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[PhD Thesis] An Algebraic Analysis of Storage Fragmentation
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Dr Terry Betteridge
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Storage fragmentation, the splitting of available computer memory space into separate gaps by allocations and deallocations of various sized blocks with consequent loss of utilisation due to reduced ability to satisfy requests, has proved 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 schedule of a kind common in many computing systems is described. Requests from a saturated first come first served queue for varying 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 available space becomes fragmented because allocated requests are not relocated or moved around in storage. The model's behaviour and in particular 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 striking 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 the location is allowed.
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