A doctor has scheduled two appointments, one at 1pm and the other at 1:30pm. The amount...

A doctor has scheduled two appointments, one at 1pm and the other at 1:30pm. The amount of time the doctor spends with the patient is a constant 20 minutes plus a random amount of time which is distributed as exponential with mean 8 minutes. Assume that both patients will be on time for their appointments.

What is the chance the doctor will be late for her 1:30 appointment because she spends more than 30 minutes with her 1pm appointment?
On average, how long will the 1:30 appointment be at the doctor’s office?

The time the 1:30 appointment spends in the office is the sum of 3 parts: the random waiting time W, the constant 20 minutes of examination time and the additional random examination time T.

We seek E[W + 20 + T] = E[W] + 20 + E[T]

To determine the E[W], condition on whether or not the 1:00pm appointment is still going on at 1:30pm.

Explanations with answers please

Solutions

Expert Solution

Let T1 be the random amount of time taken by patient appointed at 1 pm. Then T1 ~ Exp( = 1/8)

The doctor spends more than 30 minutes with her 1pm appointment when the random time is more than 10 minutes.

Probability that the doctor will be late for her 1:30 appointment = P(T1 > 10) = exp(-10/8)

= 0.2865048

The time the 1:30 appointment spends in the office is the sum of 3 parts: the random waiting time W, the constant 20 minutes of examination time and the additional random examination time T.

We seek E[W + 20 + T] = E[W] + 20 + E[T]

E[T] = 8 minutes

The random waiting time W is the time when T1 - 10 when T1 > 10. For T1 < 10, W = 0

= 2.292039 minutes

Expected time at doctor’s office = E[W] + 20 + E[T] = 2.292039 + 20 + 8

= 30.29204 minutes

milcah answered 10 months ago

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