Theory of Probability and Mathematical Statistics
Scaling limits for some Mittag-Leffler queues
Giacomo Ascione and Luigia Caputo
Link
Abstract: In this paper, we consider five models of heavy-tailed queues involving Mittag-Leffler distributions that generalize the classical M/M/1 queues. These models are suitable modifications of previously defined models in such a way that the classical M/M/1 queue can be recovered by a suitable selection of parameters. We provide the distribution of inter-arrival and service times of both the original and modified queueing models. We then study the scaling limits of all the proposed models, and we argue that the behaviour of the limiting processes can be used to characterise the traffic regime of the queues.
Keywords: Mittag-Leffler distribution, stable subordinator, queueing system
Bibliography: G. Ascione, N. Leonenko, and E. Pirozzi, Fractional queues with catastrophes and their transient behaviour, Mathematics 6 (2018), no. 9, 159.
G. Ascione, N. Leonenko, and E. Pirozzi, Fractional immigration-death processes, J. Math. Anal. Appl. 495 (2021), no. 2, 124768.
G. Ascione, N. Leonenko, and E. Pirozzi, On the transient behaviour of fractional M/M/∞ queues, Nonlocal and Fractional Operators, Springer, 2021, pp. 1–22.
G. Ascione, N. Leonenko, and E. Pirozzi, Skorokhod reflection problem for delayed Brownian motion with applications to fractional queues, Symmetry 14 (2022), no. 3, 615.
Giacomo Ascione, Yuliya Mishura, and Enrica Pirozzi, Fractional deterministic and stochastic calculus, De Gruyter Series in Probability and Stochastics, vol. 4, De Gruyter, Berlin, [2023] ©2023. MR 4760754, DOI 10.1515/9783110780017
Søren Asmussen, Extreme value theory for queues via cycle maxima, Extremes 1 (1998), no. 2, 137–168. MR 1814621, DOI 10.1023/A:1009970005784
Boris Baeumer and Mark M. Meerschaert, Stochastic solutions for fractional Cauchy problems, Fract. Calc. Appl. Anal. 4 (2001), no. 4, 481–500. MR 1874479
N. H. Bingham, Limit theorems for occupation times of Markov processes, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 17 (1971), 1–22. MR 281255, DOI 10.1007/BF00538470
Jacob Butt, Nicos Georgiou, and Enrico Scalas, Queuing models with Mittag-Leffler inter-event times, Fract. Calc. Appl. Anal. 26 (2023), no. 4, 1465–1503. MR 4623217, DOI 10.1007/s13540-023-00161-4
Dexter O. Cahoy, Federico Polito, and Vir Phoha, Transient behavior of fractional queues and related processes, Methodol. Comput. Appl. Probab. 17 (2015), no. 3, 739–759. MR 3377858, DOI 10.1007/s11009-013-9391-2
E. Carlson and R. Felder, Simulation and queueing network modeling of single-product production campaigns, Comput. Chem. Eng. 16 (1992), no. 7, 707–718.
A. K. Erlang, The theory of probabilities and telephone conversations, Nyt Tidsskrift for Matematik (1909), 33–39.
Iosif I. Gikhman and Anatoli V. Skorokhod, The theory of stochastic processes. I, Classics in Mathematics, Springer-Verlag, Berlin, 2004. Translated from the Russian by S. Kotz; Reprint of the 1974 edition. MR 2058259, DOI 10.1007/978-3-642-61921-2
Rudolf Gorenflo, Anatoly A. Kilbas, Francesco Mainardi, and Sergei Rogosin, Mittag-Leffler functions, related topics and applications, Springer Monographs in Mathematics, Springer, Berlin, [2020] ©2020. Second edition [of 3244285]. MR 4179587, DOI 10.1007/978-3-662-61550-8
Boris Harlamov, Continuous semi-Markov processes, Applied Stochastic Methods Series, ISTE, London; John Wiley & Sons, Inc., Hoboken, NJ, 2008. MR 2376500, DOI 10.1002/9780470610923
Carlos M. Hernández-Suárez, Carlos Castillo-Chavez, and Osval Montesinos López, An application of queueing theory to SIS and SEIS epidemic models, Math. Biosci. Eng. 7 (2010), no. 4, 809–823. MR 2739991, DOI 10.3934/mbe.2010.7.809
H. Kaspi and B. Maisonneuve, Regenerative systems on the real line, Ann. Probab. 16 (1988), no. 3, 1306–1332. MR 942771, DOI 10.1214/aop/1176991693
Alexander Kerss, Nikolai N. Leonenko, and Alla Sikorskii, Fractional Skellam processes with applications to finance, Fract. Calc. Appl. Anal. 17 (2014), no. 2, 532–551. MR 3181070, DOI 10.2478/s13540-014-0184-2
A. Khintchine, Mathematical theory of a stationary queue, Matematicheskii Sbornik 39 (1932), no. 4, 73–84.
Anatoly A. Kilbas, Hari M. Srivastava, and Juan J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006. MR 2218073
J. F. C. Kingman, The single server queue in heavy traffic, Proc. Cambridge Philos. Soc. 57 (1961), 902–904. MR 131298, DOI 10.1017/S0305004100036094
Nick Laskin, Fractional Poisson process, Commun. Nonlinear Sci. Numer. Simul. 8 (2003), no. 3-4, 201–213. Chaotic transport and complexity in classical and quantum dynamics. MR 2007003, DOI 10.1016/S1007-5704(03)00037-6
Nikolai Leonenko, Enrico Scalas, and Mailan Trinh, Limit theorems for the fractional nonhomogeneous Poisson process, J. Appl. Probab. 56 (2019), no. 1, 246–264. MR 3981156, DOI 10.1017/jpr.2019.16
Francesco Mainardi, Rudolf Gorenflo, and Enrico Scalas, A fractional generalization of the Poisson processes, Vietnam J. Math. 32 (2004), no. Special Issue, 53–64. MR 2120631
Francesco Mainardi, Rudolf Gorenflo, and Alessandro Vivoli, Renewal processes of Mittag-Leffler and Wright type, Fract. Calc. Appl. Anal. 8 (2005), no. 1, 7–38. MR 2179226
L. Mayhew and D. Smith, Using queuing theory to analyse completion times in accident and emergency departments in the light of the Government 4-hour target, (2006).
Mark M. Meerschaert, Erkan Nane, and P. Vellaisamy, The fractional Poisson process and the inverse stable subordinator, Electron. J. Probab. 16 (2011), no. 59, 1600–1620. MR 2835248, DOI 10.1214/EJP.v16-920
M. M. Meerschaert and P. Straka, Inverse stable subordinators, Math. Model. Nat. Phenom. 8 (2013), no. 2, 1–16. MR 3049524, DOI 10.1051/mmnp/20138201
Jigen Peng and Kexue Li, A note on property of the Mittag-Leffler function, J. Math. Anal. Appl. 370 (2010), no. 2, 635–638. MR 2651683, DOI 10.1016/j.jmaa.2010.04.031
Felix Pollaczek, Über eine Aufgabe der Wahrscheinlichkeitstheorie. I, Math. Z. 32 (1930), no. 1, 64–100 (German). MR 1545152, DOI 10.1007/BF01194620
Tilak Raj Prabhakar, A singular integral equation with a generalized Mittag Leffler function in the kernel, Yokohama Math. J. 19 (1971), 7–15. MR 293349
Ronald Pyke, Markov renewal processes: definitions and preliminary properties, Ann. Math. Statist. 32 (1961), 1231–1242. MR 133888, DOI 10.1214/aoms/1177704863
Enrico Scalas, Rudolf Gorenflo, Hugh Luckock, Francesco Mainardi, Maurizio Mantelli, and Marco Raberto, Anomalous waiting times in high-frequency financial data, Quant. Finance 4 (2004), no. 6, 695–702. MR 2238629, DOI 10.1080/14697680500040413
T. Radivojević, J. Anselmi, and E. Scalas, Ergodic transition in a simple model of the continuous double auction, PloS one 9 (2014), no. 2, e88095.
L. Sabatelli, S. Keating, J. Dudley, and P. Richmond, Waiting time distributions in financial markets, Eur. Phys. J. B Condens. Matter Phys. 27 (2002), no. 2, 273–275. MR 1930869, DOI 10.1140/epjb/e20020151
Stefan G. Samko, Anatoly A. Kilbas, and Oleg I. Marichev, Fractional integrals and derivatives, Gordon and Breach Science Publishers, Yverdon, 1993. Theory and applications; Edited and with a foreword by S. M. Nikol′skiĭ; Translated from the 1987 Russian original; Revised by the authors. MR 1347689
Gennady Samorodnitsky and Murad S. Taqqu, Stable non-Gaussian random processes, Stochastic Modeling, Chapman & Hall, New York, 1994. Stochastic models with infinite variance. MR 1280932
O. P. Sharma, Markovian queues, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood, New York, 1990. MR 1096304
A. Silberschatz, P. Galvin, and G. Gagne, Operating system concepts essentials, Wiley Publishing, 2013.
Hiroshi Tanaka, Stochastic differential equations with reflecting boundary condition in convex regions, Hiroshima Math. J. 9 (1979), no. 1, 163–177. MR 529332
Ward Whitt, Some useful functions for functional limit theorems, Math. Oper. Res. 5 (1980), no. 1, 67–85. MR 561155, DOI 10.1287/moor.5.1.67
Mihaela D. Iftime, Space-time distributions, Southwest J. Pure Appl. Math. 1 (1999), 23–28. MR 1717584
W. Whitt, Stochastic-process limits: an introduction to stochastic-process limits and their application to queues, Springer Science & Business Media, 2002.
