Queueing Models with Maxima of Service Times

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Abstract

We develop a queueing model that approximates the effect of synchronisations at parallel service completion instants. We first obtain exact results for the maxima of independent exponential random variables with arbitrary parameters and follow this with an approximation for general random variables, which reduces to the exact result in the exponential case. We use this in a queueing model of RAID (Redundant Array of Independent Disks) systems, in which accesses to multiple disks occur concurrently and complete only when every disk involved has completed. The random variables to be maximised are therefore disk response times which are modelled by the waiting times in an M/G/1 queue. To compute the mean value of their maximum requires the second moment of queueing time and we obtain this in terms of the third moment of disk service time, itself a function of seek time, rotational latency and block transfer time. These quantities are analysed individually in detail. Validation by simulation, with realistic hardware parameters and block sizes, shows generally good agreement at all traffic intensity levels, including the threshold above which performance deteriorates sharply.