Routing without Flow Control Costas Busch Rensselaer Polytechnic - - PowerPoint PPT Presentation

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Routing without Flow Control Costas Busch Rensselaer Polytechnic - - PowerPoint PPT Presentation

Jan 11, 2023 261 likes •770 views

Routing without Flow Control Costas Busch Rensselaer Polytechnic Institute Maurice Herlihy Brown University Roger Wattenhofer Microsoft Research 1 The network: n mesh n n n Discrete time Bi-directional links At most one

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Routing without Flow Control

Costas Busch

Rensselaer Polytechnic Institute

Maurice Herlihy

Brown University

Roger Wattenhofer

Microsoft Research

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n n

The network:

mesh

n n

  • 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 when there is a free link: A link direction is empty

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Most dynamic routing algorithms use 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:

) (n O ) (n O

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Talk Outline

The Algorithm

Time Analysis

Stability Future 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

Active

with probability

       n 1 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

Excited

with probability

        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 Analysis

Stability Future Work

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Good condition for a column:

at most non-sleeping packets with destination in the column

n 10

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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 steps becomes active

 

n 

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

 

n O

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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 packets with destinations in same column

m

p) 1 ( 

Excitement probability

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No interruption probability:

c p m   ) 1 (

       n 1

 

n 

constant

(when the destination column is in good condition)

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Probability of success after a deflection:

         n c p 1

Expected number of deflections until success:

 

n 

Expected delivery time for an active packet:

 

n O

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Talk Outline

The Algorithm

Time Analysis

Stability Future Work

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Divide time in time periods:

n 6 t

Examine the condition of a column

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Good condition Bad condition

n

e  1

n

e n m 10  n m 10 

1 time period

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Good condition Bad condition

n

e  1

n

e n m 10 

n

ne  1

n

ne n m 10 

4n time periods 1 time period

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Good condition Bad condition

n

e  1

n

e n m 10  n m 10 

1 time period

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Proof Outline

In a time period:

  • At most new non-sleeping

packets are generated with destinations in the column

n 2

  • At least non-sleeping packets

are delivered (if )

n 2 n m 8 

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Good condition Bad condition

n m 10 

n

ne  1

n

ne n m 10 

4n time periods

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Proof Outline

In a time period:

  • At most new non-sleeping

packets are generated with destinations in the column

n 2

  • At least non-sleeping packets

are delivered

n 3

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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 Analysis

Stability Future Work

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  • Arbitrary network topologies
  • De-randomization:

Determistic destinations No randomized algorithm

  • Small number of packets