Numerical analysis for bulk-arrival queueing system root-finding and steady-state probabilities in GIr/M/1 queues

Queueing theorists have presented, as solutions to many queueing models, probability generating functions in which state probabilities are expressed as functions of the roots of characteristic equations, evaluation of the roots in particular cases being left to the reader. Many users have complained that such solutions are inadequate. Some queueing theorists, in particular Neuts, rather than use Rouché’s theorem to count roots and an equation-solver to find them, have developed new algorithms to solve queueing problems numerically, without explicit calculation of roots. Powell has shown that in many bulk service queues arising in transportation models, characteristic equations can be solved and state probabilities can be found without serious difficulty, even when the number of roots to be found is large. The authors have slightly modified Powell’s method, and have extended his work to cover a number of bulk-service queues discussed by Chaudhry et al. and a number of bulk-arrival queues discussed in the present paper.