Discrete time queue with interruptions and repeat/resumption of interrupted service

Here we consider an infinite-capacity, single-server, discrete-time queue in which customers arrive according to a PH-distributed interarrival process. If the system is idle, an arriving customer begins service immediately. During service, one or more interruptions may occur, and the interarrival times of the interruption process are geometrically distributed with parameter p. When an interruption occurs, a threshold random clock starts ticking. If the duration of the interruption exceeds the threshold random variable, the interrupted customer must restart the service from the beginning upon completion of the interruption; otherwise, the service is resumed. Several performance measures are evaluated, and numerical illustrations of the system behavior are also provided.