Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-23T05:26:09.887Z Has data issue: false hasContentIssue false
  • English
  • Français

Loss rate for a general Lévy process with downward periodic barrier

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Zbigniew Palmowski  and
Przemysław Świa̧tek
Zbigniew Palmowski
Affiliation:
University of Wrocław, Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland. Email address: [email protected]
Przemysław Świa̧tek
Affiliation:
University of Wrocław, Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland. Email address: [email protected]
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.

In this paper we consider a general Lévy process X reflected at a downward periodic barrier At and a constant upper barrier K, giving a process VKt=Xt +LAt−LKt. We find the expression for a loss rate defined by lK=ELK1 and identify its asymptotics as K→∞ when X has light-tailed jumps and EX1<0.

Keywords

Loss rateLévy processbarrierqueueing process

MSC classification

Primary: 60K05: Renewal theory 60K25: Queueing theory
Secondary: 60K10: Applications (reliability, demand theory, etc.)
Type
Part 2. Lévy Processes
Information
Journal of Applied Probability , Volume 48 , Issue A: New Frontiers in Applied Probability (Journal of Applied Probability Special Volume 48A) , August 2011 , pp. 99 - 108
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] Asmussen, S., (2003). Applied Probability and Queues, 2nd edn. Springer, New York.Google Scholar
[2] Asmussen, S. and Pihlsgård, M., (2007). Loss rates for Lévy processes with two reflecting barriers. Math. Operat. Res. 32, 308–321.CrossRefGoogle Scholar
[3] Bekker, R. and Zwart, B., (2003). On an equivalence between loss rates and cycle maxima in queues and dams. Prob. Eng. Inf. Sci. 19, 241–255.Google Scholar
[4] Cooper, W. L., Schmidt, V. and Serfozo, R. F., (2001). Skorohod–Loynes characterizations of queueing, fluid, and inventory processes. Queueing Systems 37, 233–257.Google Scholar
[5] Dupuis, P. and Ramanan, K., (1999). Convex duality and the Skorokhod problem. I. Prob. Theory Relat. Fields 115, 153–195.CrossRefGoogle Scholar
[6] Dupuis, P. and Ramanan, K., (1999). Convex duality and the Skorokhod problem. II. Prob. Theory Relat. Fields 115, 197–236.Google Scholar
[7] Jelenković, P. R., (1999). Subexponential loss rates in a GI/GI/1 queue with applications. Queueing Systems 33, 91–123.Google Scholar
[8] Kella, O. and Stadje, W., (2004). A Brownian motion with two reflecting barriers and Markov-modulated speed. J. Appl. Prob. 41, 1237–1242.Google Scholar
[9] Kella, O. and Whitt, W., (1992). Useful martingales for stochastic storage processes with Lévy input. J. Appl. Prob. 29, 396–403.Google Scholar
[10] Kella, O., Boxma, O. and Mandjes, M., (2006). A Lévy process reflected at a Poisson age process. J. Appl. Prob. 43, 221–230.CrossRefGoogle Scholar
[11] Kim, H. S. and Shroff, N. B., (2001). On the asymptotic relationship between the overflow probability and the loss ratio. Adv. Appl. Prob. 33, 836–863.Google Scholar
[12] Pihlsgård, M., (2004). Loss rate asymptotics in a GI/G/1 queue with finite buffer. Stoch. Models 21, 913–931.Google Scholar
[13] Pistorius, M. R., (2003). On doubly reflected completely asymmetric Lévy processes. Stoch. Process. Appl. 107, 131–143.CrossRefGoogle Scholar
[14] Sakama, Y. and Miyazawa, M., (2009). Asymptotic behaviors of the loss rate for Markov modulated fluid queue with a finite buffer. Submitted.Google Scholar
[15] Siegmund, D., (1976). The equivalence of absorbing and reflecting barrier problems for stochastically monotone Markov processes. Ann. Prob. 4, 914–924.Google Scholar
[16] Tanaka, H., (1979). Stochastic differential equations with reflecting boundary conditions in convex regions. Hiroshima Math. J. 9, 163–177.Google Scholar
[17] Zwart, A. P., (2000). A fluid queue with a finite buffer and subexponential input. Adv. Appl. Prob. 32, 221–243.Google Scholar