Guaranteed Smooth Scheduling in Packet Switches

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Guaranteed Smooth Scheduling in Packet Switches
Isaac Keslassy (Stanford University), Murali
Kodialam, T.V. Lakshman, Dimitri Stiliadis
(Bell-Labs)
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Outline

1. Frame Scheduling
2. GLJD Algorithm
3. GLJD Guarantees
4. Simulations
5. Conclusion

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Packet Switch Scheduling
Crossbar
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. . . .
N
1
N
. . . .
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Frame-Based Scheduling
Example
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Frame-Based Scheduling

• Traffic Demand Matrix
• Given a frame size of F, and an NxN integer
  demand matrix R of row and column sum equal to F
• Can we send Rij amount of traffic from i to j
  during any frame?
• Birkhoff-von Neumann (BvN) Decomposition
• We can decompose R as a sum of K lt N2 weighted
  permutations, with the weight sum equal to F.
• Frame-Scheduling
• Apply BvN decomposition, then cycle through the
  permutations (C.S Chang et al., 2000).

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Problems in BvN Decomposition

• Storage
• N512 ? Nlog2N 4.6 Kbits per permutation ?
  N3log2N 1.1 Gbits total
• Too many permutations to store on a chip
• Speed
• Time complexity in O(N4.5) (when F N2)
• Jitter (variable delay)

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What is jitter?
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Why smooth scheduling?

• Low-jitter guaranteed-bandwidth traffic
• For instance Expedited Forwarding in Diffserv
• Typically, 10 of the traffic
• Less burstiness
• Bursty TCP traffic results in multiple losses
• Increased short-term fairness
• Less buffering (or delay lines) for smoothly
  arriving flows

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Outline

1. Frame Scheduling
2. GLJD Algorithm
3. GLJD Guarantees
4. Simulations
5. Conclusion

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Smooth scheduling idea

1. Decomposition find a decomposition of R into
   matches such that each entry of R appears in at
   most one match.
2. Scheduling use a scheduling algorithm to
   smoothly schedule the matches (matches are
   independent).

Our algorithm
Known method
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Smooth scheduling example

1. Decomposition
2. Scheduling

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Optimal Smooth Decomposition

• Theorem Optimal decomposition is NP-hard
• ? Need to find a provably good approximation
  algorithm

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Smooth Decomposition Example

• Idea group together close coefficients

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GLJD (Greedy Low-Jitter Decomposition)
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Outline

1. Frame Scheduling
2. GLJD Algorithm
3. GLJD Guarantees
4. Simulations
5. Conclusion

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GLJD guarantees

• Theorem 1 (matrices)
• GLJD needs at most K2N-1 matrices
• Proof outline Consider the union of a row i and
  a column j. It has at most 2N-1 non-zero
  elements. At each iteration, at least one of
  these elements is scheduled and removed from
  further consideration.

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GLJD guarantees

• Theorem 2 (upper bound)
• Assume R of sum 1. Both D and GLJD need a
  bandwidth ? 2HN-1, i.e. O(ln N)
• Proof outline upper-bound each matrix weight and
  sum the bounds
• Theorem 3 (lower bound)
• Both D and GLJD need a bandwidth of ?(ln N)
• Proof outline use a specific matrix as a
  counter-example

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GLJD guarantees

• Theorem 4 (approximation ratio)
• GLJD is a (2-1/N) bandwidth approximation
  algorithm to D
• Proof outline upper-bound bandwidth needed by
  GLJD and lower-bound D based on matrix structure

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Outline

1. Frame Scheduling
2. GLJD Algorithm
3. GLJD Guarantees
4. Simulations
5. Conclusion

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GLJD Simulation Summary (N64)

• Gain with respect to BvN
• Smoothness
• Storage
• GLJD needs 10 times less matrices than BvN
• Complexity
• GLJD is 100 times faster than BvN
• Trade-off with
• Bandwidth Efficiency
• Bandwidth ratio guarantee 2HN-1 8.5
• Simulation 1.55

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Simulations jitter
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Conclusion

• BvN decomposition is an optimal but impractical
  algorithm
• Practical smooth decomposition with
• Lower storage requirements
• Lower complexity
• ?(ln N) bandwidth approximation ratio guarantee