Mathematical Problems in Engineering

Volume 2018, Article ID 3018758, 15 pages

https://doi.org/10.1155/2018/3018758

## Traffic Intensity Estimation in Finite Markovian Queueing Systems

^{1}Departamento de Estatística, Universidade Federal de Minas Gerais, 31270-901 Belo Horizonte, MG, Brazil^{2}Pró-Reitoria de Planejamento e Desenvolvimento, Universidade Federal do Pará, 66075-110 Belém, PA, Brazil^{3}Departamento de Ciência da Computação, Universidade Estadual de Montes Claros, 39401-089 Montes Claros, MG, Brazil^{4}Department of Industrial Engineering & Innovation Sciences, Eindhoven University of Technology, 5600 MB Eindhoven, Netherlands

Correspondence should be addressed to Frederico R. B. Cruz; rb.gmfu.tse@zurcf

Received 5 January 2018; Revised 10 April 2018; Accepted 16 May 2018; Published 26 June 2018

Academic Editor: Jason Gu

Copyright © 2018 Frederico R. B. Cruz et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In many everyday situations in which a queue is formed, queueing models may play a key role. By using such models, which are idealizations of reality, accurate performance measures can be determined, such as traffic intensity (), which is defined as the ratio between the arrival rate and the service rate. An intermediate step in the process includes the statistical estimation of the parameters of the proper model. In this study, we are interested in investigating the finite-sample behavior of some well-known methods for the estimation of for single-server finite Markovian queues or, in Kendall notation, queues, namely, the maximum likelihood estimator, Bayesian methods, and bootstrap corrections. We performed extensive simulations to verify the quality of the estimators for samples up to 200. The computational results show that accurate estimates in terms of the lowest mean squared errors can be obtained for a broad range of values in the parametric space by using the Jeffreys’ prior. A numerical example is analyzed in detail, the limitations of the results are discussed, and notable topics to be further developed in this research area are presented.

#### 1. Introduction

Queueing models are idealizations of many real systems. However, they enable the accurate determination of performance measures as long as a previous step has been fulfilled; that step is the statistical estimation of its parameters [1, 2]. It is impossible to discuss inference in queues without mentioning the pioneering work of Clarke in the 1950s, who describes maximum likelihood estimators for the arrival rates and service times in simple queues, and the work of Schruben and Kulkarni in the 1980s, who consider the problem of bias in queue estimation. It is also important to mention the Bayesian papers on the subject, including that of Muddapur in the 1970s, who published one of the first results that extended Clark’s methodology, the series of papers from Armero and Bayarri [3, 4], Armero and Conesa [5–8], Choudhury and Borthakur [9], and, more recently, Chowdhury and Mukherjee [10, 11], Cruz et al. [12], and Quinino and Cruz [13]. These are only a few examples of the papers in this important research area.

The purpose of this paper is to evaluate the behavior of a traffic intensity estimator (), which is defined as the ratio between the arrival rate () and the service rate (), specifically for single-server finite Markovian queues, which model various real systems [14]. The type of queue researched in this paper is typically seen in many service-oriented settings, where there is a finite queue in front of a server. Think of a gas station, where cars can queue up for a limited amount of space, and the traffic intensity should be not too high, as in this case cars might not join the queue. An alternative example is observed within a supermarket context where customers line up for the check out. Having a good understanding of the traffic intensity is crucial in these situations. Good estimations are needed to properly design the system (*e.g.,* to install the gas pump buffer sizes needed) or to properly manage the system (*e.g.,* to set waiting spaces for the store clerks at check out).

In summary, previous results obtained for* infinite* Markovian queues are extended here for* finite* Markovian queues. To reach this goal this paper combines some techniques and approaches from the work of Almeida and Cruz [15] (*i.e.*, Bayesian inference and Monte Carlo simulation for evaluation of estimators under finite samples) with other classical tools (*e.g.*, Gibbs sampling and bootstrapping).

The remainder of the paper is organized as follows. Section 2 details the queue equations and estimators for . The computational results are presented and discussed in Section 3, followed by Section 4, which concludes the text with some final remarks and topics for future research in the area.

#### 2. Material and Methods

When you have Poisson arrivals, exponential service times, a single server, and limited waiting space, you have an queue; in Kendall notation, represents the number of customers simultaneously allowed in the queueing system. The probability of a number of users of the system, for , is given by [14]:where is the traffic intensity. Estimating traffic intensity is important as it is a key design parameter in production network design, routing of products, and so on.

##### 2.1. Maximum Likelihood Estimator

Maximum likelihood estimation for the truncated geometric model is known since Thomasson and Kapadia [16]. Consider the stationary probability distribution given by (1). Next, consider a random sample of size , , where is the number of customers an outside observer finds in the system. In this case, a maximum of customers are allowed in the queueing system at once. Therefore, the likelihood function iswhere . Note that the likelihood is a function of traffic intensity and sample , although only its size, , and its sum, , which is a sufficient statistic for , are necessary.

Needless to say that, for the implementation of maximum likelihood estimator (MLE), any bounded optimization algorithm could be used. However for the sake of simplicity, the implementation used in this study was encoded in R [17] and can be seen in Listing 1. For convenience, the logarithm of the likelihood function was considered because it allows products to be turned into sums. Maximizing the log-likelihood is done numerically through an R internal function. However, tests (not shown) were conducted with the original likelihood function; they indicated that the results did not change significantly in terms of accuracy or computational effort.