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Basic tandem model

NUMERICAL ALGORITHMS FOR TRANSIENT ANALYSIS OF FLUID QUEUES Stéphane Mocanu Laboratoire d’Automatique de Grenoble FRANCE. Basic tandem model. Two machines separated by a finite buffer Unreliable machines Deterministic service times Infinite arrivals an machine M 1

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Basic tandem model

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  1. NUMERICAL ALGORITHMS FOR TRANSIENT ANALYSIS OF FLUIDQUEUESStéphaneMocanuLaboratoire d’Automatique de GrenobleFRANCE

  2. Basic tandem model • Two machines separated by a finite buffer • Unreliable machines • Deterministic service times • Infinite arrivals an machine M1 • Infinite available places at the exit of M2

  3. Fluid (continuous) modem

  4. Non-blocking, Time Dependent Failures Communication systems (Mitra) Blocking, Operation Dependent Failures Production systems (Gershwin) Versions

  5. Suppose M1 slowed down by M2 (U1>U2, x=C) Operation depending failures The failure rate is reduced to: A completely blocked (starved) machine cannot fail !

  6. Internal equations Continuous transition • Not an ordinary Markov chain • Continuous transitions on the “fluid direction” • Infinitesimal variation of the probability mass Discrete state Discrete transitions

  7. State = {M1 state, M2 state, buffer level} An example: homogeneous case Machines driven by two state Markov chains U U 0 0

  8. Joint evolution

  9. Evolution equations A PDE system Markov chain generator Drift matrix In the example

  10. Boundary conditions for ODF systems Discontinuities of the probability distribution PC(t) P0(t)

  11. Difficulties Boundary condition does NOT verify the PDE • Some boundary states are of 0 probability • Some transitions are modified (due to ODF) M0 on lower boundary MC on upper boundary

  12. Homogeneous case Example: state (0,0,C) Matrix form

  13. Initial conditions Specify Example : machine state (1,1) (both ON), buffer empty

  14. The problem Find an integration algorithm for • under boundary conditions b.c. • with initial conditions i.c.

  15. Decompose the system in Linear evolution Wave evolution Apply b.c. The integration scheme

  16. Recurrent solution • Linear transform • Wave transform

  17. Recurrent form of the b.c.

  18. Numerical results Initial state : (0,1) buffer half full

  19. Numerical results First starvation Initial state : (0,1) buffer half full

  20. Numerical results

  21. Needs compatible i.c. Warning : machine state (1,1), buffer empty is NOT compatible But : machine state (1,1), buffer = Dx, it IS Some boundaries propagates bad For the instance we need explicit analysis of boundary conditions Actual numerical implementation is limited to ON/OFF machines Some limitations

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