A small optical shop is staffed by an optometrist and an optician. Customers may arrive at the shop every 30 min, immediately after the hour and immediately after the half-hour. Two stages of service are provided in the following sequence. A customer is given an eye examination by the optometrist at stage 1 followed by the fitting of glasses by the optician at stage 2. Only one chair is provided by the optometrist for a customer at stage 1, and only one chair is provided by the optician for a customer at stage 2. Both the optometrist and the optician may be idle if they have no customer, or busy if they are working with a customer. In addition, the optometrist may be blocked if she has completed work on a customer at stage 1 before the optician has completed work on a customer at stage 2.

A person who arrives at the beginning of a 30 min interval during which the optometrist is busy or blocked does not enter the shop. A customer enters the shop with probability p at the beginning of a 30 min interval during which the optometrist is idle. The optometrist begins her eye examination promptly but may not complete it during the current 30 min. interval. If the optometrist is giving an eye examination at the beginning of a 30 min interval, she has a probability r of completing it and passing the customer to the optician at the beginning of the next 30 min interval if the optician is idle. Otherwise, the customer remains with the optometrist who is blocked until the optician completes work with a customer at stage 2. If the optician is working with a customer at the beginning of a 30 min interval, she has a probability q of completing her work during the current 30 min interval so that the customer departs. To model the operation of the optical shop as a Markov chain, let the state X_n be the pair (u, v). The quantity u is I if the optometrist is idle at the beginning of the nth 30 min interval, W if the optometrist is working at the beginning of the nth 30 min interval, and B if the optometrist is blocked at the beginning of the nth 30 min interval. The quantity v is I if the optician is idle at the beginning of the nth 30 min interval, and W if the optician is working at the beginning of the nth 30 min interval. The five states are indexed below:  

Construct the transition probability matrix.

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State, X, Optometrist Optician 1 = (I,I) Idle Idle 2 = (I, W) Idle Working 3 = (W, I) Working Idle 4=(W, W) Working Working 5= (B, W) Blocked Working