Abstract
In this paper, we show that a positive recurrent fluid queue is automatically V-uniformly ergodic for some function V ≥ 1 but never uniformly ergodic. This reveals a similarity of ergodicity between a fluid queue and a quasi-birth-and-death process. As a byproduct of V-uniform ergodicity, we derive computable bounds on the exponential moments of the busy period.
Similar content being viewed by others
References
S Asmussen. Busy period analysis, rare events and transient behavior in fulidflow models, J Appl Math Stoch Anal, 1994, 7(3): 269–299.
A Badescu, L Breuer, G Latouche. Risk processes analyzed asfluid queues, J Scand Actuarial, 2005, 2: 127–141.
B Cloez, M Hairer. Exponential ergodicity for Markov processes with random switching, Bernoul-li, 2015, 21(1): 505–536.
A da Silva Soares. Fluid queues: building upon the analogy with QBD processes. Doctoral Dissertation, Available at: http://theses.ulb.ac.be:8000/ETD-db/collection/available/ULBetd-03102005-160113/, 2005.
M Davis. Markov Models and Optimization, Chapman and Hall, London, 1993.
S Dendievel, G Latouche, Y Liu. Poisson equation for discrete-time quasi-birth-and-death pro-cesses, Perform Eval, 2013, 70: 564–577.
D Down, S P Meyn, R L Tweedie. Exponential and uniform ergodicity of Markov Processes, Ann Probab, 1995, 23(4): 1671–1691.
A I Elwalid, D Mitra. Analysis and design of rate-based congestion control of high speed net-works, I: stochasticfluid models, access regulation, Queueing Systems Theory Appl, 1991, 9: 19–64.
M Govorun, G Latouche, MA Remiche. Stability forfluid queues: characteristic inequalities, Stoch Models, 2013, 29: 64–88.
S Jiang, Y Liu, S Yao. Poisson equation for discrete-time single-birth processes. Statistics and Probability Letters, 2014, 85, 78–83.
V G Kulkarni, E Tzenova. Mean first passage times influid queues, Oper Res Lett, 2002, 30: 308–318.
J Kushner. Stochastic Stability and Control Volume 33 of Mathematics in Science and Engi-neering, New York: Academic Press, 1967.
G Latouche, V Ramaswami. Introduction t. Matrix Analytic Methods in Stochastic Modeling, ASA-SIAM Series on Statistics and Applied Probability, SIAM, Philadelphia PA, 1999.
Y Liu, Z Hou. Several types of ergodicity for M/G/1-type Markov chains and Markov processes, J Appl Probab, 2006, 43(1): 141–158.
Y Liu, Z Hou. Exponential and strong ergodicity for Jump processes with application to queuing theory, Chin Ann Math, 2008, 29B(2): 199–206.
Y Liu, Y Zhang. Central limit theorems for ergodic ontinuous-time Markov chains with appli-cations to single birth processes, Front Math China, 2015, 10(4): 933–947.
Y Mao, Y Tai, Y Q Zhao, J Zou. Ergodicity for the GI/G/1-type Markov chain, J Appl Probab Statist, 2014, 9(1): 1–44.
S P Meyn, R L Tweedie. Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes, Adv in Appl Probab, 1993, 25: 518–548.
V Ramaswami. Matrix analytic methods for stochasticfluidflows, Elsevie. Science B V, Edin-burgh, UK, pages, 1019–1030, 1999.
L C G Rogers. Fluid models in queueing theory and Wiener-Hopf factorization of Markov chains, Ann Appl Probab, 1994, 4(2): 390–413.
J Shao. Criteria for transience and recurrence of regime-switching diffusion processes, Electron J Probab, 2015, 20(63): 1–15.
J Shao, F B Xi. Strong ergodicity of the regime-switching diffusion processes, Stoch Proc Appl, 2013, 123: 3903–3918.
T E Stern, A I Elwalid. Analysis of separable Markovmodulated rate models for information-handling systems, Adv Appl Probab, 1991, 23: 105–139.
N Va Foreest, M Mandjes, W Scheinhardt. Analysis of feedbackfluid model for heterogeneous TCP sources, Comm Statist Stoch Models, 2003, 19: 299–324.
G Yin, F B Xi. Stability of regime-switching jump diffusions, SIAM J Contr Optim, 2010, 48: 4525–4549.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by the National Natural Science Foundation of China (11571372, 11771452) and the Innovation Program of Central South University (10900-50601010).
Rights and permissions
About this article
Cite this article
Liu, Yy., Li, Y. V-uniform ergodicity for fluid queues. Appl. Math. J. Chin. Univ. 34, 82–91 (2019). https://doi.org/10.1007/s11766-019-3543-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11766-019-3543-2