1. AIC05 - S. Mocanu 1 NUMERICAL ALGORITHMS FOR TRANSIENT ANALYSIS OF FLUID QUEUES Stéphane Mocanu Laboratoire d’Automatique de Grenoble FRANCE

2. AIC05 - S. Mocanu 2 Basic tandem model Two machines separated by a finite buffer Unreliable machines Deterministic service times Infinite arrivals an machine M 1 Infinite available places at the exit of M 2

3. AIC05 - S. Mocanu 3 Fluid (continuous) modem

4. AIC05 - S. Mocanu 4 Versions Non-blocking, Time Dependent Failures Communication systems (Mitra) Blocking, Operation Dependent Failures Production systems (Gershwin)

5. AIC05 - S. Mocanu 5 Operation depending failures Suppose M 1 slowed down by M 2 (U 1 >U 2, x=C) The failure rate is reduced to: A completely blocked (starved) machine cannot fail !

6. AIC05 - S. Mocanu 6 Internal equations Not an ordinary Markov chain Continuous transitions on the “fluid direction” Infinitesimal variation of the probability mass Discrete transitions Discrete state Continuous transition

7. AIC05 - S. Mocanu 7 An example: homogeneous case State = {M 1 state, M 2 state, buffer level} U U 0 0 Machines driven by two state Markov chains

8. AIC05 - S. Mocanu 8 Joint evolution

9. AIC05 - S. Mocanu 9 Evolution equations A PDE system Markov chain generator Drift matrix In the example

10. AIC05 - S. Mocanu 10 Boundary conditions for ODF systems Discontinuities of the probability distribution P 0 (t) P C (t)

11. AIC05 - S. Mocanu 11 Difficulties Boundary condition does NOT verify the PDE –Some boundary states are of 0 probability –Some transitions are modified (due to ODF) M 0 on lower boundary M C on upper boundary

12. AIC05 - S. Mocanu 12 Homogeneous case Example: state (0,0,C) Matrix form

13. AIC05 - S. Mocanu 13 Initial conditions Specify Example : machine state (1,1) (both ON), buffer empty

14. AIC05 - S. Mocanu 14 The problem Find an integration algorithm for –under boundary conditions b.c. –with initial conditions i.c.

15. AIC05 - S. Mocanu 15 The integration scheme Decompose the system in –Linear evolution –Wave evolution Apply b.c.

16. AIC05 - S. Mocanu 16 Recurrent solution Linear transform Wave transform

17. AIC05 - S. Mocanu 17 Recurrent form of the b.c.

18. AIC05 - S. Mocanu 18 Numerical results Initial state : (0,1) buffer half full

19. AIC05 - S. Mocanu 19 Numerical results Initial state : (0,1) buffer half full First starvation

20. AIC05 - S. Mocanu 20 Numerical results

21. AIC05 - S. Mocanu 21 Some limitations Needs compatible i.c. –Warning : machine state (1,1), buffer empty is NOT compatible –But : machine state (1,1), buffer =  x, 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