more accurate estimates of estimator variance than the regenerative method. Similar results hold when the almost regenerative method is compared to the frequently used fixed time increment method for event-oriented simulations.

The Effects of Competition and Regulation on Hospital Bed Supply and the Reservation Quality of the Hospital

This article uses a simple queuing model to examine several factors that affect hospital bed supply decisions and the reservation quality of the hospital. The model takes account of demand uncertainty, the internal organization of the hospital, nonprice competition among hospitals, and hospital regulation. The empirical work indicates that hospital reserve margins vary directly with the extent of interhospital competition and inversely with the intensity of hospital regulation. Estimates of the reservation quality of the hospital are shown to depend critically on assumptions about internal organization of hospitals and the ways in which hospital utilization varies over the year.

Variance reduction techniques for the simulation of Markov process

Let X n, n?0 be an irreducible, aperiodic, Markov chain with finite state space E, transition matrix P, and stationary distribution ?. Let f be a real valued function on E and define r = ?f. A method of reducing the variance of simulation estimates for r is presented. The method combines the techniques of numerical analysis and simulation by partially solving an appropriate system of linear equations using some matrix iterative procedure and then estimating the difference between the true and partial solutions via simulation. After k iterations of the iterative procedure, functions f ? ? = 0, ..., k are defined so that r = ?f gn for each ?. Let $$\hat x_v (N) = \sum\limits_{n = 0}^N {f_v (x_n )/(N + 1)} $$ and $$\hat x_\beta (N) = \sum\limits_{v = 0}^k {\beta (v)\hat x_v (N)} $$ where $$\sum\limits_{v = 0}^k {\beta (v) = 1} $$ . Then $$\hat x_\beta (N) \to r$$ a.s. as N ? ? and β is chosen to minimize the asymptotic variance of $$\hat x_\beta (N)$$ . Numerical results for a simple queueing model are presented.

Optimal bus scheduling and fleet selection: A programming approach

A set of nonlinear programming models is developed here to determine an optimal vehicle-mix and fleet selection. The institutional framework assumed is as follows: a given number of n vehicles is available for dispatch from some source to serve passengers along fixed routes; the arriving passengers follow a simple queueing process, i.e. if a passenger cannot be served in a given time interval i, he has to wait until the next interval, thus forming a queue. One wishes to choose the number of buses n i , to assign in interval i ( i = 1, 2, …, 9) so as to optimize a criterion function, which includes such components as costs of operating the fleet, gross returns per vehicle per trip and the implicit cost of passengers' waiting time. Stochastic aspects of travel demand are handled through some formulations of stochastic programming. It is shown that under suitable simplifying assumptions, stochastic linear programming methods could provide good approximations to the more general nonlinear programming models.

An exact solution method for the general class of closed separable queueing networks

In this paper we present a convolution algorithm for the full class of closed, separable queueing networks. In particular, the algorithm represents an alternative method to those already known for the solution of networks with class changes, and is the first efficient algorithm to deal with Lam-type networks [11]. As an application of the algorithm, we study a simple queueing network with disk I/O devices connected to a single CPU through a single channel. The algorithm is then used to develop a simple, accurate approximation for the blocking of disk devices that takes place when a customer using a disk is waiting for or in service at the channel.

An exact solution method for the general class of closed separable queueing networks

In this paper we present a convolution algorithm for the full class of closed, separable queueing networks. In particular, the algorithm represents an alternative method to those already known for the solution of networks with class changes, and is the first efficient algorithm to deal with Lam-type networks [11]. As an application of the algorithm, we study a simple queueing network with disk I/O devices connected to a single CPU through a single channel. The algorithm is then used to develop a simple, accurate approximation for the blocking of disk devices that takes place when a customer using a disk is waiting for or in service at the channel.

Importance sampling in systems simulation: a practical failure?

A network of servers, known as a grading in telecommunication engineering, is simulated in order to estimate the probability of a customer being “blocked”: all servers busy. Since blocking is a very rare event (1‰ to 5% chance), importance sampling was considered for reduction of the simulation variance. The basic idea of importance sampling is first explained by means of a non-dynamic system. For dynamic systems a method was proposed by Bayes in 1970, which is related to the “virtual measures” published by Carter and Ignall in 1975. For simple queuing systems, we derive the resulting variance, using the renewal or regenerative property of such systems. For our practical “grading” system several alternative importance regions are investigated. For practical reasons we choose to start an importance region immediately after a call gets blocked (not a renewal state). The analysis and simulation experiments for the resulting estimator yielded the estimated optimal length of the importance region and the optimal number of replications of the region. Unfortunately, a net increase in variance resulted.

A Simple Method for Reducing Queuing Times in M/G/1

In many queuing situations customers are served in order of arrival; yet if customers with shorter service times were served first, it is well known that the mean queuing time could be reduced considerably. In a comparison with the “optimal” shortest service-time rule, a simple queue discipline involving only two non-preemptive priority classes fares very well. Numerical results illustrate the practical effects of implementing this simple alternative to service in order of arrival in a variety of queuing situations.

An Analytic and Experimental Study of Multiple Channel Controllers

A multiple channel controller (MCC) is a controller which switches a given number of channels among a larger number of input/output devices and permits simultaneous access to as many devices as there are channels available. A simple queuing model for multiple channel controllers is created, and an approximate solution for this model is generated. The approximate solution is found, using simulation, to be very close for those cases examined to the actual behavior of the model. Further simulations indicate that the approximate solution of the model appears to be robust with respect to changes in some of the assumptions used in making the approximation. Trace data taken from a real system are analyzed and they confirm the predicted utility of MCC's. The problem of optimal scheduling of MCC's is briefly discussed. Alternative system configurations are compared with the objective of minimizing queuing delays.

The impact of distributions and disciplines on multiple processor systems

Simple queueing models are used to study the performance tradeoffs of multiple processor systems. Issues considered include the impact of CPU service disciplines and distributions, level of multiprogramming, multitasking, and job priorities.

A Queuing Model for the Utilisation of Machines under Random Parallel Loading

In a system consisting of many identical machines, any set of machines may be reserved from any other machine in order that the set may carry out a production job simultaneously. A simple queuing model, entailing various approximations, is used to give formulae for utilisation, holding time and other parameters of the system. The model suggests that a practical system designed on these lines would become congested at between one-tenth and one-fifth of full loading.

How Useful are Queue Models in Port Investment Decisions for Container Berths?

The operational characteristics of U.K. container terminals and shipping services are reviewed in relation to their effect on the use of queue models in decisions on investment in berth construction and cargo handling equipment. Some simple queue models for a new container service are evaluated, showing the importance of interpreting the choice of parameters and results of these models in relation to the particular costs and requirements of container operations and how this ranks investment options differently to the traditional application of queue models to port investment. This ranking is compared with the actual decision made and the effects of some other factors not included in the models noted.

Comparative effectiveness of certain queueing systems with adaptive demand and service mechanisms

This paper sets out to evaluate simple queueing models embodying the concept of adaptation of service to demand and vice versa, systems which hold out the promise of improved operational characteristics by comparison with conventional non-adaptive systems. Over the many years during which research in queueing theory has been in progress there has been little concern with models of this kind, yet it is in the direction of environmental adaptation that one must look for operational improvements. The models studied are mainly of birth and death type. Some consideration is given to renewal models which, in a certain sense, are the equivalents of birth and death types. The emphasis in the paper is placed on the requirements of an operational assessment and on its realization. The use of digital computers in research of this kind is underlined. For illustration extracts from [9] are used to describe the performance of three fundamental adaptive systems.

A Practical Optimisation Technique for a Queueing System

A scheduling problem in an outpatient clinic is considered. First the system is represented by means of a simple queueing model. The problem of minimising the patients' total queueing time is then formulated as a non-linear program and solved by applying the Kuhn-Tucker conditions. The results obtained enable the person who manages the appointment system to allocate appointment dates to patients who have different priorities.