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The busy cycle of the reflected superposition of Brownian motion and a compound Poisson process

Published online by Cambridge University Press:  14 July 2016

David Perry*
Affiliation:
University of Haifa
Wolfgang Stadje*
Affiliation:
Universität Osnabrück
*
Postal address: University of Haifa, Department of Statistics, Haifa 31905, Israel.
∗∗ Postal address: Department of Mathematics and Computer Science, University of Osnabrück, 49069 Osnabrück, Germany. Email address: wolfgang@mathematik.uni-osnabrueck.de

Abstract

We consider a reflected superposition of a Brownian motion and a compound Poisson process as a model for the workload process of a queueing system with two types of customers under heavy traffic. The distributions of the duration of a busy cycle and the maximum workload during a cycle are determined in closed form.

Type
Short Communications
Copyright
Copyright © by the Applied Probability Trust 2001 

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References

Asmussen, S. (1987). Applied Probability and Queues. John Wiley, New York.Google Scholar
Bardhan, I. (1993). Diffusion approximation for GI/M/s queues with service interruptions. Operat. Res. Lett. 13, 175182.Google Scholar
Bardhan, I., and Chao, X. (1996). On martingale measures when asset returns have unpredictable jumps. Stoch. Proc. Appl. 63, 3354.Google Scholar
Borovkov, A. A. (1964). On the first passage time for one class of processes with independent increments. Theory Prob. Appl. 10, 331334.Google Scholar
Kella, O., and Whitt, W. (1990). Diffusion approximations for queues with server vacations. Adv. Appl. Prob. 22, 706729.CrossRefGoogle Scholar
Kella, O., and Whitt, W. (1991). Queues with server vacations and Lévy processes with secondary jump input. Ann. Appl. Prob. 1, 104117.Google Scholar
Kella, O., and Whitt, W. (1992). Useful martingales for stochastic storage processes with Lévy input. J. Appl. Prob. 29, 396403.Google Scholar
Möller, C. M. (1995). Stochastic differential equations for ruin probabilities. J. Appl. Prob. 32, 7489.CrossRefGoogle Scholar
Perry, D., and Stadje, W. (1999). Heavy traffic analysis of a queueing system with bounded capacity and two types of customers. J. Appl. Prob. 36, 11551166.Google Scholar
Perry, D., and Stadje, W. (2000). Risk analysis for a stochastic cash management model with two types of customers. Insurance: Math. Econ. 26, 2536.Google Scholar
Schäl, M. (1993). On hitting times for jump-diffusion processes with past dependence local characteristics. Stoch. Proc. Appl. 47, 131142.Google Scholar
Yor, M. (1992). Some Aspects of Brownian Motion, Part I: Some Special Functionals. Birkhäuser, Basel.Google Scholar