{sorma, petel, zebpe}@ida.liu.se Performance Analysis of Applications with Stochastic Task Execution Times Sorin Manolache, Petru Eles, Zebo Peng University of Linköping, Sweden {sorma, petel, zebpe}@ida.liu.se

Overview Toolset consisting of three tools Computation model: set of annotated task graphs Main performance indicator: expected deadline miss ratio per task or task graph

Computation Model Period 2 4 6 12 9 3 Execution times execution time probab E Deadlines B C Late task policy F Scheduling policy D P1 P2

Internals Concurrently constructs and analyses the GSMP underlying the system. Method of supplementary variables + tricks for memory reduction. Constructs the GSMP. Approximates the GSMP with a much larger CTMC. Exploits the regular structure of the CTMC for on-the-fly generation of its infinitesimal generator. Recursively computes the finishing time distribution, ignoring some dependencies among some of the random variables modelling the system.

Limitations We assume non-preemptive execution in all three methods First method is efficient only in the case of monoprocessor applications Third method is limited to fixed-priority scheduling

Scalability Third method is polynomial O(NLCM/h|ETPDF|/h) First two methods build the GSMP underlying the application  Analysis visits each state that described the behaviour of the system  Theoretically exponential, but practically…

…Practically First method (for monoprocessors) 20 independent tasks

…Practically Second method (for multiprocessors) 60 dependent tasks on 2 processors 18 dependent tasks on 6 processors

Accuracy First method gives exact results Second method relies on approximating generalised distributions with Coxian distributions  problems with non-smooth distributions Third method gives approximate results. Its accuracy is experimentally investigated

Case Study Tasks A and B have exponentially distributed execution times, average rate = 1/7 and 1/2 respectively Task C, D, and E execute for 4, 5, and 6 time units respectively Interprocessor communication takes 0.5 time units Tasks on the orange processor in descending order of priorities: E, C, D. A B E C D

Case Study (cont’d)

Conclusions Three performance analysis approaches for applications with stochastic task execution times Multiproc. Accuracy Exponential Design space exploration Exact approach No Exact Theoretically yes, practically not a problem ETPDF approx. Yes Good for smooth distributions Yes, in the number of processors Equation-based Coarser than ETPDF approx. No. Linear in the number of tasks.