Consider a system where a single product is produced at a facility. Items are produced

one unit at a time with exponentially distributed production times with mean $\frac{1}{}.$ The

production facility can produce ahead of demand in a "make$-$to$-$stock" fashion. However, items

in inventory incur a holding cost $h$ per unit per unit time. Customers place orders according

to a Poisson process with rate $.$ Upon arrival, an order can be fulfilled from inventory, can

be backordered, or can be rejected. Orders that are backordered incur a backordering cost $b$

per unit per unit time, whereas orders that are rejected incur a per unit lost sales cost $c.$

Production and order fulfillment are managed according to the following policy. The production

facility produces units as long as there are backorders and as long as the inventory level is

less than a level $S.$ Orders from customers are always fulfilled from on$-$hand inventory if

there is inventory in stock. If there is no inventory in stock, then orders from customers are

backordered if the backorder level is strictly less than $B.$ Otherwise, orders from customers

are rejected. $($Here$,$ both $S$ and $B$ are integer valued.$)$

$($a$)$ How would you define the state of the production facility at any time, if the goal is to

model it as a birth$-$death chain? What are the values that the state can take? Provide the

state$-$transition diagram for this chain, and write down the detailed balance equations.

$($b$)$ For fixed $S$ and $B,$ solve for the steady state distribution of the production facility.

$($c$)$ For fixed $S$ and $B,$ provide expressions for the average inventory level, average backorder

level and the average lost sales.

$($d$)$ For $S=5,B=4,=4$ and $=5$ compute the probability that a customer order is

backordered.

$($e$)$ For the same parameter values as above, compute the average waiting time for an order

conditioned on being backordered.