1. Overflow Queueing Networks: Limiting Traffic Equations, Trajectories and Sojourn Times Stijn Fleuren, Yoni Nazarathy, Erjen Lefeber Open Problem Session EURANDOM October 28, 2010 * Supported by NWO-VIDI Grant 639.072.072

2. Overview Overflow queueing networks Large buffer fluid scaling Limiting: traffic equations, trajectories, sojourn times (conjectures 1, 2, 3) Some items for discussion (problem session): – Related work? Where to take this? – Approaches for the limit proofs? – Generalizing DPH distributions? – “Almost Discrete” Sojourn Time Phenomenon Disclaimer: Conjectures 1,2,3 are rough…

3. 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

4. Modification: Finite Buffers and Overflows Wolff, 1988, Chapter 8 & references there in & after Exact Traffic Equations: Problem Data: Explicit Solutions: Generally No Assume: open, no “dead” nodes, no “jam” (open overflows)

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

6. Large Buffer Fluid Scaling And maybe scale space and initial conditions when needed

7. Limiting Traffic Equations or

8. Properties of the Limiting Traffic Equations Proposition: Unique solution exists under certain non- singularity assumptions of P and Q Proposition: An algorithm in at most iterations (as opposed to ) Conjecture 1: Under general processing time and arrival assumptions

9. Limiting Trajectories In similar spirit to the traffic equations, limiting trajectories,, may be calculated… Conjecture 2: Under general assumptions,

10. Sojourn Times

11. Construction of Limiting 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

12. Sojourn Times Scale to a Discrete Distribution!!! Conjecture 3: If,,

13. 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

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

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

16. Discussion – Related work? Where to take this? – Approaches for the limit proofs? – Generalizing DPH distributions? – “Almost Discrete” Sojourn Time Phenomenon: modeling call-centers using overflow networks and variations