Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T21:17:19.488Z Has data issue: false hasContentIssue false
  • English
  • Français

Large-scale join-idle-queue system with general service times

Part of: Special processes Operations research and management science

Published online by Cambridge University Press:  30 November 2017

S. Foss  and
A. L. Stolyar
S. Foss*
Affiliation:
Heriot-Watt University and Novosibirsk State University
A. L. Stolyar*
Affiliation:
University of Illinois at Urbana-Champaign
*
* Postal address: Department of Actuarial Mathematics and Statistics, Heriot-Watt University, Edinburgh EH14 4AS, UK. Email address: [email protected]
** Postal address: Department of Industrial and Enterprise Systems Engineering, University of Illinois at Urbana-Champaign, 104 S. Mathews Avenue, Office 201C, Urbana, IL 61801, USA. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A parallel server system with n identical servers is considered. The service time distribution has a finite mean 1 / μ, but otherwise is arbitrary. Arriving customers are routed to one of the servers immediately upon arrival. The join-idle-queue routeing algorithm is studied, under which an arriving customer is sent to an idle server, if such is available, and to a randomly uniformly chosen server, otherwise. We consider the asymptotic regime where n → ∞ and the customer input flow rate is λn. Under the condition λ / μ < ½, we prove that, as n → ∞, the sequence of (appropriately scaled) stationary distributions concentrates at the natural equilibrium point, with the fraction of occupied servers being constant at λ / μ. In particular, this implies that the steady-state probability of an arriving customer waiting for service vanishes.

Keywords

Large-scale service systempull-based load distributionjoin-idle-queueload balancingfluid limitstationary distributionasymptotic optimality

MSC classification

Primary: 90B15: Network models, stochastic
Secondary: 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 4 , December 2017 , pp. 995 - 1007
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.)

References

[1] Badonnel, R. and Burgess, M. (2008). Dynamic pull-based load balancing for autonomic servers. In Network Operations and Management Symposium, NOMS 2008, IEEE, pp. 751–754. Google Scholar
[2] Billingsley, P. (1995). Probability and Measure, 3rd edn. John Wiley, New York. Google Scholar
[3] Bramson, M., Lu, Y. and Prabhakar, B. (2012). Asymptotic independence of queues under randomized load balancing. Queueing Systems 71, 247–292. Google Scholar
[4] Bramson, M., Lu, Y. and Prabhakar, B. (2013). Decay of tails at equilibrium for FIFO join the shortest queue networks. Ann. Appl. Prob. 23, 1841–1878. Google Scholar
[5] Eschenfeldt, P. and Gamarnik, D. (2015). Join the shortest queue with many servers. The heavy traffic asymptotics. Preprint. Available at https://arxiv.org/abs/1502.00999. Google Scholar
[6] Lu, Y. et al. (2011). Join-idle-queue: a novel load balancing algorithm for dynamically scalable web services. Performance Evaluation 68, 1056–1071. Google Scholar
[7] Mitzenmacher, M. (2001). The power of two choices in randomized load balancing. IEEE Trans. Parallel Distributed Systems 12, 1094–1104. Google Scholar
[8] Mukherjee, D., Borst, S. C., van Leeuwaarden, J. S. H. and Whiting, P. A. (2016). Universality of load balancing schemes on the diffusion scale. J. Appl. Prob. 53, 1111–1124. Google Scholar
[9] Stolyar, A. L. (2015). Pull-based load distribution in large-scale heterogeneous service systems. Queueing Systems 80, 341–361. Google Scholar
[10] Stolyar, A. L. (2017). Large-scale heterogeneous service systems with general packing constraints. Adv. Appl. Prob. 49, 61–83. CrossRefGoogle Scholar
[11] Stolyar, A. L. (2017). Pull-based load distribution among heterogeneous parallel servers: the case of multiple routers. Queueing Systems 85, 31–65. Google Scholar
[12] Vvedenskaya, N. D., Dobrushin, R. L. and Karpelevich, F. I. (1996). A queueing system with a choice of the shorter of two queues—an asymptotic approach. Problems Information Transmission 32, 15–27. Google Scholar