On the asymptote of the optimal routing policy for two service stations

It has been previously proved that the optimal routing control of a two-station Markovian network with linear cost is described by a monotone switching curve. With the discounted cost objective function, it is proven in this paper that the optimal switching curve has a finite asymptotic limit when c₁≠c₂, where c_i is the unit inventory cost at station i . Whereas, for the case with c₁=c₂, as well as the case with the long-run average objective function, the switching curve does not have a finite asymptote