Queueing Models with MAP Arrivals Useful in Service Sectors

Abstract

Queueing models have found applications in many fields, notably in service sectors. In this paper, we study queueing models that have significant applications in service sectors. We look at multi-server systems with MAP arrivals. We assume phase type services for single server systems and exponential services when dealing with multi-server systems. All arriving customers finding no idle server will not wait in the system to receive services but rather leave their information in a registry list. These customers will be reached out on a first-come-first-served basis (FCFS) by an idle server soon after completing its current service. The reach out time is assumed to be exponential and at the end of this time, with a certain probability the reached out customer is available for service; with complementary probability the customer is not reachable due to various reasons including the customer not picking up the call from the service system to receive a service. In the case when the reach out is unsuccessful, the server will remain idle should there be no customer in the registry list. However, if there is at least one customer in the registry, then the server will start another reach out. The classical approach using matrix-analytic methods is employed and discuss a few illustrative examples that bring out the qualitative nature of the models in steady-state. When dealing with MAP/G/c queues we resort to simulation and present a few examples. Some concluding remarks including a few extensions to the models studied here are presented.