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A continuous-time treatment of certain queues and infinite dams

Published online by Cambridge University Press:  09 April 2009

R. M. Loynes
R. M. Loynes
Affiliation:
Statistical Laboratory, University of Cambridge.
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Summary

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The continuous-time behaviour of a model which represents certain queues and infinite dams with correlated inputs is considered. It is shown how the transient behaviour may be investigated, and the asymptotic behaviour is obtained. Finally the methods are illustrated for a queue whose input consists of two superimposed renewal processes.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 2 , Issue 4 , December 1962 , pp. 484 - 498
Copyright
Copyright © Australian Mathematical Society 1962

References

Bartlett, M. S. (1956), Stochastic Processes. Cambridge University Press.Google Scholar
Beneš, V. (1957), “On Queues with Poisson Arrivals”, Ann. Math. Statist., 28, 670–677.CrossRefGoogle Scholar
Cox, D. R. (1955), “A Use of Complex Probabilities in the Theory of Stochastic Processes”, Proc. Camb. Phil. Soc., 51, 313–319.CrossRefGoogle Scholar
Loynes, R. M. (1961a), “The Stability of a Queue with Non-Independent Inter-Arrival and Service Times”. Proc. Camb. Phil. Soc. (to be published).CrossRefGoogle Scholar
Loynes, R. M. (1961b), “Stationary Waiting-Times for Single-Server Queues”, Ann. Math. Statist. (to be published).CrossRefGoogle Scholar
McShane, E. J. (1947), Integration. Princeton University Press.Google Scholar
Smith, W. L. (1953). “On the Distribution of Queueing Times”, Proc. Camb. Phil. Soc., 49, 449–461.CrossRefGoogle Scholar
Smith, W. L. (1955), “Regenerative Stochastic Processes”, Proc. Roy. Soc. A, 232, 6–31.Google Scholar
Takács, L. (1955), “Investigation of Waiting Time Problems by Reduction to Markov Processes”, Acta Math. Acad. Sci. Hungaricae, 6, 101–129.CrossRefGoogle Scholar
Takács, L. (1961), “Transient Behaviour of Single-Server Queueing Processes with Erlang Input”, Trans. Am. Math. Soc., 100, 1–28.CrossRefGoogle Scholar
Titchmarsh, E. C. (1939), Theory of Functions. Oxford University Press.Google Scholar
Zygmund, A. (1951), “A Remark on Characteristic Functions”, Second Berkeley Symposium on Mathematical Statistics and Probability. Berkeley: University of California Press.Google Scholar