Robustness of Rootfinding in Single-Server Queueing Models

There has been frequent controversy over the years regarding the use of numerical rootfinding for the solution of queueing problems. It has been said that such problems quite often present computational difficulties. However, it turns out that rootfinding in queueing is so well structured that problems rarely occur. There are fundamental properties possessed by the well-known queueing models that eliminate classical rootfinding problems. Most importantly, we show that distinctness of roots is common within simply determined regions in the complex plane and provide conditions under which the characteristic equations for the G/E_K/1 and E_K/G/1 models have easily found, distinct roots. Furthermore, we show that the characteristic equation for the more general G/GE_K/1 model has a collection of real and complex roots which are effectively distinct and located in clearly defined regions of the complex domain. Extensive computational results are given to support our contentions.

INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.