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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.

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## Finite Buffer Fluid Networks with Overflows

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**Finite Buffer Fluid Networks with Overflows**Yoni Nazarathy, Swinburne University of Technology, Melbourne. StijnFleuren and ErjenLefeber, Eindhoven University of Technology, the Netherlands. The University of Sydney, Operations Management and Econometrics Seminar, July 29, 2011.**“Almost Discrete” Sojourn Time Phenomena**Taken from seminar of AviMandelbaum, MSOM 2010 (slide 82).**Outline**• Background: Open Jackson networks • Introducing overflows • Fluid networks as limiting approximations • Traffic equations and their solution • Discrete sojourn times**Open Jackson NetworksJackson 1957, Goodman & Massey 1984,**Chen & Mandelbaum 1991 Problem Data: Assume: open, no “dead” nodes Traffic Equations (Stable Case): Traffic Equations (General Case):**Open Jackson NetworksJackson 1957, Goodman & Massey 1984,**Chen & Mandelbaum 1991 Problem Data: Assume: open, no “dead” nodes Traffic Equations (Stable Case): Product Form “Miracle”:**Modification: Finite Buffers and Overflows**Problem Data: Assume: open, no “dead” nodes, no “jam” (open overflows) Explicit Solutions: Generally No Exact Traffic Equations: Generally No A practical (important) model: We say Yes**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:**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.**Traffic Equations**or or**LCP**(Linear Complementarity Problem)**Existence, Uniqueness and Solution**Immediate naive algorithm with 2Msteps We essentially assume that our matrix ( ) is a “P”-Matrix We have an algorithm (for our G) taking M2steps**Back To Sojourn Times….**Taken from seminar of AviMandelbaum, MSOM 2010 (slide 82).**The “Fast” Chain and “Slow” Chain**start 1’ 1 0 “Fast” chain on {0, 1, 2, 1’, 2’, 3’, 4’}: 2’ 2 3’ “Slow” chain on {0, 1, 2} DPH distribution (hitting time of 0) transitions based on “Fast” chain 4’ E.g: Moshe Haviv (soon) book: Queues, Section on “Shortcutting states”**The DPH Parameters (Details)**“Fast” chain “Slow” chain**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)**“Molecule” Sojourn Times**Observe, For job at entrance of buffer : A job at entrance of buffer : routed almost immediately according to A “fast” chain and “slow” chain…

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