Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-10T15:36:26.922Z Has data issue: false hasContentIssue false
  • English
  • Français

Queues with Delays in Two-State Strategies and Lévy Input

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

R. Bekker ,
O. J. Boxma  and
O. Kella
R. Bekker*
Affiliation:
Vrije Universiteit
O. J. Boxma*
Affiliation:
EURANDOM and Eindhoven University of Technology
O. Kella*
Affiliation:
The Hebrew University of Jerusalem
*
∗Postal address: Department of Mathematics, Vrije Universiteit, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands. Email address: rbekker@few.vu.nl
∗∗Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands.
∗∗∗Postal address: Department of Statistics, The Hebrew University of Jerusalem, Mount Scopus, Jerusalem, 91905, Israel.
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 consider a reflected Lévy process without negative jumps, starting at the origin. When the reflected process first upcrosses level K, a timer is activated. After D time units, the timer expires and the Lévy exponent of the Lévy process is changed. As soon as the process hits zero again, the Lévy exponent reverses to the original function. If the process has reached the origin before the timer expires then the Lévy exponent does not change. Using martingale techniques, we analyze the steady-state distribution of the resulting process, reflected at the origin. We pay special attention to the cases of deterministic and exponential timers, and to the following three special Lévy processes: (i) a compound Poisson process plus negative drift (corresponding to an M/G/1 queue), (ii) Brownian motion, and (iii) a Lévy process that is a subordinator until the timer expires.

Keywords

M/G/1 queueworkload processstorage processreflected Lévy processLévy exponenttwo-state strategydelayed feedback control

MSC classification

Secondary: 60K25: Queueing theory
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 2 , June 2008 , pp. 314 - 332
Copyright
Copyright © Applied Probability Trust 2008 

Footnotes

Supported by grant 964/06 from the Israel Science Foundation.

References

[1] Alili, L. and Kyprianou, A. E. (2005). Some remarks on first passage of Lévy processes, the American put and pasting principles. Ann. Appl. Prob. 15, 2062–2080.Google Scholar
[2] Altman, E., Basar, T. and Srikant, R. (1997). Multi-user rate-based flow control with action delays: a team-theoretic approach. In Proc. 36th IEEE Conf. Decision Control (San Diego, 1997), IEEE, New York, pp. 2387–2392.Google Scholar
[3] Altman, E., Kofman, D. and Yechiali, U. (1995). Discrete time queues with delayed information. Queueing Systems 19, 361–376.Google Scholar
[4] Artalejo, J.R. (2001). On the M/G/1 queue with D-policy. Appl. Math. Modelling 25, 1055–1069.Google Scholar
[5] Asmussen, S. (2003). Applied Probability and Queues, 2nd edn. Springer, New York.Google Scholar
[6] Avram, F., Kyprianou, A. E. and Pistorius, M. R. (2004). Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options. Ann. Appl. Prob. 14, 215–238.Google Scholar
[7] Bekker, R. (2005). Queues with state-dependent rates. , Department of Mathematics and Computer Science, Eindhoven University of Technology.Google Scholar
[8] Bekker, R. (2007). Queues with Lévy input and hysteretic control. Preprint WS2008-1, Department of Mathematics, Vrije Universiteit Amsterdam.Google Scholar
[9] Bekker, R., Boxma, O. J. and Resing, J.A.C. (2007). Lévy processes with adaptable exponent. Res. Rep. 2007-014, EURANDOM.Google Scholar
[10] Bekker, R., Borst, S. C., Boxma, O. J. and Kella, O. (2004). Queues with workload-dependent arrival and service rates. Queueing Systems 46, 537–556.Google Scholar
[11] Bertoin, J. (1996). Lévy Processes, Cambridge University Press.Google Scholar
[12] Bertoin, J. (1997). Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval. Ann. Appl. Prob. 7, 156–169.Google Scholar
[13] Bingham, N. H. (1975). Fluctuation theory in continuous time. Adv. Appl. Prob. 7, 705–766.Google Scholar
[14] Brockwell, P. J., Resnick, S. I. and Tweedie, R. L. (1982). Storage processes with general release rule and additive inputs. Adv. Appl. Prob. 14, 392–433.Google Scholar
[15] Cohen, J. W. (1976). On the optimal switching level for an M/G/1 queueing system. Stoch. Process. Appl. 4, 297–316.Google Scholar
[16] Cohen, J. W. (1982). The Single Server Queue. North-Holland, Amsterdam.Google Scholar
[17] Denteneer, D. and van Leeuwaarden, J. S. H. (2005). The delayed bulk service queue: a model for a reservation process. In Proc. 19th ITC, North-Holland, Amsterdam, pp. 909–918.Google Scholar
[18] Denteneer, D., van Leeuwaarden, J. S. H. and Adan, I. J. B. F. (2007). The acquisition queue. Queueing Systems 56, 229–240.Google Scholar
[19] De Turck, K. and Wittevrongel, S. (2005). Delay analysis of the go-back-N ARQ protocol over a time-varying channel. In Proc. 2nd Europ. Perf. Eval. Workshop (Lecture Notes Comput. Sci. 3670), Springer, Berlin, pp. 124–138.Google Scholar
[20] De Vuyst, S., Wittevrongel, S. and Bruneel, H. (2004). Delay analysis of the stop-and-wait ARQ protocol over a correlated channel. In Proc. HET-NETs '04, pp. P21/1–P21/11.Google Scholar
[21] Dshalalow, J. H. (1997). Queueing systems with state dependent parameters. In Frontiers in Queueing: Models and Applications in Science and Engineering, CRC, Boca Raton, FL, pp. 61–116.Google Scholar
[22] Feinberg, E. A. and Kella, O. (2002). Optimality of D-policies for an M/G/1 queue with a removable server. Queueing Systems 42, 355–376.Google Scholar
[23] Gaver, D. P. and Miller, R. G. (1962). Limiting distributions for some storage problems. In Studies in Applied Probability and Management Science, Stanford University Press, pp. 110–126.Google Scholar
[24] Harrison, J. M. (1985). Brownian Motion and Stochastic Flow Systems. John Wiley, New York.Google Scholar
[25] Jacobson, V. (1988). Congestion avoidance and control. In Proc. ACM SIGCOMM 1988, pp. 314–329.Google Scholar
[26] Kella, O. (1998). An exhaustive Lévy storage process with intermittent output. Stoch. Models 14, 979–992.Google Scholar
[27] Kella, O. and Whitt, W. (1992). Useful martingales for stochastic storage processes with Lévy input. J. Appl. Prob. 29, 396–403.Google Scholar
[28] Kyprianou, A. E. and Palmowski, Z. (2005). A martingale review of some fluctuation theory for spectrally negative Lévy processes. In Séminaire de Probabilités XXXVIII (Lecture Notes Math. 1857), Springer, Berlin, pp. 16–29.Google Scholar
[29] Lee, E. Y. and Ahn, S. K. (1998). PM_λ-policy for a dam with input formed by a compound Poisson process. J. Appl. Prob. 35, 482–488.Google Scholar
[30] Lee, J. and Kim, J. (2007). Workload analysis of an M/G/1 queue under the P_{λ}M policy with a set-up time. Appl. Math. Modelling 31, 236–244.Google Scholar
[31] Malhotra, R., van Haalen, R., Mandjes, M. R. H. and Núñez-Queija, R. (2005). Modeling the interaction of IEEE 802.3x hop-by-hop flow control and TCP end-to-end flow control. In Proc. NGI 2005 Conf. Next Generation Internet Networks Traffic Eng. (Rome).Google Scholar
[32] Nguyen-Ngoc, L. and Yor, M. (2005). Some martingales associated to reflected Lévy processes. In Séminaire de Probabilités XXXVIII (Lecture Notes Math. 1857), Springer, Berlin, pp. 42–69.Google Scholar
[33] Padhye, J., Firoiu, V., Towsley, D. and Kurose, J. (1998). Modeling TCP throughput: a simple model and its empirical validation. In Proc. ACM SIGCOMM 1998, ACM, New York, pp. 303–314.Google Scholar
[34] Prabhu, N. U. (1965). Queues and Inventories. John Wiley, New York.Google Scholar
[35] Prabhu, N. U. (1980). Stochastic Storage Processes. Springer, New York.Google Scholar
[36] Sharma, V. (2001). Queues with service rate controlled by a delayed feedback. Queueing Systems 39, 303–315.Google Scholar
[37] Tijms, H. C. (1976). Optimal control of the workload in an M/G/1 queueing system with removable server. Math. Operat. Statist. 7, 933–944.Google Scholar