### Nonparametric inference for queueing networks of GEOMX/G/∞ queues in discrete time

#### Abstract

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.

#### Article information

Source
Adv. in Appl. Probab., Volume 46, Number 3 (2014), 790-811.

Dates
First available in Project Euclid: 29 August 2014

https://projecteuclid.org/euclid.aap/1409319560

Digital Object Identifier
doi:10.1239/aap/1409319560

Mathematical Reviews number (MathSciNet)
MR3254342

Zentralblatt MATH identifier
1306.60136

Subjects
Secondary: 62M05: Markov processes: estimation

#### Citation

Edelmann, Dominic; Wichelhaus, Cornelia. Nonparametric inference for queueing networks of GEOM X /G/∞ queues in discrete time. Adv. in Appl. Probab. 46 (2014), no. 3, 790--811. doi:10.1239/aap/1409319560. https://projecteuclid.org/euclid.aap/1409319560

#### References

• \item[] Bhat, U. N. and Subba Rao, S. (1986/87). Statistical analysis of queueing systems. Queueing Systems 1, 217–247.
• \item[] Bingham, N. H. and Pitts, S. M. (1999). Non-parametric estimation for the $M/G/\infty$ queue. Ann. Inst. Statist. Math. 51, 71–97.
• \item[] Brown, M. (1970). An $M/G/\infty$ estimation problem. Ann. Math. Statist. 41, 651–654.
• \item[] Conti, P. L. (1999). Large sample Bayesian analysis for Geo/$G$/1 discrete-time queueing models. Ann. Statist. 27, 1785–1807.
• \item[] Conti, P. L. (2002). Nonparametric statistical analysis of discrete-time queues, with applications to ATM teletraffic data. Stoch. Models 18, 497–527.
• \item[] Daley, D. J. (1976). Queueing output processes. Adv. Appl. Prob. 8, 395–415.
• \item[] Hall, P. and Park, J. (2004). Nonparametric inference about service time distribution from indirect measurements. J. R. Statist. Soc. B 66, 861–875.
• \item[] Hansen, M. B. and Pitts, S. M. (2006). Nonparametric inference from the $M$/$G$/1 workload. Bernoulli 12, 737–759.
• \item[] Ke, J.-C. and Chu, Y.-K. (2006). Nonparametric and simulated analysis of intensity for a queueing system. Appl. Math. Comput. 183, 1280–1291.
• \item[] 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.
• \item[] Pickands, J., III and Stine, R. A. (1997). Estimation for an $M/G/\infty$ queue with incomplete information. Biometrika 84, 295–308.
• \item[] Pitts, S. M. (1994). Nonparametric estimation of the stationary waiting time distribution function for the $GI/G/1$ queue. Ann. Statist. 22, 1428–1446.
• \item[] Proakis, J. G. and Manolakis, D. G. (1992). Digital Signal Processing: Principles, Algorithms, and Applications. Macmillan, New York.
• \item[] Ross, S. M. (1970). Identifiability in $GI/G/k$ queues. J. Appl. Prob. 7, 776–780.
• \item[] 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. \endharvreferences