Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-18T05:40:11.205Z Has data issue: false hasContentIssue false
  • English
  • Français

Factorization Identities for Reflected Processes, with Applications

Part of: Special processes Stochastic processes

Published online by Cambridge University Press:  30 January 2018

Brian H. Fralix ,
Johan S. H. van Leeuwaarden  and
Onno J. Boxma
Brian H. Fralix*
Affiliation:
Clemson University
Johan S. H. van Leeuwaarden*
Affiliation:
Eindhoven University of Technology
Onno J. Boxma*
Affiliation:
Eindhoven University of Technology
*
∗ Postal address: Department of Mathematical Sciences, Clemson University, O-110 Martin Hall, Box 340975, Clemson, SC 29634, USA. Email address: bfralix@clemson.edu
∗∗ Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands.
∗∗ Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands.
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 derive factorization identities for a class of preemptive-resume queueing systems, with batch arrivals and catastrophes that, whenever they occur, eliminate multiple customers present in the system. These processes are quite general, as they can be used to approximate Lévy processes, diffusion processes, and certain types of growth‒collapse processes; thus, all of the processes mentioned above also satisfy similar factorization identities. In the Lévy case, our identities simplify to both the well-known Wiener‒Hopf factorization, and another interesting factorization of reflected Lévy processes starting at an arbitrary initial state. We also show how the ideas can be used to derive transforms for some well-known state-dependent/inhomogeneous birth‒death processes and diffusion processes.

Keywords

Lévy processPalm distributionrandom walktime-dependent behaviorWiener‒Hopf factorization

MSC classification

Primary: 60G50: Sums of independent random variables; random walks 60G51: Processes with independent increments; L'evy processes 60G55: Point processes 60K25: Queueing theory
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 3 , September 2013 , pp. 632 - 653
Copyright
© Applied Probability Trust 

References

Abate, J. and Whitt, W. (1987). Transient behavior of regulated Brownian motion. II. Nonzero initial conditions. Adv. Appl. Prob. 19, 599–631.Google Scholar
Abate, J. and Whitt, W. (1987). Transient behavior of the M/M/1 queue: starting at the origin. Queueing Systems 2, 41–65.Google Scholar
Abate, J. and Whitt, W. (1988). Transient behavior of the M/M/1 queue via Laplace transforms. Adv. Appl. Prob. 20, 145–178.Google Scholar
Abate, J. and Whitt, W. (1995). Numerical inversion of Laplace transforms of probability distributions. ORSA J. Comput. 7, 36–43.CrossRefGoogle Scholar
Abate, J. and Whitt, W. (1998). Calculating transient characteristics of the Erlang loss model by numerical transform inversion. Commun. Statist. Stoch. Models 14, 663–680.CrossRefGoogle Scholar
Asmussen, S. (2003). Applied Probability and Queues, 2nd edn. Springer, New York.Google Scholar
Baccelli, F. and Brémaud, P. (2003). Elements of Queueing Theory, 2nd edn. Springer, Berlin.CrossRefGoogle Scholar
Bailey, N. T. J. (1954). A continuous time treatment of a simple queue using generating functions. Proc. R. Statist. Soc. B 16, 288–291.Google Scholar
Bekker, R., Boxma, O. J. and Resing, J. A. C. (2009). Lévy processes with adaptable exponent. Adv. Appl. Prob. 41, 177–205.CrossRefGoogle Scholar
Bingham, N. H. (1975). Fluctuation theory in continuous time. Adv. Appl. Prob. 7, 705–766.Google Scholar
Brémaud, P. (1981). Point Processes and Queues. Springer, New York.Google Scholar
Brémaud, P. (1999). Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues. Springer, New York.CrossRefGoogle Scholar
Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, Vol. I, 2nd edn. Springer, New York.Google Scholar
Daley, D. J. and Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes, Vol. II, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Darling, D. A. and Siegert, A. J. F. (1953). The first passage problem for a continuous Markov process. Ann. Math. Statist. 24, 624–639.Google Scholar
Debicki, K., Kosiński, K. and Mandjes, M. (2012). On the infimum attained by a reflected Lévy process. Queueing Systems 70, 23–35.Google Scholar
Flajolet, P. and Guillemin, F. (2000). The formal theory of birth-and-death processes, lattice-path combinatorics and continued fractions. Adv. Appl. Prob. 32, 750–778.Google Scholar
Fralix, B. H. (2013). On the time-dependent moments of Markovian queues with reneging. To appear in Queueing Systems.CrossRefGoogle Scholar
Fralix, B. H. and Riaño, G. (2010). A new look at transient versions of Little's law, and M/G/1 preemptive last-come–first-served queues. J. Appl. Prob. 47, 459–473.Google Scholar
Fralix, B. H., Riaño, G. and Serfozo, R. F. (2007). Time-dependent Palm probabilities and queueing applications. EURANDOM Rep. 2007–041. Available at http://www.eurandom.nl/reports/index.htm.Google Scholar
Fralix, B. H., van Leeuwaarden, J. S. H. and Boxma, O. J. (2011). A new Wiener–Hopf identity for a general class of reflected processes. EURANDOM Rep. 2011–024. Available at http://www.eurandom.nl/reports/index.htm.Google Scholar
Garnett, O., Mandelbaum, A. and Reiman, M. (2004). Designing a call center with impatient customers. Manufact. Service Operat. Manag. 4, 208–227.CrossRefGoogle Scholar
Greenwood, P. and Pitman, J. (1980). Fluctuation identities for Lévy processes and splitting at the maximum. Adv. Appl. Prob. 12, 893–902.Google Scholar
Gusak, D. V. and Korolyuk, V. S. (1968). On the first passage time across a given level for processes with independent increments. Theory Prob. Appl. 13, 448–456.Google Scholar
Kallenberg, O. (1983). Random Measures, 3rd edn. Akademie, Berlin.Google Scholar
Kella, O. and Mandjes, M. (2013). Transient analysis of reflected Lévy processes. Statist. Prob. Lett. 83, 2308–2315.CrossRefGoogle Scholar
Kuznetsov, A. E. (2010). An analytical proof of the Pecherskii–Rogozin identity and the Wiener–Hopf factorization. Theory Prob. Appl. 55, 432–443.Google Scholar
Marcellán, F. and Pérez, G. (2003). The moments of the M/M/s queue-length process. Queueing Systems 44, 281–304.Google Scholar
Millar, P. W. (1978). A path decomposition for Markov processes. Ann. Prob. 6, 345–348.Google Scholar
Palmowski, Z. and Vlasiou, M. (2011). A Lévy input model with additional state-dependent services. Stoch. Process. Appl. 121, 1546–1564.Google Scholar
Percheskii, E. A. and Rogozin, B. A. (1969). On the Joint distribution of random variables associated with fluctuations of a process with independent increments. Theory Prob. Appl. 14, 410–423.Google Scholar
Saaty, T. L. (1960). Time-dependent solution of the many-server Poisson queue. Operat. Res. 8, 755–772.Google Scholar