A Unified Method to Analyze Overtake Free Queueing Systems

In this paper we demonstrate that the distributional laws that relate the number of customers in the system (queue), L(Q) and the time a customer spends in the system (queue), S(W) under the first-in-first-out (FIFO) discipline are special cases of the H = AG law and lead to a complete solution for the distributions of L, Q, S, W for queueing systems which satisfy distributional aws for both L and Q (overtake free systems). Moreover, in such systems the derivation of the distributions of L, Q, S, W can be done in a unified way. Consequences of the distributional aws include a generalization of PASTA to queueing systems with arbitrary renewal arrivals under heavy traffic conditions, a generalization of the Pollaczek-Khinchine formula to the GI/G/1 queue, an extension of the Fuhrmann and Cooper decomposition for queues with generalized vacations under mixed generalized Erlang renewal arrivals, approximate results for the distributions of L, S in a GI/G/c queue, and exact results for the distributions of L, Q, S, W in priority queues with mixed generalized Erlang renewal arrivals. PERFORMANCE ANALYSIS; DISTRIBUTIONAL LAWS; PRIORITY QUEUES; GENERALIZED VACATIONS; HEAVY TRAFFIC; MIXED GENERALIZED ERLANG ARRIVALS AMS 1991 SUBJECT CLASSIFICATION: PRIMARY 60K25; 90B22