Scheduling Algorithms to Minimize Session Delays

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Scheduling Algorithms to Minimize Session Delays
Nandita Dukkipati David Gutierrez
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Introduction

• Provision of QoS in the Internet
• QoS measure for short flows Response time seen
  by the user
• Important to minimize average session delays for
  short flows
• Scheduling algorithms provide QoS at a single
  node
• Coordinate QoS at each network element to provide
  end-to-end QoS

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Problem

• Optimal schedule of sessions in the single node
  case and the network case for a known traffic
  arrival pattern
• Weight (or ?) allocations for sessions in
  Generalized Processor Sharing schedulers to
  minimize delays
• Comparison of average session delays in
  Generalized Processor Sharing and Shortest Job
  First schedulers

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Background

• Arrival process deterministic and known
• Fluid arrivals and fluid service
• Session delay, di Di - Ai
• Average session delay,
• Optimal scheduling policy

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Minimizing Delays in Single Node

• N Sessions
• Ai(t) Arrival rate of session i at time t
• Si(t) Size of session i
• Optimal schedule ?

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Shortest Job First

• Optimal policy when all jobs are present in the
  server at once in the system
• Not optimal in our system of interest

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Shortest Time to Finish First (STFF)

• 2 sessions
• Non unique optimal schedule

• STFF-2
• TAi Maximum time to arrival of last bit
• Compute the minimum time to finish, Tfi
  max(TAi, Si / C)
• Determine the priorities of the sessions

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Optimality of STFF-2

• Lemma 1 The STFF-2 discipline minimizes the
  session delay of 2 sessions at a single node
  server
• Sketch of Proof

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STFF - N

• Definition Pair wise STFF
• Lemma 2 Every optimal discipline must be pair
  wise STFF
• STFF-N

• Initialization
• Np Set of all sessions whose priorities have
  been decided Ø
• j 1 to N
• Compute the minimum time to finish, TFi, for
  unprioritized sessions
• Assign priority j to the session with min(TFi)
• Update Np and remaining capacity
• C(t) C(t) (capacity taken by sessions
  belonging to Np)

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STFF - N

• Not optimal
• Example

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Minimizing Delays in GPS

• Problem Find weights Fi in GPS to minimize
  session delays
• i1, i2, ..., iN Increasing order of session
  sizes
• Find Fi such that

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Minimizing Delays in GPS

• GPS can realize any strict priority based schedule

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Scheduling Sessions - Network Case

• Equivalent to Job Shop Scheduling problem
  NP-hard
• A lower bound on average session delay

1. For every session identify the bottleneck
router along its path 2. For a session i Ni
Set of sessions bottlenecked at router i Ci
Capacity of router 3. 4. Minimum average delay











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Heuristic Schedule - Network Case

• STFF - Network
• 1. j 1 to N
• 2.
• 3. Assign priority j to the session with min(Fi)

• Motivation
• Little's law
• Blocking Factor

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Summary

• STFF - 2 Optimal scheduling discipline for 2
  sessions at a single node
• STFF - N A heuristic for N sessions at a single
  node
• Generalized Process Sharing can realize any
  strict priority based schedule
• Weight allocation in GPS schedulers to minimize
  delays
• Lower bound on delay of sessions in the network
  case
• STFF - Network A heuristic for the network case

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Questions
June 05, 2002