The Annals of Probability
- Ann. Probab.
- Volume 34, Number 2 (2006), 493-527.
On the maximum queue length in the supermarket model
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).
Ann. Probab., Volume 34, Number 2 (2006), 493-527.
First available in Project Euclid: 9 May 2006
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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]
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