Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-26T18:18:15.640Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Shoichi Nishimura
Alexander Dudin*
Affiliation:
Belarus State University
Shoichi Nishimura*
Affiliation:
Science University of Tokyo
*
∗Postal address: Department of Applied Mathematics and Informatics, Belarus State University, 4, F.Skorina Ave, Minsk-50, 220050 Belarus.
∗∗Postal address: Department of Applied Mathematics, Science University of Tokyo, 1–3, Kagurazaka, Shinjuku-ku, Tokyo, 162–8601 Japan. Email address: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

Disaster arrival in a queuing system with negative arrivals causes all customers to leave the system instantaneously. Here we obtain a queue-length and virtual waiting (sojourn) time distribution for the more complicated system BMAP/SM/1 with MAP input of disasters.

Keywords

Negative arrivalsBMAP/SM/1-type queuemodified multidimensional quasi-Toeplitz Markov chain

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 3 , September 1999 , pp. 868 - 881
Copyright
Copyright © Applied Probability Trust 1999 

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

Bayer, N., and Boxma, O. J. (1996). Wiener–Hopf analysis of an M/G/1 queue with negative customers and of a related class of random walks. Queueing Systems 23, 301–316.CrossRefGoogle Scholar
Chen, A., and Renshaw, E. (1997). The M/M/1 queue with mass exodus and mass arrivals when empty. J. Appl. Prob. 34, 192–207.CrossRefGoogle Scholar
Dudin, A. N., and Klimenok, V. I. (1997). Characteristics calculation for the single server queueing system, which operates in the synchronized Markov random environment. Autom. Remote Control 58, 74–84.Google Scholar
Dudin, A. N., and Klimenok, V. I. (1999). Multi-dimensional quasitoeplitz Markov chains. Accepted by J. Appl. Math. Stoch. Anal.CrossRefGoogle Scholar
Gail, H. R., Hantler, S. L., Sidi, M., and Taylor, B. A. (1995). Linear independence of root equations for M/G/1 type of Markov chains. Queueing Systems 20, 321–339.CrossRefGoogle Scholar
Gail, H. R., Hantler, S. L., and Taylor, B. A. (1996). Spectral analysis of M/G/1 and G/M/1 type Markov chains. Adv. Appl. Prob. 28, 114–165.CrossRefGoogle Scholar
Gelenbe, E. (1991). Product form networks with negative and positive customers. J. Appl. Prob. 28, 655–663.CrossRefGoogle Scholar
Graham, A. (1981). Kronecker Products and Matrix Calculus with Applications. Ellis Horwood, Chichester, UK.Google Scholar
Harrison, P. G., and Pitel, E. (1995). Response time distributions in tandem G-networks. J. Appl. Prob. 32, 224–247.CrossRefGoogle Scholar
Harrison, P. G., and Pitel, E. (1996). The M/G/1 queue with negative customers. Adv. Appl. Prob. 28, 540–566.CrossRefGoogle Scholar
Jain, G., and Sigman, K. (1996). A Pollaczeck–Khinchine formula for M/G/1 queues with disasters. J. Appl. Prob. 33, 1191–1200.CrossRefGoogle Scholar
Lucantoni, D. M. (1991). New results on the single server queue with a batch Markovian arrival process. Stoch. Mod. 7, 1–46.Google Scholar
Lucantoni, D. M., and Neuts, M. F. (1994). Some steady-state distributions for the MAP/SM/1 queue. Stoch. Mod. 10, 575–598.Google Scholar
Neuts, M. F. (1989). Structured Stochastic Matrices of M/G/1 Type and Their Applications. Marcel Dekker, New York.Google Scholar
Nishimura, S., and Sato, H. (1997). Eigenvalue expression for a batch Markovian arrival process. J. Operat. Res. Soc. Japan 40, 122–132.Google Scholar
Skorokhod, X. (1980). Probability Theory and random Processes, High School, Kiev.Google Scholar