Bounds for the mean system size in M/G/1/K-queues

Contrary to their infinite capacity counterparts, the moments of the distribution of the number in a M/G/1/K-system cannot be determined by means of the Pollaczek-Khinchine equation. If the finite capacity K is small the distribution under study can be obtained as the steady-state probability distribution related to the transition probability matrix. For larger capacities, we derive upper and lower bounds on the mean system size in an M/G/1/K-queue for which the first two moments of the number in the system of the infinite capacity queue are known. Numerical examples for the M/D/1/1-and M/D/1/3-queues are given.