On some belations between stationary distributions of queue lengths and imbedded queue lengths in g/g/s queueing systems

Abstract

In the present paper we consider the stationary distributions of the number of customers in G/G/s queuing systems at arbitrary points in time and at arrival or departure epochs. Using the theory of stationary marked point processes it can be shown that these distributions are comparable with respect to the stochastic order relation ≦. By this means we get, as a special case, the results of MARSHALL-WOLFF [9] concerning first moments of the mentioned distributions. For GI/G/I queues we give and estimation for the difference between expected queue lengths at arbitray points in time and just before and arrival epoch.