CONDITIONS FOR THE PRODUCT FORM OF THE STATIONARY PROBABILITY DISTRIBUTION OF MARKOVIAN RESOURCE LOSS SYSTEMS

In classical queueing systems, servers and waiting places play the role of resources required for service of customers. In resource queuing systems in addition to servers and waiting places, customers may require some additional resources. This may be some random amount of resource occupied for the duration of the waiting time, service time, or residence time. In this paper, we consider Markovian resource queueing systems in which an arriving customer is lost if the system does not have enough available resources. First, the class of Markovian resource loss systems considered is described. Then, the necessary and sufficient conditions for the product form of the stationary probability distribution of the system state and the volumes of the resources occupied by the customers are derived. As an example, the results obtained are applied to the analysis of the stationary distribution of resource loss system MAP/M/L/0 with a Markovian arrival process and exponentially distributed service times. It is shown that for this system the stationary probability distribution has product form only if the arrival process is Poisson.