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Load Balanced Birkhoff-von Neumann Switches

Cheng-Shang Chang, Duan-Shin Lee and Chi-Yao Yue

presented by

Prashanth Pappu

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High Performance Switches

• Non-blocking crossbar• Fixed time slot, fixed size cell • Parallelism, memory speed =

line rate.• Quadratic complexity but

concentrated in a single chip set.

• Centralized scheduler

Controller

IPP

IPP

IPP

OPP

OPP

OPP

. . .

. . .

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Centralized Schedulers

• VOQs to avoid HOL blocking. • Equivalent to finding a matching on a bipartite

graph (Anderson et al)• McKeown et al. – 100% throughput with MWM.• 10Gb/s line rate implies 40 ns for scheduling.• Maximal size matching algorithms (PIM, iSLIP)• More ports and faster line rates makes it harder

to implement scheduling algorithms.

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Overview

• New scheduling algorithm (based on Birkhoff-von Neumann decomposition) – “On service guarantees for input buffered crossbar switches: a capacity decomposition approach by Birkhoff and von Neumann”, IEEE IWQoS’99.

• Birkhoff-von Neumann switches are not practical.• Load balanced Birkhoff-von Neumann switch – “Load Balanced

Birkhoff-von Neumann Switches, Part I: One-stage Buffering”, Computer Communications. • Mis-sequencing problem and solutions – “Load Balanced Birkhoff-

von Neumann Switches, Part II: Multi-stage Buffering”, Computer Communications and I. Keslassy and N. Mckweon “Maintaining Packet Order in Two Stage Switches”, IEEE Infocom, 2002.

• Providing guaranteed rate services (The actual paper!) – “Providing guaranteed rate services in the load balanced Birkhoff-von Neumann Switches”, IEEE Infocom 2003.

• Talk presents only algorithms + results – proofs.

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Birkhoff-von Neumann Switch

0 0 0 1

1 0 0 0

0 1 0 0

0 0 1 0

• Crossbar configuration is a permutation matrix, P.

• 4x4 switch, input 1-output 4, input 2-output 1 etc.

• Input rate matrix is admissible.

1),( j

jir

1),( i

jir

)),(( jirR

1

2

3

4

1 2 3 4

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Birkhoff-von Neumann Switch• Can any admissible rate

matrix be serviced?• How do we map rate matrix to

a sequence of permutation matrices? (Change in pov as opposed to finding matching on bipartite graph)

• Express as convex combination of permutation matrices.

• Obtain the decomposition and schedule each permutation matrix proportional to its weight.

K

kkkPR

1

K

kk

1

1

KkPk ,,1,

222 NNK

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Birkhoff-von Neumann Switch

• (von Neumann 1953) Transform the doubly substochastic rate matrix to a doubly stochastic matrix.

• (Birkhoff 1946) Decompose doubly stochastic rate matrix to weighted sum of permutation matrices.

• (PGPS) Use simple packet scheduling algorithm (WFQ) to determine which permutation matrix should be used to configure crossbar.

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Example

• von-Neumann conversion.

• Pivots around (1,2), (2,1), (2,2) etc.

• There are other (fairer) ways to obtain this conversion.

0 0.3 0.2 0.4

0.2 0.3 0 0.2

0.4 0.1 0.3 0

0.2 0 0.2 0.3

R =

0 0.4 0.2 0.4

0.4 0.4 0 0.2

0.4 0.2 0.4 0

0.2 0 0.4 0.4

R’ =

)( 3NO

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Example0 0.4 0.2 0.4

0.4 0.4 0 0.2

0.4 0.2 0.4 0

0.2 0 0.4 0.4

0 1 0 0

1 0 0 0

0 0 1 0

0 0 0 1

0 0 0 1

0 1 0 0

1 0 0 0

0 0 1 0

0 0 1 0

0 0 0 1

0 1 0 0

1 0 0 0

R’ = = 0.4 +0.4

+0.2

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Not practical

• Birkhoff-von Neumann decomposition is non-trivial with O(N4.5) complexity, though required only when rates change.

• Need to know rate matrix.

• Memory : O(N2) permutation matrices.

• Does not support multicast.

• Solution – Load balanced Birkhoff-von Neumann switch.

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Load balanced BvN switch

• We know decomposition is easy for uniform Bernouli i.i.d traffic.

• Use a first stage that load balances traffic to second stage!• First stage uses permutation matrices generated from a

one-cycle permutation matrix. (Input i connects to output (n+i) modulo N at time n.)

. . .

. . .

Load balancing stage BvN switch

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Second stage (Switching)

• Traffic from first stage is instantly transferred to buffers at second stage.

• With balanced traffic, second stage can also use a deterministic sequence of cyclical permutation matrices. (Input j is connected to output ((n-j) modulo N) at time n.)

• Both stages are identical, complexity of scheduling algorithm O(1).

• Low hardware complexity.• 100% throughput (under a mild technical

condition)

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1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E+04

1.00E+05

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99

Ave

rage

De

lay

Output-buffered Load Balanced Birkhoff-von NeumannBirkhoff-von Neumann 4-SLIP

Uniform pareto bursty traffic, N=16

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Load balanced BvN switch (multi-buffered)

• Problem of mis-sequencing of packets.• Packets are distributed on arrival times – no bound on a

resequencing buffer.• Use load-balancing and re-sequencing buffers.• Load-balancing based on flows and not according to arrival times.

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FCFS with jitter control

• Flow splitter sends packets from same flow in round robin fashion to the N VOQs.

• Causes packets of same flow to be split almost evenly among inputs of second stage.

• Jitter control at second stage delays each packet to its maximum delay (targeted departure time is obtained from corresponding OQ switch)

• Flows entering second stage are time-shifted versions of original ones.

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FCFS with jitter control

• Delay of a packet is bounded by sum of delay through the corresponding OQ switch and (N-1)Lmax + NMmax.

• Essentially delay < 2N for unicast traffic.

• Size of load balancing buffer bounded by NLmax.

• Size of re-sequencing buffer bounded by NMmax.

• Lmax (Mmax) is the maximum number of flows at an input (output).

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Guaranteed Rate Services

• Load balanced BvN switch provides best effort service.

• How do we provide service guarantees?

• Earliest Deadline First (EDF) based scheme.

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EDF based scheme.

• Same architecture as FCFS scheme with jitter control.• Targeted departure time is departure time of

corresponding link with capacity equal to the guaranteed rate of the flow.

• Packets served in EDF order at output buffer.

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EDF scheme

• Every packet of a guaranteed rate flow has a delay bound – targeted departure time + (N-1)Lmax + NMmax.

• Resequencing and load balancing buffer bounded by NMmax.

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Not discussed…

• Full Frames First – an algorithm that prevents packets from being mis-sequenced. (Will be discussed in next paper presentation – “Scaling Internet routers using optics”)

• Frame based scheme for guaranteed rate services – algorithm based on FFF for providing rate guarantees.