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.

Keywords: Mean max response times; Fork join processes; Multi RAID levels; IO requests; Storage systems

Article Outline

1. Introduction

2. Maximum of random variables

2.1. Moments

2.2. Clusters of identical exponential random variables

2.3. Mean of the maximum of general random variables

2.4. Accuracy of the approximation

3. RAID storage system

3.1. RAID levels

3.2. Multi-level RAID storage system

3.3. RAID analytical models

4. The multi-level RAID analytical model

4.1. Mean response times

4.1.1. Mean values

4.1.2. Higher moments

4.2. Mean response time on RAID0-1

4.2.1. One-block read requests

4.2.2. Multiple-blocks read requests

4.2.3. One-block write requests

4.2.4. Multiple blocks write requests

4.2.5. Overall mean response time on RAID0-1

4.3. Mean response time on RAID5

4.3.1. Read requests

4.3.2. Write requests

Remarks

5. Results and discussion

6. Sources of approximation

6.1. Moment estimation at individual disks

6.1.1. Moments of service time

6.1.2. Moments of queueing time at a single disk

6.2. Dependence of parallel queues

6.3. Non-Poisson arrivals

7. Conclusion

References

Vitae