A Rate of Convergence Problem in the Theory of Queues

In [5] D. G Kendall discussed the rate-of-convergence properties of the embedded Markov chains associated with the queueing systems $M/G/1$ and $GI/M/1$, and determined conditions for which convergence to their equilibrium values of the transition probabilities $p_{ij}^{(n)}$ is of geometric type. The present paper is a sequel to his work. In it we shall apply the more powerful theorems developed in [7] to show that when geometric convergence takes place it is uniform in $i$ and $j$, and that the best common ratio of geometric convergence can be simply calculated from a knowledge of the elements of the system. The results are extended to the $\chi$-squared systems $E_k /G/1$ and $GI/E_k /1$.