Geodesic
A~Rate of Convergence Problem in the Theory of Queues
Teoriâ veroâtnostej i ee primeneniâ, Tome 9 (1964) no. 1, pp. 104-112

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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 $GI/E_k /1$. 
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     author = {D. Vere-Jones},
     title = {A~Rate of {Convergence} {Problem} in the {Theory} of {Queues}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {104--112},
     publisher = {mathdoc},
     volume = {9},
     number = {1},
     year = {1964},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TVP_1964_9_1_a9/}
}
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D. Vere-Jones. A~Rate of Convergence Problem in the Theory of Queues. Teoriâ veroâtnostej i ee primeneniâ, Tome 9 (1964) no. 1, pp. 104-112. http://geodesic.mathdoc.fr/item/TVP_1964_9_1_a9/