The Annals of Probability

On the maximum queue length in the supermarket model

Malwina J. Luczak and Colin McDiarmid

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Abstract

There are n queues, each with a single server. Customers arrive in a Poisson process at rate λn, where 0<λ<1. Upon arrival each customer selects d≥2 servers uniformly at random, and joins the queue at a least-loaded server among those chosen. Service times are independent exponentially distributed random variables with mean 1. We show that the system is rapidly mixing, and then investigate the maximum length of a queue in the equilibrium distribution. We prove that with probability tending to 1 as n→∞ the maximum queue length takes at most two values, which are lnlnn/lnd+O(1).

Article information

Source
Ann. Probab., Volume 34, Number 2 (2006), 493-527.

Dates
First available in Project Euclid: 9 May 2006

Permanent link to this document
https://projecteuclid.org/euclid.aop/1147179980

Digital Object Identifier
doi:10.1214/00911790500000710

Mathematical Reviews number (MathSciNet)
MR2223949

Zentralblatt MATH identifier
1102.60083

Subjects
Primary: 60C05: Combinatorial probability
Secondary: 68R05: Combinatorics 90B22: Queues and service [See also 60K25, 68M20] 60K25: Queueing theory [See also 68M20, 90B22] 60K30: Applications (congestion, allocation, storage, traffic, etc.) [See also 90Bxx] 68M20: Performance evaluation; queueing; scheduling [See also 60K25, 90Bxx]

Keywords
Supermarket model join the shortest queue random choices power of two choices maximum queue length load balancing equilibrium concentration of measure

Citation

Luczak, Malwina J.; McDiarmid, Colin. On the maximum queue length in the supermarket model. Ann. Probab. 34 (2006), no. 2, 493--527. doi:10.1214/00911790500000710. https://projecteuclid.org/euclid.aop/1147179980


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References

  • Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • Graham, C. (2000). Kinetic limits for large communication networks. In Modelling in Applied Sciences (N. Bellomo and M. Pulvirenti, eds.) 317--370. Birkhäuser, Boston.
  • Graham, C. (2000). Chaoticity on path space for a queueing network with selection of the shortest queue among several. J. Appl. Probab. 37 198--201.
  • Graham, C. (2004). Functional central limit theorems for a large network in which customers join the shortest of several queues. Probab. Theory Related Fields 131 97--120.
  • Grimmett, G. R. and Stirzaker, D. R. (2001). Probability and Random Processes, 3rd ed. Oxford Univ. Press.
  • Luczak, M. J. (2003). A quantitative law of large numbers via exponential martingales. In Stochastic Inequalities and Applications (E. Giné, C. Houdré and D. Nualart, eds.) 93--111. Birkhäuser, Basel.
  • Luczak, M. J. and McDiarmid, C. (2005). On the power of two choices: Balls and bins in continuous time. Ann. Appl. Probab. 15 1733--1764.
  • Luczak, M. J. and McDiarmid, C. (2005). Asymptotic distributions and chaos for the supermarket model. Unpublished manuscript.
  • Luczak, M. J. and McDiarmid, C. (2005). Balls and bins in continuous time: Long-term asymptotics and chaos. Unpublished manuscript.
  • Luczak, M. J. and Norris, J. R. (2004). Strong approximation for the supermarket model. Ann. Appl. Probab. 15 2038--2061.
  • Martin, J. B. and Suhov, Y. M. (1999). Fast Jackson networks. Ann. Appl. Probab. 9 854--870.
  • McDiarmid, C. (1998). Concentration. In Probabilistic Methods for Algorithmic Discrete Mathematics (M. Habib, C. McDiarmid, J. Ramirez and B. Reed, eds.) 195--248. Springer, Berlin.
  • Mitzenmacher, M. (1996). Load balancing and density dependent jump Markov processes. In Proc. 37th Ann. Symp. Found. Comp. Sci. 213--222. IEEE Comput. Soc. Press, Los Alamitos, CA.
  • Mitzenmacher, M. (1996). The power of two choices in randomized load-balancing. Ph.D. dissertation, Berkeley.
  • Mitzenmacher, M., Richa, A. W. and Sitaraman, R. (2001). The power of two random choices: A survey of techniques and results. In Handbook of Randomized Computing (S. Rajasekaran, P. M. Pardalos, J. H. Reif and J. D. P. Rolim, eds.) 1 255--312. Kluwer, Dordrecht.
  • Turner, S. R. E. (1998). The effect of increasing routing choice on resource pooling. Probab. Engrg. Inform. Sci. 12 109--124.
  • Vvedenskaya, N. D., Dobrushin, R. L. and Karpelevich, F. I. (1996). Queueing system with selection of the shortest of two queues: An asymptotic approach. Problems Inform. Transmission 32 15--27.