Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-20T04:02:05.002Z Has data issue: false hasContentIssue false
  • English
  • Français

Matchmaking and Testing for Exponentiality in the M/G/∞ Queue

Part of: Special processes Inference from stochastic processes

Published online by Cambridge University Press:  14 July 2016

Rudolf Grübel  and
Hendrik Wegener
Rudolf Grübel*
Affiliation:
Leibniz Universität Hannover
Hendrik Wegener*
Affiliation:
Leibniz Universität Hannover
*
∗Postal address: Institut für Mathematische Stochastik, Leibniz Universität Hannover, Postfach 6009, D-30060 Hannover, Germany.
∗Postal address: Institut für Mathematische Stochastik, Leibniz Universität Hannover, Postfach 6009, D-30060 Hannover, Germany.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Customers arrive sequentially at times x1 < x2 < · · · < xn and stay for independent random times Z1, …, Zn > 0. The Z-variables all have the same distribution Q. We are interested in situations where the data are incomplete in the sense that only the order statistics associated with the departure times xi + Zi are known, or that the only available information is the order in which the customers arrive and depart. In the former case we explore possibilities for the reconstruction of the correct matching of arrival and departure times. In the latter case we propose a test for exponentiality.

Keywords

AsymptoticsKendall's τlog-concave densitylog-convex densityqueuepredictionpermutation

MSC classification

Secondary: 60K25: Queueing theory 62M07: Non-Markovian processes 62M20: Prediction
Type
Research Article
Information
Journal of Applied Probability , Volume 48 , Issue 1 , March 2011 , pp. 131 - 144
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] An, M. Y. (1998). Logconcavity versus logconvexity: a complete characterization. J. Econom. Theory 80, 350–369.Google Scholar
[2] Asmussen, S. (2003). Applied Probability and Queues, 2nd edn. Springer, New York.Google Scholar
[3] Bagnoli, M. and Bergstrom, T. (2005). Log-concave probability and its applications. Econom. Theory 26, 445–469.CrossRefGoogle Scholar
[4] Bai, Z. and Hsing, T. (2005). The broken sample problem. Prob. Theory Relat. Fields 131, 528–552.CrossRefGoogle Scholar
[5] Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York.Google Scholar
[6] Bäuerle, N. and Grübel, R. (2005). Multivariate counting processes: copulas and beyond. ASTIN Bull. 35, 379–408.Google Scholar
[7] Billingsley, P. (1986). Probability and Measure, 2nd edn. John Wiley, New York.Google Scholar
[8] Bingham, N. H. and Pitts, S. M. (1999). Non-parametric estimation for the M/G/∞ queue. Ann. Inst. Statist. Math. 51, 71–97.Google Scholar
[9] Buchmann, B. and Grübel, R. (2003). Decompounding: an estimation problem for Poisson random sums. Ann. Statist. 31, 1054–1074.Google Scholar
[10] Chung, K. L. (1967). Markov Chains With Stationary Transition Probabilities. Springer, New York.Google Scholar
[11] DeGroot, M. H. and Goel, P. K. (1980). Estimation of the correlation coefficient from a broken random sample. Ann. Statist. 8, 264–278.Google Scholar
[12] DeGroot, M. H., Feder, P. L. and Goel, P. K. (1971). Matchmaking. Ann. Math. Statist. 42, 578–593.Google Scholar
[13] Diaconis, P. (1988). Group Representations in Probability and Statistics (IMS Lecture Notes Monogr. Ser. 11). Institute of Mathematical Statistics, Hayward, CA.Google Scholar
[14] Hall, P. and Park, J. (2004). Nonparametric inference about service time distribution from indirect measurements. J. R. Statist. Soc. B 66, 861–875.Google Scholar
[15] Jungnickel, D. (1999). Graphs, Networks and Algorithms. Springer, Berlin.CrossRefGoogle Scholar
[16] Lehmann, E. L. (1959). Testing Statistical Hypotheses. John Wiley, New York.Google Scholar
[17] Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. John Wiley, Chichester.Google Scholar
[18] Takács, L. (1962). Introduction to the Theory of Queues. Oxford University Press, New York.Google Scholar