#### Asymptotically Optimal Production Policies in Dynamic
Stochastic Jobshops with Limited Buffers

**Abstract: **We consider a
production planning problem for a jobshop with unreliable machines producing a
number of products. There are upper and lower bounds on intermediate parts
and an upper bound on finished parts. The machine capacities are modelled
as finite state Markov chains. The objective is to choose the rate of
production so as to minimize the total discounted cost of inventory and
production. Finding an optimal control policy for this problem is difficult.
Instead, we derive an asymptotic
approximation by letting the rates of change of the machine states approach
infinity. The asymptotic analysis leads to a limiting problem in which the
stochastic machine capacities are replaced by their equilibrium mean capacities.
The value function for the original problem is shown to converge to the value
function of the limiting problem. The convergence rate of the value
function together with the error estimate for the constructed asymptotic optimal
production policies are established.