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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 Ф) 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 { Φi } in GPS to minimize session delays i1, i2, ..., iN: Increasing order of session sizes Find { Φi } 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
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