Optimal use of an extra server in a two station tandem queueing network

A two station tandem queueing system with given numbers of customers initially at each station and no arrivals is considered. There is a fixed server at each station, but also an additional server that can be dynamically allocated to wherever its use will do most good. There are differing linear holding costs at each station, and the aim is to use the extra server to minimize the expected total holding cost incurred until the system empties. It is shown that if either the extra server can be switched between the two stations at any time, or if it is restricted in use to just one station, where it can be turned on or off, then the optimal use of the server is such that after a service completion at one station, the effort devoted there never increases, and the effort devoted to the other station never decreases