An approximation method for a processor sharing queue with controlled arrivals and a waitbuffer

Consider a stored program controlled telephone exchange that is offered two different types of jobs, viz. call request and operator tasks. The main objective of the telephone company would be to handle as many call requests as possible. Accepting all incoming call requests however, would lead to very long sojourn times when the arrival rate of these jobs is high. One can prevent this by restricting the maximum number of call requests that can be in service. Call requests that arrive when the number of call requests in service is equal to the maximum, are rejected and considered lost. At the same time the exchange has to deal with the operator tasks that are necessary for the correct operation of the exchange. The jobs can not be rejected but delayed for some time. To guarantee a correct behaviour of the exchange this delay must not be too large. An approximation method is presented for a queueing model in which operator tasks are not allowed to be rejected. Approximate formulas are derived for the probability that an operator task is delayed, and for the mean and variance of the waiting times for delayed tasks. Numerical results are presented, showing that the approximation is accurate (relative errors smaller than 5%) under certain conditions