Ph. D. candidate Scuola Superiore S.Anna Pisa - Italy Designing Real-Time Software visiting prof. Baruah Univ. of North Carolina Chapel Hill - NC

Designing is optimizing where:  the application is represented by a tuple of design variables x ;  x is in a design space  ;  the reward function f express our goal. The designer has to find the application x, such that:

Application model  Application model: tasks set {  1,  2, …,  n }.   i = (C i, T i, D i ), where:  C i is the max computation time of  i (WCET).  T i is the min interarrival time (or period).  D i is the relative deadline.  U i = C i / T i is the utilization factor of  i. Di  Ti.Di  Ti.Di  Ti.Di  Ti. Task  x = (2, 4, 3) absolute deadlines activation instants

Design variables/Reward function  The designer of RT software can select among:  many implementations of an algorithm ( C i );  different periods on the programmable timers ( T i );  different task deadlines ( D i ).  The reward function f(x) typically:  pushes against the schedulability;  increases for: bigger C i (better algorithm), smaller T i (the task is run more often), smaller D i (greater responsiveness).

The design space   There may be some external constraint:  task activation periods depend on external events;  only a bunch of implementation is available. The selected tasks set needs to be schedulable! We need to define a schedulability condition/constraint

Una linea Attempt 1: Response Time An iterative algorithm applied to the task set/application x x YESNO It does not provide informative result:  distance to the boundary.

Using Response Time for optimizing NO designVar2 designVar1 reward function Initial solution ? ? ? ? ? ? ? ? ? ? ? Where? A “YES/NO” test is not well suited for optimization algorithms!!

Attempt 2: Simpler Formulation The RT literature offers many simpler only sufficient test: Liu & Layland test Hyperbolic Bound More restrictive Only a suboptimum is found Opt. methods may be applied Closed Formulation

Attempt 3 * : Closed formulation + iff * the good one a closed (OK for optimization) necessary and sufficient (no suboptimum) formulation of the schedulable application domain  The goal To make it easier, we consider:  is a domain of C i, function of T i, D i.

First little break

The time demand Y i (t) It occurs when all the task are simultaneously activated at t 0, they require the maximum possible C i, every T i. It is: Time Demand Y i (t 0, t 1 ) “Max computation time required by the first i higher priority tasks in the interval [t 0, t 1 ] ”

iff schedulability condition  When the tasks from 1 to i-1 are running, the task  i is schedulable iff :  Is  3 schedulable? 1111 2222 3333

Shrinking the points set Checking the: on the infinity of points in [0, D i ] is not feasible! By arbitrarily eliminating points in the set, results in more stringent condition: By arbitrarily eliminating points in the set, results in more stringent condition: ONLY SUFFICIENT! What is the smallest set of points, such that the condition keeps being necessary and sufficient?

Previous result ? [1989. Lehoczky, Sha, Ding] 1111 2222 3333

Our result [2002. Bini, Buttazzo] 1111 2222 3333

Comparing the results 1111 2222 3333 LSD BB [0,D i ] infinity of points LSD finite number unbounded, grow for large D i / T j BB bounded by 2 i-1 ProsCons

Schedulability of  1 Schedulability of  C1C1C1C1 C2C2C2C2 Task periods are: T 1 = 3, T 2 =8, T 3 =20, D i = T i. Applying BB

Schedulability of  Applying BB (cont’d)

C1C1 C2C2 C3C3  1 condition  2 condition  3 condition Drawing the Schedulability Region

Second little break

Given the tasks set Build the schedulabilityregion YESNO WCETs C i T i and D i Check C i in the region the the Hyperplanes Hyperplanes Exact Exact Test Test Hyperplanes Exact Test

We compared the number of steps of the test with respect to: Response Time Analysis (by Audsley et al., Dec. 1993) Response Time analysis Improved (by Sjödin, Hansson, Dec. 1995) Performance of the HET

number of steps density n tasks average steps number RTA=Response Time Analysis RTI=Response Time analysis Improved HET=Hyperplanes Exact Test Performance of the HET

Putting everything together When the design variables are: The design problem is so formulated: The design domain is NOT convex! BAD NEWS

Max performance on C i : the 2 tasks case C1C1C1C1 C2C2C2C

Future work  Implementing the optimization algorithm by Branch & Bound.  To find the schedulability region in the periods/deadlines domain.  Developing a complete tool for the tuning of RT systems.

The End