The shorter queue problem: A numerical study using the matrix-geometric solution
The paper is devoted to a numerical study of a shorter queue system (SQS) with two queues where customers, on arrival, join the shorter one. As a truncation of SQS, the MSQS problem is considered. MSQS has an additional property: if the difference between the longer and the shorter queue reaches some level n, then one customer leaves the longer queue for the shorter one. For large n MSQS gives an approximation to SQS. The infinitesimal generator of a Markov process describing MSQS under standard assumptions can be presented in the form of a three-diagonal infinite block matrix. To find the stationary probability vector the matrix-geometric approach of \textit{M. F. Neuts} [''Matrix-geometric solutions in stochastic models'' (1981; Zbl 0469.60002)] is used. This approach is a matrix generalization of the well-known way to investigate the scalar three-diagonal case and requires to solve a matrix quadratic equation. One can obtain the desired minimal nonnegative solution of this equation (minimal ''matrix eigenvalue'') by an iteration scheme investigated by Neuts. This paper contains the results of computations of the stationary distributions and average waiting times for the various values of the traffic intensity \(\rho\). It is shown that SQS is slower than the corresponding M/M/2 system. The relative differences in waiting times changes from 0.5 for \(\rho =0.1\) to 0.05 for \(\rho =0.9\).
scientific article; zbMATH DE number 3870999
scientific article