Abstract
A method for a quantitative comparison of wide sense regenerative processes is discussed. The main idea appears to be to make assumptions on the processes being studied that permit one to construct so-called crossing times which are simultaneous regeneration times for another pair of regenerative processes (called crossing), each element of the pair coinciding in distribution with one of the initial processes. Provided that intercrossing times have proper moments (higher, than of the first order), the problem of uniform-in-time comparison is reduced (using renewal-type arguments) to obtaining comparison estimates over finite horizons only. Respective estimates are formulated in terms of probability metrics. Possible applications include continuity of queues, approximation of Markov chains, etc.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anichkin, S.: Rate of convergence estimates in Blackwell's theorem,J. Soviet Math. 40(4) (1988), 449–453.
Kalashnikov, V.: An uniform estimate of the rate of convergence in the discrete time renewal theorem,Probab. Theory Appl. 22(2) (1977), 399–403.
Kalashnikov, V.:Qualitative Analysis of the Behaviour of Complex Systems by the Test Functions Method, Nauka, Moscow (1978) (in Russian).
Kalashnikov, V.: Approximation of stochastic models, inSemi-Markov Models: Theory and Applications, Plenum Press, New York, (1986), 319–336.
Kalashnikov, V.: A complete metric in the space D(0,∞),J. Soviet Math. 47(5) (1989), 2725–2729.
Kalashnikov, V.: Regenerative queueing processes and their qualitative and quantitative analysis,Queueing Systems and Their Applications 6 (1990), 113–136.
Kalashnikov, V.:Mathematical Methods in Queueing Theory, Kluwer Acad. Publ., Dordrecht, 1994.
Kalashnikov, V.:Topics on Regenerative Processes, CRC Press, Boca Raton, 1994.
Kalashnikov, V.: and Rachev, S.:Mathematical Methods for Construction of Queueing Models, Wadsworth & Brooks/Cole, 1990.
Kennedy, D.: The continuity of the single-server queue,J. Appl. Probab. 9(2) (1972), 370–381.
Lindvall, T.: Weak convergence of probability measures and random functions in the function space D(0, ∞),J. Appl. Probab. 10(1) (1973), 109–121.
Lindvall, T.: A probabilistic proof of Blackwell's renewal theorem,Ann. Probab. 5(2) (1977), 482–485.
Lindvall, T.: On coupling of continuous time renewal processes,J. Appl. Probab. 19 (1982), 82–89.
Meyer, P.:Probability and Potentials, Blaisdell Publ. Co., Waltham, 1966.
Shiryaev, A.:Probability, Springer-Verlag, New York, 1984.
Sigman, K. and Wolff, R.: A review of regenerative processesSIAM Rev. 35(2) (1993), 269–288.
Strassen, V.: The existence of probability measures with given marginals,Ann. Math. Statist. 36(2) (1965), 423–439.
Thorisson, H.: The coupling of regenerative processes,Adv. Appl. Probab. 15 (1983), 531–561.
Thorisson, H.: Construction of a stationary regenerative process,Stoch. Proc. Appl. 42(2) (1992), 237–253.
Whitt, W.: The continuity of queues,Adv. Appl. Probab. 6(1) (1974), 175–183.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kalashnikov, V.V. Crossing and comparison of regenerative processes. Acta Applicandae Mathematicae 34, 151–172 (1994). https://doi.org/10.1007/BF00994263
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00994263