1. Resource Augmentation for Fault-Tolerance Feasibility of Real-time Tasks under Error Bursts Abhilash Thekkilakattil, Radu Dobrin, Sasikumar Punnekkat and Huseyin Aysan

2. Real-time tasks Dependable Hard Real-time Systems web images Fault tolerance requirements Timing characteristics of the physical components Event occurrences

3. Fault Tolerance in Real-time Systems Nature of errors -Normally continuous events e.g., a vehicle passing through electromagnetic fields -Occur continuously over a period of time: error bursts Challenging to handle in classical FT-feasibility analysis -Need to map continuous events into singleton events Temporal redundancy for transient faults faults/errors as singleton events error Burst classical FT-feasibility analysis

4. Why FT-feasibility? FT-feasibility analysis simplifies system design process: ●Can check for the existence of a schedule even before finding the scheduler ●If a schedule exists: ●Use optimal scheduling policies e.g., EDF ●If a schedule does not exist: ●Adjust task attributes ●Use a faster processor Use of a faster processor ●Can we find upper-bounds? ●Will the search for the optimal speed eventually terminate? ●Is the speed-up required affordable?

5. Problem Description Questions: ●How can we perform an FT-feasibility analysis for a given set of temporally redundant real-time tasks under a specified error burst length? ●If the real-time task set is not found to be FT-feasible, what is the lowest processor speed that guarantees its FT-feasibility under the error burst?

6. Problem Description Questions: ●How can we perform an FT-feasibility analysis for a given set of temporally redundant real-time tasks under a specified error burst length? ●If the real-time task set is not found to be FT-feasible, what is the lowest processor speed that guarantees its FT-feasibility under the error burst?

7. Definitions t T length worst case error overhead (E t ) ε ε Worst Case Temporal Wastage (W err (t)) = maximum wasted execution time ∑

8. Assumptions ● Objective: Find the Worst Case Temporal Wastage at time t ● Strategy: we assume that there are no deadline misses under the error burst and derive the sufficient condition for this to be true t’t’ t abs. deadline of τ i Error detection τiτi

9. Case 1 t’t’ t dkdk W err (d k ) = W err (t) W err (d k ) = W err (d (k-1) ) Under EDF, the tasks released in the interval [t’,t] having a deadline greater than t are not hit by the error burst τjτj τiτi Worst Case Temporal Wastage at the absolute deadline of a job that is not hit by the error burst abs. deadline of τ i t < d (k-1) ≤ d k

10. Case 2 τiτi τjτj τkτk t’t’t τiτi τjτj τkτk t’t’t τiτi τjτj τkτk t’t’t Worst Case Temporal Wastage at t when the error burst hits only a single job scenario a: W err (t)=2(C k – ε) scenario b: W err (t)= 2(C j – ε) scenario c: W err (t)= 2(C i – ε)

11. ε Case 3 ε ε ε τiτi τjτj τkτk τlτl t’t’ t ε T length Worst Case Temporal Wastage at t when the error burst hits more than one job (C l – ε) (C k – ε) (C j – ε) (C i – ε) W err (t) =2(C i – ε)+ ∑ (C k – ε), D k ≤ D i

12. Worst Case Temporal Wastage: General Case W err (t) = Max Case 1 : WCTW at the previous absolute deadline If τ i is not hit by the error burst Case 2 : max{2(C k – ε)}, D k ≤ D i When the error burst hits only a single job Case 3 : 2(C i – ε) + ∑ (C k – ε), D k ≤ D i When the error burst hits more than one job t is the absolute deadline of a task τ i

13. FT-Feasibility: A Sufficient Condition DBF(t)T length t EtEt E t + DBF(t) ≤ t E t = W err (t) + T length

14. Problem Description Questions: ●How can we perform an FT-feasibility analysis for a given set of temporally redundant real-time tasks under a specified error burst length? ●If the real-time task set is not found to be FT-feasible, what is the lowest processor speed that guarantees its FT-feasibility under the error burst?

15. Resource Augmentation t DBF(t) deadline miss T length T length + W err (t) + DBF(t) ≤ t [W err (t) =x+y] If, T length + W err (t) + DBF(t) > t then, speed-up by S, T length + W err (t) /S+ DBF(t)/S ≤ t Hence, S ≤ (DBF(t)+ W err (t) )/ (t - T length ) =>S = max{(DBF(t)+ W err (t) )/ (t -T length )} S = max { (DBF(t) + W err (t) )/ (t - T length )} Is S bounded ? X + Y’ = W err (t) /S S ≤ 6 [y ≥ 2 => T length ≤ D min /2] X + Y = W err (t) DBF(t)DBF(t)/S Put W err (t) = 2DBF(t) and DBF(t) = t S ≤ 3t/(t- T length ) S ≤ 3y/(y-1) y=t/ T length

16. Conclusions ● Fault tolerance feasibility analysis of real-time tasks ●A sufficient condition for FT-feasibility ● Resource augmentation bounds for FT-feasibility ●Speed-up ≤ 6 if T length ≤ shortest deadline/2 ● Future work: ●More severe error models ●Utilization based test for FT-feasibility

17. Thank you ! Questions ?