Memory Management Algorithms

1
Memory Management Algorithms for DSM Switches
Huan Liu, Damon Mosk-Aoyama
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Distributed Shared Memory Switch

• What is known (for FIFO)
• 2N is necessary
• 3N-1 is sufficient
• Questions
• Can we close the gap?
• Possible to have an implementable algorithm?

3
Our main results

• On the gap
• Counter example to show 2.25N is necessary
• A heuristic that uses 2.5N memories
• On practical algorithms
• Simulation results on simple algorithms under
  Bernoulli i.i.d. traffic

4
The counter example

• Consider 4x4 switch
• Can generalize to arbitrary N

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Output 1
5
6
6

7
7
8
8
Output 4
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If we have N/4 more
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Output 1
1
1
1
6

7
2
2
5
7

8
3
5
3
8
9
4
4
2
9
Output 4
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Observation

• Greedily minimizing the number of memories used
  can lead to trouble
• Need to reuse memories later as time slot fills up

1
2
3
4
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Heuristics with 2.5N memories

• Minimize intersection between adjacent time slots
  
• Minimize intersection between neighboring pairs
• After N/2 cells arrived in a time slot, reuse
  memories already assigned to the adjacent time
  slot.
• Simulation has been running for 100M cycles with
  no problem

Minimize intersection
1
2
3
4
2
1
6
4
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Random algorithm

• Assign memories to arriving cells randomly
• Drop if another cell using the memory is
• departing now
• departing in the future in the same time slot

S1
S2
Si

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Upper bound on Drop Rate

• Suppose there are memories. The drop
  probability is

• The drop rate can now be computed as

• Use Si distribution from M/M/1

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Fixed Arrival Rate
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Fixed Number of Memories
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Distributed random algorithm

• Each packet makes independent decision
• Pick a random memory that is NOT
• departing now
• departing in the same time slot in the future
• If two arriving packets pick the same memory, we
  drop one

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Distributed random algorithm - simulation
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Centralized random algorithm

• Assign each packet in turn
• Randomly pick a memory that is NOT
• departing now
• departing in the same time slot in the future
• assigned for other packets arriving at the same
  time

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Centralized random algorithm - simulation
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Conclusion

• Still gap ? more work
• Better counter example?
• Prove 2.5N is sufficient
• Also gap between theory and practical algorithm