A conditional response time of the M/M/1 processor-sharing queue

In this paper we examine the distribution of the response time of an arriving customer conditioned on the number of customers present in an M/M/1 processor-sharing queue. We show that the rth moment of the distribution is a polynomial in the number of customers present, and obtain a recursion for its determination. The Laplace-Stieltjes transform of the conditional distribution is also obtained. We find an expansion in powers of the arrival rate, which permits accurate computation when the utilization is not too close to one, and also give an asymptotic expansion when the number of customers in the system is large. This permits assessment of response time in a heavily loaded system. We present numerical results using these methods and discuss their relative merits.