Completion times of jobs on two-state service processes and their asymptotic behavior

We consider the task completion time of a repairable server system in which a server experiences randomly occurring service interruptions during which the server works slowly. Every service-state change preempts the task that is being processed. The server may then resume the interrupted task, it may replace the task with a different one, or it may restart the same task from the beginning, under the new service-state. The total time that the server takes to complete a task of random size including interruptions is called completion time . We study the completion time of a task under the last two cases as a function of the task size distribution, the service interruption frequency/severity, and the repair frequency. We derive closed form expressions for the completion time distribution in Laplace domain under replace and restart recovery disciplines and present their asymptotic behavior. In general, the heavy tailed behavior of completion times arises due to the heavy tailedness of the task time. However, in the preempt-restart service discipline, even in the case that the server still serves during interruptions albeit at a slower rate, completion times may demonstrate power tail behavior for exponential tail task time distributions. Furthermore, we present an $M/G/\infty$ queue with exponential service time and Markovian service interruptions. Our results reveal that the stationary first order moments, that is, expected system time and expected number in the system are insensitive to the way the service modulation affects the servers; system-wide modulation affecting every server simultaneously vs identical modulation affecting each server independently.