Data-Driven Fitting of the G/G/1 Queue
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ISSN: 1004-3756 (paper), 1861-9576 (online) CN 11-2983/N
Data-Driven Fitting of the G/G/1 Queue Nanne A. Dieleman Department of Econometrics and Operations Research, Vrije Universiteit Amsterdam, Amsterdam, The Netherlands [email protected] ()
Abstract. The Maximum Likelihood Estimation (MLE) method is an established statistical method to es-
timate unknown parameters of a distribution. A disadvantage of the MLE method is that it requires an analytically tractable density, which is not available in many cases. This is the case, for example, with applications in service systems, since waiting models from queueing theory typically have no closed-form solution for the underlying density. This problem is addressed in this paper. MLE is used in combination with Stochastic Approximation (SA) to calibrate the arrival parameter θ of a G/G/1 queue via waiting time data. Three different numerical examples illustrate the application of the proposed estimator. Data sets of an M/G/1 queue, G/M/1 queue and model mismatch are considered. In a model mismatch, a mismatch is present between the used data and the postulated queuing model. The results indicate that the estimator is versatile and can be applied in many different scenarios. Keywords: G/G/1 queue, maximum likelihood estimation, stochastic approximation, data-driven fitting
1. Introduction
the density of modeled phenomenon X should
In this paper, a simulation-based estimator is derived for estimating the parameters of the
be continuous and the input random variables should have support on the whole space. This means that the estimator cannot be readily applied to service systems as waiting times in
arrival process of a G/G/1 queue. This estimator is based on the Maximum Likelihood Estimation (MLE) technique, which is a wellknown technique for fitting probabilistic models to data. The basic philosophy of MLE is that one considers a parametric family of densities, denoted by f θ (x), where θ denotes a parameter. Given an observation X of a random phenomenon one finds the value for θ that maximizes f θ (X), i.e., that maximizes the rate at which the actual observation X occurs, see for example Millar (2011) for more details. A disadvantage of the MLE method is that it requires an analytically tractable density, which is not available in many cases. In Peng et al. (2019) and Peng et al. (2018), a simulation-based Maximum Likelihood (ML) estimator is derived for estimating densities and their θ-derivatives from observed data. It is not necessary to have an analytically tractable density to use this method. However,
service systems have a point mass in zero and the input random variables including interarrival and service times have distributions supported on [0, ∞). In this paper, a ML estimator is provided that addresses these particular aspects. Stochastic Approximation (SA) is used to calibrate the arrival parameter θ of a G/G/1 queue via the derived estimator and waiting time data. The results for the general setting are stated, and various applications of the estimator
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