An M/M/1 Queue with n Undesired Services and a Desired Service

We study an M/M/1 queue with a set of services in undesired states and a desired state. Arrivals are according to a Poisson process with mean rate \(\lambda \). An arrival enters into desired service with probability \(\theta \), otherwise goes to one of the undesired services with complementary probability depending on the environmental factors. When the service time in undesired states exceeds a threshold random variable, the customer is pushed out of the system. Otherwise the customer, after recognizing that the current service is not the right one moves to desired service state. We discuss the case of n undesired services and a desired service.