Stationary distributions and convergence for M/M/1 queues in interactive random environment

We study a Markovian single-server queue in interactive random environments. The arrival and service rates of the queue depend on the environment, while the transition dynamics of the random environment depends on the queue length. We consider two types of Markov random environments: a pure jump process and a reflected jump-diffusion. In both cases, the joint dynamics is constructed so that the stationary distribution can be explicitly found in a product form. We derive an explicit estimate for exponential rate of convergence to stationarity via coupling.

[1]  Guodong Pang,et al.  Ergodicity of a Lévy-driven SDE arising from multiclass many-server queues , 2017, The Annals of Applied Probability.

[2]  Richard Cornez.  Birth and death processes in random environments with feedback , 1987, Journal of Applied Probability.

[3]  S. Asmussen,et al.  Applied Probability and Queues , 1989 .

[4]  G. J. K. Regterschot,et al.  The Queue M|G|1 with Markov Modulated Arrivals and Services , 1986, Math. Oper. Res..

[5]  T. Kurtz,et al.  Stationary Solutions and Forward Equations for Controlled and Singular Martingale Problems , 2001 .

[6]  Tetsuya Takine.  Single-Server Queues with Markov-Modulated Arrivals and Service Speed , 2005, Queueing Syst. Theory Appl..

[7]  Yuri M. Suhov,et al.  Random walks in a queueing network environment , 2014, Journal of Applied Probability.

[8]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[9]  Sanjay K. Bose,et al.  Fundamentals of Queueing Networks , 2002 .

[10]  Huseyin Topaloglu,et al.  Dynamic service rate control for a single-server queue with Markov-modulated arrivals , 2013 .

[11]  Elizabeth L. Wilmer,et al.  Markov Chains and Mixing Times , 2008 .

[12]  Baris Ata,et al.  An Optimal Callback Policy for General Arrival Processes: A Pathwise Analysis , 2020, Oper. Res..

[13]  J. Blanchet,et al.  Rates of Convergence to Stationarity for Multidimensional RBM , 2016, 1601.04111.

[14]  Frank Kelly,et al.  Reversibility and Stochastic Networks , 1979 .

[15]  Ruth J. Williams.  Semimartingale reflecting Brownian motions in the orthant , 1995 .

[16]  Marcel F. Neuts,et al.  Matrix-geometric solutions in stochastic models - an algorithmic approach , 1982 .

[17]  M. Kelbert,et al.  Probability and Statistics by Example: Contents , 2008 .

[18]  Antonis Economou,et al.  Generalized product-form stationary distributions for Markov chains in random environments with queueing applications , 2005, Advances in Applied Probability.

[19]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[20]  Philippe Robert.  Stochastic Networks and Queues , 2003 .

[21]  François Baccelli,et al.  Elements Of Queueing Theory , 1994 .

[22]  A. Sarantsev.  Lyapunov Functions and Exponential Ergodicity for Reflected Brownian Motion in the Orthant and Competing Brownian Particles , 2013 .

[23]  A. Sarantsev.  Reflected Brownian Motion in a Convex Polyhedral Cone: Tail Estimates for the Stationary Distribution , 2015, 1509.01781.

[24]  R. Cogburn,et al.  Markov Chains in Random Environments: The Case of Markovian Environments , 1980 .

[25]  P. Lions,et al.  Stochastic differential equations with reflecting boundary conditions , 1984 .

[26]  A. Sarantsev.  Explicit Rates of Exponential Convergence for Reflected Jump-Diffusions on the Half-Line , 2015, 1509.01783.

[27]  R. Douc,et al.  Subgeometric rates of convergence of f-ergodic strong Markov processes , 2006, math/0605791.

[28]  V. Climenhaga.  Markov chains and mixing times , 2013 .

[29]  W. Kendall,et al.  Efficient Markovian couplings: examples and counterexamples , 2000 .

[30]  Guodong Pang,et al.  Uniform Polynomial Rates of Convergence for A Class of Lévy-Driven Controlled SDEs Arising in Multiclass Many-Server Queues , 2019, Modeling, Stochastic Control, Optimization, and Applications.

[31]  Nico M. van Dijk.  Queueing networks and product forms - a systems approach , 1993, Wiley-Interscience series in systems and optimization.

[33]  Rates of convergence to the stationary distribution for k-dimensional diffusion processes , 1986 .

[34]  S. Meyn,et al.  Computable exponential convergence rates for stochastically ordered Markov processes , 1996 .

[35]  Mu-Fa Chen,et al.  Coupling Methods for Multidimensional Diffusion Processes , 1989 .

[36]  Birth and death processes with random environments in continuous time. , 1981, Journal of applied probability.

[37]  G. Falin,et al.  A heterogeneous blocking system in a random environment , 1996, Journal of Applied Probability.

[38]  J. Michael Harrison,et al.  Dynamic Routing and Admission Control in High-Volume Service Systems: Asymptotic Analysis via Multi-Scale Fluid Limits , 2005, Queueing Syst. Theory Appl..

[39]  Ruslan K. Krenzler,et al.  Jackson networks in nonautonomous random environments , 2016, Advances in Applied Probability.

[40]  Danna Zhou,et al.  d. , 1934, Microbial pathogenesis.

[41]  Elena Yudovina,et al.  Stochastic networks , 1995, Physics Subject Headings (PhySH).

[42]  Ioannis Karatzas,et al.  Brownian Motion and Stochastic Calculus , 1987 .

[43]  Ruth J. Williams.  Reflected Brownian motion with skew symmetric data in a polyhedral domain , 1987 .

[44]  J. Hunter.  Coupling and mixing times in a Markov chain , 2009 .

[45]  Uri Yechiali.  A Queuing-Type Birth-and-Death Process Defined on a Continuous-Time Markov Chain , 1973, Oper. Res..

[46]  Y. Peres,et al.  Markov Chains and Mixing Times: Second Edition , 2017 .

[47]  Tsuyoshi Murata,et al.  {m , 1934, ACML.

[48]  A. Sarantsev.  PR ] 1 4 A pr 2 01 8 CONVERGENCE AND STATIONARY DISTRIBUTIONS FOR , 2018 .

[49]  Xiongzhi Chen.  Brownian Motion and Stochastic Calculus , 2008 .

[50]  Y. Suhov,et al.  Models of Markov processes with a random transition mechanism , 2015, 1508.05598.

[51]  J. Michael Harrison,et al.  Design and Control of a Large Call Center: Asymptotic Analysis of an LP-Based Method , 2006, Oper. Res..

[52]  K. Waldmann,et al.  Optimal control of arrivals to multiserver queues in a random environment , 1984 .

[53]  Yiqiang Q. Zhao,et al.  Subgeometric ergodicity for continuous-time Markov chains , 2010 .

[54]  S. Sawyer.  A FORMULA FOR SEMIGROUPS, WITH AN APPLICATION TO BRANCHING DIFFUSION PROCESSES , 1970 .

[55]  Srinivas R. Chakravarthy,et al.  Queues with interruptions: a survey , 2014 .

[56]  Mark Kelbert,et al.  Markov Chains : a primer in random processes and their applications , 2008 .

[57]  David D. Yao,et al.  Fundamentals of Queueing Networks , 2001 .

[58]  A. Borodin,et al.  Handbook of Brownian Motion - Facts and Formulae , 1996 .

[59]  S. Meyn,et al.  Stability of Markovian processes II: continuous-time processes and sampled chains , 1993, Advances in Applied Probability.