Queues with interruption in Markovian environment

Abstract

This paper carries out an analysis of an M/M/1 queue with service interruption, where the interruptions are caused by different environmental factors which are the states of a Markov chain. Interruption due to only one factor is allowed at a time and during an interruption period, no more interruption is allowed. The server is unaware of the interruption until a random amount of time elapses. The duration of unidentified interruption is assessed with the help of the random clock that started ticking at the onset of the interruption. The interruption is identified only when the random clock is realized. A new service is commenced to the interrupted customer after fixing the interruption. The kind of service to be started after fixing the interruption depends on the environmental factor that caused the interruption. Even though any number of interruptions can occur during the service of a customer, the maximum number of interruptions is restricted to a finite number K and if the number of interruptions exceeds this maximum, the customer goes out of the system without completing the service. The superclock measures the total duration of interruption in the service of a customer. This clock starts ticking at the onset of the first interruption to a customer in service. When interruption is fixed, this clock stops. If another interruption strikes the same customer’s service, the superclock starts from the earlier position where it stopped ticking and so on. Finally, when the customer leaves the system the superclock is reset to the zero position. On realization of superclock the customer goes out of the system without completing service. We analysed service process to obtain stability condition. Then we performed steady state analysis and important performance measures are obtained.