Design variability due to within-die and die-to-die variations has potential to significantly increase the maximum operating clock period and the leakage power of the system in future process technology generations. when minimizing total energy of an MPSoC system, the variations in both the clock period and the leakage power of the multiple cores have to be taken into account. This paper targets system level task allocation to stochastically minimize the total energy of the system running multiple applications on MPSoC under process variation. In this work, we first introduce an integrated model for total energy of the system which incorporates all the factors - dynamic energy, variations in leakage energy and variations in clock frequency. Our model shows very close correlation with the energy of the system computed using empirical methods. We provide formal theorems of the optimality of the solution in simple scenarios. For identical cores, the proposed load assignment can be found optimally using stochastic ordering. We then propose intuitive matching technqiues to perform load assignments on non-identical processing cores. The proposed energy model and proposed techniques enable efficient computation of task distribution without requiring highly expensive stochastic analysis. We apply this model in allocating tasks in the embedded system benchmark suites on MPSoC. In addition, we propose a technique for allocating tasks that maximizes the energy yield of the system over an interval of energy constraints rather than a single energy constraint. In our experiments, we show that using our technique, we improve the average yield of the system significantly. Our experiments show that we can reach up to 5.3 times improvement in average yield of the energy constraints which translates into 0.6 increase in yield of the energy constraints on average.