Delay, Memory, and Messaging Tradeoffs in Distributed Service Systems

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 associa...

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Bibliographic Details
Main Authors: Gamarnik, David (Contributor), Tsitsiklis, John N. (Author), Zubeldia, Martin (Author)
Other Authors: Massachusetts Institute of Technology. Laboratory for Information and Decision Systems (Contributor), Sloan School of Management (Contributor)
Format: Article
Language:English
Published: Institute for Operations Research and the Management Sciences (INFORMS), 2019-03-04T20:28:40Z.
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Summary: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)).
National Science Foundation (U.S.) (Grant CMMI-1234062)