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Markovian bulk-arriving queues with state-dependent control at idle time

Part of: Markov processes Special processes

Published online by Cambridge University Press:  01 July 2016

Anyue Chen  and
Eric Renshaw
Anyue Chen*
Affiliation:
University of Greenwich
Eric Renshaw*
Affiliation:
University of Strathclyde
*
∗ Postal address: School of Computing and Mathematical Sciences, University of Greenwich, Maritime Greenwich Campus, Old Royal Naval College, Park Row, Greenwich, London SE10 9LS, UK
∗∗ Postal address: Department of Statistics and Modelling Science, Livingstone Tower, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, UK. Email address: [email protected]
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Abstract

This paper considers a Markovian bulk-arriving queue modified to allow both mass arrivals when the queue is idle and mass departures which allow for the possibility of removing the entire workload. Properties of queues which terminate when the server becomes idle are developed first, since these play a key role in later developments. Results for the case of mass arrivals, but no mass annihilation, are then constructed with specific attention being paid to recurrence properties, equilibrium queue-size structure, and waiting-time distribution. A closed-form expression for the expected queue size and its Laplace transform are also established. All of these results are then generalised to allow for the removal of the entire workload, with closed-form expressions being developed for the equilibrium size and waiting-time distributions.

Keywords

Bulk arrivalbusy period distributionequilibrium distributionidle timequeue sizerecurrencestate-dependent inputwaiting-time distribution

MSC classification

Primary: 60J35: Transition functions, generators and resolvents
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 60K25: Queueing theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 2 , June 2004 , pp. 499 - 524
Copyright
Copyright © Applied Probability Trust 2004 

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References

Anderson, W. J. (1991). Continuous Time Markov Chains. Springer, Berlin.CrossRefGoogle Scholar
Asmussen, S. (2003). Applied Probability and Queues, 2nd edn. John Wiley, New York.Google Scholar
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
Chaudhry, M. L. and Templeton, J. G. C. (1983). A First Course in Bulk Queues. John Wiley, New York.Google Scholar
Chen, A. and Renshaw, E. (1990). Markov branching processes with instantaneous immigration. Prob. Theory Relat. Fields 87, 209–240.CrossRefGoogle Scholar
Chen, A. and Renshaw, E. (1993a). Existence and uniqueness criteria for conservative uni-instantaneous denumerable Markov processes. Prob. Theory Relat. Fields 94, 427–456.CrossRefGoogle Scholar
Chen, A. and Renshaw, E. (1993b). Recurrence of Markov branching processes with immigration. Stoch. Process Appl. 45, 231–242.CrossRefGoogle Scholar
Chen, A. and Renshaw, E. (1995). Markov branching processes regulated by emigration and large immigration. Stoch. Process Appl. 57, 339–359.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
Chen, A. and Renshaw, E. (2000). Existence, recurrence and equilibrium properties of Markov branching processes with instantaneous immigration. Stoch. Process Appl. 88, 177–193.CrossRefGoogle Scholar
Chung, K. L. (1967). Markov Chains with Stationary Transition Probabilities, 2nd edn. Springer, New York.Google Scholar
Dudin, A. and Nishimura, S. (1999). A BMAP/SM/1 queueing system with Markovian arrival input of disasters. J. Appl. Prob. 36, 868–881.CrossRefGoogle Scholar
Gelenbe, E. (1991). Product-form queueing networks with negative and positive customers. J. Appl. Prob. 28, 656–663.CrossRefGoogle Scholar
Gelenbe, E., Glynn, P. and Sigman, K. (1991). Queues with negative arrivals. J. Appl. Prob. 28, 245–250.CrossRefGoogle Scholar
Gross, D. and Harris, C. M. (1985). Fundamentals of Queueing Theory. John Wiley, New York.Google Scholar
Harrison, P. G. and Pitel, E. (1993). Sojourn times in single-server queues with negative customers. J. Appl. Prob. 30, 943–963.CrossRefGoogle Scholar
Henderson, W. (1993). Queueing networks with negative customers and negative queue lengths. J. Appl. Prob. 30, 931–942.CrossRefGoogle Scholar
Jain, G. and Sigman, K. (1996). A Pollaczek–Khintchine formula for M/G/1 queues with disasters. J. Appl. Prob. 33, 1191–1200.CrossRefGoogle Scholar
Kleinrock, I. (1975). Queueing Systems, Vol. 1. John Wiley, 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.CrossRefGoogle Scholar
Lucantoni, D. M. and Neuts, M. F. (1994). Some steady-state distributions for the MAP/SM/1 queue. Stoch. Models 10, 575–598.CrossRefGoogle Scholar
Medhi, J. (1991). Stochastic Models in Queueing Theory. Academic Press, San Diego, CA.Google Scholar
Neuts, M. F. (1979). A versatile Markovian point process. J. Appl. Prob. 16, 764–779.CrossRefGoogle Scholar
Neuts, M. F. (1981). Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. Johns Hopkins University Press, Baltimore, MD.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
Parthasarathy, P. R. and Krishna Kumar, B. (1991). Density-dependent birth and death processes with state-dependent immigration. Math. Comput. Modelling 15, 11–16.CrossRefGoogle Scholar
Renshaw, E. and Chen, A. (1997). Birth–death processes with mass annihilation and state-dependent immigration. Stoch. Models 13, 239–254.CrossRefGoogle Scholar
Stadje, W. (1989). Some exact expressions for the bulk-arrival queue MX/M/1. Queueing Systems 4, 85–92.CrossRefGoogle Scholar
Yang, W. Q. (1990). The Construction Theory of Denumerable Markov Processes. John Wiley, New York.Google Scholar