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By J.G. Dai Presented by Shengbo Chen Oct 22nd

On Positive Harris Recurrence of Multiclass Queueing Networks: A Unified Approach Via Fluid Limit Models. By J.G. Dai Presented by Shengbo Chen Oct 22nd. An Example. Poisson arrival Deterministic service time. 6(0.897). 5(0.001). A. 4(0.001). B. 3(0.001). 1(0.001). 2(0.897).

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By J.G. Dai Presented by Shengbo Chen Oct 22nd

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  1. On Positive Harris Recurrence of Multiclass Queueing Networks: A Unified Approach Via Fluid Limit Models By J.G. Dai Presented by Shengbo Chen Oct 22nd

  2. An Example • Poisson arrival • Deterministic service time 6(0.897) 5(0.001) A 4(0.001) B 3(0.001) 1(0.001) 2(0.897)

  3. An Example(cont) • Intuition: the network would be stable • However, simulation result tells that queue length tends to infinity

  4. Main objective of this paper • Establish a sufficient condition for the stability of a multiclass network

  5. Clue of the proof in this paper • If the underlying Markov process describing the network dynamics is positive Harris recurrent, then the queueing discipline for the network is stable. • If the corresponding fluid limit model is stable, then the Markov process is positive Harris recurrent

  6. System Model • open queueing network • d stations(servers) • K classes of customers (or packets) • interarrival time for class k • service time for class k • probability of class k becomes class l • Bernoulli routing • is convergent

  7. System Model(cont) • Assumptions: • iid and mutually independent • State description: • queue length • remaining time before next arrival • remaining service time

  8. Positive Harris Recurrent • A process Xt is Harris recurrent if there exists some measure , such that whenever , where

  9. Theorm 3.1 • If there exists , such that where and x is the starting point then X is positive Harris recurrent

  10. Fluid Limit Model • Fluid limit model is developed to check Thm 3.1 • Notations: • total number of exogenous arrivals to class l by time t • total number of service completions for class l given time t • cumulative service time that server spent on class l given time t • cumulative idle time for server i

  11. Fluid Limit Model(cont) • For any sequence of initial states there is a subsequence such that • Furthermore, the limit satisfies the following

  12. Main Theorem: Thm 4.2 • If the fluid limit model of the queueing discipline is stable, then the Markov chain describing the dynamics of the network is positive Harris recurrent. • Fluid limit model is stable if there exists a constant δ>0, such that for any fluid limit with

  13. An example to use FLM • Generalized Jackson Network • Only one class of customers served at each station • General interarrival and service time distributions • We want to show that ρ<1 is a sufficient condition that the network is stable.

  14. An example to use FLM(cont) • Let • Combined with the FLM, • It can be seen that f(t) has negative derivative when (when ), which means when δis large

  15. Thank you

  16. Simulation result

  17. Intuition for Thm 3.1 Back

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