Hostname: page-component-54dcc4c588-r5qjk Total loading time: 0 Render date: 2025-09-11T04:46:20.438Z Has data issue: false hasContentIssue false
  • English
  • Français

Tail Asymptotics for a Random Sign Lindley Recursion

Part of: Special processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Maria Vlasiou  and
Zbigniew Palmowski
Maria Vlasiou*
Affiliation:
Eindhoven University of Technology
Zbigniew Palmowski*
Affiliation:
University of Wrocław
*
∗ Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB, Eindhoven, The Netherlands. Email address: m.vlasiou@tue.nl
∗∗ Postal address: Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland. Email address: zpalma@math.uni.wroc.pl
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 investigate the tail behaviour of the steady-state distribution of a stochastic recursion that generalises Lindley's recursion. This recursion arises in queueing systems with dependent interarrival and service times, and includes alternating service systems and carousel storage systems as special cases. We obtain precise tail asymptotics in three qualitatively different cases, and compare these with existing results for Lindley's recursion and for alternating service systems.

Keywords

Stochastic recursionCramér conditionssubexponential distributionintermediate tailsharp asymptotics

MSC classification

Secondary: 60F10: Large deviations 60K25: Queueing theory

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 1 , March 2010 , pp. 72 - 83
Copyright
Copyright © Applied Probability Trust 2010 

References

Asmussen, S. (2003). Applied Probability and Queues. Springer, New York.Google Scholar
Bertoin, J. and Doney, R. A. (1996). Some asymptotic results for transient random walks. Adv. Appl. Prob. 28, 207–226.CrossRefGoogle Scholar
Boxma, O. J. and Vlasiou, M. (2007). On queues with service and interarrival times depending on waiting times. Queueing Systems 56, 121–132.Google Scholar
Cline, D. B. H. (1986). Convolution tails, product tails and domains of attraction. Prob. Theory Relat. Fields 72, 529–557.Google Scholar
Cohen, J. W. (1982). The Single Server Queue. North-Holland, Amsterdam.Google Scholar
Denisov, D. and Zwart, B. (2007). On a theorem of Breiman and a class of random difference equations. J. Appl. Prob. 44, 1031–1046.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events (Appl. Math. 33). Springer, Berlin.Google Scholar
Foss, S. and Zachary, S. (2003). The maximum on a random time interval of a random walk with long-tailed increments and negative drift. Ann. Appl. Prob. 13, 37–53.Google Scholar
Goldie, C. M. (1991). Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Prob. 1, 126–166.Google Scholar
Korshunov, D. (1997). On distribution tail of the maximum of a random walk. Stoch. Process. Appl. 72, 97–103.Google Scholar
Lindley, D. V. (1952). The theory of queues with a single server. Proc. Camb. Phil. Soc. 48, 277–289.CrossRefGoogle Scholar
Park, B. C., Park, J. Y. and Foley, R. D. (2003). Carousel system performance. J. Appl. Prob. 40, 602–612.Google Scholar
Pitman, E. J. G. (1980). Subexponential distribution functions. J. Austral. Math. Soc. A 29, 337–347.Google Scholar
Veraverbeke, N. (1977). Asymptotic behaviour of Wiener–Hopf factors of a random walk. Stoch. Process. Appl. 5, 27–37.Google Scholar
Vlasiou, M. (2007). A non-increasing Lindley-type equation. Queueing Systems 56, 41–52.Google Scholar
Vlasiou, M. and Adan, I. J. B. F. (2005). An alternating service problem. Prob. Eng. Inf. Sci. 19, 409–426.CrossRefGoogle Scholar
Vlasiou, M. and Adan, I. J. B. F. (2007). Exact solution to a Lindley-type equation on a bounded support. Operat. Res. Let. 35, 105–113.Google Scholar
Vlasiou, M., Adan, I. J. B. F. and Wessels, J. (2004). A Lindley-type equation arising from a carousel problem. J. Appl. Prob. 41, 1171–1181.Google Scholar
Whitt, W. (1990). Queues with service times and interarrival times depending linearly and randomly upon waiting times. Queueing Systems 6, 335–351.Google Scholar
Zachary, S. (2004). A note on Veraverbeke's theorem. Queueing Systems 46, 9–14.Google Scholar