What is the expected waiting time for a patient?

A physician schedules 32 patient appointments per 8-hour work day at 15 minute intervals starting at 0700.

The expected duration of each appointment is also 15 minutes, although the actual duration of an appointment ranges from 5 to 25 minutes (uniformly distributed).

The physician always completes the current appointment before moving on to the next appointment.

Patients always arrive exactly on time for their appointments.

---> On Average, how long should a patient expect to wait before his/her appointment commences? (That is, what is the expected value of waiting time over the population of patients in a given day?)

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The distribution of each appointment length is uniform over 5 to 25 mins. The lengths of appointments are assumed to be independent of one another.

The distribution of wait for the second patient w2 is then uniform over 0 to 10 mins. The expected wait for the second patient is then E[w2] = 5 mins.

The distribution of when the doctor will next be ready is uniform over 07.20 to 07.50, so they are expected to be ready at 07.35. Thus, since their appointment was at 07.30 the third patient is expected to wait E[w3] = 5 mins.

The distribution of when the doctor will next be ready is uniform over 07.25 to 08.15, so they are expected to be ready at 07.50. Since their appointment was at 07.45 the fourth patient is expected to wait E[w4] = 5 mins.

Similarly, all patients can expect to wait 5 mins, regardless of their appointment time.

**The average waiting time for each patient over the day is 5 mins**.

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