Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-23T02:44:34.726Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonparametric Inference for Queueing Networks of GeomX/G/∞ Queues in Discrete Time

Part of: Special processes Inference from stochastic processes

Published online by Cambridge University Press:  22 February 2016

Dominic Edelmann  and
Cornelia Wichelhaus
Dominic Edelmann*
Affiliation:
University of Heidelberg
Cornelia Wichelhaus*
Affiliation:
University of Heidelberg
*
∗ Postal address: Institute of Applied Mathematics, University of Heidelberg, Im Neuenheimer Feld 294, 69120 Heidelberg, Germany.
∗ Postal address: Institute of Applied Mathematics, University of Heidelberg, Im Neuenheimer Feld 294, 69120 Heidelberg, 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.

We study nonparametric estimation problems for discrete-time stochastic networks of GeomX/G/∞ queues. We assume that we are only able to observe the external arrival and external departure processes at the nodes over a stretch of time. Based on such incomplete information of the system, we aim to construct estimators for the unknown general service time distributions at the nodes without imposing any parametric condition. We propose two different estimation approaches. The first approach is based on the construction of a so-called sequence of differences, and a crucial relation between the expected number of external departures at a node and specific sojourn time distributions in the network. The second approach directly utilizes the structure of the cross-covariance functions between external arrival and departure processes at the nodes. Both methods lead to deconvolution problems which we solve explicitly. A detailed simulation study illustrates the numerical performances of our estimators and shows their advantages and disadvantages.

Keywords

Cross-covariancediscrete deconvolutiondiscrete-time queueing systemnonparametric statistics for queuessojourn time estimation

MSC classification

Primary: 60K25: Queueing theory
Secondary: 62M05: Markov processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 46 , Issue 3 , September 2014 , pp. 790 - 811
Copyright
© Applied Probability Trust 

References

Bhat, U. N. and Subba Rao, S. (1986/87). Statistical analysis of queueing systems. Queueing Systems 1, 217–247.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
Brown, M. (1970). An M/G/∞ estimation problem. Ann. Math. Statist. 41, 651–654.Google Scholar
Conti, P. L. (1999). Large sample Bayesian analysis for Geo/G/1 discrete-time queueing models. Ann. Statist. 27, 1785–1807.Google Scholar
Conti, P. L. (2002). Nonparametric statistical analysis of discrete-time queues, with applications to ATM teletraffic data. Stoch. Models 18, 497–527.CrossRefGoogle Scholar
Daley, D. J. (1976). Queueing output processes. Adv. Appl. Prob. 8, 395–415.Google Scholar
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
Hansen, M. B. and Pitts, S. M. (2006). Nonparametric inference from the M/G/1 workload. Bernoulli 12, 737–759.Google Scholar
Ke, J.-C. and Chu, Y.-K. (2006). Nonparametric and simulated analysis of intensity for a queueing system. Appl. Math. Comput. 183, 1280–1291.Google Scholar
Liu, Z., Wynter, L., Xia, C. H. and Zhang, F. (2006). Parameter inference of queueing models for IT systems using end-to-end measurements. Performance Evaluation 63, 36–60.Google Scholar
Pickands, J., III and Stine, R. A. (1997). Estimation for an M/G/∞ queue with incomplete information. Biometrika 84, 295–308.Google Scholar
Pitts, S. M. (1994). Nonparametric estimation of the stationary waiting time distribution function for the GI/G/1 queue. Ann. Statist. 22, 1428–1446.CrossRefGoogle Scholar
Proakis, J. G. and Manolakis, D. G. (1992). Digital Signal Processing: Principles, Algorithms, and Applications. Macmillan, New York.Google Scholar
Ross, S. M. (1970). Identifiability in GI/G/k queues. J. Appl. Prob. 7, 776–780.CrossRefGoogle Scholar
Wichelhaus, C. and Langrock, R. (2012). Nonparametric inference for stochastic feedforward networks based on cross-spectral analysis of point processes. Electron. J. Statist. 6, 1670–1714.Google Scholar