On More Combinatorial Methods in the Theory of Queues
In a series of papers and in his book , Takács has given a new direction to the research work in the theory of queues, by introducing a combinatorial approach to a certain class of problems. It is seen that the two basic combinatorial results (ref: Theorem 1 and Theorem 2 in ) of Takacs are directly connected with the distributions of Kolmogorov-Smirnov two-sample and one-sample statistics as derived by Steck , . (See , , ). It is rather significant to mention that the combinatorial structure of these problems has been considered by Narayana  in the context of occupancy problems. Using his technique, we will generalize both Narayana’s and Steck’s results. One of these (viz. the first one) involves counting lattice paths in the space under some given restrictions. We also note that Yadin’s treatment of queues with alternating priorities through random walk in  can be essentially viewed as path enumeration problems. For instance, his results (2.13). on zero switch rule can be obtained as a special case from Combinatorial Lemma 2 in , where the author is dealing with queues involving batches.
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- 1.Mohanty, S. G. (1971), “A short proof of Steck’s result on two-sample Smirnov statistics”, Annals of Mathematical Statistics, Vol. 42, 413–414.Google Scholar