Abstract
In this paper we investigate an M/M/∞ queue whose parameters depend on an external random environment that we assume to be a semi-Markovian process with finite state space. For this model we show a recursive formula that allows to compute all the factorial moments for the number of customers in the system in steady state. The used technique is based on the calculation of the raw moments of the measure of a bidimensional random set. Finally the case when the random environment has only two states is deeper analyzed. We obtain an explicit formula to compute the above mentioned factorial moments when at least one of the two states has sojourn time exponentially distributed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1964)
Baykal-Gursoy, M., Xiao, W.: Stochastic decomposition in M/M/∞ queues with Markov-modulated service rates . Queueing Syst. 48, 75–88 (2004)
Brémaud, P.: Markov Chains . Springer, New York (1999)
Daley, D.J., Vere-Jones, D.: An Introduction to the Theory of Point Processes, 1st edn. Springer Series in Statistics. Springer, New York (1988)
D’Auria, B.: Stochastic decomposition of the M/G/∞ queue in a random environment. Oper. Res. Lett. 35, 805–812 (2007)
D’Auria, B., Resnick, S.: Data network models of burstiness. Adv. Appl. Probab. 38, 373–404 (2006)
Keilson, J., Servi, L.: The matrix M/M/∞ system: Retrials models and Markov modulated sources. Adv. Appl. Probab. 25, 453–471 (1993)
Neuts, M.: Matrix–Geometric Solutions in Stochastic Models: An Algorithmic Approach. Johns Hopkins University Press, Baltimore (1981)
O’Cinneide, C., Purdue, P.: The M/M/∞ queue in a random environment. J. Appl. Probab. 23, 175–184 (1986)
Ozawa, T.: Analysis of queues with Markovian service processes. Stoch. Models 20, 391–413 (2004)
Resnick, S.: Extreme Values, Regular Variation and Point Processes. Springer, New York (1987)
Seneta, E.: Non-Negative Matrices and Markov Chains, 2nd edn. Springer Series in Statistics. Springer, New York (1981)
Sengupta, B.: A queue with service interruptions in an alternating random environment . Oper. Res. 38, 308–318 (1990)
Takine, T.: Single-server queues with Markov-modulated arrivals and service speed. Queueing Syst. Theory Appl. 49, 7–22 (2005)
Takine, T., Sengupta, B.: A single server queue with service interruptions. Queueing Syst. 26, 285–300 (1997)
Author information
Authors and Affiliations
Corresponding author
Additional information
Part of this research took place while the author was still post-doc at EURANDOM, Eindhoven, The Netherlands. The work was supported by the Spanish Ministry of Education and Science by the Grant MTM2007-63140.
Rights and permissions
About this article
Cite this article
D’Auria, B. M/M/∞ queues in semi-Markovian random environment. Queueing Syst 58, 221–237 (2008). https://doi.org/10.1007/s11134-008-9068-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11134-008-9068-7