Scheduling Jobs Subject to Nonhomogeneous Poisson Shocks

Consider n tasks which must be attempted in some order. To successfully complete any given task requires a random amount of time. However, we suppose that there are external events, called shocks, which occur according to a nonhomogeneous Poisson process. If no shocks occur while a task is being performed, then that task is considered a success. If a shock does occur, then work on that task ends and work on the next one begins. We consider such problems as maximizing

1. The expected number of successful tasks;

2. The length of time until no tasks remain;

3. The expected total reward earned; where a reward R₁ is obtained upon successful completion of task i.