1. Imprecise Computation September 7, 2006
JWS Liu, KJ Lin, WK Shih, AC. S. Yu, JY Chung, and W. Zhao, Algorithms for scheduling imprecise computations, IEEE Copmuter, pages 58-68, 1991.

2. Problem: Possibility of Timing Failure
Use WCET for hard real-time tasks
Meet all deadlines
Hard RT systems can be overloaded or disrupted by the environment
Too many targets too track at the same time
Faulty sensors in a cruise control system
Deriving WCET is not trivial
What about context switch time ignored in schedulability analysis?
Leave some margin in your schedule
Take an iterative approach
Measurement: No guarantee on accuracy
Analysis: Need to consider architectural details, clock spped, cache, instruction pipelining, multithreading, etc.

3. Problem: Possibility of Timing Failure
Use average case execution time for soft real-time tasks
Usually, i.e., in normal cases, meet deadlines
Under transient overload, e.g., due to abnormal sensor values that take more time for processing, deadlines can be missed

4. Imprecise Computation Model
Avoid timing faults during transient overloads
Prefer timely, imprecise results to late, precise results
A task consists of a mandatory and an optional part
The mandatory part must complete before the task deadline
The optional part improves the quality
Can be skipped under overload
Tradeoff between quality and timeliness

5. Examples
Radar tracking: Get estimated target locations in a timely fashion than accurate location info too late
Video streaming: Transmit a low quality image in time rather than missing the deadline, e.g., to meet the 30 frames/s requirement
Control loop: Produce an approximate result by a control law as long as the controlled system, e.g., cruise control system, remain stable

6. Imprecise Computation Model
A task produces a precise result if both the mandatory and optional parts are executed
A task produces an imprecise, i.e., approximate, result if only the mandatory part is executed
Monotone: Quality monotonically increases as a task executes longer
0/1: Either fully execute the optional part or not at all

7. Imprecise Task Model Task Ti(mi, oi, di)
mi: exec time of mandatory part
oi: exec time of optional part
di: deadline of Ti
mi = 0 for all task Ti in traditional soft real-time systems
oi = 0 for all task Ti in traditional hard real-time systems

8. Error Function Monotone task model 0/1 task model
Error ei = fi(oi – ci) where ci is the completed optional part of Ti
0/1 task model
ci = oi or ci = 0 for task Ti

9. Imprecise Scheduling Problems
Minimize total error
Total error E = ∑i=1, N wiei where wi > 0 is the weight
Find a schedule that is optimal in that it is feasible and minimizes the total error
Refer to F algorithm in the paper
Minimize max error
Find a feasible schedule minimizing the max error rather total error
Minimize #discarded optional tasks
Minimize #tardy tasks

10. Error Charasteristics of Periodic Tasks
Cumulative Error
Tracking & control applications
Total error can be infinite
Take average over π, i.e., multiple of all the periods
Non-cumulative Error
Audio & video
Efficient huerisitic:
Execute optional parts only after executing all ready mandatory tasks
Assign high priority to mandatory tasks
Apply RMS to find a feasible schedule for mandatory tasks

11. (m, k) Firm Deadline Model
[1] Moncef Hamdaoui, Parameswaran Ramanathan, “A Dynamic Priority Assignment Technique for Streams with (m,k)-Firm Deadlines,” IEEE Transactions on Computers, Vol. 44, Issue 12, Dec
[2] P. Ramanathan, “Overload Management in real-time control applications using (m,k)-firm guarantee”, IEEE Transactions on Parallel and Distributed System, Vol. 10, Issue 6, June 1999.

12. (m, k) Firm Deadlines
For a task Ti, meet at least mi deadlines out of ki consecutive jobs, i.e., task (Ti) instances
Example
Associate a video stream with (2,3) firm deadlines
Transmit at least 2 out of 3 consecutive video frames within the deadline
The other two frames can be dropped under overload → Different from imprecise computation

13. (m, k) Firm Real-Time Task Model
Ti = (Ci, Pi, mi, ki)
Ci: Computation time
Pi: Period
Maximum allowable loss rate = (ki-mi)/ki
Dynamic Failure
Fewer than mi jobs in the window of ki consecutive jobs of Ti meet the deadline

14. Task Ti’s instance Ti,a released at aPi cannot be dropped if:
a = ami/ki ki/mi where a = 0, 1, 2, ...
Example: (mi, ki) = (2, 3)
Instances 0 and 1 of Ti cannot be dropped according to the equation above

15. Distance Based Priority (DBP) Scheme [1]
Assign the highest priority to the task that is closest to a dynamic failure
System Model

16. MK-RMS Utilization Bound [2]
Sufficient (but not necessary)
A task set consisted of N periodic (m,k) tasks is schedulable if:
Ut = ∑i=1, N (Ci/Pi)(mi/ki) ≤ n(21/n – 1)
Algorithm:
Follow RMS
Under overload, skip task instances if no dynamic failure will occur

17. Any Question?
We will cover “Elastic Task Model” next time (and some power-aware RT scheduling if time permits)