Often, parallel and distributed computing systems must operate in an environment replete with uncertainty. Determining a resource allocation that accounts for this uncertainty in a way that can provide a probabilistic guarantee that a given level of quality of service (QoS) is achieved is an important research problem. This paper defines a stochastic methodology for quantifiably determining a resource allocation?s ability to satisfy QoS constraints in the midst of uncertainty in system parameters. Uncertainty in system parameters and its impact on system performance are modeled stochastically. This stochastic model is then used to derive a quantitative expression for the robustness of a resource allocation. The paper investigates the utility of the proposed stochastic robustness metric by applying the metric to resource allocations in a simulated distributed system. The simulation results are then compared with deterministically defined metrics from the literature.