Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-28T08:23:07.645Z Has data issue: false hasContentIssue false
  • English
  • Français

Large-scale heterogeneous service systems with general packing constraints

Part of: Operations research and management science Special processes

Published online by Cambridge University Press:  17 March 2017

A. L. Stolyar
A. L. Stolyar*
Affiliation:
Lehigh University
*
* Current address: Department of Industrial and Enterprise Systems Engineering, University of Illinois at Urbana-Champaign, 104 S. Mathews Avenue, Urbana, IL 61801, USA. Email address: stolyar@illinois.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

A service system with multiple types of customers, arriving according to Poisson processes, is considered. The system is heterogeneous in that the servers can also be of multiple types. Each customer has an independent, exponentially distributed service time, with the mean determined by its type. Multiple customers (possibly of different types) can be placed for service into one server, subject to `packing' constraints, which depend on the server type. Service times of different customers are independent, even if served simultaneously by the same server. The large-scale asymptotic regime is considered such that the customer arrival rates grow to ∞. We consider two variants of the model. For the infinite-server model, we prove asymptotic optimality of the greedy random (GRAND) algorithm in the sense of minimizing the weighted (by type) number of occupied servers in steady state. (This version of GRAND generalizes that introduced by Stolyar and Zhong (2015) for homogeneous systems, with all servers of the same type.) We then introduce a natural extension of the GRAND algorithm for finite-server systems with blocking. Assuming subcritical system load, we prove existence, uniqueness, and local stability of the large-scale system equilibrium point such that no blocking occurs. This result strongly suggests a conjecture that the steady-state blocking probability under the algorithm vanishes in the large-scale limit.

Keywords

Queueing networkstochastic bin packingheterogeneous service systempacking constraintblockinglossgreedy random (GRAND) algorithmfluid limitcloud computing

MSC classification

Primary: 90B15: Network models, stochastic
Secondary: 60K25: Queueing theory
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 1 , March 2017 , pp. 61 - 83
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] Bansal, N.,Caprara, A. and Sviridenko, M. (2009).A new approximation method for set covering problems, with applications to multidimensional bin packing.SIAM J. Comput. 39,1256–1278.Google Scholar
[2] Bramson, M.,Lu, Y. and Prabhakar, B. (2012).Asymptotic independence of queues under randomized load balancing.Queueing Systems 71,247–292.CrossRefGoogle Scholar
[3] Bramson, M.,Lu, Y. and Prabhakar, B. (2013).Decay of tails at equlibrium for FIFO join the shortest queue networks.Ann. Appl. Prob. 23,1841–1878.CrossRefGoogle Scholar
[4] Csirik, J. et al. (2006).On the sum-of-squares algorithm for bin packing.J. Assoc. Comput. Mach. 53,1–65.Google Scholar
[5] Ghaderi, J.,Zhong, Y. and Srikant, R. (2014).Asymptotic optimality of bestfit for stochastic bin packing.Performance Evaluation Rev. 42,64–66.Google Scholar
[6] Gulati, A. et al. (2012).VMware distributed resource management: design, implementation, and lessons learned.VMware Tech. J. 1,45–64.Google Scholar
[7] Guo, Y.,Stolyar, A. L. and Walid, A. (2013).Shadow-routing based dynamic algorithms for virtual machine placement in a network cloud.In Proc. IEEE INFOCOM 2013,IEEE,pp.620–628.Google Scholar
[8] Gupta, V. and Radovanovic, A. (2012).Online stochastic bin packing.Preprint. Available at http://arxiv.org/abs/1211.2687v1.Google Scholar
[9] Maguluri, S. T. and Srikant, R. (2013).Scheduling jobs with unknown duration in clouds.IEEE/ACM Trans. Networking 22,1938–1951.Google Scholar
[10] Maguluri, S. T.,Srikant, R. and Ying, L. (2012).Stochastic models of load balancing and scheduling in cloud computing clusters.In Proc. IEEE INFOCOM 2012,IEEE,pp.702–710.Google Scholar
[11] Mitzenmacher, M. (2001).The power of two choices in randomized load balancing.IEEE Trans. Parallel Distrib. Systems 12,1094–1104.Google Scholar
[12] Stolyar, A. L. (2013).An infinite server system with general packing constraints.Operat. Res. 61,1200–1217.CrossRefGoogle Scholar
[13] Stolyar, A. L. (2015).Pull-based load distribution in large-scale heterogeneous service systems.Queueing Systems 80,341–361.Google Scholar
[14] Stolyar, A. L. and Zhong, Y. (2013).A large-scale service system with packing constraints: minimizing the number of occupied servers.In Proc. SIGMETRICS `13,Association for Computing Machinery,New York,pp.41–52.Google Scholar
[15] Stolyar, A. L. and Zhong, Y. (2015).Asymptotic optimality of a greedy randomized algorithm in a large-scale service system with general packing constraints.Queueing Systems 79,117–143.Google Scholar
[16] Stolyar, A. L. and Zhong, Y. (2015).A service system with packing constraints: greedy randomized algorithm achieving sublinear in scale optimality gap.Preprint. Available at https://arxiv.org/abs/1511.03241v1.Google Scholar
[17] Vvedenskaya, N.,Dobrushin, R. and Karpelevich, F. (1996).Queueing system with selection of the shortest of two queues: an asymptotic approach.Probl. Peredachi Inf. 32,20–34 (in Russian).English translation: Probl. Inf. Trans. 32,15–27.Google Scholar
[18] Xie, Q.,Dong, X.,Lu, Y. and Srikant, R. (2015).Power of d choices for large-scale bin packing: a loss model.Performance Evaluation Rev. 43,321–334.Google Scholar