Hostname: page-component-7479d7b7d-jwnkl Total loading time: 0 Render date: 2024-07-11T00:06:33.706Z Has data issue: false hasContentIssue false
  • English
  • Français

Decay property of stopped Markovian bulk-arriving queues

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Junping Li  and
Anyue Chen
Junping Li*
Affiliation:
Central South University
Anyue Chen*
Affiliation:
University of Liverpool and University of Hong Kong
*
∗ Postal address: School of Mathematical Science and Computing Technology, Central South University, Changsha, 410075, P. R. China. Email address: jpli@mail.csu.edu.cn
∗∗ Postal address: Division of Statistics and Probability, Department of Mathematical Sciences, The University of Liverpool, Liverpool, L69 7ZL, UK. Email address: achen@liv.ac.uk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider decay properties including the decay parameter, invariant measures, invariant vectors, and quasistationary distributions of a Markovian bulk-arriving queue that stops immediately after hitting the zero state. Investigating such behavior is crucial in realizing the busy period and some other related properties of Markovian bulk-arriving queues. The exact value of the decay parameter λC is obtained and expressed explicitly. The invariant measures, invariant vectors, and quasistationary distributions are then presented. We show that there exists a family of invariant measures indexed by λ ∈ [0, λC]. We then show that, under some conditions, there exists a family of quasistationary distributions, also indexed by λ ∈ [0, λC]. The generating functions of these invariant measures and quasistationary distributions are presented. We further show that a stopped Markovian bulk-arriving queue is always λC-transient and some deep properties are revealed. The clear geometric interpretation of the decay parameter is explained. A few examples are then provided to illustrate the results obtained in this paper.

Keywords

Markovian bulk-arriving queuedecay parameterinvariant measureinvariant vectorquasistationary distribution

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J35: Transition functions, generators and resolvents
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 40 , Issue 1 , March 2008 , pp. 95 - 121
Copyright
Copyright © Applied Probability Trust 2008 

References

Anderson, W. (1991). Continuous-Time Markov Chains: An Applications-Oriented Approach. Springer, New York.CrossRefGoogle Scholar
Asmussen, S. (2003). Applied Probability and Queues, 2nd edn. Springer, 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. Y. 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. Y. and Renshaw, E. (2004). Markovian bulk-arriving queues with state-dependent control at idle time. Adv. Appl. Prob. 36, 499–524.CrossRefGoogle Scholar
Chen, A. Y., Li, J. P. and Ramesh, N. I. (2005). Uniqueness and extinction of weighted Markov branching processes. Method. Comput. Appl. Prob. 7, 489–516.CrossRefGoogle Scholar
Chen, M. F. (1992). From Markov Chains to Non-Equilibrium Particle Systems. World Scientific, Singapore.CrossRefGoogle Scholar
Chen, M. F. (2004). Eigenvalues, Inequalities, and Ergodic Theory. Springer, London.Google Scholar
Chung, K. L. (1967). Markov Chains with Stationary Transition Probabilities, 2nd edn. Springer, New York.Google Scholar
Daley, D. J. (1969). Quasi-stationary behaviour of a left-continuous random walk. Ann. Math. Statist. 40, 532–539.CrossRefGoogle Scholar
Darroch, J. N. and Seneta, E. (1967). On quasi-stationary distributions in absorbing continuous-time finite Markov chains. J. Appl. Prob. 4, 192–196.CrossRefGoogle 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
Flaspohler, D. C. (1974). Quasi-stationary distributions for absorbing continuous-time denumerable Markov chains. Ann. Inst. Statist. Math. 26, 351–356.CrossRefGoogle Scholar
Gelenbe, E. (1991). Product form 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
Kelly, F. P. (1983). Invariant measures and the generator. In Probability, Statistics, and Analysis (London Math. Soc. Lecture Notes Ser. 79), eds Kingman, J. F. C. and Reuter, G. E., Cambridge University Press, pp. 143–160.CrossRefGoogle Scholar
Kijima, M. (1993). Quasi-limiting distributions of Markov chains that are skip-free to the left in continuous-time. J. Appl. Prob. 30, 509–517.CrossRefGoogle Scholar
Kingman, J. F. C. (1963). The exponential decay of Markov transition probability. Proc. London Math. Soc. 13, 337–358.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, 575–589.CrossRefGoogle Scholar
Medhi, J. (1991). Stochastic Models in Queuing Theory. Academic Press, San Diego, CA.Google Scholar
Nair, M. G. and Pollett, P. K. (1993). On the relationship between μ-invariant measures and quasistationary distributions for continuous-time Markov chains. Adv. Appl. Prob. 25, 82–102.CrossRefGoogle Scholar
Neuts, M. F. (1979). A versatile Markovian point process. J. Appl. Prob. 16, 764–779.CrossRefGoogle Scholar
Neuts, M. F. (1981). Matrix-Geometric Solution 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
Norris, J. R. (1997). Markov Chains. Cambridge University Press.CrossRefGoogle 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
Pollett, P. K. (1988). Reversibility, invariance and μ-invariance. Adv. Appl. Prob. 20, 600–621.CrossRefGoogle Scholar
Pollett, P. K. (1995). The determination of quasi-instationary distribution directly from the transition rates of an absorbing Markov chain. Math. Comput. Modelling 22, 279–287.CrossRefGoogle Scholar
Pollett, P. K. (1999). Quasi-stationary distributions for continuous time Markov chains when absorption is not certain. J. Appl. Prob. 36, 268–272.CrossRefGoogle Scholar
Stadje, W. (1989). Some exact expressions for the bulk-arrival queue MX/M/1. Queuing Systems 4, 85–92.CrossRefGoogle Scholar
Syski, R. (1992). Passage Times for Markov Chains. IOS Press, Amsterdam.Google Scholar
Tweedie, R. L. (1974). Some ergodic properties of the Feller minimal process. Quart. J. Math. Oxford Ser. (2) 25, 485–495.CrossRefGoogle Scholar
Van Doorn, E. A. (1985). Conditions for exponential ergodicity and bounds for the decay parameter of a birth-death process. Adv. Appl. Prob. 17, 514–530.CrossRefGoogle Scholar
Van Doorn, E. A. (1991). Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes. Adv. Appl. Prob. 23, 683–700.CrossRefGoogle Scholar
Vere-Jones, D. (1962). Geometric ergidicity in denumerable Markov chains. Quart. J. Math. Oxford Ser. (2) 13, 7–28.CrossRefGoogle Scholar
Yaglom, A. M. (1947). Certain limit theorems of the theory of branching random processes. Dokl. Akad. Nauk SSSR (N. S.) 56, 795–798.Google Scholar