Worst-Case Execution Time Problem: Given a program and an architecture, find the worst-case execution time (WCET) of that program on the given architecture Fundamental Limitation: Given an arbitrary program, the problem of deciding if it will ever stop is undecidable (known as the Halting Problem) By restricting the programming constructs used, we can, in many cases, get an upper bound

Approaches to WCET Estimation Program and architecture analysis Easy to do when the architecture is simple No cache No pipelining Single instruction issue width Very difficult in modern speculative architectures Deep pipelines with interlocks between stages Multiple levels of caching Multiple instructions issued per clock Speculative execution and branch prediction

WCET Estimation Approaches Profiling-based statistical approach Estimate, based on experimental results, the confidence associated with some estimate of WCET Technique uses the statistics of extremes

Gumbel Distribution Function Given a set of sample data {x1, x2, …, xn}, define m = E[x] - ls s=(sqrt(6)/p).StDev(x) l = 0.5772156649... Mean and standard deviation are estimated from the data using the standard statistical formulae Then, G(x) = exp(-exp(-(x-m)/s)), x>s is the Gumbel distribution function

Using the Gumbel PDF Run the software under application conditions: make a total of n such profiling runs for some selected n Calculate the parameters for the Gumbel distribution Condition the Gumbel distribution on the maximum observed value Read from the conditioned distribution the WCET estimate for the desired confidence level