Hostname: page-component-89b8bd64d-n8gtw Total loading time: 0 Render date: 2026-05-08T06:49:43.784Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite-pool queueing with heavy-tailed services

Part of: Computer system organization Special processes Operations research and management science

Published online by Cambridge University Press:  15 September 2017

Gianmarco Bet ,
Remco van der Hofstad  and
Johan S. H. van Leeuwaarden
Gianmarco Bet*
Affiliation:
Eindhoven University of Technology
Remco van der Hofstad*
Affiliation:
Eindhoven University of Technology
Johan S. H. van Leeuwaarden*
Affiliation:
Eindhoven University of Technology
*
* Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands.
* Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands.
* Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the Δ(i)/G/1 queue, in which a total of n customers join a single-server queue for service. Customers join the queue independently after exponential times. We consider heavy-tailed service-time distributions with tails decaying as x -α, α ∈ (1, 2). We consider the asymptotic regime in which the population size grows to ∞ and establish that the scaled queue-length process converges to an α-stable process with a negative quadratic drift. We leverage this asymptotic result to characterize the head start that is needed to create a long period of uninterrupted activity (a busy period). The heavy-tailed service times should be contrasted with the case of light-tailed service times, for which a similar scaling limit arises (Bet et al. (2015)), but then with a Brownian motion instead of an α-stable process.

Keywords

Heavy-traffic approximationheavy-tailed distributionfunctional central limit theoremSkorokhod reflection map

MSC classification

Primary: 60K25: Queueing theory
Secondary: 90B22: Queues and service 68M20: Performance evaluation; queueing; scheduling

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 3 , September 2017 , pp. 921 - 942
Copyright
Copyright © Applied Probability Trust 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1] Aïdékon, E., van der Hofstad, R., Kliem, S. and van Leeuwaarden, J. S. H. (2016). Large deviations for power-law thinned Lévy processes. Stoch. Process. Appl. 126, 1353–1384. CrossRefGoogle Scholar
[2] Aldous, D. (1997). Brownian excursions, critical random graphs and the multiplicative coalescent. Ann. Prob. 25, 812–854. CrossRefGoogle Scholar
[3] Asmussen, S. (2003). Applied Probability and Queues, 2nd edn. Springer, New York. Google Scholar
[4] Bet, G., van der Hofstad, R. and van Leeuwaarden, J. S. H. (2015). Heavy-traffic analysis through uniform acceleration of queues with diminishing populations. Preprint. Available at https://arxiv.org/abs/1412.5329v2. Google Scholar
[5] Bet, G., van der Hofstad, R. and van Leeuwaarden, J. S. H. (2017). Big jobs arrive early: from critical queues to random graphs. Preprint. Available at https://arxiv.org/abs/1704.03406. Google Scholar
[6] Bhamidi, S., van der Hofstad, R. and van Leeuwaarden, J. S. H. (2012). Novel scaling limits for critical inhomogeneous random graphs. Ann. Prob. 40, 2299–2361. CrossRefGoogle Scholar
[7] Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York. CrossRefGoogle Scholar
[8] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press. CrossRefGoogle Scholar
[9] Boxma, O. J. and Cohen, J. W. (1998). The M/G/1 queue with heavy-tailed service time distribution. IEEE J. Selected Areas Commun. 16, 749–763. CrossRefGoogle Scholar
[10] David, H. A. and Nagaraja, H. N. (2003). Order Statistics, 3rd edn. John Wiley, Hoboken, NJ. CrossRefGoogle Scholar
[11] Dhara, S., van der Hofstad, R., van Leeuwaarden, J. S. H. and Sen, S. (2016). Heavy-tailed configuration models at criticality. Preprint. Available at https://arxiv.org/abs/1612.00650. Google Scholar
[12] Dhara, S., van der Hofstad, R., van Leeuwaarden, J. S. H. and Sen, S. (2017). Critical window for the configuration model: finite third moment degrees. Electron. J. Prob. 22, 16. CrossRefGoogle Scholar
[13] Groeneboom, P. (1989). Brownian motion with a parabolic drift and Airy functions. Prob. Theory Relat. Fields 81, 79–109. CrossRefGoogle Scholar
[14] Groeneboom, P. (2010). The maximum of Brownian motion minus a parabola. Electron. J. Prob. 15, 1930–1937. CrossRefGoogle Scholar
[15] Honnappa, H., Jain, R. and Ward, A. R. (2014). On transitory queueing. Preprint. Available at https:// arxiv.org/abs/1412.2321. Google Scholar
[16] Honnappa, H., Jain, R. and Ward, A. R. (2015). A queueing model with independent arrivals, and its fluid and diffusion limits. Queueing Systems 80, 71–103. CrossRefGoogle Scholar
[17] Iglehart, D. L. and Whitt, W. (1970). Multiple channel queues in heavy traffic. I. Adv. Appl. Prob. 2, 150–177. CrossRefGoogle Scholar
[18] Iglehart, D. L. and Whitt, W. (1970). Multiple channel queues in heavy traffic. II. Sequences, networks, and batches. Adv. Appl. Prob. 2, 355–369. CrossRefGoogle Scholar
[19] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin. CrossRefGoogle Scholar
[20] Janson, S., Louchard, G. and Martin-Löf, A. (2010). The maximum of Brownian motion with parabolic drift. Electron. J. Prob. 15, 1893–1929. CrossRefGoogle Scholar
[21] Joseph, A. (2014). The component sizes of a critical random graph with given degree sequence. Ann. Appl. Prob. 24, 2560–2594. CrossRefGoogle Scholar
[22] Klenke, A. (2008). Probability Theory: A Comprehensive Course. Springer, London. CrossRefGoogle Scholar
[23] Louchard, G. (1994). Large finite population queueing systems. The single-server model. Stoch. Process. Appl. 53, 117–145. CrossRefGoogle Scholar
[24] Mandelbaum, A. and Massey, W. A. (1995). Strong approximations for time-dependent queues. Math. Operat. Res. 20, 33–64. CrossRefGoogle Scholar
[25] Massey, W. A. (1981). Non-stationary queues. Doctoral thesis. Stanford University. Google Scholar
[26] Massey, W. A. (1985). Asymptotic analysis of the time dependent M/M/1 queue. Math. Operat. Res. 10, 305–327. CrossRefGoogle Scholar
[27] Newell, G. F. (1968). Queues with time-dependent arrival rates. III. A mild rush hour. J. Appl. Prob. 5, 591–606. CrossRefGoogle Scholar
[28] Pittel, B. (2001). On the largest component of the random graph at a nearcritical stage. J. Combinatorial Theory B 82, 237–269. CrossRefGoogle Scholar
[29] Roberts, M. I. (2016). The probability of unusually large components in the near-critical Erdős-Rényi graph. Preprint. Available at https://arxiv.org/abs/1610.05485. Google Scholar
[30] Roberts, M. I. and Sengul, B. (2017). Exceptional times of the critical dynamical Erdős-Rényi graph. Preprint. Available at https://arxiv.org/abs/1610.06000. Google Scholar
[31] Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman & Hall, New York. Google Scholar
[32] Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press. Google Scholar
[33] Simon, T. (2011). Hitting densities for spectrally positive stable processes. Stochastics 83, 203–214. CrossRefGoogle Scholar
[34] Skorokhod, A. V. (1956). Limit theorems for stochastic processes. Theory Prob. Appli. 1, 261–290. CrossRefGoogle Scholar
[35] Van der Hofstad, R., Janssen, A. J. E. M. and van Leeuwaarden, J. S. H. (2010). Critical epidemics, random graphs, and Brownian motion with a parabolic drift. Adv. Appl. Prob. 42, 1187–1206. CrossRefGoogle Scholar
[36] Van der Hofstad, R., Kliem, S. and van Leeuwaarden, J. S. H. (2014). Cluster tails for critical power-law inhomogeneous random graphs. Preprint. Available at https://arxiv.org/abs/1404.1727. Google Scholar
[37] Van der Hofstad, R., van Leeuwaarden, J. S. H. and Stegehuis, C. (2016). Mesoscopic scales in hierarchical configuration models. Preprint. Available at https://arxiv.org/abs/1612.02668. Google Scholar
[38] Whitt, W. (2002). Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer, New York. CrossRefGoogle Scholar
[39] Whitt, W. (2016). Heavy-traffic limits for a single-server queue leading up to a critical point. Operat. Res. Lett. 44, 796–800. CrossRefGoogle Scholar
[40] Whitt, W. and You, W. (2017). Time-varying robust queueing. Submitted. Google Scholar