Advances in Applied Probability

Nonparametric estimation of the service time distribution in the M/G/∞ queue

Alexander Goldenschluger

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The subject of this paper is the problem of estimating the service time distribution of the M/G/∞ queue from incomplete data on the queue. The goal is to estimate G from observations of the queue-length process at the points of the regular grid on a fixed time interval. We propose an estimator and analyze its accuracy over a family of target service time distributions. An upper bound on the maximal risk is derived. The problem of estimating the arrival rate is considered as well.

Article information

Adv. in Appl. Probab., Volume 48, Number 4 (2016), 1117-1138.

First available in Project Euclid: 24 December 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation
Secondary: 60K25: Queueing theory [See also 68M20, 90B22] 62M09: Non-Markovian processes: estimation

M/G/∞ queue nonparametric estimation minimax risk stationary process covariance function rates of convergence


Goldenschluger, Alexander. Nonparametric estimation of the service time distribution in the M/G/∞ queue. Adv. in Appl. Probab. 48 (2016), no. 4, 1117--1138.

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  • Beneš, V. E. (1957). Fluctuations of telephone traffic. Bell System Tech. J. 36, 965–973.
  • Bingham, N. H. and Dunham, B. (1997). Estimating diffusion coefficients from count data: Einstein–Smoluchowski theory revisited. Ann. Inst. Statist. Math. 49, 667–679.
  • Bingham, N. H. and Pitts, S. M. (1999). Non-parametric estimation for the M/G/$\infty$ queue. Ann. Inst. Statist. Math. 51, 71–97.
  • Blanghaps, N., Nov, Y. and Weiss, G. (2013). Sojourn time estimation in an M/G/$\infty$ queue with partial information. J. Appl. Prob. 50, 1044–1056.
  • Borovkov, A. A. (1967). Limit laws for queueing processes in multichannel systems. Siberian Math. J. 8, 746–763.
  • Brillinger, D. R. (1974). Cross-spectral analysis of processes with stationary increments including the stationary $G/G/\infty$ queue. Ann. Prob. 2, 815–827.
  • Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods, 2nd edn. Springer, New York.
  • Brown, M. (1970). An $M/G/\infty$ estimation problem. Ann. Math. Statist. 41, 651–654.
  • Goldenshluger, A. (2015). Nonparametric estimation of service time distribution in the $M/G/\infty$ queue and related estimation problems. Preprint. Available at
  • Goldenshluger, A. and Nemirovski, A. (1997). On spatially adaptive estimation of nonparametric regression. Math. Meth. Statist. 6, 135–170.
  • Grübel, R. and Wegener, H. (2011). Matchmaking and testing for exponentiality in the M/G/$\infty$ queue. J. Appl. Prob. 48, 131–144.
  • Hall, P. and Park, J. (2004). Nonparametric inference about service time distribution from indirect measurements. J. R. Statist. Soc. B 66, 861–875.
  • Iglehart, D. L. (1973). Weak convergence in queueing theory. Adv. Appl. Prob. 5, 570–594.
  • Kingman, J. F. C. (1993). Poisson Processes. Oxford University Press.
  • Lindley, D. V. (1956). The estimation of velocity distributions from counts. In Proceedings of the International Congress of Mathematicians (Amsterdam, 1954), Vol. III, Noordhoff, Groningen, pp. 427–444.
  • Milne, R. K. (1970). Identifiability for random translations of Poisson processes. Z. Wahrscheinlichkeitsth. 15, 195–201.
  • Moulines, E., Roueff, F., Souloumiac, A. and Trigano, T. (2007). Nonparametric inference of photon energy distribution from indirect measurement. Bernoulli 13, 365–588.
  • Nemirovski, A. (2000). Topics in non-parametric statistics. In Lectures on Probability Theory and Statistics (Saint-Flour, 1998; Lecture Notes Math. 1738), Springer, Berlin, pp. 85–277.
  • Parzen, E. (1962). Stochastic Processes. Holden-Day, San Francisco, CA.
  • Pickands, J., III and Stine, R. A. (1997). Estimation for an M/G/$\infty$ queue with incomplete information. Biometrika 84, 295–308.
  • Reynolds, J. F. (1975). The covariance structure of queues and related processes–-a survey of recent work. Adv. Appl. Prob. 7, 383–415.
  • Ross, S. M. (1970). Applied Probability Models with Optimization Applications. Holden-Day, San Francisco, CA.
  • Schweer, S. and Wichelhaus, C. (2015). Nonparametric estimation of the service time distribution in the discrete-time $GI/G/\infty$ queue with partial information. Stoch. Process. Appl. 125, 233–253.
  • Tsybakov, A. B. (2009). Introduction to Nonparametric Estimation. Springer, New York.
  • Whitt, W. (1985). The renewal-process stationary-excess operator. J. Appl. Prob. 22, 156–167.
  • Whitt, W. (1974). Heavy traffic limit theorems for queues: a survey. In Mathematical Methods in Queueing Theory (Proc. Conf., Western Michigan Univ., Kalamazoo, MI, 1973; Lecture Notes Econom. Math. Systems 98), Springer, Berlin, pp. 307–350. \endharvreferences