Delay, Memory, and Messaging Tradeoffs in Distributed Service Systems

Abstract

We consider the following distributed service model: jobs with unit mean, exponentially distributed, and independent processing times arrive as a Poisson process of rate λn, with 0 < λ < 1, and are immediately dispatched by a centralized dispatcher to one of n First-In-First-Out queues associated with n identical servers. The dispatcher is endowed with a finite memory, and with the ability to exchange messages with the servers.

We propose and study a resource-constrained “pull-based” dispatching policy that involves two parameters: (i) the number of memory bits available at the dispatcher, and (ii) the average rate at which servers communicate with the dispatcher. We establish (using a fluid limit approach) that the asymptotic, as n → ∞, expected queueing delay is zero when either (i) the number of memory bits grows logarithmically with n and the message rate grows superlinearly with n, or (ii) the number of memory bits grows superlogarithmically with n and the message rate is at least λn. Furthermore, when the number of memory bits grows only logarithmically with n and the message rate is proportional to n, we obtain a closed-form expression for the (now positive) asymptotic delay.

Finally, we demonstrate an interesting phase transition in the resource-constrained regime where the asymptotic delay is non-zero. In particular, we show that for any given α > 0 (no matter how small), if our policy only uses a linear message rate αn, the resulting asymptotic delay is upper bounded, uniformly over all λ < 1; this is in sharp contrast to the delay obtained when no messages are used (α = 0), which grows as 1/(1 − λ) when λ ↑ 1, or when the popular power-of-d-choices is used, in which the delay grows as log(1/(1 − λ)).