Using mixture of Gamma distributions for Bayesian analysis in an M/G/1 queue with optional second service Mohammadi, A.; Salehi-Rad, M. R.; Wit, E. C. Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Abstract The paper proposes Bayesian framework in an M/G/1 queuing system with optional second service. The semi-parametric model based on a finite mixture of Gamma distributions is considered to approximate both the general service and re-service times densities in this queuing system. A Bayesian procedure based on birth-death MCMC methodology is proposed to estimate system parameters, predictive densities and some performance measures related to this queuing system such as stationary system size and waiting time. The approach is illustrated with several numerical examples based on various simulation studies. applied queuing systems, some customers need to be re-serviced after taking their main service. For example, in a production line, some items might fail and require repair. In these kinds of problems, we must re-service some items.
KeywordsThe primary aim of this paper is to propose a Bayesian inference scheme for an M/G/1 queuing system in which some customers with probability p need re-servicing. This queuing system has a service unit, in which customers arrive according to a Poisson process and demanding service with a general distribution. A fraction p of these customers request re-service with possibly another general distribution. From a classical queuing theory perspective, this queuing system has been studied by Salehi-Rad and Mengersen (2002) and Salehi-Rad et al. (2004); they considered three alternatives for re-servicing in this queuing system and obtained the mean busy period, the probability of the idle period and the probability generating function (pgf) of the steady-state system size. More recently, this queuing system has been studied by Mohammadi and Salehi-Rad (2012) based on Bayesian approach by using a mixture of truncated Normal distributions.The main contribution of this paper is to introduce a semi-parametric model for the general density of service and re-service based on a mixture of Gamma distributions, providing an alternative Bayesian approach for approximating the general distributions in queuing systems based on former work. Secondly, we will introduce a Bayesian algorithm based on the birth-death MCMC approach of Stephens (2000a) in order to fit this model to data.The use of finite mixture distributions is very common and the Bayesian approach provides an important tool in semi-parametric density estimation, see for instance Diebolt and Robert (1994) and Robert (1996). Recently, MCMC methods fo...
