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Bin Packing with Queues

Part of: Operations research and management science

Published online by Cambridge University Press:  14 July 2016

Devavrat Shah  and
John N. Tsitsiklis
Devavrat Shah*
Affiliation:
Laboratory for Information and Decision Systems, MIT
John N. Tsitsiklis*
Affiliation:
Laboratory for Information and Decision Systems, MIT
*
∗Postal address: Laboratory for Information and Decision Systems, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139, USA.
∗Postal address: Laboratory for Information and Decision Systems, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139, USA.
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Abstract

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We study the best achievable performance (in terms of the average queue size and delay) in a stochastic and dynamic version of the bin-packing problem. Items arrive to a queue according to a Poisson process with rate 2ρ, where ρ ∈ (0, 1). The item sizes are independent and identically distributed (i.i.d.) with a uniform distribution in [0, 1]. At each time unit, a single unit-size bin is available and can receive any of the queued items, as long as their total size does not exceed 1. Coffman and Stolyar (1999) and Gamarnik (2004) have established that there exist packing policies under which the average queue size is finite for every ρ ∈ (0, 1). In this paper we study the precise scaling of the average queue size, as a function of ρ, with emphasis on the critical regime where ρ approaches 1. Standard results on the probabilistic (but static) bin-packing problem can be readily applied to produce policies under which the queue size scales as O(h2), where h = 1 / (1 - ρ), which raises the question of whether this is the best possible. We establish that the average queue size scales as Ω(hlogh), under any policy. Furthermore, we provide an easily implementable policy, which packs at most two items per bin. Under that policy, the average queue size scales as O(hlog3/2h), which is nearly optimal. On the other hand, if we impose the additional requirement that any two items packed together must have near-complementary sizes (in a sense to be made precise), we show that the average queue size must scale as Θ(h2).

Keywords

Bin packingqueueingheavy traffic

MSC classification

Secondary: 90B36: Scheduling theory, stochastic 90B22: Queues and service
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 4 , December 2008 , pp. 922 - 939
Copyright
Copyright © Applied Probability Trust 2008 

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