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Poisson Hypothesis for Mean-Field Models of Generalised Jackson Networks with Countable Set of Nodes

This document explores the Poisson Hypothesis in infinite Jackson networks, open Jackson networks, and symmetrical Jackson networks. It discusses the properties of non-homogenous systems, non-linear Markov processes, and stationary measures. The paper also poses open questions related to the Poisson Hypothesis in generalized Jackson networks.

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Poisson Hypothesis for Mean-Field Models of Generalised Jackson Networks with Countable Set of Nodes

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  1. Poisson hypothesis for mean-field models of generalised Jackson networks with countable set of nodes Alexander Rybko Joint work with S.Shlosman

  2. Poisson Hypothesis

  3. Infinite Jackson Networks • Open Jackson network with countable set of nodes J • A pair (V,P) defines an open Jackson Network • vector of rates of Poisson input flows in nodes • Suppose that service times of all customers are i.i.d exponentially distributed with mean 1 • is the routing stochastic matrix; is a probability of the event: a customer comes to the node j after being serviced in the node i • Restriction: the matrix P is twice substochastic:

  4. Let be the minimal solution of (vector) equation (1) • The product of probability measures on we shall name the multiplicative phases, where • Let be a countable (breaking) Markov chain with the phase space J and the substochastic matrix • Let be a probability of the event that the trajectory of starting from an initial state will never break

  5. Lemma 1. a) If , there is no more then one multiplicative phase for Jackson network. b) If where is the minimal solution of (1), then the multiplicative phase of Jackson network is unique iff

  6. Theorem 2. Let , then • the minimal multiplicative phase is a unique invariant measure of the Markov process describing the evolution of the infinite Jackson network; • for any initial state the measure weakly converges to when : for any finite subset and any vector where is a projection of the process on the subset

  7. Self-averaging property of non-homogenous systems Let’s consider the system with Poisson input flow of variable intensity and with stationary ergodic sequence of service times with mean value 1. Let the function satisfy the non-overload condition Let also satisfy the conditions

  8. Theorem 3. Let be the rate of an output flow of the system . Then satisfies the equation: and the kernels depend on by restriction only. And more over, the kernels are stochastic: for any t where when

  9. Poisson hypothesis for symmetrical Jackson networks r

  10. A sequence of finite generalized Jackson networks • Set of nodes : for each the network contains identical nodes. So the total number of nodes is Let’s denote by • In each node of class j the service time distribution is

  11. In each node of class j the input rate of Poisson flow is equal to : • For a routing matrix we have • Let the increasing sequence converges to and the ratios converge to the limiting uniformly boundered ratios

  12. Non-linear Markov Processes (Linear) Markov chain Configurations = points in S. State = probability measure on S. Transition matrix State is transformed to by Non-linear Markov chain: Transition probability to go from s to t depends also on the state The Non-linear Markov chain is defined by the collection of transition matrices and state is transformed to by Evolution:

  13. Limiting dynamical system as the nonlinear Markov process Dynamical system :evolution of probability measure of nonlinear Markov process: countable set J of systems Additional equations: where

  14. Theorem 3. For any and any function weakly continuous on the equation holds. The convergence is uniform on any finite interval

  15. Theorem 4. Stationary measure Where is a stationary measure of J independent systems Where is the unique solution of countable set of equations:

  16. Closed networks, J-finite (2) (3)

  17. Proposition Theorem 4 where is the solution of a system of equations

  18. Open networks (4) (5) Stationary solution: where (6) minimal solution (6)

  19. Proposition: is unique has only trivial solution

  20. Theorem 5 Suppose that Then the Poisson Hypothesis holds. Proof: Let and then

  21. References • Kel’bertM.Ya., Kontsevich M.L., Rybko A.N. Infinite Jackson Networks, Theor.Probab. And Appl. 1988 v.33 • Stolyar P.I.T. 1989 v.25#4 • Rybko, Shlosman Moscow Mathematical Journal 2005 v.5#3, v.5#4, 2008 v.8#1 • Dobrushin, Karpelevich, Vvedenskaya P.I.T. 1996 v.32#1 • Karpelevich, Rybko P.I.T. 2000 v.36#2 • Rybko, Shlosman, Vladimirov P.I.T. 2006 v.42#4 • Rybko, Shlosman P.I.T. 2005 #3 • Rybko, Shlosman, VladimirovJ.ofStat.Physics 2009 v.134#1

  22. Open Questions • Is Poisson Hypothesis true for generalized Jackson networks with several types of customers? For example in the case when their service times are exponentially distributed with mean values depending on their types. We can not prove Poisson Hypothesis in this situation even in the case of a complete graph with an increasing number of nodes. • Is Poisson Hypothesis true for non-FIFO service discipline? • What kind of self-averaging properties between inputs and outputs are true in these situations?

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