Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-10T21:21:50.329Z Has data issue: false hasContentIssue false
  • English
  • Français

Sojourn Time Estimation in an M/G/∞ Queue with Partial Information

Part of: Operations research and management science Inference from stochastic processes

Published online by Cambridge University Press:  30 January 2018

Nafna Blanghaps ,
Yuval Nov  and
Gideon Weiss
Nafna Blanghaps*
Affiliation:
University of Haifa
Yuval Nov*
Affiliation:
University of Haifa
Gideon Weiss*
Affiliation:
University of Haifa
*
∗ Postal address: Department of Statistics, University of Haifa, Haifa, 31905, Israel.
∗ Postal address: Department of Statistics, University of Haifa, Haifa, 31905, Israel.
∗ Postal address: Department of Statistics, University of Haifa, Haifa, 31905, Israel.
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.

We propose an estimator for the cumulative distribution function G of the sojourn time in a steady-state M/G/∞ queueing system, when the available data consists of the arrival and departure epochs alone, without knowing which arrival corresponds to which departure. The estimator generalizes an estimator proposed in Brown (1970), and is based on a functional relationship between G and the distribution function of the time between a departure and the rth latest arrival preceding it. The estimator is shown to outperform Brown's estimator, especially when the system is heavily loaded.

Keywords

M/G/∞sojourn time estimationSmoluchowski processsemiparametric estimation

MSC classification

Primary: 62M09: Non-Markovian processes
Secondary: 90B22: Queues and service
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 4 , December 2013 , pp. 1044 - 1056
Copyright
© Applied Probability Trust 

References

Ayesta, U. and Mandjes, M. (2009). Bandwidth sharing networks under a diffusion scaling. Ann. Operat. Res. 170, 41–58.CrossRefGoogle Scholar
Bingham, N. H. and Dunham, B. (1997). Estimating diffusion coefficients from count data: Einstein–Smoluchowski theory revisited. Ann. Inst. Statist. Math. 49, 667–679.CrossRefGoogle Scholar
Bingham, N. H. and Pitts, S. M. (1999). Non-parametric estimation for the M/G/∞ queue. Ann. Inst. Statist. Math. 51, 71–97.CrossRefGoogle Scholar
Breiman, L. (1992). Probability. Society for Industrial and Applied Mathematics, Philadelphia, PA.CrossRefGoogle Scholar
Brenner, S. L., Nossal, R. J. and Weiss, G. H. (1978). Number fluctuation analysis of random locomotion. Statistics of a Smoluchowski process. J. Statist. Phys. 18, 1–18.CrossRefGoogle Scholar
Brown, M. (1970). An M/G/∞ estimation problem. Ann. Math. Statist. 41, 651–654.CrossRefGoogle Scholar
Chandrasekhar, S. (1943). Stochastic processes in physics and astronomy. Rev. Modern Phys. 15, 1–89.CrossRefGoogle Scholar
De Leeuw, J., Hornik, K. and Mair, P. (2009). Isotone optimization in R: pool-adjacent-violators algorithm {(PAVA)} and active set methods. J. Statist. Software 32.CrossRefGoogle Scholar
Duffey, E. and Watt, A. S. (eds) (1971). The Scientific Management of Animal and Plant Communities for Conservation. Blackwell, Oxford.Google Scholar
Grübel, R. and Wegener, H. (2011). Matchmaking and testing for exponentiality in the M/G/∞ queue. J. Appl. Prob. 48, 131–144.CrossRefGoogle Scholar
Hall, P. and Park, J. (2004). Nonparametric inference about service time distribution from indirect measurements. J. R. Statist. Soc. B, 66, 861–875.CrossRefGoogle Scholar
Kac, M. (1959). Probability and Related Topics in Physical Sciences. Interscience, London.Google Scholar
Lindley, D. V. (1956). The estimation of velocity distributions from counts. In Proceedings of the International Congress of Mathematicians, 1954, Amsterdam, Noordhoff, Groningen, pp. 427–444.Google Scholar
Massey, F. J. Jr. (1951). The Kolmogorov–Smirnov test for goodness of fit. J. Amer. Statist. Assoc. 46, 68–78.CrossRefGoogle Scholar
Nelgabatz, N. (2012). Estimation for the service time distribution in an M/G/∞ system with partial information. Master's Thesis, University of Haifa.Google Scholar
Park, J. (2007). On the choice of an auxiliary function in the M/G/∞ estimation. Comput. Statist. Data Anal. 51, 5477–5482.CrossRefGoogle Scholar
Parzen, E. (1962). Stochastic Processes. Holden-Day, San Fransisco, CA.Google Scholar
Pickands, J., III and Stine, R. A. (1997). Estimation for an M/G/∞ queue with incomplete information. Biometrika 84, 295–308.CrossRefGoogle Scholar
Robertson, T., Wright, F. T. and Dykstra, R. L. (1988). Order Restricted Statistical Inference. John Wiley, Chichester.Google Scholar
Ross, S. M. (2010). Introduction to Probability Models, 10th edn. Academic Press, Amsterdam.Google Scholar
Rothschild, V. (1953). A new method of measuring the activity of spermatozoa. J. Experimental Biol. 30, 178–199.CrossRefGoogle Scholar
Ruben, H. (1963). The estimation of a fundamental interaction parameter in an emigration–immigration process. Ann. Math. Statist. 34, 238–259.CrossRefGoogle Scholar
Smoluchowski, M. (1906). {Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen}. Ann. Physik 326, 756–780.CrossRefGoogle Scholar
Smoluchowski, M. (1916). {Drei Vorträge über Diffusion, Brownschen Bewegung und Koagulation von Kolloid-teilchen}. Physik. Z. 17, 557–585.Google Scholar