Achieving Stability in a Network of IQ SwitchesAchieving Stability in a Network of IQ Switches

Neha KumarShubha 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]

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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

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 = (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

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-jif (voqij is matched & congested)

n = # non-empty voqxjs;block voqij for n cycles;

else if (cyclesij > 0)cyclesij--;

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 SLLNlimn ! 1 Ai(n)/n = i 8 i

• Arrivals are admissibleIf fx is the set of flows that go through port x, then

i 2 fx 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 limn ! 1 Xn/n =

limn ! 1 1/n i(Ai – Di) = 0 w.p.1

Xn – queue lengths vector at time n

Di – departure vector at time i

Ai - arrival vector at time i

Single-Server Switches

Claim: Fair-LQF is stable

Proof (1)

Lemma 1:For flow i at switch S,

if limn ! 1 Ai(n)/n = i and i < 1/N

then Fair-LQF ensures that limn ! 1 Di(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 S1 Sk

• From traffic model and Lemma 1, limn ! 1 Di

S1(n)/n exists and is i

Proof (3)

• Observe that limn ! 1 Ai

S2(n)/n =

limn ! 1 DiS1(n)/n = i

• Repeatedly applying Lemma 1, limn ! 1 Ai

Sj(n)/n =

limn ! 1 DiSj(n)/n = i 8j · k

Proof (4)

• Remove flow i from consideration

• Reduce service rates for S1 Sk accordingly

• Repeat above for reduced network while flows exist

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Fair-LQF on Counterexample

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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.