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Stochastic Models of Queue Storage

We study a model of queue storage in which items (requests for single units of storage) arrive in a Poisson stream and are accommodated by the first available location in a linear scan of storage. The processing times of items are independent, exponentially distributed random variables. The set of occupied locations (identified by their indices) at time t forms a random subset Si, of [1,2,.…]. The extent of the fragmentation in Si, i.e., the alternating holes and occupied regions of storage, is measured by Wt, = max St, – |St|.