1. 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

2. 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

3. 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

4. Gumbel Distribution Function
Given a set of sample data {x1, x2, …, xn}, define
m = E[x] - ls
s=(sqrt(6)/p).StDev(x)
l =
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

5. 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