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 = λ G law and lead to a complete solution for the distributions of L , Q, S , W for queueing systems which satisfy distributional laws 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 laws 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 G//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 /∞ queue, and exact results for the distributions of L, Q , S, W in priority queues with mixed generalized Erlang renewal arrivals.