Finite Buffer Fluid Networks with Overflows Yoni Nazarathy, Swinburne University of Technology, Melbourne. Stijn Fleuren and Erjen Lefeber, Eindhoven University of Technology, the Netherlands.

Talk Outline Background: Open Jackson networks Introducing finite buffers and overflows – Interlude: How I got to this problem Fluid networks as limiting approximations Traffic equations and their solution Almost discrete sojourn times

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

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”:

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: Yes

Our Contribution (in progress) Efficient Algorithm for Unique Solution: Limiting Traffic Equations: Limiting Sojourn Time Distribution Limiting Deterministic Trajectories

Interlude: How I got to this problem Output process, D(t), asymptotic variance: Control of queueing networks: BRAVO effect for M/M/1/K load Server 2 Server 1 PUSH PULL PUSH

When K is Big, Things are “Simpler”

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

Traffic Equations (at equib. point) or

LCP (Linear Complementarity Problem)

Min-Linear Equations as LCP

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 type of G) taking M 2 steps

Sojourn Times Scale to a Discrete Distribution!!!

“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

The “Fast” Chain and “Slow” Chain 1’ 2’ 3’ 4’ “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”

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

Sojourn Times Scale to a Discrete Distribution!!!

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

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.