1. Finite Buffer Fluid Networks with Overflows Yoni Nazarathy, Swinburne University of Technology, Melbourne. Stijn Fleuren and Erjen Lefeber, Eindhoven University of Technology, the Netherlands. The University of Sydney, Operations Management and Econometrics Seminar, July 29, 2011.

2. “Almost Discrete” Sojourn Time Phenomena Taken from seminar of Avi Mandelbaum, MSOM 2010 (slide 82).

3. Outline Background: Open Jackson networks Introducing overflows Fluid networks as limiting approximations Traffic equations and their solution Discrete sojourn times

4. Open Jackson Networks Jackson 1957, Goodman & Massey 1984, Chen & Mandelbaum 1991 Traffic Equations (Stable Case): Traffic Equations (General Case): Problem Data: Assume: open, no “dead” nodes

5. Open Jackson Networks Jackson 1957, Goodman & Massey 1984, Chen & Mandelbaum 1991 Traffic Equations (Stable Case): Problem Data: Assume: open, no “dead” nodes Product Form “Miracle”:

6. Modification: Finite Buffers and Overflows Exact Traffic Equations: Problem Data: Explicit Solutions: Generally No Assume: open, no “dead” nodes, no “jam” (open overflows) A practical (important) model: We say Yes

7. When K is Big, Things are “Simpler”

8. Scaling Yields a Fluid System A sequence of systems: Make the jobs fast and the buffers big by taking The proposed limiting model is a deterministic fluid system:

9. Fluid Trajectories as an Approximation Not proved in this current work, yet similar statement appears in a different model (and rigorously proved). Come to 14:00 Stats Seminar, Carslaw 173.

10. Traffic Equations or

11. LCP (Linear Complementarity Problem)

12. Min-Linear Equations as LCP

13. Existence, Uniqueness and Solution Immediate naive algorithm with 2 M steps We essentially assume that our matrix ( ) is a “P”-Matrix We have an algorithm (for our G) taking M 2 steps

15. Back To Sojourn Times…. Taken from seminar of Avi Mandelbaum, MSOM 2010 (slide 82).

17. Sojourn Times Scale to a Discrete Distribution!!!

18. The “Fast” Chain and “Slow” Chain 1’ 2’ 3’ 4’ 1 2 0 “Fast” chain on {0, 1, 2, 1’, 2’, 3’, 4’}: “Slow” chain on {0, 1, 2} start DPH distribution (hitting time of 0) transitions based on “Fast” chain E.g: Moshe Haviv (soon) book: Queues, Section on “Shortcutting states”

19. The DPH Parameters (Details) “Fast” chain “Slow” chain

20. Sojourn Times Scale to a Discrete Distribution!!!

21. Summary – Trend in queueing networks in past 20 years: “When don’t have product-form…. don’t give up: try asymptotics” – Limiting traffic equations and trajectories – Molecule sojourn times (asymptotic) – Discrete!!! – Future work on the limits: More standard: E.g. convergence of trajectories (2:00 talk) Hi-tech (I don’t know how to approach): Weak convergence of sojourn times (we will leave it as a conjecture for now)

22. “Molecule” Sojourn Times Observe, For job at entrance of buffer : A “fast” chain and “slow” chain… A job at entrance of buffer : routed almost immediately according to