Reliability of Systems with Limited Repairs

Reliability is the probability that a system functions according to specifications over a given period of time. During this period, system specifications may allow failures and repairs to occur. This paper considers systems with specifications which limit the repair process. Such systems place a limitation on either the repair duration or the number of repairs. For example, a system controlling a real-time process may go down, be repaired, and continue proper control as long as the repair duration does not exceed a specified bound. Otherwise, the system fails. We model and analyze systems with three different types of limited repairs: 1) Bounded repair time, 2) Bounded cumulative repair time, and 3) Bounded number of repairs. Examples of such models exist in real-time process control, shock models, transaction processing, and maintenance models. For each of the three types of systems with limited repairs, we derive the distributions and the mean values of the system lifetime, the cumulative operational time, and the largest continuous operational time before a complete system failure. We also consider the execution of a task on such systems. The task is preempted upon the occurence of a failure, and is resumed or repeated after repair. The probability of completion of a task with a given work requirement in the three limited downtime scenarios is derived. We study the effect of preemptive-resume versus preemptive-repeat failures on the probability of task completion.