Optimal Computer Crash Performance Precaution

We thus define a tolerance level r meaning that we only tolerate that the completion time of a parallel program after a crash is at most a factor r + 1 larger than if we use optimal allocation on m− 1 computers. This is an r-dependent restriction of the set of allocations of a program. Then, what is the worst-case ratio of the optimal r-dependent completion time in the case of no crash and the unrestricted optimal completion time of the same parallel program? We denote the maximal ratio of completion times f(r,m) i.e., the ratio for worst-case programs. In the paper we establish upper and lower bounds of the worst-case cost function f(r,m) and characterize worst-case programs.