On the maximum queue length in the supermarket model

Luczak, Malwina J.; McDiarmid, Colin

doi:10.1214/00911790500000710

Cited by 75 publications

(145 citation statements)

References 15 publications

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“…Its path space evolution was studied by Graham [11] who moreover showed that, starting from independent initial states, as N → ∞, the queues of the limiting process evolve independently. Luczak and McDiarmid [15] showed that the length of the longest queue scales as (log log N )/ log D + O (1). Certain generalizations have also been explored.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic independence of queues under randomized load balancing

2012

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Randomized load balancing greatly improves the sharing of resources while being simple to implement. In one such model, jobs arrive according to a rate-αN Poisson process, with α < 1, in a system of N rate-1 exponential server queues. In Vvedenskaya et al. [19], it was shown that when each arriving job is assigned to the shortest of D, D ≥ 2, randomly chosen queues, the equilibrium queue sizes decay doubly exponentially in the limit as N → ∞. This is a substantial improvement over the case D = 1, where queue sizes decay exponentially.The reasoning in [19] does not easily generalize to jobs with nonexponential service time distributions. A modularized program for treating randomized load balancing problems with general service time distributions was introduced in Bramson et al. [5]. The program relies on an ansatz that asserts that, for a randomized load balancing scheme in equilibrium, any fixed number of queues become independent of one another as N → ∞. This allows computation of queue size distributions and other performance measures of interest.In this article, we demonstrate the ansatz in several settings. We consider the least loaded balancing problem, where an arriving job is assigned to the queue with the smallest workload. We also consider the more difficult prob- lem, where an arriving job is assigned to the queue with the fewest jobs, and demonstrate the ansatz when the service discipline is FIFO and the service time distribution has a decreasing hazard rate. Last, we show the ansatz always holds for a sufficiently small arrival rate, as long as the service distribution has 2 moments.

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Section: Introductionmentioning

confidence: 99%

Asymptotic independence of queues under randomized load balancing

2012

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“…In order to reduce the feedback cost, and yet to keep the delay 'small', JSQ has been generalized to SQ(d), whereby the dispatcher runs JSQ only for a subset of d randomly sampled servers from the uniform distribution (see Mitzenmacher [5] and Luczak and McDiarid [3]). Note that SQ(d) reduces to a simple uniform random selection when d = 1, and to JSQ when d = N , where N is the total number of servers.…”

Section: Introductionmentioning

confidence: 99%

Stochastic bounds for randomized load balancing

Godtschalk

Ciucu

2012

SIGMETRICS Perform. Eval. Rev.

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Randomized load balancing is a cost efficient policy for job scheduling in parallel server queueing systems whereby, with every incoming job, a central dispatcher randomly polls some servers and selects the one with the smallest queue. By exactly deriving the jobs' delay distribution in such systems, in explicit and closed form, Mitzenmacher [5] proved the socalled 'power-of-two' result, which states that by randomly polling only two servers yields an exponential improvement in delay over randomly selecting a single server. Such a fundamental result, however, was obtained in an asymptotic regime in the total number of servers, and does do not necessarily provide accurate estimates for practical finite regimes with small or moderate number of servers. In this paper we obtain stochastic lower and upper bounds on the jobs' average delay in non-asymptotic regimes, by borrowing ideas for analyzing the particular case of the Join-the-ShortestQueue (JSQ) policy. Numerical illustrations indicate not only that the obtained bounds are remarkably accurate, but also that the existing exact but asymptotic results can be largely misleading in some finite regimes.

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“…If there is more than one chosen server with a shortest queue, then the customer goes to the first such queue in her list of d. Service times are independent unit mean exponentials, and customers are served according to the first-come first-served discipline. Recent work on the supermarket model includes [5,6,7,14,16,17,25]. The survey [22] gives several applications and related results.…”

Section: Introductionmentioning

confidence: 99%

“…Further, again for a fixed positive integer k 0 , as n tends to infinity, in the equilibrium distribution the proportion of queues with length at least k 0 converges in probability to λ (d k 0 −1)/(d−1) , and thus the probability that a given queue has length at least k 0 also converges to λ (d k 0 −1)/(d−1) . Recent results in [16] include rapid mixing and two-point concentration for the maximum queue length in equilibrium. The main contribution of the present paper is to give quantitative versions of the convergence results for the supermarket model mentioned above, and to extend them to hold uniformly over all times.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic distributions and chaos for the supermarket model

Luczak

McDiarmid

2007

Electron. J. Probab.

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In the supermarket model there are n queues, each with a unit rate server. Customers arrive in a Poisson process at rate λn, where 0 < λ < 1. Each customer chooses d ≥ 2 queues uniformly at random, and joins a shortest one.It is known that the equilibrium distribution of a typical queue length converges to a certain explicit limiting distribution as n → ∞. We quantify the rate of convergence by showing that the total variation distance between the equilibrium distribution and the limiting distribution is essentially of order n −1 ; and we give a corresponding result for systems starting from quite general initial conditions (not in equilibrium). Further, we quantify the result that the systems exhibit chaotic behaviour: we show that the total variation distance between the joint law of a fixed set of queue lengths and the corresponding product law is essentially of order at most n −1 .

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On the maximum queue length in the supermarket model

Cited by 75 publications

References 15 publications

Asymptotic independence of queues under randomized load balancing

Asymptotic independence of queues under randomized load balancing

Stochastic bounds for randomized load balancing

Asymptotic distributions and chaos for the supermarket model

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