## Bernoulli

• Bernoulli
• Volume 24, Number 4A (2018), 2531-2568.

### The $M/G/\infty$ estimation problem revisited

Alexander Goldenshluger

#### Abstract

The subject of this paper is the $M/G/\infty$ estimation problem: the goal is to estimate the service time distribution $G$ of the $M/G/\infty$ queue from the arrival–departure observations without identification of customers. We develop estimators of $G$ and derive exact non-asymptotic expressions for their mean squared errors. The problem of estimating the service time expectation is addressed as well. We present some numerical results on comparison of different estimators of the service time distribution.

#### Article information

Source
Bernoulli, Volume 24, Number 4A (2018), 2531-2568.

Dates
Revised: December 2016
First available in Project Euclid: 26 March 2018

https://projecteuclid.org/euclid.bj/1522051217

Digital Object Identifier
doi:10.3150/17-BEJ936

Mathematical Reviews number (MathSciNet)
MR3779694

Zentralblatt MATH identifier
06853257

#### Citation

Goldenshluger, Alexander. The $M/G/\infty$ estimation problem revisited. Bernoulli 24 (2018), no. 4A, 2531--2568. doi:10.3150/17-BEJ936. https://projecteuclid.org/euclid.bj/1522051217

#### References

• [1] Beneš, V.E. (1957). Fluctuations of telephone traffic. Bell Syst. Tech. J. 36 965–973.
• [2] Bingham, N.H. and Dunham, B. (1997). Estimating diffusion coefficients from count data: Einstein–Smoluchowski theory revisited. Ann. Inst. Statist. Math. 49 667–679.
• [3] Bingham, N.H. and Pitts, S.M. (1999). Non-parametric estimation for the $M/G/\infty$ queue. Ann. Inst. Statist. Math. 51 71–97.
• [4] Blanghaps, N., Nov, Y. and Weiss, G. (2013). Sojourn time estimation in an ${M}/{G}/\infty$ queue with partial information. J. Appl. Probab. 50 1044–1056.
• [5] Brillinger, D.R. (1974). Cross-spectral analysis of processes with stationary increments including the stationary $G/G/\infty$ queue. Ann. Probab. 2 815–827.
• [6] Brillinger, D.R. (1975). Statistical Inference for Stationary Point Processes 55–99. New York: Academic Press.
• [7] Brillinger, D.R. (1975). The identification of point process systems. Ann. Probab. 3 909–929.
• [8] Brillinger, D.R. (1976). Estimation of the second-order intensities of a bivariate stationary point process. J. Roy. Statist. Soc. Ser. B 38 60–66.
• [9] Brown, M. (1970). An $M/G/\infty$ estimation problem. Ann. Math. Stat. 41 651–654.
• [10] Cox, D.R. and Lewis, P.A.W. (1972). Multivariate point processes. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. III: Probability Theory 401–448. Berkeley: Univ. California Press.
• [11] Daley, D.J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes. Vol I: Elementary Theory and Methods, 2nd ed. Probability and Its Applications (New York). New York: Springer.
• [12] Doob, J.L. (1953). Stochastic Processes. New York: Wiley.
• [13] Goldenshluger, A. (2016). Nonparametric estimation of the service time distribution in the $M/G/\infty$ queue. Adv. in Appl. Probab. 48 1117–1138.
• [14] Hall, P. and Park, J. (2004). Nonparametric inference about service time distribution from indirect observations. J. R. Statist. Soc. B 66 861–875.
• [15] Kingman, J.F.C. (1993). Poisson Processes. Oxford: Clarendon Press.
• [16] Lindley, D.V. (1956). The estimation of velocity distributions from counts. In Proceedings of the International Congress of Mathematicians, 1954, Amsterdam, Vol. III 427–444. North-Holland: Erven P. Noordhoff N.V.
• [17] Mandjes, M. and Zuraniewski, P. (2011). $M/G/\infty$ transience, and its applications to overload detection. Perform. Eval. 68 507–527.
• [18] Milne, R.K. (1970). Identifiability for random translations of Poisson processes. Z. Wahrsch. Verw. Gebiete 15 195–201.
• [19] Milne, R.K. and Westcott, M. (1972). Further results for Gauss–Poisson processes. Adv. in Appl. Probab. 4 151–176.
• [20] Mori, T. (1975). Ergodicity and identifiability for random translations of stationary point processes. J. Appl. Probab. 12 734–743.
• [21] Newman, D.S. (1970). A new family of point processes which are characterized by their second moment properties. J. Appl. Probab. 7 338–358.
• [22] Petty, K.F., Bickel, P., Ostland, M., Rice, J., Schoenberg, F. and Ritov, Y. (1998). Accurate estimation of travel times from single–loop detectors. Transp. Res., Part A Policy Pract. 32 1–17.
• [23] Pickands, J. III and Stine, R.A. (1997). Estimation for an $M/G/\infty$ queue with incomplete information. Biometrika 84 295–308.
• [24] Rothschild, L. (1953). A new method for measuring the activity of spermatozoa. J. Exp. Biol. 30 178–199.
• [25] Schweer, S. and Wichelhaus, C. (2015). Nonparametric estimation of the service time distribution in the discrete-time $GI/G/\infty$ queue with partial information. Stochastic Process. Appl. 125 233–253.