Guaranteed Smooth Scheduling in Packet Switches

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Guaranteed Smooth Scheduling in Packet Switches
Isaac Keslassy, Murali Kodialam, T.V. Lakshman,
Dimitri Stiliadis
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Outline

• Short History of Packet Switching
• Smooth Scheduling
• GLJD Algorithm
• GLJD Guarantees
• Simulations
• Bursty Scheduling
• Jitter-Insensitive Scheduling

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Input Queueing
Input 1
Scheduler
?
?
Input N
Output 1
Output N
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Input QueueingHead of Line Blocking
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Virtual Output Queueing
Scheduling ? finding matchings
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Other applications

• Packets scheduled using matchings in
• Routers (best-effort) as well as SONET TDM
  switches (periodic)
• Optics Broadcast-and-select WDM TTFR
• Wireless N-to-N FDMA

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Frame-Based Scheduling
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Birkhoff-von Neumann (BvN) Theorem

• Theorem if R is an NxN doubly-stochastic matrix,
  we can decompose it as a sum of K weighted
  permutations, with K lt N2

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Problems in BvN Decomposition

• Scalability
• Too many permutations for an SRAM chip
• N512 gt Nlog2N4,608bits/permutation gt
  N3log2N150Mbytes total
• Speed
• Time complexity in O(N4.5)
• Jitter (variable delay)

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Outline

• Short History of Packet Switching
• Smooth Scheduling
• GLJD Algorithm
• GLJD Guarantees
• Simulations
• Bursty Scheduling
• Jitter-Insensitive Scheduling

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What is smooth scheduling?
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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 (buffer of 1 instead of 25)

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Smooth scheduling idea

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

Our algorithm
Known method
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Smooth scheduling example

• Decomposition
• Scheduling

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Optimal Smooth Decomposition

• Theorem Optimal decomposition is NP-hard
• ? Need to find an approximation algorithm

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Smooth Decomposition Example

• Idea group together close coefficients

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GLJD (Greedy Low-Jitter Decomposition)

• Algorithm at each iteration, pick biggest
  coefficient left, then biggest non-conflicting,
  etc., until convergence

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Outline

• Short History of Packet Switching
• Smooth Scheduling
• GLJD Algorithm
• GLJD Guarantees
• Simulations
• Bursty Scheduling
• Jitter-Insensitive Scheduling

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GLJD guarantees

• Theorem 1 (matrices)
• GLJD needs at most K2N-1 matrices
• Theorem 2 (upper bound)
• Assume R of sum 1. Both D and GLJD need a
  bandwidth ? 2HN-1, i.e. O(ln N)
• Theorem 3 (lower bound)
• Both D and GLJD need a bandwidth of ?(ln N)
• Theorem 4 (approximation ratio)
• GLJD is a (2-1/N) bandwidth approximation
  algorithm to D

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Outline

• Short History of Packet Switching
• Smooth Scheduling
• GLJD Algorithm
• GLJD Guarantees
• Simulations
• Bursty Scheduling
• Jitter-Insensitive Scheduling

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Simulations jitter
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Simulations complexity

• Simulations typically show that
• GLJD needs 10 times less matrices than BvN
• GLJD is 100 times faster than BvN

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Simulations bandwidth efficiency

• GLJD vs. BvN
• Guarantee ?(log N).
• Simulation 1.55

(N64)
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Outline

• Short History of Packet Switching
• Smooth Scheduling
• GLJD Algorithm
• GLJD Guarantees
• Simulations
• Bursty Scheduling
• Jitter-Insensitive Scheduling

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Bursty scheduling

• Assume optical transmissions with slow tuning
  times

• Then we prefer a very bursty scheduling

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Bursty scheduling

• Objective find a decomposition that
• Minimizes the number of permutations
• Minimizes the bandwidth needed
• We can use again GLJD!
• Compared with GLJD smooth scheduling same
  decomposition, different scheduling
• Guarantees
• K?2N-1,
• BW ?(ln N)

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Outline

• Short History of Packet Switching
• Smooth Scheduling
• GLJD Algorithm
• GLJD Guarantees
• Simulations
• Bursty Scheduling
• Jitter-Insensitive Scheduling

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Problems in BvN Decomposition

• Scalability
• ?(N2) permutations, problem for SRAM chip
• Speed
• Time complexity in O(N4.5)
• Jitter

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Pigeonhole Scheduling

• 1) Assume some decomposition already exists for R
  integer of sum ?M, with at most 2M-1 matrices.

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Pigeonhole Scheduling

• 2) Assume a new request. There is always a slot
  to schedule it without moving the other requests.

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Pigeonhole Scheduling Results

• For R with integer coefficients of sum M
• 2M-1 matrices
• 2-1/M BW-approximation ratio to BvN
• (Similar to Clos network)
• For R with real coefficients
• 2dN-1 matrices
• 2(N-1)/d BW-approximation ratio
• (d is some parameter).
• Example d4N gt 10N matrices, 2.25 BW

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Conclusion

• BvN decomposition is an optimal but impractical
  result
• Practical smooth decomposition with ?(ln N)
  bandwidth approximation ratio
• Practical bursty decomposition with ?(ln N)
  bandwidth approximation ratio
• Practical pigeonhole decomposition with 2?
  bandwidth approximation ratio, trade-off with
  number of matrices