1. Scheduling Algorithms to Minimize Session Delays
Nandita Dukkipati
David Gutierrez

2. 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

3. 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

4. Background Arrival process: deterministic and known
Fluid arrivals and fluid service
Session delay, di = Di - Ai
Average session delay,
Optimal scheduling policy

5. 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 ?

6. 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

7. 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

8. 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 :

9. 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)

10. STFF - N
Not optimal
Example:

11. 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:

12. Minimizing Delays in GPS• •
GPS can realize any strict priority based schedule

13. 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

14. 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

15. 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

16. Questions
June 05, 2002