Abstract
We consider the following stochastic bin packing process: the items arrive continuously over time to a server and are packed into bins of unit size according to an online algorithm. The unpacked items form a queue. The items have random sizes with symmetric distribution. Our first contribution identifies some monotonicity properties of the queueing system that allow to derive bounds on the queue size for First Fit and Best Fit algorithms. As a direct application, we show how to compute the stability region under very general conditions on the input process. Our second contribution is a study of the queueing system under heavy load. We show how the monotonicity properties allow one to derive bounds for the speed at which the stationary queue length tends to infinity when the load approaches one. In the case of Best Fit, these bounds are tight. Our analysis shows connections between our dynamic model, average-case results on the classical bin packing problem and planar matching problems.
Highlights
Consider the following bin packing process where items arrive continuously over time to a server and have random sizes in [0, 1]
The server contains an infinite number of bins of unit capacity that leaves the system at integer time t = 0, 1, . . . Bin number t leaves the system at time t + 1 with items in the queue that arrived in the system before time t + 1, i.e. the t-th bin is packed with items present in the queue at time t and any item that arrive in the t-th slot
This method is generic in the sense that it does not depend on the stochastic assumptions and is valid for both First Fit and Best Fit
Summary
Consider the following bin packing process where items arrive continuously over time to a server and have random sizes in [0, 1]. Our first contribution gives a method to compute an upper bound on the queue size of the queueing system This method is generic in the sense that it does not depend on the stochastic assumptions and is valid for both First Fit and Best Fit. As an application, we show that it allows to derive the stability region for symmetric distribution (discrete or continuous) for stationary and ergodic inputs: the average load being less than one is a necessary and sufficient condition for the stability of algorithms Best Fit and First Fit. To the best of our knowledge this result was only known for First Fit and for discrete distributions of the items. It allows to compare the performance of algorithms which have the same stability region
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