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The untraceable events method for absorbing processes

Part of: Special processes Probability theory on algebraic and topological structures

Published online by Cambridge University Press:  14 July 2016

Toshinao Nakatsuka
Toshinao Nakatsuka*
Affiliation:
Tokyo Metropolitan University
*
∗Postal address: Faculty of Urban Liberal Arts, Tokyo Metropolitan University, 192-0397 Tokyo, Japan. Email address: tnaka@comp.metro-u.ac.jp
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Abstract

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In this paper we propose a new method of determining the stability of queueing systems. We attain it using the absorbing process and introduce the untraceable events method to show the existence of the absorbing process. The advantage of our method is that we are able to discuss the stability of various variables for both discrete and continuous parameters in a general framework with nonstationary input. An untraceable event has the property that the state loses the memory of its origin. In a concrete model, we use the boundedness of the state at an epoch in time with respect to the initial condition and choose the form of the untraceable event corresponding to the input distribution.

Keywords

Absorbing processuntraceable eventBorovkov eventcontinuous-time parameterasymptotic mean stationarityqueueing system

MSC classification

Primary: 60B10: Convergence of probability measures
Secondary: 60K25: Queueing theory
Type
Research Article
Information
Journal of Applied Probability , Volume 43 , Issue 3 , September 2006 , pp. 652 - 664
Copyright
© Applied Probability Trust 2006 

References

Altman, E. and Hordijk, A. (1997). Applications of Borovkov's renovation theory to non-stationary stochastic recursive sequences and their control. Adv. Appl. Prob. 29, 388–413.CrossRefGoogle Scholar
Borovkov, A. A. (1972). Continuity theorems for multichannel systems with refusals. Theory Prob. Appl. 17, 434–444.CrossRefGoogle Scholar
Borovkov, A. A. (1976). Stochastic Processes in Queueing Theory. Springer, Berlin.CrossRefGoogle Scholar
Borovkov, A. A. (1978). Ergodicity and stability theorems for a class of stochastic equation and their applications. Theory Prob. Appl. 23, 227–247.CrossRefGoogle Scholar
Borovkov, A. A. (1998). Ergodicity and Stability of Stochastic Processes. John Wiley, New York.Google Scholar
Borovkov, A. A. and Foss, S. G. (1992). Stochastically recursive sequences and their generalizations. Siberian Adv. Math. 2, 16–81.Google Scholar
Brandt, A., Franken, P. and Lisek, B. (1990). Stationary Stochastic Models. John Wiley, New York.Google Scholar
Dai, J. G. (1995). On positive Harris recurrence of multiclass queueing networks: a unified approach via fluid limit models. Ann. Appl. Prob. 5, 49–77.CrossRefGoogle Scholar
Foss, S. G. (1992). On the ergodicity conditions for stochastically recursive sequences. Queueing Systems 12, 287–296.CrossRefGoogle Scholar
Foss, S. G. and Kalashnikov, V. V. (1991). Regeneration and renovation in queues. Queueing Systems 8, 211–224.CrossRefGoogle Scholar
Kalähne, U. (1976). Existence, uniqueness and some invariance properties of stationary distributions for general single-server queues. Math. Operationsforsch. Statist. 7, 557–575.Google Scholar
Krengel, U. (1985). Ergodic Theorems. De Gruyter, Berlin.CrossRefGoogle Scholar
Loynes, R. M. (1962). The stability of a queue with non-independent inter-arrival and service times. Proc. Camb. Phil. Soc. 58, 497–520.CrossRefGoogle Scholar
Nakatsuka, T. (1986). The substability and ergodicity of complicated queueing systems. J. Appl. Prob. 23, 193–200.CrossRefGoogle Scholar
Nakatsuka, T. (1987). Substability and ergodicity of queue series. Stoch. Models 3, 227–250.Google Scholar
Nakatsuka, T. (1998). Absorbing process in recursive stochastic equations. J. Appl. Prob. 35, 418–426.CrossRefGoogle Scholar
Rolski, T. (1981). Queues with non-stationary input stream: Ross's conjecture. Adv. Appl. Prob. 13, 603–618.CrossRefGoogle Scholar
Szpankowski, W. (1990). Towards computable stability criteria for some multidimensional stochastic processes. In Stochastic Analysis of Computer and Communication Systems, ed. Takagi, H., North-Holland, Amsterdam, pp. 131–172.Google Scholar