1 Real-Time Queueing Theory Presented by: John Lehoczky Carnegie Mellon SAMSI Workshop Congestion Control and Heavy Traffic.

1

Real-Time Queueing TheoryReal-Time Queueing Theory

Presented by:

John Lehoczky

Carnegie Mellon

SAMSI Workshop

Congestion Control and Heavy Traffic

2

BackgroundBackground

• Real-time systems refer to computer and communication systems in which the applications/tasks/jobs/packets have explicit timing requirements (deadlines).

• These arise in (e.g.):

– voice and video transmission (e.g. teleconferencing)

– control systems (e.g. automotive)

– avionics systems

3

GoalsGoals

• For a given workload model we want:

– to predict the fraction of the workload that will meet its deadline (end-to-end in the network case),

– to design workload scheduling and control policies that will ensure service guarantees (e.g. a suitably small fraction miss their deadline),

– to investigate network design issues, e.g.:

• Number of priority bits needed

• Cost/benefit from flow tables

• Cost/benefit from keeping lead-time information

4

ModelModel

• Multiple streams in a multi-node acyclic network.

• Independent streams of jobs.

• Jobs in a stream form a renewal process and have independent computational requirements at each node

• For a given stream, each job has an i.i.d. deadline (different for different streams)

• Node processing is EDF (Q-EDF), FIFO, PS, Fixed Priority.

5

Analysis: 1Analysis: 1

• In addition to tracking the workload at each node, we need to track the lead-time (= time until deadline elapses) for each task.

• The dimensionality becomes unbounded, and exact analysis is impossible.

• We resort to a heavy traffic analysis. This is appropriate for real-time problems. If we can analyze and control under heavy traffic, moderate traffic will be better.

6

Analysis: 2Analysis: 2

• Heavy traffic analysis (traffic intensity on each node converges to 1)

• One node – workload converges to Brownian motion. Multiple nodes, workload may converge to RBM.

• Conditional on the workload, lead-time profile converges to a deterministic form depending upon – stream deadline distributions,– scheduling policy– traffic intensity

• Combining the lead-time profile with the equilibrium distribution of the workload process, we can determine the lateness fraction for each flow.

7

Processor Sharing – Exp. DeadlinesProcessor Sharing – Exp. Deadlines

8

Processor Sharing – Exp. DeadlinesProcessor Sharing – Exp. Deadlines

9

Processor Sharing – Exp. DeadlinesProcessor Sharing – Exp. Deadlines

10

Processor Sharing – Exp. DeadlinesProcessor Sharing – Exp. Deadlines

11

Processor Sharing–Const. DeadlinesProcessor Sharing–Const. Deadlines

12

Processor Sharing-Const. DeadlinesProcessor Sharing-Const. Deadlines

13

Processor Sharing-Const. DeadlinesProcessor Sharing-Const. Deadlines

14

EDF Miss Rate PredictionEDF Miss Rate Prediction=0.95EDF schedulingUniform(10,x) deadlines

EDF Deadline Miss Rate:

_

DEDF e Internet

Exponential

Uniform

: computed from the first two moments of task inter-arrival times and service times.

: Mean Deadline_

D