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Asymptotic behaviour of a stopping time related to cumulative sum procedures and single-server queues

Published online by Cambridge University Press:  14 July 2016

Wolfgang Stadje
Wolfgang Stadje*
Affiliation:
Universität Osnabrück
*
∗Postal address: Universität Osnabrück, Fachbereich Mathematik/Informatik, 45 Osnabrück, Postfach 4469, W. Germany.
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Abstract

For a sequence ξ1, ξ2, · ·· of i.i.d. random variables let X0 = 0 and Xk = max(Xk–1 + ξ k, 0) for k = 1, 2, ···. Let . These stopping times are used in Page's (1954) one-sided cusum procedures and are also important in queueing theory. Various asymptotic properties of Nx are derived.

Keywords

CUSUM PROCEDURESQUEUE LENGTH PROCESSGI/M/1: ASYMPTOTIC BEHAVIOUR
Type
Research Papers
Information
Journal of Applied Probability , Volume 24 , Issue 1 , March 1987 , pp. 200 - 214
Copyright
Copyright © Applied Probability Trust 1987 

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References

Blasi, A. (1976) On a random walk between a reflecting and an absorbing barrier. Ann. Prob. 4, 695–696.10.1214/aop/1176996041Google Scholar
Cohen, J. W. (1967) The distribution of the maximum number of customers present simultaneously during a busy period for the queueing systems M/G/1 and G/M/1. J. Appl. Prob. 4, 162–179.10.2307/3212309Google Scholar
Cohen, J. W. (1968) Extreme value distribution for the M/G/1 and the G/M/1 queueing systems. Ann. Inst. H. Poincaré 4, 83–98.Google Scholar
Cohen, J. W. (1982) The Single Server Queue. North-Holland, Amsterdam.Google Scholar
Enns, E. G. (1969) The trivariate distribution of the maximum queue length, the number of customers served and the duration of the busy period for the M/G/1 queueing system. J. Appl. Prob. 6, 154–161.10.2307/3212282Google Scholar
Enns, E. G. (1972) The distribution of a number of random variables associated with the absorbing Markov chain. Austral. J. Statist. 14, 79–83.10.1111/j.1467-842X.1972.tb00341.xGoogle Scholar
Feller, W. (1949) Fluctuation theory of recurrent events. Trans. Amer. Math. Soc. 67, 98–119.10.1090/S0002-9947-1949-0032114-7Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and Its Applications, volume II. Wiley, New York.Google Scholar
Gelfand, A. O. (1964) An estimate for the remainder term in a limit theorem for recurrent events. Theory Prob. Appl. 9, 299–302.10.1137/1109042Google Scholar
Gradshteyn, I. S. and Rychik, I. M. (1980) Tables of Integrals, Series, and Products, 4th edn. Academic Press, New York.Google Scholar
Heathcote, C. R. (1965) On the maximum of the queue GI/M/1. J. Appl. Prob. 2, 206–214.10.2307/3211885Google Scholar
Heyde, C. C. (1971) On the growth of the maximum queue length in a stable queue. Operat. Res. 19, 447–452.10.1287/opre.19.2.447Google Scholar
Iglehart, D. L. (1972) Extreme values in the GI/G/1 queue. Ann. Math. Statist. 43, 627–635.10.1214/aoms/1177692642Google Scholar
Kennedy, D. P. (1976) Some martingales related to cumulative sum tests and single-server queues. Stoch. Proc. Appl. 4, 261–269.10.1016/0304-4149(76)90014-4Google Scholar
Khan, R. A. (1979) Some first passage problems related to cusum procedures. Stoch. Proc. Appl. 9, 207–215.10.1016/0304-4149(79)90032-2Google Scholar
Khan, R. A. (1984) On cumulative sum procedures and the SPRT applications. J. R. Statist. Soc. B 46, 79–85.Google Scholar
Oberhettinger, F. and Baldi, L. (1973) Tables of Laplace Transforms. Springer-Verlag, Berlin.10.1007/978-3-642-65645-3Google Scholar
Page, E. S. (1954) Continuous inspection schemes. Biometrika 41, 100–114.10.1093/biomet/41.1-2.100Google Scholar
Robbins, N. B. (1976) Convergence of some expected first passage times. Ann. Prob. 4, 1027–1029.10.1214/aop/1176995948Google Scholar
Seneta, E. (1967) On the maximum of absorbing Markov chains. Austral. J. Statist. 9, 93–102.10.1111/j.1467-842X.1967.tb00187.xGoogle Scholar
Spitzer, F. (1976) Principles of Random Walk. Springer-Verlag, Berlin.10.1007/978-1-4684-6257-9Google Scholar
Takács, L. (1962) Introduction to the Theory of Queues. Oxford University Press, New York.Google Scholar
Weesakul, B. (1961) The random walk between a reflecting and an absorbing barrier. Ann. Math. Statist. 32, 765–769.10.1214/aoms/1177704971Google Scholar