The Number of States of a Dynamic Storage Allocation System

Abstract

In the standard model for dynamic storage allocation with immovable blocks, a state is a partition of an arena of length n into blocks of positive integral lengths, where each block is marked either busy or idle. Betteridge and Benes considered the state model in analysing the behavior of allocation policies under Poisson inputs and were able to get exact results for small arenas; Robson's minimax results provide qualitative insight about the effectiveness of certain allocation policies in large arenas. Knuth and Reeves have made preliminary progress in the statistical mechanics of large arenas. This paper considers the sizes of various pertinent state spaces. It has proved convenient to study circular arenas, where all cells are inherently similar. Indeed, the artificial construct of a small circular arena may, by avoiding end effects, model the behavior of a large arena better than would a similarly small linear arena. We may reduce the size of models still further by assuming homogeneity and treating circular states that are rotated images of each other as identical. In the same way, Benes profitably combined mirror-image states of linear arenas. Thus it will be useful to understand equivalence classes of states under various automorphism groups— reflection, rotation, or both. The internal layout of a contiguous run of idle blocks does not matter; for convenience we shall let all idle blocks have length 1. In mathematical terms, a state is an ordered partition of n in which two species of 1 may occur. States invariant under an automorphism—in particular reflection—will be called symmetric. States not known to be symmetric under any but the identity automorphism will be called unrestricted. In making proofs, it is helpful to recognize certain special states called closed. The closed states of a linear arena are those states that have a busy block at a given end. In a circular arena, some cell boundary may be chosen as a cut or origin for numbering the cells. Then the closed states of the circular arena are those states in which a block spans the cut. The spanning block, which must be busy, is called the cut block. Any state of an arena of length n will be called an n-state for short, and a block of length k will be called a k-block. The term n-state may be further qualified as linear or circular according to the kind of arena. A rotation of a circular arena throughy cell positions will be I I l | l 2