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In a production facility, items are produced at a rate of 5 per hour, where the time to produce one item is exponentially distributed.
In a production facility, items are produced at a rate of 5 per hour, where the time to produce one item is exponentially distributed. The facility can hold at most 5 items in its inventory due to capacity limitations. Demands arrive according to a Poisson process with a rate of 3 per hour. The production will stop if the inventory is in full capacity. If there is no item in the inventory, the customer will go away empty handed.
(a). Formulate the problem as a birth-and-death process, i.e., denote the states, transition
rates, and limiting probabilities (if they exist?)
(b). What is the probability that an arriving customer goes away empty handed?
(c). What fraction of the time the production is stopped?
(d). Consider a holding cost of $2i if there are i items in the inventory, $6 for the cost of lost
demand, and $7 as the cost of production interruption. What is the expected total cost
of the system?
