Bayesian sample sizes in an M/M/1 queueing systems

Abstract

Although the broad field of applications amply justifies an intensive study of queueing models, it turned out that these models are also of great use outside the field of queueing theory, e.g., in inventory and maintenance theory very important for production systems. In these applications, the queue model M/M/1 has several works that illustrate its importance, but usually, they adopt the parameters known. The literature presents estimation methods such as maximum likelihood and Bayesian that could be directly applied. However, there is no discussion about planning the sample size to guarantee the precision of the parameters that are estimated. An accurate estimate of the model parameters can avoid wrong decisions on inventory control and maintenance. The focus here is on sample size determination for estimates in single-server Markovian infinity queues, considered the simplest and yet most useful queueing model for its accuracy to model many problems of practical interest. In this article, a Bayesian method is described for sample size determination for traffic intensity estimation, one of the most important parameters for performance evaluation of queueing systems. Numerical results are presented to attest to the efficiency and efficacy of the approach.

Traffic Intensity Estimation in Finite Markovian Queueing Systems

“…Therefore, the inference process for estimating starts with (2) and assumes an a priori beta distribution, that is, ( ) ∼ Beta( , ), which has been successfully used in inference in other Markovian queues [12,13] and results in the following a posteriori distribution:…”

“…Thus, combining the likelihood, (2), and Jeffreys prior, given by (13), it is possible to find the following posterior probability distribution for :…”

“…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][6][7][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.…”

Sample Size Determination for Markovian Queueing Models

“…117) had an explicit demonstration of relations between posterior distributions and the exponential family. For the second point, Quinino and Cruz (2017) accomplished this in their paper. Quinino and Cruz (2017) only provided an approach for determining the sample size for the standard M/M/1 queueing system.…”

“…For the second point, Quinino and Cruz (2017) accomplished this in their paper. Quinino and Cruz (2017) only provided an approach for determining the sample size for the standard M/M/1 queueing system. In this thesis, we study what kind of change will happen for determining the sample size for the M/M/1 queue with balking, and for the M/M/c queueing system, and whether or not we could use the same method to obtain the required sample size for the M/M/1 with multiple vacations, and for the M/M/c queue with balking.…”