Patient appointments at an outpatient clinic are scheduled every 15 minutes, starting a t9:00am. We assume that a patient's waiting time starts to count from their scheduled appointment time (if they arrive too early, it's not the clinic's fault). Patients wait their turn to be seen by a single medical doctor (MD). The time the MD spends treating each patient is exponentially distributed, with a mean of 13 minutes, and the times the MD spends with dierent patients are mutually independent. Management wants to use simulation to describe the lengths of time that patients spend at the clinic, from their appointment time to the moment at which treatment is complete, during an 8-hour day. Each patient experiences a time in the queue (possibly zero), followed by a time being treated. If we number the patients 1,2,3,..., we can dene the following quantities, all in minutes: Q(n) = time the n-th patient waits for treatment to begin (a.k.a. time in queue). S(n) = time the n-th patient spends being treated. T(n) = total time the n-th patient spends in the clinic. The value of S(n) is given by the exponential distribution with rate (patients per minute) equal to 1/13 (RandExponential(1/13) in Excel),and T(n) is simply Q(n)+S(n). The only missing piece of the simulation model is the calculation of Q(n),which works as follows. The rst patient does not have to wait, so Q(1)=0. Because patients are seen in sequence rather than in parallel, if I am the n-th patient (n > 1), two things can happen: either the patient before me (person n1) spent less than 15 minutes in the clinic, or they spent more than 15 minutes in the clinic. If they spent less than 15 minutes (T(n1) < 15), because they arrived 15 minutes before my appointment, they were nished before my appointment and my Q(n)=0. Otherwise, if T(n1) > 15, I will have to wait T(n1)15 minutes before the doctor is free to see me. (I was not there for the rst 15 minutes of my predecessor's time in the clinic.) Hence, Q(n)=max{0, T(n1)15}.

(a) Build and run a simulation model (1,000 trials) to estimate the average Q(n) and average T(n) of one day's patients. Assume one day is 8 hours long (32 patients), starting at 9:00am.

(b) What is the expected time at which the day's work is done? In the scheduling literature, this is known as the clinic's makespan, and it equals the last patient's departure time.

(c) Compute the 95% condence intervals for the three performance metrics you estimated in parts (a) and (b).