Real-time systems usually operate in an environment that changes continuously. These changes cause the performance of the system to vary during run time. An allocation of resources in this environment must be robust. Using the amount of load variation that the allocation can accommodate as a measure of robustness, we develop a mathematical formulation for the problem of robust resource allocation. Due to the complexity of the models used to represent the problem, the formulation is non-linear. We propose a linearization technique based on variable substitution to reduce the mathematical formulation to a mixed integer programming formulation, called SMIP. Compared with existing techniques, the search space of SMIP is not restricted. Thus, if a feasible allocation exists, SMIP will always produce an optimal allocation.