4.2.1 Consider a discrete-time periodic review inventory model (see Chapter 3, Section 3.3.1), and let n be the total demand in period n. Let Xn be the inventory quantity on hand at the end of period n. Instead of following an .s;S/ policy, a

.q;Q/ policy will be used: If the stock level at the end of a period is less than or equal to q D 2 units, then Q D 2 additional units will be ordered and will be available at the beginning of the next period. Otherwise, no ordering will take place. This is a .q;Q/ policy with q D 2 and Q D 2. Assume that demand that is not filled in a period is lost (no back ordering).

(a) Suppose that X0 D 4 and that the period demands turn out to be 1 D 3;

2 D 4; 3 D 0; 4 D 2. What are the end-of-period stock levels for periods n D 1;2;3;4?

(b) Suppose that 1; 2; : : : are independent random variables, each having the probability distribution where

Then, X0;X1; : : : is a Markov chain. Determine the transition probability distribution and the limiting distribution.

(c) In the long run, during what fraction of periods are orders placed?

k Pr{=k} 0 1 2 3 4 0.3 0.2 0.1 Pr{ k) = 0.1. 0.3