Exponential risk-sensitive optimal scheduling

We study a problem of scheduling jobs with random processing times under a risk sensitive optimality criterion. We assume that risk sensitivity is given by an exponential disutility function. We present in detail the job scheduling model we consider and present its standard formulation as a controlled Markov process. To facilitate comparisons with the results we derive in this paper, we include the analysis of the stochastic optimal control problem corresponding to the risk null performance criterion given by an expected total weighted completion time. We present both a detailed dynamic programming (DP) algorithm as well as an interchange argument. We introduce risk-sensitivity by considering the minimization of the expected exponential utility of the total weighted completion time. We develop the corresponding DP algorithm from which the risk-sensitive optimal policies (schedules) are obtained. It is interesting to note, that for the risk-sensitive criterion a simple interchange argument is not applicable, and thus the only general computational and analytical tool for this situation is the DP algorithm that we develop. Finally, by means of a simple example, we illustrate how the optimal schedule depends on the risk sensitivity coefficient