Queueing models of RAID systems with maxima of waiting times

Abstract

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.

Introduction

Traditional, e.g. product-form, queueing networks cannot model synchronizations at parallel service completion instants. We approximate this effect in a queueing model of RAID (Redundant Array of Independent Disks) systems, derived by considering the explicit flow of control in the physical architecture. The contention in each parallel phase of processing is represented using an approach based on the $M/G/1$ queue. The synchronization time is then the maximum of a collection of $M/G/1$ queue sojourn times (also called waiting times or response times). We assume these sojourn times to be independent; initially, exponential random variables, as in an $M/M/1$ queue, and then general.

Based on initial work in [12], Section 2 derives an exact recurrence formula for the Laplace transform of the probability density function of the maximum of a set of independent exponential random variables, from which the mean and higher moments follow. In the special case that all the constituent exponential distributions are identical, the well-known result for the mean value of the maximum in terms of harmonic numbers follows immediately. The recurrence is then generalized to approximate the mean of the maximum of independent, generally distributed random variables. This simplifies to the previous exact result when the constituent distributions are exponential but in general requires their second moments. The accuracy of the approximation is assessed by comparison with simulation results obtained for Erlang and Pareto constituent distributions, which typify the cases of small and large variances respectively.

RAID storage systems and existing analytical models are briefly reviewed in Section 3 and the results of Section 2 are then used in our new multi-level RAID performance model in Section 4. This model assumes Poisson external requests but allows general disk seek, latency and transfer times. We determine the higher moments of the queueing time in the $M/G/1$ queue by differentiating its Laplace–Stieltjes transform at the origin. The second moment is then given in terms of the third moment of the service time, which is obtained in turn from the assumed distributions of seek time, rotational latency and block transfer time. Detailed studies of their principles of operation show that RAID levels 0–1 and 5 produce quite different demands on the disks in the array for each type of input-output access. This difference is amplified in the corresponding queueing times; it is seen in both the explicit simulation of the physical systems’ operation and in the calculation of mean and variance of queueing time in the analytical model.

The accuracy of the model is assessed in Section 5 by comparing the analytical predictions with a simulation of the actual system at the operational level. The quantitative results are presented as graphs of mean system response time against traffic intensity, showing generally good agreement and hence providing justification for our approach. The validity of the assumption of Poisson arrivals was tested numerically by comparing with simulation models with non-Poisson input; the simulation output (mean response time) shows little change. This is consistent with the commonly observed robustness of the Poisson assumption for external arrivals. In addition, in Section 6 we further investigate possible causes of inaccuracy in the model’s approximations. We isolate two possible sources, apart from the precision of the mean–max algorithm assessed in Section 2.4: (a) the representation of the delay at a single disk as the response time in an $M/G/1$ queue; and (b) the effect of assuming such response times are independent when arrivals actually occur simultaneously. The paper concludes in Section 7 with a summary of the present contribution, open questions and suggestions for further research.