Achieving Stability in a Network of IQ Switches
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Achieving Stability in a Network of IQ SwitchesAchieving Stability in a Network of IQ Switches
Neha KumarShubha 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
1/30
1/30
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 = (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
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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-jif (voqij is matched & congested)
n = # non-empty voqxjs;block voqij for n cycles;
else if (cyclesij > 0)cyclesij--;
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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 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
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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 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
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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 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
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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
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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
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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
1/3
1/3
1/3
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.