Question

After Pooling Q1 Consider the same urgent care center with 4 doctors where the average arrival rate for each doctor is 24 patients per hour and the average service time of each patient is 2 minutes. Both inter-arrival times and service times have a coefficient of variation equal to one. But now, the doctors are pooled so that a doctor can serve any patient. After pooling, how many minutes does a patient spend on average in the urgent care center (including the time waiting in line as well as the actual service)? Round your answer to two decimal points.

Answer 1

On average, a patient would spend 2.5 minutes in the urgent care center after pooling of doctors, which includes both waiting in line and service time.

Step-by-step explanation:

To calculate the average time a patient spends in an urgent care center after the doctors have been pooled, we need to understand the system's arrival and service rates. In queuing theory, this type of system can be modeled using an M/M/1 queue, where 'M' stands for memoryless, which is a property of an exponential distribution. However, here we have multiple doctors, so it is more akin to an M/M/c queue where 'c' stands for the number of servers, in this case, doctors.

The arrival rate for each doctor is 24 patients per hour, so for 4 doctors, it is 24 patients/hour * 4 = 96 patients per hour. This means the arrival rate (λ) is 96 patients per hour, or 1.6 patients per minute. Each patient's service time is 2 minutes, which means the service rate (μ) is 1 patient every 2 minutes, or 0.5 patients per minute per doctor. For 4 doctors, the combined service rate would be 0.5 * 4 = 2 patients per minute.

Using the formula for the average time a customer (patient) spends in the system (W), which is W = 1 / (μ - λ), we can calculate:

W = 1 / (2 - 1.6) = 1 / 0.4 = 2.5 minutes.

This would mean, on average, a patient would spend 2.5 minutes in the urgent care center, including waiting in line and the actual service.