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Regenerative derivatives of regenerative sequences

Part of: Probabilistic methods, simulation and stochastic differential equations Special processes Markov processes

Published online by Cambridge University Press:  01 July 2016

Paul Glasserman
Paul Glasserman*
Affiliation:
Columbia University
*
∗Postal address: 403 Uris Hall, Columbia Business School, New York, NY 10027, USA. E-mail address: pglasser@research.gsb.columbia.edu.
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Abstract

Given a parametric family of regenerative processes on a common probability space, we investigate when the derivatives (with respect to the parameter) are regenerative. We primarily consider sequences satisfying explicit, Lipschitz recursions, such as the waiting times in many queueing systems, and show that derivatives regenerate together with the original sequence under reasonable monotonicity or continuity assumptions. The inputs to our recursions are i.i.d. or, more generally, governed by a Harris-ergodic Markov chain. For i.i.d. input we identify explicit regeneration points; otherwise, we use coupling arguments. We give conditions for the expected steady-state derivative to be the derivative of the steady-state mean of the original sequence. Under these conditions, the derivative of the steady-state mean has a cycle-formula representation.

Keywords

REGENERATIVE PROCESSDERIVATIVE ESTIMATIONPERTURBATION ANALYSISQUEUESHARRIS CHAINSCOUPLING

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60J05: Discrete-time Markov processes on general state spaces 65C99: None of the above, but in this section
Type
Research Article
Information
Advances in Applied Probability , Volume 25 , Issue 1 , March 1993 , pp. 116 - 139
Copyright
Copyright © Applied Probability Trust 1993 

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References

[1] Asumussen, S. (1987) Applied Probability and Queues. Wiley, New York.Google Scholar
[2] Asmussen, S. and Thorisson, H. (1987) A Markov chain approach to periodic queues. J. Appl. Prob. 24, 215–225.Google Scholar
[3] Athreya, K. B. and Ney, P. (1978) A new approach to the limit theory of recurrent Markov chains. Trans. Amer. Math. Soc. 245, 493–501.CrossRefGoogle Scholar
[4] Baccelli, F., Makowski, A. and Shwartz, A. (1989) The fork-join queue and related systems with synchronization constraints: stochastic ordering and computable bounds. Adv. Appl. Prob. 21, 629–660.CrossRefGoogle Scholar
[5] Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
[6] Borovkov, A. A. (1987) Limit theorems for queueing networks, II Theory Prob. Appl. 32, 257–272.Google Scholar
[7] Brandt, A. (1986) The stochastic difference equation Yn + 1 =An Yn + Bn with stationary coefficients. Adv. Appl. Prob. 18, 211–220.Google Scholar
[8] Charlot, F., Ghidouche, M. and Hamami, M. (1978) Irréducibilité et recurrence au sens de Harris des ‘Temps d'attente’ des files GI/G/q. Z. Wahrscheinklichkeitsth. 43, 187–203.CrossRefGoogle Scholar
[9] Chen, H. and Yao, D. D. (1990) Derivatives of the expected delay in the GI/G/1 queue. J. Appl. Prob. 27, 899–907.CrossRefGoogle Scholar
[10] Fu, M. C. and Hu, J. Q. (1991) Consistency of infinitesimal perturbation analysis for the GI/G/m queue. Eur. J. Operat. Res. 54, 121–139.CrossRefGoogle Scholar
[11] Glasserman, P. (1991) Stationary waiting-time derivatives. QUESTA. To appear.Google Scholar
[12] Glasserman, P., Hu, J. Q. and Strickland, S. G. (1991) Strongly consistent steady-state derivative estimates. Prob. Eng. Inf. Sci. 5, 415–428.Google Scholar
[13] Heidelberger, P., Cao, X. R., Zazanis, M. and Suri, R. (1988) Convergence properties of infinitesimal perturbation analysis estimates. Management Sci. 34, 1281–1302.Google Scholar
[14] Hu, J. Q. (1990) Strong consistency of infinitesimal perturbation analysis for the G/G/1 queue. Technical Report, Harvard University Division of Applied Sciences.Google Scholar
[15] Hu, J. Q. (1992) Convexity of sample path performance and strong consistency of infinitesimal perturbation analysis. IEEE Trans. Autom. Control 37, 258–262.Google Scholar
[16] Kiefer, J. and Wolfowitz, J. (1955) On the theory of queues with many servers. Trans. Amer. Math. Soc. 78, 1–18.Google Scholar
[17] Konstantopoulos, T. and Walrand, J. (1989) Stationarity and stability of fork-join queues. J. Appl. Prob. 26, 604–614.Google Scholar
[18] Loynes, R. M. (1962) The stability of a queue with non-independent interarrival and service times. Proc. Camb. Phil. Soc. 58, 497–520.CrossRefGoogle Scholar
[19] Nummelin, E. (1978) A splitting technique for Harris recurrent Markov chains. Z. Wahrscheinlichkeitsth. 43, 309–318.CrossRefGoogle Scholar
[20] Nummelin, E. (1981) Regeneration in tandem queues. Adv. Appl. Prob. 13, 221–230.Google Scholar
[21] Nummelin, E. (1984) General Irreducible Markov Chains and Non-negative Operators. Cambridge University Press.CrossRefGoogle Scholar
[22] Rockafellar, R. T. (1972) Convex Analysis. Princeton University Press.Google Scholar
[23] Royden, H. L. (1968) Real Analysis, 2nd edn. Macmillan, New York.Google Scholar
[24] Sigman, K. (1988) Queues as Harris recurrent Markov chains. QUESTA 3, 179–198.Google Scholar
[25] Sigman, K. (1988) Regeneration in tandem queues with multiserver stations. J. Appl. Prob. 25, 391–403.CrossRefGoogle Scholar
[26] Sigman, K. (1989) Notes on the stability of closed queueing networks. J. Appl. Prob. 26, 678–682. Correction 27 (1990), 735.Google Scholar
[27] Suri, R. and Zazanis, M. (1988) Perturbation analysis gives strongly consistent estimates for the M/G/1 queue. Management Sci. 34, 39–64.Google Scholar
[28] Thorisson, H. (1983) The coupling of regenerative processes. Adv. Appl. Prob. 15, 531–561.Google Scholar
[29] Zazanis, M. (1987) Weak convergence of sample path derivatives for the waiting time in a single-server queue. Proc. 25th Allerton Conf., 297–304.Google Scholar
[30] Zazanis, M. and Suri, R. (1986) Perturbation analysis of the GI/G/1 queue. Working Paper, Northwestern University.Google Scholar