Abstract
We consider a system of [Formula: see text] parallel single-server queues with unit exponential service rates and a single dispatcher where tasks arrive as a Poisson process of rate [Formula: see text]. When a task arrives, the dispatcher assigns it to a server with the shortest queue among [Formula: see text] randomly selected servers ([Formula: see text]). This load balancing strategy is referred to as a JSQ([Formula: see text]) scheme, noting that it subsumes the celebrated Join-the-Shortest Queue (JSQ) policy as a crucial special case for [Formula: see text]. We construct a stochastic coupling to bound the difference in the queue length processes between the JSQ policy and a JSQ([Formula: see text]) scheme with an arbitrary value of [Formula: see text]. We use the coupling to derive the fluid limit in the regime where [Formula: see text] as [Formula: see text] with [Formula: see text], along with the associated fixed point. The fluid limit turns out not to depend on the exact growth rate of [Formula: see text] and in particular coincides with that for the JSQ policy. We further leverage the coupling to establish that the diffusion limit in the critical regime where [Formula: see text] as [Formula: see text] with [Formula: see text] corresponds to that for the JSQ policy. These results indicate that the optimality of the JSQ policy can be preserved at the fluid level and diffusion level while reducing the overhead by nearly a factor O([Formula: see text]) and O([Formula: see text]), respectively.
Highlights
In this paper, we establish a universality property for a broad class of randomized load balancing schemes in manyserver systems
As mentioned in Subsection 2.4, the fluid limit for the ordinary Join-the-Shortest Queue (JSQ) policy is provided in Subsection 4.1, and in Subsection 4.2 we prove a universality result establishing that under the condition that d(N) → ∞ as N → ∞, the fluid limit for the JSQ(d(N)) scheme coincides with that for the ordinary JSQ policy
We developed a novel stochastic coupling construction to bound the difference in the queue length processes between the JSQ policy (d N) and a scheme with an arbitrary value of d
Summary
We establish a universality property for a broad class of randomized load balancing schemes in manyserver systems. We focus on a basic scenario of sending tasks from a single dispatcher to N parallel queues with identical servers, exponentially distributed service requirements, and a service discipline at each individual server that is oblivious to the actual service requirements (e.g., first-come first-served) In this canonical case, the socalled Join-the-Shortest-Queue (JSQ) policy has several strong optimality properties and, in particular, minimizes the overall mean delay among the class of nonanticipating load balancing policies that do not have any advance knowledge of the service requirements (Winston 1977, Weber 1978, Ephremides et al 1980). As a result, obtaining a stochastic coupling bound in this paper becomes analytically more challenging, and in contrast with the infiniteserver scenario, involves the cumulative loss terms and tail sums of the occupancy states of the ordinary JSQ policy.
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