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Hitting Times and the Running Maximum of Markovian Growth-Collapse Processes

Part of: Limit theorems Markov processes Special processes

Published online by Cambridge University Press:  14 July 2016

Andreas Löpker  and
Wolfgang Stadje
Andreas Löpker*
Affiliation:
Eindhoven University of Technology and EURANDOM
Wolfgang Stadje*
Affiliation:
University of Osnabrück
*
∗Current address: Department of Economics and Social Sciences, Helmut Schmidt University, PO Box 700822, 22008 Hamburg, Germany. Email address: lopker@hsu-hh.de
∗∗Postal address: Department of Mathematics and Computer Science, University of Osnabrück, 49069 Osnabrück, Germany. Email address: wolfgang@mathematik.uos.de
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Abstract

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We consider the level hitting times τy = inf{t ≥ 0 | Xt = y} and the running maximum process Mt = sup{Xs | 0 ≤ s ≤ t} of a growth-collapse process (Xt)t≥0, defined as a [0, ∞)-valued Markov process that grows linearly between random ‘collapse’ times at which downward jumps with state-dependent distributions occur. We show how the moments and the Laplace transform of τy can be determined in terms of the extended generator of Xt and give a power series expansion of the reciprocal of Ee−sτy. We prove asymptotic results for τy and Mt: for example, if m(y) = Eτy is of rapid variation then Mt / m-1(t) →w 1 as t → ∞, where m-1 is the inverse function of m, while if m(y) is of regular variation with index a ∈ (0, ∞) and Xt is ergodic, then Mt / m-1(t) converges weakly to a Fréchet distribution with exponent a. In several special cases we provide explicit formulae.

Keywords

Growth-collapse processpiecewise deterministic Markov processhitting timerunning maximumasymptotic behaviorregular variationseparable jump measure

MSC classification

Secondary: 60K30: Applications (congestion, allocation, storage, traffic, etc.) 60J25: Continuous-time Markov processes on general state spaces 60J75: Jump processes 60F05: Central limit and other weak theorems
Type
Research Article
Information
Journal of Applied Probability , Volume 48 , Issue 2 , June 2011 , pp. 295 - 312
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] Altman, E., Avrachenkov, K., Kherani, A. A. and Prabhu, B. J. (2005). Performance analysis and stochastic stability of congestion control protocols. In INFOCOM '05, Vol. 2 (March 2005), pp. 1316–1327.Google Scholar
[2] Asmussen, S. (1998). Extreme value theory for queues via cycle maxima. Extremes 1, 137–168.Google Scholar
[3] Asmussen, S. (2003). Applied Probability and Queues, 2nd edn. Springer, New York.Google Scholar
[4] Bertoin, J. and Yor, M. (2005). Exponential functionals of Lévy processes. Prob. Surveys 2, 191–212.Google Scholar
[5] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation (Encyclopedia Math. Appl. 27). Cambridge University Press.Google Scholar
[6] Borovkov, K. and Last, G. (2008). On level crossings for a general class of piecewise-deterministic Markov processes. Adv. Appl. Prob. 40, 815–834.Google Scholar
[7] Borovkov, K. and Vere-Jones, D. (2000). Explicit formulae for stationary distributions of stress release processes. J. Appl. Prob. 37, 315–321.CrossRefGoogle Scholar
[8] Boxma, O., Perry, D., Stadje, W. and Zacks, S. (2006). A Markovian growth-collapse model. Adv. Appl. Prob. 38, 221–243.Google Scholar
[9] Carmona, P., Petit, F. and Yor, M. (2001). Exponential functionals of Lévy processes. In Lévy Processes, Barndorff-Nielsen, O. E. et al. Birkhäuser, Boston, MA, pp. 41–55.Google Scholar
[10] Davis, M. H. A. (1984). Piecewise deterministic Markov processes: a general class of non-diffusion stochastic models. J. R. Statist. Soc. B 46, 353–388.Google Scholar
[11] Davis, M. H. A. (1993). Markov Models and Optimization (Monogr. Statist. Appl. Prob. 49). Chapman and Hall, London.Google Scholar
[12] Djurčić, D. and Torgašev, A. (2007). Some asymptotic relations for the generalized inverse. J. Math. Anal. Appl. 335, 1397–1402.Google Scholar
[13] Dumas, V., Guillemin, F. and Robert, P. (2002). A Markovian analysis of additive-increase multiplicative-decrease algorithms. Adv. Appl. Prob. 34, 85–111.CrossRefGoogle Scholar
[14] Eliazar, I. and Klafter, J. (2004). A growth-collapse model: Lévy inflow, geometric crashes, and generalized Ornstein-Uhlenbeck dynamics. Physica A 334, 1–21.Google Scholar
[15] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. John Wiley, New York.Google Scholar
[16] Guillemin, F., Robert, P. and Zwart, B. (2004). AIMD algorithms and exponential functionals. Ann. Appl. Prob. 14, 90–117.Google Scholar
[17] Kalashnikov, V. (1997). Geometric Sums: Bounds for Rare Events with Applications. Kluwer, Dordrecht.Google Scholar
[18] Keilson, J. (1966). A limit theorem for passage times in ergodic regenerative processes. Ann. Math. Statist. 37, 866–870.CrossRefGoogle Scholar
[19] Kella, O. and Stadje, W. (2001). On hitting times for compound Poisson dams with exponential Jumps and linear release rate. J. Appl. Prob. 38, 781–786.Google Scholar
[20] Löpker, A. H. and van Leeuwaarden, J. S. H. (2008). Transient moments of the TCP window size process. J. Appl. Prob. 45, 163–175.Google Scholar
[21] Maulik, K. and Zwart, B. (2006). Tail asymptotics for exponential functionals of Lévy processes. Stoch. Process. Appl. 116, 156–177.Google Scholar
[22] Ott, T. J. and Kemperman, J. H. B. (2008). Transient behavior of processes in the TCP paradigm. Prob. Eng. Inf. Sci. 22, 431–471.Google Scholar
[23] Ott, T. J. and Swanson, J. (2007). Asymptotic behavior of a generalized TCP congestion avoidance algorithm. J. Appl. Prob. 44, 618–635.Google Scholar
[24] Palmowski, Z. and Rolski, T. (2002). A technique for exponential change of measure for Markov processes. Bernoulli 8, 767–785.Google Scholar
[25] Rootzén, H. (1988). Maxima and exceedances of stationary Markov chains. Adv. Appl. Prob. 20, 371–390.CrossRefGoogle Scholar
[26] Van Leeuwaarden, J. S. H., Löpker, A. H. and Janssen, A. J. E. M. (2010). Connecting renewal age processes with M/D/1 and M/D/∞ queues through stick breaking. Stoch. Models 26, 141–163.Google Scholar
[27] Yafaev, D. R. (1985). On the asymptotics of solutions of Volterra integral equations. Ark. Mat. 23, 185–201.Google Scholar
[28] Zheng, X. G. (1991). Ergodic theorems for stress release processes. Stoch. Process. Appl. 37, 239–258.Google Scholar