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Achieving Stability in a Network of IQ Switches Neha Kumar Shubha U. Nabar
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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
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The Problem Can we ensure stability in networks of IQ switches using a simple local and online scheduling policy?
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LQF is Unstable [AZ ‘01] 1/30
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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
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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
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Max-Min Fairness Given server capacity C and n flows with rates 1 n, rate allocation R = (r 1 r n ) is max-min fair iff 1. n r i · C, r i · i 2. any r i can be increased only by reducing r j s.t. r j · r i
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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--;
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Fair-MWM [KPS ‘04] if (voq_size > threshold) add voq to congested list; MWM-schedule unblocked voqs; for all i-j if (voq ij is matched & congested) n = # non-empty voq xj s; block voq ij for n cycles; else if (cycles ij > 0) cycles ij --;
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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
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Our Model: Traffic Arrivals for each flow satisfy SLLN lim n ! 1 A i (n)/n = i 8 i Arrivals are admissible If f x is the set of flows that go through port x, then i 2 f x i < 1
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Our Model: Flows A flow is a set of packets that traverse the same path within the network Per-Flow Queueing Deterministic Routing
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Our Model: Stability A network of switches is rate stable if lim n ! 1 X n /n = lim n ! 1 1/n i (A i – D i ) = 0 w.p.1 X n – queue lengths vector at time n D i – departure vector at time i A i - arrival vector at time i
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Single-Server Switches Claim: Fair-LQF is stable
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Proof (1) Lemma 1: For flow i at switch S, if lim n ! 1 A i (n)/n = i and i < 1/N then Fair-LQF ensures that lim n ! 1 D i (n)/n exists and is i regardless of other arrivals at S. Work in Progress
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Proof (2) Consider flow i with smallest injection rate, that passes through switches S 1 S k From traffic model and Lemma 1, lim n ! 1 D i S1 (n)/n exists and is i
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Proof (3) Observe that lim n ! 1 A i S2 (n)/n = lim n ! 1 D i S1 (n)/n = i Repeatedly applying Lemma 1, lim n ! 1 A i Sj (n)/n = lim n ! 1 D i Sj (n)/n = i 8j · k
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Proof (4) Remove flow i from consideration Reduce service rates for S 1 S k accordingly Repeat above for reduced network while flows exist ▪
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Fair-LQF on Counterexample 1/3
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N x N Switches Claim: Fair-MWM is stable Work in Progress
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Simulation Results
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Fair-LQF vs LQF (1) LQF causes packets to grow unboundedly in system Number of packets stays bounded under Fair-LQF
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Fair-LQF vs LQF (2) LQF causes packets to grow unboundedly in system Number of packets stays bounded under Fair-LQF
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Fair-MWM vs MWM (1) Bad guys are punished As they ask for higher rates
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Fair-MWM vs MWM (2) Good guys continue to get their fair share As bad guys grow in rate
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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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