Asymptotic analysis of stochastic manufacturing system with slow and fast machines

A dynamic flow-shop is considered consisting of one slow and one fast machine. Production capacities of both machines vary randomly according to a Markov chain whose transition rates are consistent with the time scale of the fast machine. The problem of minimization of a discounted cost of manufacturing is formulated. A conjecture is presented regarding the asymptotic behaviour of the value function for the above problem when the separation of slow and fast time scales becomes singular. Natural conditions are formulated under which the conjecture might be satisfied.