Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-28T05:31:59.840Z Has data issue: false hasContentIssue false
  • English
  • Français

A stable algorithm for stationary distribution calculation for a BMAP/SM/1 queueing system with Markovian arrival input of disasters

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Alexander Dudin  and
Olga Semenova
Alexander Dudin*
Affiliation:
Belarusian State University
Olga Semenova*
Affiliation:
Belarusian State University
*
∗ Postal address: Laboratory of Applied Probabilistic Analysis, Department of Applied Mathematics and Computer Sciences, Belarusian State University, 4 F. Skorina Avenue, 220050 Minsk 50, Belarus.
∗ Postal address: Laboratory of Applied Probabilistic Analysis, Department of Applied Mathematics and Computer Sciences, Belarusian State University, 4 F. Skorina Avenue, 220050 Minsk 50, Belarus.
Get access Rights & Permissions [Opens in a new window]

Abstract

Disaster arrival into a queueing system causes all customers to leave the system instantaneously. We present a numerically stable algorithm for calculating the stationary state distribution of an embedded Markov chain for the BMAP/SM/1 queue with a MAP input of disasters.

Keywords

BMAP/SM/1-type queuedisastercensored Markov chainstable algorithm

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 2 , June 2004 , pp. 547 - 556
Copyright
Copyright © Applied Probability Trust 2004 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Artalejo, J. (2000). G-networks: a versatile approach for work removal in queueing networks. Europ. J. Operat. Res. 126, 233–249.Google Scholar
Bellman, R. (1960). Introduction to Matrix Analysis. McGraw Hill, New York.Google Scholar
Chakravarthy, S. (2001). The batch Markovian arrival process: a review and future work. In Advances in Probability and Stochastic Processes, eds Krishnamoorthy, A. et al., Notable Publications, Neshanic Station, NJ, pp. 21–49.Google Scholar
Dudin, A. N. (1998). Optimal multithreshold control for a BMAP/G/1 queue with N service modes. Queueing Systems 30, 273–287.Google Scholar
Dudin, A. N., and Nishimura, S. (1999). A BMAP/SM/1 queueing system with Markovian arrival of disasters. J. Appl. Prob. 36, 868–881.CrossRefGoogle Scholar
Gantmakher, F. R. (1966). Theory of Matrices, 2nd edn. Nauka, Moscow (in Russian).Google Scholar
Gelenbe, E. (1991). Product form networks with negative and positive customers. J. Appl. Prob. 28, 655–663.Google Scholar
Gómez-Corral, A. (2002). On a tandem G-network with blocking. Adv. Appl. Prob. 34, 626–661.Google Scholar
Grassmann, W. K., and Heyman, D. P. (1990). Equilibrium distribution of block-structured Markov chains with repeated rows. J. Appl. Prob. 27, 557–576.Google Scholar
Kemeni, J. G., Snell, J. L., and Knapp, A. W. (1966). Denumerable Markov Chains. Van Nostrand, New York.Google Scholar
Lucantoni, D. M. (1991). New results on the single server queue with a batch Markovian arrival process. Stoch. Models 7, 1–46.Google Scholar
Neuts, M. F. (1989). Structured Stochastic Matrices of M/G/1 Type and Their Applications. Marcel Dekker, New York.Google Scholar
Ramaswami, V. (1988). A stable recursion for the steady-state vector in Markov chains of M/G/1 type. Stoch. Models. 4, 183–188.Google Scholar
Ye, Q. (2000). High accuracy algorithms for solving nonlinear matrix equations in queueing models. In Advances in Algorithmic Methods for Stochastic Models, eds Latouche, G. and Taylor, P., Notable Publications, Neshanic Station, NJ, pp. 401–415.Google Scholar