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Limit theorems for the single server queue with traffic intensity one

Published online by Cambridge University Press:  14 July 2016

N. U. Prabhu
N. U. Prabhu*
Affiliation:
Cornell University
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Extract

We consider a single server queueing system with inter-arrival times {un, n ≧ 1}, and service times {υn, n ≧ 1} and the queue-discipline, ‘first-come, first-served’. It is assumed that {un} and {υn} are two independent renewal processes, and 0 > E(un) =a < ∞, 0 < E(υn) = b < ∞. The traffic intensity is P ρ b/a(0 > ρ > ∞). This paper is concerned with the case ρ = 1, where it is known that the various queueing processes such as the queue-length Q(t) and waiting time W(t) diverge to + ∞ in distribution as t → ∞. Borovkov [1], [2] and Brody [3] have obtained limit distributions for Q(t) and W(t) with appropriate location and scale parameters in the cases P ≧ 1. Here we investigate random variables related to the busy and idle periods in the system. To explain our approach, we consider the random variables Xn = υn – un (n ≧ 1). Let S0 ≡ 0, Sn = X1 + X2 + ··· + Xn (n ≧ 1), and define the sequence {Nk, k ≧ 0} as

Type
Research Papers
Information
Journal of Applied Probability , Volume 7 , Issue 1 , April 1970 , pp. 227 - 233
Copyright
Copyright © Applied Probability Trust 

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References

[1] Borovkov, S. P. (1964) Some limit theorems in the theory of mass service, I. Theor. Probability Appl. 9, 550–565. (English Translation).Google Scholar
[2] Borovkov, S. P. (1965) Some limit theorems in the theory of mass service, II. Theor. Probability Appl. 10, 375–400. (English Translation).Google Scholar
[3] Brody, S. M. (1963) On a limit theorem of the theory of queues. Ukrain Mat. Z. 15, 76–79.Google Scholar
[4] Darling, D. A. and Kac, M. (1957) On the occupation times for Markov processes. Trans. Amer. Math. Soc. 84, 444–458.CrossRefGoogle Scholar
[5] ErdöS, P. and Kac, M. (1946) On certain limit theorems of the theory of probability. Bull. Amer. Math. Soc. 52, 292–302.Google Scholar
[6] Feller, W. (1966) An Introduction to Probability Theory and its Applications, Vol. II. John Wiley, New York.Google Scholar
[7] Heyde, C. C. (1967) A limit theorem for random walks with drift. J. Appl. Prob. 4, 144–150.Google Scholar
[8] Iglehart, D. L. (1965) Limit theorems for queues with traffic intensity one. Ann. Math. Statist. 36, 1437–1449.Google Scholar
[9] Kendall, D. G. (1957) Some problems in the theory of dams. J. R. Statist. Soc. B19, 207–212.Google Scholar
[10] Prabhu, N. U. (1968) Some new results in storage theory. J. Appl. Prob. 5, 452–460.Google Scholar
[11] Prabhu, N. U. (1969) The queue G//M/1 with traffic intensity one (to appear).Google Scholar
[12] Sinai, Y. G. (1957) On the distribution of the first positive partial sum for a sequence of independent random variables. Theor. Probability Appl. 2, 122–129. (English translation).Google Scholar
[13] Whitt, W. (1968) Weak Convergence Theorems for Queues in Heavy Traffic. Ph. D. Thesis, Cornell University.Google Scholar