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Real-Time Queueing Theory

Real-Time Queueing Theory. Presented by: John Lehoczky Carnegie Mellon SAMSI Workshop Congestion Control and Heavy Traffic. Background. Real-time systems refer to computer and communication systems in which the applications/tasks/jobs/packets have explicit timing requirements (deadlines).

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Real-Time Queueing Theory

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

  2. Background • 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. Goals • 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. Model • 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: 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: 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. Deadlines

  8. Processor Sharing – Exp. Deadlines

  9. Processor Sharing – Exp. Deadlines

  10. Processor Sharing – Exp. Deadlines

  11. Processor Sharing–Const. Deadlines

  12. Processor Sharing-Const. Deadlines

  13. Processor Sharing-Const. Deadlines

  14. EDF Miss Rate Prediction EDF Deadline Miss Rate: =0.95 EDF scheduling Uniform(10,x) deadlines Internet Exponential : computed from the first two moments of task inter-arrival times and service times. : Mean Deadline Uniform

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