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Routing without Flow Control

Costas BuschRensselaer Polytechnic Institute

Maurice HerlihyBrown University

Roger WattenhoferMicrosoft Research

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n

n

The network: meshnn

• Discrete time• Bi-directional links• At most one packet per link direction

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Dynamic Routing:Packets are injected continuously

destination

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A new packet can be injected whenthere is a free link:

A link direction is empty

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Most dynamic routing algorithmsuse flow control:

Don’t utilize all the free links

Disadvantage: Network is under-utilized

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Our Routing Algorithm:

•No flow control

•Utilizes all the free links

Advantage: Network is fully-utilized

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Features of our algorithm:• Dynamic

• Hot potato

• Optimal delivery time:

• Injection time guaranty: )(nO

)(nO

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Talk Outline The Algorithm

Time AnalysisStabilityFuture Work

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Hot-Potato Routing:

• Nodes are buffer-less

• Packets are immediately forwarded

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Conflicts

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Conflict

Conflicts

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Deflected

Conflicts

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Packet states:

Running

Excited

Active

Sleeping

Priorities:

high

low

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Sleeping packet

destination

Random destination

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Sleeping packet

Follows a path to destination

destination

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Sleeping packet

becomes Activewithprobability

n1

n

n

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Active packet

Follows a greedy path

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Active packet

Follows a greedy path

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Active packet

A conflict situation

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Conflict

Active packet

A conflict situation

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Deflected

Active packet

A conflict situation

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Deflected

Active packet

A conflict situation

becomes Excitedwithprobability

n

p 1

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Excited packet

Follows a one-bend path

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Excited packet

Follows a one-bend path

becomes Running

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Running packet

Follows a one-bend path

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Talk Outline The Algorithm

Time AnalysisStabilityFuture Work

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Good condition for a column: at most non-sleeping packets with destination in the column

n10

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Expected delivery time for one packet: n

(when the destination column is in good condition)

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Initially a packet is sleeping

In expected time stepsbecomes active

n

We will show: An active packet is delivered in expected time steps nO

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Interrupting a one-bend path

Excited

Time 1

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Interrupting a one-bend path

Running

Time 2

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Interrupting a one-bend path

Running

Excited

Time 2

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Interrupting a one-bend path

Running

Running

Time 3

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Interrupting a one-bend path

Running

Running

Time 4

conflict

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Interrupting a one-bend path

Active

Running

Time 5

deflected

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No interruption probability:

Number of non-sleeping packetswith destinations in same column

mp)1(

Excitement probability

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No interruption probability:

cp m )1(

n1

n

constant

(when the destination column is in good condition)

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Probability of success after a deflection:

n

cp 1

Expected number of deflections until success: n

Expected delivery timefor an active packet: nO

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Talk Outline The Algorithm

Time AnalysisStabilityFuture Work

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Divide time in time periods:

n6t

Examine the condition of a column

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Good condition Bad condition

ne1

ne

nm 10 nm 10

1 time period

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Good condition Bad condition

ne1

ne

nm 10 nne1

nne

nm 10

4n time periods

1 time period

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Good condition Bad condition

ne1

ne

nm 10 nm 10

1 time period

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Proof OutlineIn a time period:

•At most new non-sleeping packets are generated with destinations in the column

n2

•At least non-sleeping packets are delivered (if )

n2nm 8

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Good condition Bad conditionnm 10 nne1

nne

nm 10

4n time periods

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Proof OutlineIn a time period:

•At most new non-sleeping packets are generated with destinations in the column

n2

•At least non-sleeping packets are delivered

n3

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•Most of the time, the columns are in good condition

•Each packet is delivered in expected time

Consequences:

n

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Talk Outline The Algorithm

Time AnalysisStabilityFuture Work

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•Arbitrary network topologies

•De-randomization:Determistic destinationsNo randomized algorithm

• Small number of packets