Some results on a Bayesian sequential scheduling on two identical parallel processors

Suppose that there are two types J(0) of m jobs and J(1) of n jobs, which are processed by either one of two identical machines. The processing time: of one job of class J(0) or J(1) is the random variable from the exponential distribution with parameter u or v, respectively. The true value of v is unknown, but it has the gamma distribution with parameters w and alpha as the prior distribution. Preemption is not allowed in this problem. The objective is to minimize the expected total flowtime. From the Bayesian point of view, the problem is formulated by the principle of optimality of dynamic programming and the optimal solutions are obtained for some special cases.