Stochastic control of a processor sharing queue with partial state information

A queuing model of a telephone switch is considered. The model consists of one server where two different types of customers arrive according to two independent Poisson processes with arrival rates λ₁ and λ₂, respectively. A type-1 customer responds with a call request and a type-2 customer with an operator or maintenance task. The customers are attended to by a single server that works according to the processor-sharing discipline. The author´s goal is to design a control law that regulates the admission of new customers in order to keep the mean sojourn times at a reasonable level. His attention is restricted to control laws that use only partial state information, i.e. the decision to admit a new customer of type i depends only on the number of type-i customers present in the queue. A lemma is presented to determine the existence of an optimal critical-level policy