Queueing Analysis of Fault-Tolerant Computer Systems ™ V. F. Nicolas, V. G. Kulkarni ™ and K. S. Trivedit Ezfcndcd Absfroct Queueing models provide a useful tool for predicting the performance of many service systems including computer systems, telecommunication systems, computer/communication networks and flexible manufacturing systems. Traditional queueing models predict system performance under the assumption that all service facilities provide failure-free service. It must, however, be acknowledged that service facilities do experience faiiures and that they get repaired. In recent years, it has been increasingly recognized that separation of this performance and reliability/availability models is no longer adequate. An exact steady-state queueing analysis of such systems is considered by several authors and is carried out by means of generating functions, supplementary variables, imbedded Markov process and renewal theory, or probabilistic techniques [1,2,7,8]. Another approach is approximate, in which it is assumed that the time to reach the steady-state is much smaller than the times to failures/repairs. Therefore, it is reasonable to associate a performance measure (reward) with each state of the underlying Markov (or semi-Markov) model describing the failure/repair behavior of the system. Each of these performance measures is obtained from the steady-state queueing analysis of the system in the corresponding state [3,5]. Earlier we have developed models to derive the distribution of job completion time in a failure-prone environment [3,4]. In these models, we need to consider a possible loss of work due to the occurrence of a failure, i.e., the interrupted job may be resumed or restarted upon service resumption. Note that the job completion time analysis includes the delays due to failures and repairs. The purpose of this paper [Q] is to extend our earlier analysis so as to account for the queueing delays. In effect, we consider an exact queueing analysis of fault-tolerant systems in order to obtain the steady-state distribution and the mean of the number of jobs in the system. In particular, we study a system in which jobs arrive in a Poisson fashion and are serviced according The service requirements of the incoming to FCFS discipline. Research under grants AFOSR-840132 ™ This research was supported in part by the Air Force O&x of Scientific and AFOSR-84-0140, by the Army Research OKice under grant DAAGZB-84K-0046 and by the National Science Foundation under grant NSF-MCS-830200. t Curriculum in Operations Research and Systems Analysis, University North Carolina, Chapel Hill, NC 27514. t Department of Computer Science, Duke University, Durham, NC 27706. of jobs form a sequence of independent and identically distributed random variables. The failure/repair behaviour of the system is modelled by an irreducible continuous-time Markov chain, which is independent of the number of jobs in the system. Let the statespace be {1,2,..., n ). When the computer system is in state i it delivers service at rate ri 1 0. Furthermore, depending on the type of the state, the work done on the job is preserved or lost upon entering that state. The actual time required to complete a job depends in a complex way upon the service requirement of the job and the evolution of the state of the system. Note that even though the service requirements of jobs are independent and iden& ically distributed, the actual times required to complete these jobs are neither independent nor identically distributed, and hence the model cannot be reduced to a standard M/G/l queue [8]. As loss of work due to failures and interruptions is quite a common phenomenon in faulttolerant computer systems, the model pr+ posed here is of obvious interest. Using our earlier results on the distribution of job completion time we set up a queueing model and show that it has the block M/G /1 structure. Queueing models with such a structure have been studied by Neuts, Lucantoni and others 161. We demonstrate the usefulness of our approach by performing the numerical analysis for a system with two processors subject to failures and repairs.

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