Approximations for the Moments of Nonstationary and State Dependent Birth-Death Queues

In this paper we propose a new method for approximating the nonstationary moment dynamics of one dimensional Markovian birth-death processes. By expanding the transition probabilities of the Markov process in terms of Poisson-Charlier polynomials, we are able to estimate any moment of the Markov process even though the system of moment equations may not be closed. Using new weighted discrete Sobolev spaces, we derive explicit error bounds of the transition probabilities and new weak a priori estimates for approximating the moments of the Markov processs using a truncated form of the expansion. Using our error bounds and estimates, we are able to show that our approximations converge to the true stochastic process as we add more terms to the expansion and give explicit bounds on the truncation error. As a result, we are the first paper in the queueing literature to provide error bounds and estimates on the performance of a moment closure approximation. Lastly, we perform several numerical experiments for some important models in the queueing theory literature and show that our expansion techniques are accurate at estimating the moment dynamics of these Markov process with only a few terms of the expansion

Performance analysis of time-dependent queueing systems: survey and classification

Many queueing systems are subject to time-dependent changes in system parameters, such as the arrival rate or number of servers. Examples include time-dependent call volumes and agents at inbound call centers, time-varying air traffic at airports, time-dependent truck arrival rates at seaports, and cyclic message volumes in computer systems.There are several approaches for the performance analysis of queueing systems with deterministic parameter changes over time. In this survey, we develop a classification scheme that groups these approaches according to their underlying key ideas into (i) numerical and analytical solutions,(ii)approaches based on models with piecewise constant parameters, and (iii) approaches based on mod-ified system characteristics. Additionally, we identify links between the different approaches and provide a survey of applications that are categorized into service, road and air traffic, and IT systems

On the Timing of the Peak Mean and Variance for the Number of Customers in an M(t)/M(t)/1 Queueing System

Revised October 1994This paper examines the time lag between the peak in the arrival rate and the peaks in the mean and variance for the number of customers in an M(t)/M(t)/1l system. We establish a necessary condition for the time at which the peak in the mean is achieved. In cases in which system utilization exceeds one during some period, we show that the peak in the mean occurs after the end of this period

LETRIS: Staffing service systems by means of simulation

Purpose: This paper introduces a procedure for solving the staffing problem in a service system (i.e., determining the number of servers for each staffing period). Design/methodology: The proposed algorithm combines the use of queueing theory to find an initial solution with the use of simulation to adjust the number of servers to meet previously specified target non-delay probabilities. The basic idea of the simulation phase of the procedure is to successively fix the number of servers from the first staffing period to the last, without backtracking. Findings: Under the assumptions that the number of servers is not upper-bounded and there are no abandonments and, therefore, no retrials, the procedure converges in a finite number of iterations, regardless of the distributions of arrivals and services, and requires a reasonable amount of computing time. Originality / value: The new procedure proposed in this paper is a systematic, robust way to find a good solution to a relevant problem in the field of service management and it is very easy to implement using no more than commonly accessible tools.Peer Reviewe