Abstract
In this paper, we provide a new approach to the computation of the Laplace transform of the length of the busy period of the M/M/1 queue with constrained workload (finite dam), without the use of complex analysis.
Similar content being viewed by others
References
B.F. Baccelli and A.M. Makowski, Dynamic, transient and stationary behavior of the M/GI/1 queue via martingales, Ann. Probab. 17 (1989) 1691–1699.
J.W. Cohen, The Single Server Queue (North-Holland, Amsterdam/London, 1969).
E.B. Dynkin, Markov Processes, Vol. 1 (Wiley, New York, 1966).
W. Feller, An Introduction to Probability Theory and its Applications, Vol. 2 (Wiley, New York, 1971).
J. Gani, Problems in the probability theory of storage systems, J. Roy. Statist. Soc. B 19 (1957) 181–206.
D.P. Heyman, An approximation for the busy period of the M/G/1 queue using a diffusion model, J. Appl. Probab. 11 (1974) 159–169.
S. Karlin and H.M. Taylor, A First Course in Stochastic Processes, 2nd ed. (Academic Press, New York, 1975).
T. Kurtz, Approximation of population processes, in: CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 36 (SIAM, Philadelphia, PA, 1981).
E.Y. Lee and K.K.J. Kinateder, The expected wet period of finite dam with exponential inputs, preprint (1997).
P.A.P. Moran, The Theory of Storage (Methuen, London, 1959).
R.M. Phatarfod, Application of methods in sequential analysis to dam theory, Ann. Math. Statist. 34 (1963) 1558–1593.
R.M. Phatarfod, A note on the first emptiness problem of a finite dam with poisson type inputs, J. Appl. Probab. 6 (1969) 227–230.
R.M. Phatarfod, On some applications of Wald's identity to dams, Stochastic Process. Appl. 13 (1982) 279–292.
N.U. Prabhu, Time-dependent results in storage theory, J. Appl. Probab. 1 (1964) 1–46.
N.U. Prabhu, Stochastic Storage Processes: Qeues, Insurance Risk, Dams, and Data Communication, 2nd ed. (Springer, New York, 1998).
W.A. Rosenkrantz, Some martingales associated with queueing and storage processes, Zeit. Wahr. 58 (1981) 205–222.
W.A. Rosenkrantz, Calculation of the Laplace transform of the length of the busy period for the M/G/1 queue via martingales, Ann. Probab. 11 (1983) 817–818.
L. Takacs, Combinatorial Methods in the Theory of Stochastic Processes (Wiley, New York, 1967).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kinateder, K.K., Lee, E.Y. A new approach to the busy period of the M/M/1 queue. Queueing Systems 35, 105–115 (2000). https://doi.org/10.1023/A:1019185826015
Issue Date:
DOI: https://doi.org/10.1023/A:1019185826015