Some Results for the Actual Waiting Time in Batch Arrival Queueing Systems

Abstract

The article deals with the queueing system of GI /G /1 type with individual service and batch arrival of customers. The method of integral equations on a half-axis [0, ∞) is applied to obtain new results for the actual waiting time w n of nth arriving customer. The transient and steady state as n tends to infinity are considered. Some simplifications and numerical results for M /M /1, M /E 2 /1, and M 2 /M /1 queueing systems are derived as well.

Transient workload distribution in the $$M/G/1$$ M / G / 1 finite-buffer queue with single and multiple vacations

“…Transient results for the waiting time distribution in the G I X /G/1-type system with batch arrivals and infinite buffer are derived in Kempa (2004) for the virtual waiting time and in Kempa (2010d) for the actual waiting time, using the approach based on supplementary variables' technique, integral equations and the method of Wiener-Hopf factorization. The formulae for twofold transforms of the transient virtual waiting time distribution in the finite system with the input stream defined by a single Poisson process and by M M P P and B M AP can be found in Chydzinski (2007).…”