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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Armero C, Bayarri MJ (1994a) Bayesian prediction in M/M/1 queues. Queueing Syst 15: 401–417
Armero C, Bayarri MJ (1994b) Prior assessments for prediction in queues. Statistician 43(1): 139–153
Ausin MC, Wiper MP (2007) Bayesian control of the number of servers in a GI/M/c queueing system. J Stat Plan Infer 137: 3043–3057
Ausin MC, Wiper MP, Lillo RE (2004) Bayesian estimation for the M/G/1 queue using a phase type approximation. J Stat Plan Infer 118: 83–101
Ausin MC, Wiper MP, Lillo RE (2007) Bayesian prediction of the transient behaviour and busy period in short- and long-tailed GI/G/1 queueing systems. Comput Stat Data Anal 52: 1615–1635
Cappé O, Robert CP, Rydén T (2003) Reversible jump MCMC converging to birth-and-death MCMC and more general continuous time samplers. J R Stat Soc Ser B 65: 679–700
Celeux G, Forbes F, Robert CP, Titterington DM (2006) Titterington deviance information criteria for missing data models. Bayesian Analysis 1(4): 651–674
Claeskens G, Hjort NL (2008) Model selection and model averaging. Cambridge University Press, Cambridge
Diebolt J., Robert CP (1994) Estimation of finite mixture distributions through Bayesian sampling. J R Stat Soc B 56: 363–375
Frühwirth-Schnatter S (2006) Finite mixture and markov switching models. Springer, New York
Frühwirth-Schnatter S, Pyne S (2010) Bayesian inference for finite mixtures of univariate and multivariate skew-normal and skew-t distributions. Biostatistics 11(2): 317–336
Gilks WR, Richardson S, Spiegelhalter DJ (1996) Markov chain Monte Carlo in practice. Chapman & Hall, London
Green P (1995) Reversible jump MCMC computation and Bayesian model determination. Biometrika 82: 711–732
Green P (2003) Trans-dimensional Markov chain Monte Carlo. OUP, Oxford, pp 179–198
Hastie D, Green P (2012) Model choice using reversible jump Markov Chain Monte Carlo. Statistica Neerlandica (Special issue, All models are wrong…)
Hastings WK (1970) Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57: 97–109
Medhi J (1982) Stochastic processes. Wiley Eastern, New Delhi
Mohammadi A, Salehi-Rad MR (2012) Bayesian inference and prediction in an M/G/1 with optional second service. Commun Stat Simul Comput 41(3): 419–435
Ramirez P, Lillo RE, Wiper MP (2008) Bayesian analysis of a queuing system with a long-tailed arrival process. Commun Stat Simul Comput 4: 697–712
Richardson S, Green P (1997) On Bayesian analysis of mixtures with an unknown number of components. J R Stat Soc B 59: 731–792
Rios D, Wiper MP, Ruggeri F (1998) Bayesian analysis of M/Er/1 and M/Hk/1 queues. Queueing Syst 30: 289–308
Robert CP (1996) Mixtures of distributions: inference and estimation. In: Gilks WR, Richardson S, Spiegelhalter DJ (eds) Markov chain Monte Carlo in practice. Chapman and Hall, London, pp 441–464
Salehi-Rad MR, Mengersen K (2002) Reservicing some customers in M/G/1 queues, under two disciplines, Advances in Statestics, Combinatorics and Related Areas. World Scientific Publishing, New Jersey, pp 267–274
Salehi-Rad MR, Mengersen K, Shahkar GH (2004) Reservicing some customers in M/G/1 queues, under three disciplines. Int J Math Math Sci 32: 1715–1723
Sperrin M, Jaki T, Wit E (2010) Probabilistic relabeling strategies for the label switching problem in Bayesian mixture models. Stat Comput 20: 357–366
Stephens M (2000a) Bayesian analysis of mixture models with an unknown number of components an alternative to reversible jump methods. Ann Stat 28: 40–74
Stephens M (2000b) Dealing with label switching in mixture models. J R Stat Soc B 62(4): 795–809
Tierney L (1996) Introduction to general state-space Markov chain theory. In: Gilks WR, Richardson S, Spiegelhalter DJ (eds) Markov chain Monte Carlo in practice. Chapman & Hall, London, pp 59–74
Wiper MP, Rios D, Ruggeri F (2001) Mixtures of gamma distributions with applications. J Comput Graph Stat 10: 440–454
Acknowledgments
The authors are grateful to the anonymous reviewers for their detailed and insightful comments.
Open Access
This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Mohammadi, A., Salehi-Rad, M.R. & Wit, E.C. Using mixture of Gamma distributions for Bayesian analysis in an M/G/1 queue with optional second service. Comput Stat 28, 683–700 (2013). https://doi.org/10.1007/s00180-012-0323-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00180-012-0323-3