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Achieving Stability in a Network of IQ Switches

Achieving Stability in a Network of IQ Switches. Neha Kumar Shubha U. Nabar. Outline. The Problem Instability of LQF Prior Work Fairness in Scheduling Fair-LQF Fair-MWM Stability of Networks Single-Server Switches AZ Counterexample N x N Switches. The Problem.

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Achieving Stability in a Network of IQ Switches

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  1. Achieving Stability in a Network of IQ Switches Neha Kumar Shubha U. Nabar

  2. Outline • The Problem • Instability of LQF • Prior Work • Fairness in Scheduling • Fair-LQF • Fair-MWM • Stability of Networks • Single-Server Switches • AZ Counterexample • N x N Switches

  3. The Problem Can we ensure stability in networks of IQ switches using a simple local and online scheduling policy?

  4. LQF is Unstable [AZ ‘01] 1/30 1/30 1/30 1/30

  5. Prior Work • Longest-In-Network [AZ ‘01] • Frame-based, not local • BvN based scheduling [MGLN ’03] • Requires prior knowledge of rates • Approximate-OCF [MGLN ’03] • Involves rate estimation

  6. Outline • The Problem • Instability of LQF • Prior Work • Fairness in Scheduling • Fair-LQF • Fair-MWM • Stability of Networks • Single-Server Switches • AZ Counterexample • N x N Switches

  7. Max-Min Fairness Given server capacity C and n flows with rates 1n , rate allocation R=(r1rn) is max-min fair iff 1. n ri·C, ri·i 2. any ri can be increased only by reducing rj s.t. rj·ri

  8. Fair-LQF [KPS ‘04] if (q_size > threshold) add q to congested list; m = # congested queues; while (m != 0) round-robin on congested; m--; m = # non-empty uncongested queues; while (m != 0) lqf on uncongested; m--;

  9. Fair-MWM [KPS ‘04] if (voq_size > threshold) add voq to congested list; MWM-schedule unblocked voqs; for all i-j if (voqij is matched & congested) n = # non-empty voqxjs; block voqij for n cycles; else if (cyclesij > 0) cyclesij--;

  10. Outline • The Problem • Instability of LQF • Prior Work • Fairness in Scheduling • Fair-LQF • Fair-MWM • Stability of Networks • Single-Server Switches • AZ Counterexample • N x N Switches

  11. Our Model: Traffic • Arrivals for each flow satisfy SLLN limn!1Ai(n)/n=i8 i • Arrivals are admissible If fx is the set of flows that go through port x, then i2fxi <1

  12. Our Model: Flows A flow is a set of packets that traverse the same path within the network • Per-Flow Queueing • Deterministic Routing

  13. Our Model: Stability A network of switches is rate stable if limn!1Xn/n= limn!11/ni(Ai–Di)=0w.p.1 Xn – queue lengths vector at time n Di – departure vector at time i Ai - arrival vector at time i

  14. Single-Server Switches Claim: Fair-LQF is stable

  15. Proof (1) Lemma 1: For flow i at switch S, if limn!1Ai(n)/n=i and i<1/N then Fair-LQF ensures that limn!1Di(n)/n exists and is i regardless of other arrivals at S . Work in Progress

  16. Proof (2) • Consider flow i with smallest injection rate, that passes through switches S1 Sk • From traffic model and Lemma 1, limn!1DiS1(n)/n exists and is i

  17. Proof (3) • Observe that limn!1AiS2(n)/n = limn!1DiS1(n)/n =i • Repeatedly applying Lemma 1, limn!1AiSj(n)/n= limn!1DiSj(n)/n=i 8j·k

  18. Proof (4) • Remove flow i from consideration • Reduce service rates for S1Sk accordingly • Repeat above for reduced network while flows exist ▪

  19. Fair-LQF on Counterexample 1/3 1/3 1/3 1/3

  20. N x N Switches Claim: Fair-MWM is stable Work in Progress

  21. Simulation Results

  22. Fair-LQF vs LQF (1) LQF causes packets to grow unboundedly in system Number of packets stays bounded under Fair-LQF

  23. Fair-LQF vs LQF (2) LQF causes packets to grow unboundedly in system Number of packets stays bounded under Fair-LQF

  24. Fair-MWM vs MWM (1) Bad guys are punished As they ask for higher rates

  25. Fair-MWM vs MWM (2) Good guys continue to get their fair share As bad guys grow in rate

  26. Fair-MWM is MMF Intuition: Consider a frame-based algorithm where VOQs collect packets for T time slots. Each output independently does a MMF rate allocation. The VOQs drop all packets that cannot be scheduled. The rest of the packets are sent through. We believe that Fair-MWM does this online.

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