×

zbMATH — the first resource for mathematics

Factorization identities for reflected processes, with applications. (English) Zbl 1284.60161
The paper derives a factorization identity for a class of preemptive-resume queueing systems (PRP). These systems can be used to approximate Lévy processes, diffusion processes, and certain types of growth-collapse processes. For an arbitrary PRP system the identity is not a true factorization, but for Lévy processes it is equivalent to the Wiener-Hopf factorization. The results of the paper also provide insight into the time-dependent behavior of a number of important birth-death processes, with birth/death rates that may depend on the state of the system. For instance, it is shown how the probability mass function of the M/M/\(s\) queue length at an independent exponential time can be expressed entirely in terms of quantities from an M/M/\(1\) queue and an M/M/\(\infty \) queue. Similarly, an M/M/\(s\)/\(K\) queue (assuming that \(s < K\), and trivial otherwise) can be expressed in terms of an M/M/\(\infty \) queue and an M/M/\(1\)/\(K - s\) queue, and a similar observation may be made for a Markovian queue with reneging.

MSC:
60K25 Queueing theory (aspects of probability theory)
60G50 Sums of independent random variables; random walks
60G51 Processes with independent increments; Lévy processes
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
PDF BibTeX XML Cite
Full Text: DOI Euclid arXiv
References:
[1] Abate, J. and Whitt, W. (1987). Transient behavior of regulated Brownian motion. II. Nonzero initial conditions. Adv. Appl. Prob. 19, 599-631. · Zbl 0628.60084
[2] Abate, J. and Whitt, W. (1987). Transient behavior of the \({\mathrm M}/{\mathrm M}/1\) queue: starting at the origin. Queueing Systems 2, 41-65. · Zbl 0671.60082
[3] Abate, J. and Whitt, W. (1988). Transient behavior of the \({\mathrm M}/{\mathrm M}/1\) queue via Laplace transforms. Adv. Appl. Prob. 20, 145-178. · Zbl 0638.60097
[4] Abate, J. and Whitt, W. (1995). Numerical inversion of Laplace transforms of probability distributions. ORSA J. Comput. 7, 36-43. · Zbl 0821.65085
[5] 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. · Zbl 0905.60071
[6] Asmussen, S. (2003). Applied Probability and Queues , 2nd edn. Springer, New York. · Zbl 1029.60001
[7] Baccelli, F. and Brémaud, P. (2003). Elements of Queueing Theory , 2nd edn. Springer, Berlin. · Zbl 1021.60001
[8] Bailey, N. T. J. (1954). A continuous time treatment of a simple queue using generating functions. Proc. R. Statist. Soc. B 16, 288-291. · Zbl 0058.34801
[9] Bekker, R., Boxma, O. J. and Resing, J. A. C. (2009). Lévy processes with adaptable exponent. Adv. Appl. Prob. 41, 177-205. · Zbl 1169.60022
[10] Bingham, N. H. (1975). Fluctuation theory in continuous time. Adv. Appl. Prob. 7, 705-766. · Zbl 0322.60068
[11] Brémaud, P. (1981). Point Processes and Queues . Springer, New York.
[12] Brémaud, P. (1999). Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues . Springer, New York. · Zbl 0949.60009
[13] Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes , Vol. I, 2nd edn. Springer, New York. · Zbl 1026.60061
[14] Daley, D. J. and Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes , Vol. II, 2nd edn. Springer, New York. · Zbl 1159.60003
[15] Darling, D. A. and Siegert, A. J. F. (1953). The first passage problem for a continuous Markov process. Ann. Math. Statist. 24, 624-639. · Zbl 0053.27301
[16] Dȩbicki, K., Kosiński, K. and Mandjes, M. (2012). On the infimum attained by a reflected Lévy process. Queueing Systems 70, 23-35. · Zbl 1247.60127
[17] 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. · Zbl 0966.60069
[18] Fralix, B. H. (2013). On the time-dependent moments of Markovian queues with reneging. To appear in Queueing Systems . · Zbl 1293.60086
[19] 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. · Zbl 1217.60080
[20] 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.
[21] 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.
[22] Garnett, O., Mandelbaum, A. and Reiman, M. (2004). Designing a call center with impatient customers. Manufact. Service Operat. Manag. 4, 208-227.
[23] Greenwood, P. and Pitman, J. (1980). Fluctuation identities for Lévy processes and splitting at the maximum. Adv. Appl. Prob. 12, 893-902. · Zbl 0443.60037
[24] 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. · Zbl 0177.21304
[25] Kallenberg, O. (1983). Random Measures , 3rd edn. Akademie, Berlin. · Zbl 0544.60053
[26] Kella, O. and Mandjes, M. (2013). Transient analysis of reflected Lévy processes. Statist. Prob. Lett. 83 , 2308-2315. · Zbl 1287.60061
[27] Kuznetsov, A. E. (2010). An analytical proof of the Pecherskii-Rogozin identity and the Wiener-Hopf factorization. Theory Prob. Appl. 55, 432-443. · Zbl 1242.60048
[28] Marcellán, F. and Pérez, G. (2003). The moments of the \(M/M/s\) queue-length process. Queueing Systems 44, 281-304. · Zbl 1035.90023
[29] Millar, P. W. (1978). A path decomposition for Markov processes. Ann. Prob. 6, 345-348. · Zbl 0379.60070
[30] Palmowski, Z. and Vlasiou, M. (2011). A Lévy input model with additional state-dependent services. Stoch. Process. Appl. 121, 1546-1564. · Zbl 1219.60079
[31] 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. · Zbl 0194.49001
[32] Saaty, T. L. (1960). Time-dependent solution of the many-server Poisson queue. Operat. Res. 8, 755-772. · Zbl 0105.11702
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.