A credit counselor accepts customers who arrive at her office without appointments. Her office consists of a private room in which she counsels one customer, plus a small waiting room, which can hold up to two additional customers. Any customer who arrives when the waiting room is full does not enter. Each credit counseling session for one customer lasts 45 min. If no customers are present when she has finished counseling a customer, she takes a 45 min break until the beginning of the next 45 min period. The number of customers who arrive during a 45 min counseling period is an independent, identically distributed random variable. If A_n is the number of arrivals during the nth 45 min counseling period, then An has the following probability distribution, which is stationary in time: 

To model this problem as a regular Markov chain, let the state X_n represent the number of customers in the counselor’s office at the end of the nth 45 min counseling period, immediately after the departure of the nth customer. Construct the transition probability matrix for the Markov chain.

Transcribed Image Text:

3 4 or more P4 Number A,, of arrivals in period n, A = k 0 Probability, P = P(A = k) 1 Po P1 P2 P3