Consider an integer-time queueing system with a finite bu↵er of size 2. At the beginning of the nth time interval, the queue contains at most two customers. There is a cost of one unit for each customer in queue (i.e., the cost of delaying that customer). If there is one customer in queue, that customer is served. If there are two customers, an extra server is hired at a cost of 3 units and both customers are served. Thus the total immediate cost for two customers in queue is 5, the cost for one customer is 1, and the cost for 0 customers is 0. At the end of the nth time interval, either 0, 1, or 2 new customers arrive (each with probability 1/3).
e) Now assume that there is a decision maker who can choose whether or not to hire the extra server when there are two customers in queue. If the extra server is not hired, the 3 unit fee is saved, but only one of the customers is served. If there are two arrivals in this case, assume that one is turned away at a cost of 5 units. Find the minimum dynamic aggregate expected cost v*
i
(1), 0 ≤ i ≤, for stage 1 with the same final cost as before.
f) Find the minimum dynamic aggregate expected cost v*
i
(n, u) for stage n, 0 ≤ i ≤ 2.
g) Now assume a final cost u of one unit per customer rather than 5, and find the new minimum dynamic aggregate expected cost v*
i
(n, u), 0 ≤ i ≤ 2.