A Stochastic Model of Fragmentation in Dynamic Storage Allocation

Abstract

We study a model of dynamic storage allocation in which requests for single units of memory arrive in a Poisson stream at rate $\lambda $ and are accommodated by the first available location found in a linear scan of memory. Immediately after this first-fit assignment, an occupied location commences an exponential delay with rate parameter $\mu $, after which the location again becomes available. The set of occupied locations (identified by their numbers) at time t forms a random subset $S_t $ of $\{ 1,2, \cdots \}$. The extent of the fragmentation in $S_t $, i.e. the alternating holes and occupied regions of memory, is measured by $\max (S_t ) - |S_t |$. In equilibrium, the number of occupied locations, $|S|$, is known to be Poisson distributed with mean $\rho = \lambda/ \mu$. We obtain an explicit formula for the stationary distribution of max $(S)$, the last occupied location, and by independent arguments we show that $(E\max (S) - E|S|) /E|S| \to 0$ as the traffic intensity $\rho \to \infty $. Moreover, we verify numerically that for any $\rho $ the expected number of wasted locations in equilibrium is never more than $\frac{1}{3}$ the expected number of occupied locations. Our model applies to studies of fragmentation in paged computer systems, and to containerization problems in industrial storage applications. Finally, our model can be regarded as a simple concrete model of interacting particles [Adv. Math., 5 (1970), pp. 246–290].

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