Decay properties of Markovian bulk-arrival and bulk-service queues with state-independent control

Abstract

We consider decay properties regarding decay parameter and invariant measures of Markovian bulk-arrival and bulk-service queues with state-independent control. The exact value of the decay parameter, denoted by λ_Z, is firstly revealed. A criterion regarding λ_Z-recurrence and λ_Z-positive is obtained. The corresponding λ_Z-subinvariant/invariant measures and λ_Z-subinvariant/invariant vectors are then presented.

Keywords

Decay parameter / bulk-arrival and bulk-service queues / invariant measures / invariant vectors / λ_C-invariant measures / λ_C-invariant vectors / stateindependent control

References

[1] Anderson W. Continuous-Time Markov Chains: An Applications-Oriented Approach. New York: Springer-Verlag, 1991

[2] Chen Anyue, Li Junping, Hou Zhenting, Wang Ng Kai. Decay properties and quasistationary distributions for stopped Markovian bulk-arrival and bulk-service queues. Queueing Syst, 2010, 66: 275-311

[3] Chen Anyue, Li Junping, Ramesh N I. Uniqueness and extinction of weighted Markov branching processes. Methodol Comput Appl Probab, 2005, 7: 489-516

[4] Chen Anyue, Pollett P K, Junping Li, Hanjun Zhang. Markovian bulk-arrival and bulk-service queues with state-dependent control. Queueing Syst, 2010, 64: 267-304

[5] Kelly F P. Invariant measures and the generator. In: Kingman J F C, Reuter G E, eds. Probability, Statistics and Analysis. Lond Math Soc Lecture Notes Series 79. Cambridge: Cambridge University Press, 1983, 143-160

[6] Kingman J F C. The exponential decay of Markov transition probability. Proc Lond Math Soc, 1963, 13: 337-358

[7] Li Junping, Chen Anyue. The decay parameter and invariant measures for Markovian bulk-arrival queues with control at idle time. Methodol Comput Appl Probab, 2013, 15: 467-484

[8] Nair M G, Pollett P K. On the relationship between μ-invariant measures and quasistationary distributions for continuous-time Markov chains. Adv Appl Probab, 1993, 25: 82-102

[9] Pollett P K. The determination of quasi-instationary distribution directly from the transition rates of an absorbing Markov chain. Math Comput Modelling, 1995, 22: 279-287

[10] Pollett P K. Quasi-stationary distributions for continuous time Markov chains when absorption is not certain. J Appl Prob, 1999, 36: 268-272

[11] Van Doorn E A. Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes. Adv Appl Prob, 1991, 23: 683-700

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