Mathematical Methods in Queueing Theory pp 365-371 | Cite as

# On More Combinatorial Methods in the Theory of Queues

## Abstract

In a series of papers and in his book [9], 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 [8]) of Takacs are directly connected with the distributions of Kolmogorov-Smirnov two-sample and one-sample statistics as derived by Steck [6], [7]. (See [1], [4], [5]). It is rather significant to mention that the combinatorial structure of these problems has been considered by Narayana [3] 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 [10] 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 [2], where the author is dealing with queues involving batches.

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### References

- 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
- 2.Mohanty, S. G. (1972), “On queues involving batches”, Journal of Applied Probability, Vol. 9 430–435.CrossRefMATHMathSciNetGoogle Scholar
- 3.Narayana, T. V. (1955), “A combinatorial problem and its application to probability theory I”, Journal of Indian Society of Agricultural Statistics, Vol. 7, 169–178.MathSciNetGoogle Scholar
- 4.Pitman, E. J. G. (1972), “Simple proofs of Steck’s determinantal expressions for probabilities in the Kolmogorov and Smirnov tests”, Australian Mathematical Society Bulletin, Vol. 7, 227–232.CrossRefMATHMathSciNetGoogle Scholar
- 5.Sarkadi, K. (1973), On the exact distributions of statistics of Kolmogorov-Smirnov type“, Periodica Mathematica Hungarica, Vol. 3, 9–12.CrossRefMATHMathSciNetGoogle Scholar
- 6.Steck, G. P. (1969), The Smirnov two-sample tests as rank tests“, Annals of Mathematical Statistics, Vol. 40, 1449–1466.CrossRefMATHMathSciNetGoogle Scholar
- 7.Steck, G. P. (1971), “Rectangle probabilities for uniform order statistics and the probability that the empirical distribution function lies between two distribution functions”, Annals of Mathematical Statistics, Vol. 42, 1–11.CrossRefMATHMathSciNetGoogle Scholar
- 8.Takdcs, L. (1964), “Combinatorial methods in the theory of queues”, Review of the International Statistical Institute, Vol. 32, 207–219.CrossRefGoogle Scholar
- 9.Takâcs, L. (1967), Combinatorial Methods in the Theory of Stochastic Processes, John Wiley and Sons, New York.MATHGoogle Scholar
- 10.Yadin, M. (1970), “Queueing with alternating priorities, treated as random walks on the lattice in the plane”, Journal of Applied Probability, Vol. 7, 196–218.CrossRefMATHMathSciNetGoogle Scholar