Heterogeneous distributed computing systems often must operate in an environment where system parameters are subject to uncertainty. Robustness can be defined as the degree to which a system can function correctly in the presence of parameter values different from those assumed. We present a methodology for quantifying the robustness of resource allocations in a dynamic environment where task execution times are stochastic. The methodology is evaluated through measuring the robustness of three different resource allocation heuristics within the context of a stochastic dynamic environment. A Bayesian regression model is fit to the combined results of the three heuristics to demonstrate the correlation between the stochastic robustness metric and the presented performance metric. The correlation results demonstrated the significant potential of the stochastic robustness metric to predict the relative performance of the three heuristics given a common objective function.