×

Birth and death processes in interactive random environments. (English) Zbl 1515.60292

Summary: This paper studies birth and death processes in interactive random environments where the birth and death rates and the dynamics of the state of the environment are dependent on each other. Two models of a random environment are considered: a continuous-time Markov chain (finite or countably infinite) and a reflected (jump) diffusion process. The background is determined by a joint Markov process carrying a specific interactive mechanism, with an explicit invariant measure whose structure is similar to a product form. We discuss a number of queueing and population-growth models and establish conditions under which the above-mentioned invariant measure can be derived. Next, an analysis of the rate of convergence to stationarity is performed for the models under consideration. We consider two settings leading to either an exponential or a polynomial convergence rate. In both cases we assume that the underlying environmental Markov process has an exponential rate of convergence, but the convergence rate of the joint Markov process is determined by certain conditions on the birth and death rates. To prove these results, a coupling method turns out to be useful.

MSC:

60K25 Queueing theory (aspects of probability theory)
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60J60 Diffusion processes
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Bacaër, N.; Ed-Darraz, A., On linear birth-and-death processes in a random environment, J. Math. Biol., 69, 1, 73-90 (2014) · Zbl 1301.60111
[2] Browne, S.; Whitt, W., Piecewise-linear diffusion processes, Adv. Queueing Theory Methods Open Problems, 4, 463-480 (1995) · Zbl 0845.60087
[3] Butkovsky, O., Subgeometric rates of convergence of Markov processes in the Wasserstein metric, Ann. Appl. Probab., 24, 2, 526-552 (2014) · Zbl 1304.60076
[4] Cogburn, R., Markov chains in random environments: the case of Markovian environments, Ann. Probab., 8, 5, 908-916 (1980) · Zbl 0444.60053
[5] Cogburn, R.; Torrez, WC, Birth and death processes with random environments in continuous time, J. Appl. Probab., 18, 1, 19-30 (1981) · Zbl 0453.60073
[6] Cornez, R., Birth and death processes in random environments with feedback, J. Appl. Probab., 24, 1, 25-34 (1987) · Zbl 0619.60081
[7] Das, A., Constructions of Markov processes in random environments which lead to a product form of the stationary measure, Markov Process. Related Fields, 23, 2, 211-232 (2017) · Zbl 1379.60086
[8] Dieker, A.; Moriarty, J., Reflected Brownian motion in a wedge: sum-of-exponential stationary densities, Electron. Commun. Probab., 14, 1-16 (2009) · Zbl 1190.60077
[9] Douc, R.; Fort, G.; Guillin, A., Subgeometric rates of convergence of \(f\)-ergodic strong Markov processes, Stoch. Process. Appl., 119, 3, 897-923 (2009) · Zbl 1163.60034
[10] Economou, A., Generalized product-form stationary distributions for Markov chains in random environments with queueing applications, Adv. Appl. Probab., 37, 1, 185-211 (2005) · Zbl 1064.60163
[11] Feller, W., An Introduction to Probability Theory and Its Applications (1950), Hoboken: Wiley, Hoboken · Zbl 0039.13201
[12] Gannon, M.; Pechersky, E.; Suhov, Y.; Yambartsev, A., Random walks in a queueing network environment, J. Appl. Probab., 53, 2, 448-462 (2016) · Zbl 1344.60086
[13] Gersende, F.; Roberts, GO, Subgeometric ergodicity of strong Markov processes, Ann. Appl. Probab., 15, 2, 1565-1589 (2005) · Zbl 1072.60057
[14] Guillemin, F.; Simonian, A., Transient characteristics of an \(M/M/\infty\) system, Adv. Appl. Probab., 27, 3, 862-888 (1995) · Zbl 0829.60058
[15] Harrison, JM; Reiman, MI, On the distribution of multidimensional reflected Brownian motion, SIAM J. Appl. Math., 41, 2, 345-361 (1981) · Zbl 0464.60081
[16] Karatzas, I., Shreve, S.: Stochastic Calculus and Brownian Motion. Springer (1991) · Zbl 0734.60060
[17] Kou, SC; Kou, SG, Modeling growth stocks via birth-death processes, Adv. Appl. Probab., 35, 3, 641-664 (2003) · Zbl 1040.60057
[18] Krenzler, R.; Daduna, H., Loss systems in a random environment: steady state analysis, Queueing Syst., 80, 1, 127-153 (2015) · Zbl 1318.60094
[19] Krenzler, R.; Daduna, H.; Otten, S., Jackson networks in nonautonomous random environments, Adv. Appl. Probab., 48, 2, 315-331 (2016) · Zbl 1343.60133
[20] Kurtz, T.; Stockbridge, R., Stationary solutions and forward equations for controlled and singular martingale problems, Electron. J. Probab., 6, 1-52 (2001) · Zbl 0984.60086
[21] Lindvall, T., A note on coupling of birth and death processes, J. Appl. Probab., 16, 3, 505-512 (1979) · Zbl 0423.60034
[22] Lindvall, T., Lectures on the Coupling Method (1992), New York: Dover, New York · Zbl 0760.60078
[23] Liu, Y.; Zhang, H.; Zhao, Y., Subgeometric ergodicity for continuous-time Markov chains, J. Math. Anal. Appl., 368, 1, 178-189 (2010) · Zbl 1197.60076
[24] Mao, Y., Ergodic degrees for continuous-time Markov chains, Sci. China Ser. A Math., 47, 2, 161-174 (2004) · Zbl 1067.60069
[25] Mazumdar, RR; Guillemin, FM, Forward equation for reflected diffusions with jumps, Appl. Math. Optim., 33, 81-102 (1996) · Zbl 0841.60062
[26] Meyn, SP; Tweedie, RL, Stability of Markovian processes I: criteria for discrete-time chains, Adv. Appl. Probab., 24, 3, 542-574 (1992) · Zbl 0757.60061
[27] Meyn, SP; Tweedie, RL, Stability of Markovian processes II: continuous-time processes and sampled chains, Adv. Appl. Probab., 25, 3, 487-517 (1993) · Zbl 0781.60052
[28] Meyn, SP; Tweedie, RL, Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes, Adv. Appl. Probab., 25, 3, 518-548 (1993) · Zbl 0781.60053
[29] Norris, JR, Markov Chains (1997), Cambridge: Cambridge University Press, Cambridge · Zbl 0873.60043
[30] Otten, S., Krenzler, R., Daduna, H., Kruse, K.: Queues in a random environment. arXiv:2006.15712 (2020)
[31] Pang, G.; Sarantsev, A.; Belopolskaya, Y.; Suhov, Y., Stationary distributions and convergence for \(M/M/1\) queues in interactive random environment, Queueing Syst., 94, 3, 357-392 (2020) · Zbl 1439.60088
[32] Prodhomme, A., Strickler, É.: Large population asymptotics for a multitype stochastic SIS epidemic model in randomly switched environment. arXiv:2107.05333 (2021)
[33] Ross, SM, Introduction to Probability Models (2019), London: Academic Press, London · Zbl 1408.60002
[34] Sandrić, N.; Arapostathis, A.; Pang, G., Subexponential upper and lower bounds in Wasserstein distance for Markov processes, Appl. Math. Optim., 85, 3, 1-45 (2022) · Zbl 1497.60097
[35] Sarantsev, A., Explicit rates of exponential convergence for reflected jump-diffusions on the half-line, ALEA Latin Am. J. Probab. Math. Stat., 13, 1069-1093 (2016) · Zbl 1356.60122
[36] Sarantsev, A., Penalty method for obliquely reflected diffusions, Lith. Math. J., 61, 518-549 (2021) · Zbl 07441583
[37] Sarantsev, A., Sub-exponential rate of convergence to equilibrium for processes on the half-line, Stat. Probab. Lett., 175 (2021) · Zbl 1474.60181
[38] Soukhov, IM; Kelbert, M., Probability and Statistics by Example: Markov Chains: A Primer in Random Processes and Their Applications (2008), Cambridge: Cambridge University Press, Cambridge · Zbl 1184.62002
[39] Stadie, W., The busy period of the queueing system \(M/G/\infty \), J. Appl. Probab., 22, 697-704 (1985) · Zbl 0573.60087
[40] Stroock, DW; Varadhan, SS, Diffusion processes with boundary conditions, Commun. Pure Appl. Math., 24, 2, 147-225 (1971) · Zbl 0227.76131
[41] Tanaka, H., Stochastic differential equations with reflecting boundary condition in convex regions, Hiroshima Math. J., 9, 1, 163-177 (1979) · Zbl 0423.60055
[42] Torrez, WC, The birth and death chain in a random environment: instability and extinction theorems, Ann. Probab., 6, 6, 1026-1043 (1978) · Zbl 0392.60049
[43] Torrez, WC, Calculating extinction probabilities for the birth and death chain in a random environment, J. Appl. Probab., 16, 4, 709-720 (1979) · Zbl 0423.60070
[44] Van Doorn, EA, Conditions for exponential ergodicity and bounds for the decay parameter of a birth-death process, Adv. Appl. Probab., 17, 3, 514-530 (1985) · Zbl 0597.60080
[45] Van Doorn, EA, Rate of convergence to stationarity of the system \(M/M/N/N+ R\), Theory Probab., 19, 2, 336-350 (2011) · Zbl 1263.60081
[46] Van Doorn, EA; Zeifman, AI, On the speed of convergence to stationarity of the Erlang loss system, Queueing Syst., 63, 1, 241-252 (2009) · Zbl 1209.90122
[47] van Doorn, EA; Zeifman, AI; Panfilova, TL, Bounds and asymptotics for the rate of convergence of birth-death processes, Theory Probab. Appl., 54, 1, 97-113 (2010) · Zbl 1204.60083
[48] Ward, AR; Glynn, PW, Properties of the reflected Ornstein-Uhlenbeck process, Queueing Syst., 44, 2, 109-123 (2003) · Zbl 1026.60106
[49] Williams, RJ, Semimartingale reflecting Brownian motions in the orthant, IMA Vol. Math. Appl., 71, 125-137 (1995) · Zbl 0827.60031
[50] Zeifman, AI, Some estimates of the rate of convergence for birth and death processes, J. Appl. Probab., 28, 2, 268-277 (1991) · Zbl 0738.60088
[51] Zeifman, AI, Upper and lower bounds on the rate of convergence for nonhomogeneous birth and death processes, Stoch. Process. Appl., 59, 1, 157-173 (1995) · Zbl 0846.60084
[52] Zeifman, AI; Panfilova, TL, On convergence rate estimates for some birth and death processes, J. Math. Sci., 221, 4, 616-623 (2017) · Zbl 1373.82052
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.