Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-05T16:05:31.848Z Has data issue: false hasContentIssue false
  • English
  • Français

An approximation of stopped sums with applications in queueing theory

Published online by Cambridge University Press:  01 July 2016

Miklós Csörgő ,
Paul Deheuvels  and
Lajos Horváth
Miklós Csörgő*
Affiliation:
Carleton University
Paul Deheuvels*
Affiliation:
Université Paris VI
Lajos Horváth*
Affiliation:
Szeged University
*
∗Postal address: Department of Mathematics and Statistics, Carleton University, Ottawa, Ontario K1S 5B6, Canada.
∗∗ Postal address: Université Paris VI, t.45–55, E3, L.S.T.A., 4 Place Jussieu, 75230 Paris Cedex 05, France.
∗∗∗ Postal address: Bolyai Institute, Szeged University, H-6720 Szeged, Aradi vértanúk tere 1, Hungary.
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove strong approximations for partial sums indexed by a renewal process. The obtained results are optimal. The established probability inequalities are also used to get bounds for the rate of convergence of some limit theorems in queueing theory.

Keywords

STOPPED SUMSWIENER PROCESSRENEWAL PROCESSGI/G/1 QUEUEKINGMAN&S HEAVY TRAFFIC APPROXIMATION
Type
Research Article
Information
Advances in Applied Probability , Volume 19 , Issue 3 , September 1987 , pp. 674 - 690
Copyright
Copyright © Applied Probability Trust 1987 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Research supported by an NSERC Canada grant at Carleton University.

Research done while visiting at Carleton University, also supported by NSERC Canada grants of M. Csörgő, D. A. Dawson and J. N. K. Rao.

References

Anderson, T. W. (1960) A modification of the sequential probability ratio test to reduce the sample size. Ann Math. Statist. 31, 165–197.Google Scholar
Borovkov, K. A. (1982) On the rate of convergence in the invariance principle for generalized renewal processes. Theory Prob. Appl. 27, 461–471.Google Scholar
Chung, K. L. and Williams, R. J. (1983) Introduction to Stochastic Integration. Birkhäuser, Boston.Google Scholar
Csörgő, M., Horváth, L. and Steinebach, J. (1987) Invariance principles for renewal processes. Ann. Prob. Google Scholar
Csörgő, M., Horváth, L. and Steinebach, J. (1986) Strong approximations for renewal processes. C. R. Math. Rep. Acad. Sci. Canada. 8, 151–154.Google Scholar
Csörgő, M. and Revesz, P. (1981) Strong Approximations in Probability and Statistics. Academic Press, New York.Google Scholar
Darling, D and Siegert, A. (1953) The first passage time problem for continuous Markov processes. Ann. Math. Statist. 24, 624–639.Google Scholar
De Acosta, A. (1982) Invariance principles in probability for triangular arrays of B-valued random vectors and some applications. Ann. Prob. 10, 346–373.Google Scholar
Deheuvels, P. (1985) Pareto type distributions, strong invariance principles for risk processes and their applications to actuarial studies. Astin Bulletin.Google Scholar
Dudley, R. M. (1968) Distances of probability measures and random variables. Ann. Math. Statist. 39, 1563–1572.CrossRefGoogle Scholar
Horváth, L. (1986) Strong approximations of renewal processes and their applications. Acta Math. Hungar. 47, 13–27.Google Scholar
Horváth, L. and Willekens, E. (1986) Estimates for probability of ruin starting with a large initial reserve. Insurance Math. Econom. 5, 285–293.Google Scholar
Kennedy, D. P. (1972) Rates of convergence for queues in heavy traffic. I Adv. Appl. Prob. 4, 357–381.Google Scholar
Kingman, J. F. C. (1962) On queues in heavy traffic. J. R. Statist. Soc. B. 24, 383–392.Google Scholar
Kingman, J. F. C. (1965) The heavy traffic approximation in the theory of queues. Proc. Symp. Congestion Theory, University of North Carolina Press, Chapel Hill, pp. 137–169.Google Scholar
Komlós, J., Major, P. and Tusnády, G. (1975) An approximation of partial sums of independent R.V.&s and the sample D.F. I. Z. Wahrscheinlichkeitsth. 32, 111–131.Google Scholar
Komlós, J., Major, P. and Tusnády, G. (1976) An approximation of partial sums of independent R.V.&s and the sample D.F. II. Z. Wahrscheinlichkeitsth. 34, 33–58.Google Scholar
Major, P. (1976) The approximation of partial sums of independent r.v.&s. Z. Wahrscheinlichkeitsth. 35, 213–220.Google Scholar
Mason, D. M. and Van Zwet, W. R. (1986) A note on the strong approximation to the renewal process. Preprint.Google Scholar
Rosenkrantz, W. A. (1980) On the accuracy of Kingman&s heavy traffic approximation in the theory of queues. Z. Wahrscheinlichkeitsth. 51, 115–121.CrossRefGoogle Scholar
Strassen, V. (1967) Almost sure behaviour of sums of independent random variables and martingales. Proc. 5th Berkeley Symp. Math. Statist. Prob. 2, 315–343.Google Scholar
Takács, L. (1967) Combinatorial Methods in the Theory of Stochastic Processes. Wiley, New York.Google Scholar