An algebraic analysis of storage fragmentation

Abstract

Storage fragmentation, the splitting of available computer memory space into separate gaps by allocations and deal locations of various sized blocks with consequent loss of utilisation due to reduced ability to satisfy reque~ts, has ~roved difficult to analyse. Most previous studies rely on simulation, and nearly all of the few published analyses that do not, simplify the combinatorial complexity that arises by some averaging assumption. After a survey of these results, an exact analytical approach to the study of storage allocation and fragmentation is presented. A model of an allocation scheme of a kind common in many computing systems is described. Requests from a saturated fi rst come fi rst served queue for varyi ng amounts of contiguous storage are satisfied as soon as sufficient space becomes available in a storage memory of fixed total size. A placement algorithm decides which free locations to allocate if a choice is possible. After a variable time, allocated requests are completed and their occupied storage is freed again. In general, the avail ab 1 e space becomes fragmented because allocated requests are not relocated ~r moved around in stora~e. The model's behaviour and in particul~r the storage utilisation are studied under conditions in which the model is a finite homogeneous Markov chain. The algebraic structure of its sparse transition matrix is discovered to have a striki~g recursive pattern, allowing the steady state equation to be simplified considerably and unexpectedly to a simple and direct statement of the effect of the choice of placement algorithm on the steady state. Possible developments and uses of this simplified analysis are indicated, and some investigated. The exact probabilistic behaviour of models of relatively small memory sizes is computed, and different placement algorithms are compared with each other and with the analytic results which are derived for the corresponding model in which relocation is allowed.