Question

A clinic has one receptionist, one doctor and one nurse. The clinic opens at time zero and patients begin to arrive some time later. Patients arrive at the clinic according to the following probability distribution:

Time between arrivals (minutes)	Probability
5	0.09
10	0.12
15	0.27
20	0.26
25	0.15
30	0.11

A patient may need to see only the doctor, only the nurse, or both the doctor and the nurse together. If a patient needs to see both the doctor and nurse, the patient must wait until both the doctor and nurse are free. The attention needed by a patient who comes to the clinic is defined by the following probability distribution:

Patient Needs to See	Probability
Doctor only	0.62
Nurse only	0.10
Both Doctor and Nurse	0.28

The “service time” for a patient is the length of the time that the patient spends in consultation with the doctor/nurse/both, depending on the patient’s needs. The service time EXCLUDES the time that the patient spends waiting to see the doctor/nurse/both if they are busy. Based on whether the patient sees the doctor, the nurse, or both, the service time of each patient (in minutes) is defined by the following probability distributions.

Doctor	Probability		Nurse	Probability		Both	Probability
10	0.12		5	0.21		15	0.16
15	0.21		10	0.56		20	0.25
20	0.32		15	0.15		25	0.36
25	0.18		20	0.08		30	0.17
30	0.17					35	0.05
						40	0.01

Design a SIMULATION BY HAND of this clinic to simulate the arrival and departure of 8 patients using the following random numbers ranging 00 to 99 generated for each random variable:

• Time between arrivals

64        34        15        50        94        81        27        79

• The attention needed by the patient for each patient arriving in order 06     77        99 53        64        31        74        92

• The service time for the patient, in order of arrival

48        58        97        88        75        94        69        56

Part a

For each of the 8 patients, simulate the three random variables from above:

1. the length of time between their arrivals,
2. whom they need to see, and
3. the service time spent in consultation.

Part b

Assume that the clinic opens at 9:00am and the first arrival occurs after the “time between arrivals” of the first patient (e.g. if the time between arrivals of patient 1 is 5 mins, then patient 1 arrives at 9:05am). Calculate each patient’s arrival time and departure time by completing the following table:

Patient	1	2	3	4	5	6	7	8
Arrival Time
Departure Time

Part c

From the results in (b), determine the following performance measures based on the 8 patients:

1. The average time that a patient must wait before service starts (i.e. before seeing the doctor/nurse/both)
2. The probability that the patient needs to wait for less than 30 minutes before service starts.

Answer 1

Part a)

i) & ii)

iii)

Part b)

Patient	1	2	3	4	5	6	7	8
Attention by	Doctor Only	Both Doctor & Nurse	Both Doctor & Nurse	Doctor Only	Nurse Only	Doctor Only	Both Doctor & Nurse	Both Doctor & Nurse
RN for Service Time	48	58	97	88	75	94	69	56
Arrival Time	9.20 Am	9.35 AM	9.45AM	10.05 AM	10.35AM	11.00AM	11.15AM	11.40AM
Service Time	20	25	35	30	10	30	25	25
Service Start Time	9.20 Am	9.40 Am	10.05AM	10.40AM	11.10AM	11.20 AM	11.50 AM	12.15 PM
Departure Time	9.40 Am	10.05AM	10.40AM	11.10AM	11.20 AM	11.50 AM	12.15 PM	12.40 Pm
Patient Waiting time	0	5	20	35	35	20	35	35

Partc)

i) Average time that patient must wait = sum of patient waiting time divided by  No of patients

= 185/8 = 23.125 minutes

ii) Probabilty that the patient needs to wait for less than 30 minutes before service starts

= 4/8= 0.5

DC