Nick McKeown

Spring 2012

Lecture 4

Parallelizing an OQ Switch

EE384xPacket Switch Architectures

Scaling an OQ Switch

one output

1

k

many outputs

1

k

111

NN

Not so clear.Work conserving if memory b/w >= R(N+1)

At most two memory operations per time slot: 1 write and 1 read

Parallel OQ SwitchMay not be work-conserving

1

1

k=3

N=3

A

C

B2Time slot = 1

A5

A6

A7

A5

A6

A7

B5

B6

A8

B5

B6

A8

Time slot = 2

B6

B5

A8

C5

C6Time slot = 3

Constant size packets

ProblemHow can we design a parallel OQ work-conserving switch from slower parallel memories?

Work Conserving

Theorem (sufficiency)A parallel output-queued switch is work-conserving with 3N –1 memories, each able to perform at most one memory operation per time slot.

Re-stating the Problem

1. There are K cages which can contain an infinite number of pigeons.

2. Assume that time is slotted, and in any one time slota. At most N pigeons can arrive and at most N can

depart. b. At most 1 pigeon can enter or leave a cage via a

pigeon hole.c. The time slot at which arriving pigeons will depart

is known

3. For any switchWhat is the minimum K, such that all N pigeons can be immediately placed in a cage when they arrive, and can depart at the right time?

Only one packet can enter or leave a memory at time t

Intuition for Theorem

Only one packet can enter a memory at time t

Time = t

DT=t+X

DT=t+X

DT=t

Only one packet can enter or leave a memory at any time

Memory

Proof of Theorem

When a packet arrives in a time slot it must choose a memory not chosen by

1. The N – 1 other packets that arrive at that timeslot.

2. The N other packets that depart at that timeslot.

3. The N - 1 other packets that can depart at the same time as this packet departs (in future).

Proof

By the pigeon-hole principle, the switch can be work-conserving if there are 3N –1 memories, each able to perform at most one memory operation per time slot.

Memory

Memory

Memory

Memory

Memory

Memory

Memory

A Parallel Shared Memory Switch

C

A

Departing Packets

R

R

Arriving Packets

A5

A4

B1

C1

A1

C3

A5

A4

From theorem 1, k = 7 memories don’t suffice .. but 8 memories do

Memory

1

K=8

C3

At most one operation – a write or a read per time slot

B

B3

C1

A1

A3

B1

Distributed Shared Memory Switch

The central memories are distributed to the line cards and shared.Memory and line cards can be added incrementally.

From theorem 1, the switch is work-conserving if we have a total of 3N –1 memories, each able to perform one operation per time slot i.e. a total memory bandwidth of 3NR.

Switch Fabric

Line Card 1 Line Card 2 Line Card NR R R

Memories Memories Memories

Switch bandwidth

What switch bandwidth does the DSM switch need in order to be work-conserving?

Theorem (sufficiency)A switch bandwidth of 4NR is sufficient for a distributed shared memory switch to be work-conserving.

ProofThere are a maximum of 3 memory accesses and 1

external line access per time slot.

Switch AlgorithmWhat switching algorithm allows the DSM switch to be

work-conserving? 1. Shared bus: No algorithm needed.

2. Crossbar switch: Algorithm needed because only permutations are allowed.

Theorem

An edge coloring algorithm can switch packets for a work-conserving distributed shared memory switch

ProofKönig’s theorem: Any bipartite graph with maximum degree has an edge coloring with colors.

Summary - Switches with 100% throughput

None2NR2NR2NR/kNk

Maximal2NR6NR3R2N

MWMNR2NR2RNCrossbarIQ

None2NR2NR2NR1BusShared Mem.

Switch Algorithm

Switch BW

Total MemBW

Mem. BW

# Mem.Fabric

NoneNRN(N+1)R(N+1)RNBusOQ

PSM

C. Sets4NR2N(N+1)R2R(N+1)/kNkClosPPS - OQ

C. Sets4NR4NR4RN

C. Sets6NR3NR3RN

Edge Color4NR3NR3RNXbar

C. Sets3NR3NR3NR/kkBus

C. Sets4NR4NR4NR/kNkClos

Time Reserve*

3NR6NR3R2NCrossbar

PPS

DSMJuniper M-series

CIOQ Cisco GSR