L²-Decay rate for non-ergodic Jackson network

Abstract

We establish the additive theorem of L²-decay rate for multidimensional Markov process with independent marginal processes. Using this and the decomposition method, we obtain explicit upper and lower bounds for decay rate of non-ergodic Jackson network. In some cases, we get the exact decay rate.

Keywords

L²-Decay rate / additive theorem / decomposition method / Jackson network

References

[1] Chen M F. Equivalence of exponential ergodicity and L²-exponential convergence for Markov chains. Stochastic Process Appl, 2000, 87(2): 281-297

[2] Chen M F. Explicit bounds of the first eigenvalues. Sci China Ser A, 2000, 43: 1051-1059

[3] Chen M F. From Markov Chains to Non-equilibrium Particle Systems. Singapore: World Scientific, 2004

[4] Chen M F. Speed of stability for birth-death processes. Front Math China, 2010, 5(3): 379-515

[5] Chen M F, Mao Y H. An Introduction to Stochastic Processes. Beijing: Higher Education Press, 2007 (in Cinese)

[6] van Doorn E A. Conditions for exponential ergodicity and bounds for the decay parameter of a birth-death process. Adv Appl Probab, 1985, 17(3): 514-530

[7] Goodman J M, Massey W A. The non-ergodic Jackson network. J Appl Probab, 1984, 21: 860-869

[8] Jackson J R. Networks of waiting lines. Oper Res, 1957, 5: 518-521

[9] Jackson J R. Jobshop-like queuing systems. Manag Sci, 1963, 10: 131-142

[10] Jerrum M, Son J B, Tetali P, Vigoda E. Elementary bounds on Poincaré and log-Sobolev constant for decomposable Markov chains. Ann Appl Probab, 2004, 14: 1741-1765

[11] Kelbert M Ya, Kontsevich M L, Rybko A N. On Jackson networks on countable graphs. Veroyatn Primen, 1988, 33: 379-382

[12] Mao Y H. Nash inequalities for Markov processes in dimension on<?Pub Caret?>e. Acta Math Sin (Engl Ser), 2002, 18(1): 147-156

[13] Mao Y H. General Sobolev type inequalities for symmetric forms. J Math Anal Appl, 2008, 33: 1092-1099

[14] Mao Y H. L^p Poincaré inequality for general symmetric forms. Acta Math Sin (Engl Ser), 2009, 25: 2055-2064

[15] Mao Y H, Ouyang S X. Strong ergodicity and uniform decay for Markov processes. Math Appl (Wuhan), 2006, 19(3): 580-586

[16] Mao Y H, Xia L H. Spectral gap for jump processes by decomposition method. Front Math China, 2009, 4(2): 335-348

[17] Mao Y H, Xia L H. The specrtal gap for quasi-birth and death processes. Acta Math Sin (Engl Ser), 2012, 28(5): 1075-1090

[18] Mao Y H, Xia L H. Spectral gap for open Jackson networks. 2012, Preprint

[19] Meyn S, Tweedie R L. Markov Chains and Stochastic Stability. London: Springer-Verlag, 1993

[20] Wang F Y. Functional inequalities for the decay of sub-Markov semigroups. Potential Anal, 2003, 18: 1-23

[21] Wang F Y. Functional Inequalities, Markov Semigroups and Spectral Theory. Beijing: Science Press, 2005

[22] Yau S T, Schoen R. Lectures on Differential Geometry. Beijing: Higher Education Press, 2004 (in Chinese)

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg