Abstract
Leta ands denote the inter arrival times and service times in aGI/GI/1 queue. Let a(n), s(n) be the r.v.s. with distributions as the estimated distributions ofa ands from iid samples ofa ands of sizesn. Letw be a r.v. with the stationary distribution π of the waiting times of the queue with input(a,s). We consider the problem of estimatingE[w α], α> 0 and α via simulations when (a (n),s(n)) are used as input. Conditions for the accuracy of the asymptotic estimate, continuity of the asymptotic variance and uniformity in the rate of convergence to the estimate are obtained. We also obtain rates of convergence for sample moments, the empirical process and the quantile process for the regenerative processes. Robust estimates are also obtained when an outlier contaminated sample ofa ands is provided. In the process we obtain consistency, continuity and asymptotic normality of M-estimators for stationary sequences. Some robustness results for Markov processes are included.
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References
S. Asmussen,Applied Probability and Queues (Wiley, New York, 1987).
S. Asmussen, Queueing simulation in heavy traffic, Math. Oper. Res. 17 (1992) 84–111.
J.-P. Aubin,Set Valued Analysis (Birkhauser, Boston, 1990).
B. Bank, J. Guddat, D. Klatte, B. Kummer and K. Tammer,Nonlinear Parametric Optimization (Academic Verlag, Berlin, 1982).
I.V. Basawa, Large sample statistics for stochastic processes: some recent developments, in:Probability, Statistics and Design of Experiments, Proc. of R.C. Bose Symp. (Wiley Eastern, New Delhi, 1988).
I.V. Basawa and N.U. Prabhu, Large sample inference from single server queues, Queueing Systems 3 (1988) 298–304.
I.V. Basawa and B.L.S. Prakasa Rao,Statistical Inference for Stochastic Processes (Academic Press, New York, 1980).
S.J. Bean and C.P. Tsokos, Developments in nonparametric density estimation, Int. Stat. Rev. 48 (1980) 267–287.
R.J. Beran, L. LeCam and P.W. Millar, Convergence of stochastic empirical measures, J. Multivariate Anal. 23 (1987) 159–168.
U.N. Bhat and S.S. Rao, Statistical analysis of queueing systems, Queueing Systems 1 (1987) 217–247.
M. Czorgo, P. Deheuvels and L. Horvath, An application of stopped sums with applications in queueing theory, Adv. Appl. Prob. 19 (1987) 674–690.
M. Czorgo and Revesz,Strong Approximations in Probability and Statistics (Academic Press, New York, 1981).
D.L. Donoho and R.C. Liu, The automatic robustness of minimum distance estimators, Ann. Stat. 16 (1988) 552–586.
D.L. Donoho and R.C. Liu, Pathologies of minimum distance estimators, Ann. Stat. 16 (1988) 587–608.
S.N. Ethier and T.G. Kurtz,Markov Processes, Characterisation and Convergence (Wiley, New York, 1986).
P.W. Glynn and W. Whitt, The asymptotic efficiency of simulation estimators, Oper. Res. 40 (1992) 505–520.
A. Gut,Stopped Random Walks, Limit Theorems and Applications (Springer, New York, 1988).
F.R. Hampel, P.J. Rousseeuw, E.M. Ronchetti and W.A. Stahel,Robust Statistics (Wiley, New York, 1986).
C.R. Heathcote, Complete exponential convergence and some related topics, J. Appl. Prob. 4 (1967) 217–256.
P.J. Huber,Robust Statistics (Wiley, New York, 1981).
S. Janson, Renewal theory form-dependent variables, Ann. Prob. 11 (1983) 558–568.
V.V. Kalashnikov, Assessing the sensitivity of queueing system, Avtomat. i Telemekh. (May 1981).
V.V. Kalashnikov,Mathematical Methods in Queueing Theory (Kluwer, Dordrecht, 1994).
V.V. Kalashnikov and Rachev,Mathematical Methods for Construction of Queueing Models (Wadsworth and Brooks, Pacific Grove, 1990).
N.V. Kartashov, Inequalities in theorems of ergodicity and stability for Markov chains with common phase space I, Theory Prob. Appl. 30 (1985) 247–259.
A.J. King and R.T. Rockafellar, Asymptotic theory for solutions in statistical estimation and stochastic progress, Math. Oper. Res. 18 (1993) 148–162.
S.S. Lavenberg and P.D. Welch, A perspective on the use of control variables to increase the efficiency of Monto Carlo simulations, Manag. Sci. 28 (1981) 322–335.
Lebroux, Consistent estimation of a mixing distribution, Ann. Stat. 20 (1992) 1350–1360.
E.L. Lehmann,Theory of Point Estimation (Wiley, New York, 1983).
R. Lepage and L. Billard,Exploring Limits of Boot Strap (Wiley, New York, 1992).
R.D. Martin and V.J. Yohai, Influence functionals for time series, Ann. Stat. 14 (1986) 781–818.
K. Pawlikowski, Steady-state simulation of queueing processes: a survey of problems and solutions, ACM Comp. Surv. 22 (1990) 123–170.
W. Philipp, Invariance principles for independent and weakly dependent random variables, in:Dependence in Probability and Stat., eds. E. Eberlain and M.S. Taqqu (Obevolfach, 1985).
B.L.S. Prakasa Rao,Asymptotic Theory of Statistical Inference (Wiley, New York, 1987).
R. Redner and H.F. Walker, Mixture densities, maximum likelihood and the EM algorithm, SIAM Rev. 26 (1984) 195–239.
R. Serfozo, Convergence of Lebesgue integrals with varying measures, Sankhya, Series A 44 (1982) 380–402.
J.G. Shanthikumar, Bounds and an approximation for single server queues, J. Oper. Res. Soc. Jpn. 26 (1983)118–134.
V. Sharma, Invariance principles for regenerative and Markov processes with applications to queueing networks, Technical Report ISRO-IISc STC (1993).
D. Thiruvaiyaru and I.V. Basawa, Empirical Bayes estimation for queueing systems and networks, Queueing Systems 11 (1992) 179–202.
J. Tiefeng, Large deviations for renewal processes, Stoch. Proc. Appl. 50 (1994) 57–71.
N.M. Van Dijk, Controlled Markov processes, time discretization, CWI Tracts (1984).
J.B. Wets Roger, Constrained estimation: consistency and asymptotics, Appl. Stoch. Meth. Data Anal. 7 (1991) 17–32.
W. Whitt, Planning queueing simulations, Manag. Sci. 35 (1989) 1341–1365.
W. Whitt, Asymptotic formulas for Markov processes with applications to simulation, Oper. Res. 40 (1992) 279–291.
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Sharma, V. Reliable estimation via simulation. Queueing Syst 19, 169–192 (1995). https://doi.org/10.1007/BF01148945
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DOI: https://doi.org/10.1007/BF01148945