Peter G. Harrison, Soraya Zertal
A queueing model is developed that approximates the effect of synchronizations at parallel service completion instants. Exact results are first obtained for the maxima of independent exponential random variables with arbitrary parameters, and this is followed by a corresponding approximation for general random variables, which reduces to the exact result in the exponential case. This approximation is then used 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. We consider the two most common RAID variants, RAID0-1 and RAID5, as well as a multi-RAID system in which they coexist. This can be used to model adaptive multi-level RAID systems in which the RAID level appropriate to an application is selected dynamically. The random variables whose maximum has to be computed in these applications are disk response times, which are modelled by the waiting times in M / G / 1 queues. 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. Sub-models for these quantities are investigated and calibrated individually in detail. Validation against a hardware simulator shows good agreement at all traffic intensity levels, including the threshold for practical operation above which performance deteriorates sharply.
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