Consider a small convenience store that is open 24 h a day, 7 days a week. The store is staffed by one checkout clerk. The convenience store has space for only three customers including the one who is being served. Customers may arrive at the store every 30 min, immediately before the hour and immediately before the half-hour. Any customer who arrives to fi nd three customers already inside the store does not enter the store. The probability of an arrival at the end of a consecutive 30 min interval is p. If the store is empty, the clerk is idle. If the store is not empty, the probability of a service completion at the end of a consecutive 30 min interval is r. To model this problem as a regular Markov chain, let the state X_n represent the number of customers in the store at the end of the nth consecutive 30 min interval. Construct the transition probability matrix for the Markov chain.