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

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