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

LQF is Unstable [AZ ‘01] 1/30

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

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

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

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--;

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 --;

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

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

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

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

Single-Server Switches Claim: Fair-LQF is stable

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

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

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

Proof (4) Remove flow i from consideration Reduce service rates for S 1  S k accordingly Repeat above for reduced network while flows exist ▪

Fair-LQF on Counterexample 1/3

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

Simulation Results

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

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

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

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

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.