Find the operating characteristics of this system. An airport ticket counter has a service rate of 180 per hour, exponentially distributed, with Poisson arrivals at the rate of 120 per hour

To find the operating characteristics of this system, we need to determine some key metrics such as the average number of customers in the system, the average waiting time for a customer, and the utilization of the service facility.

1. Average number of customers in the system (L):
To calculate L, we can use Little's Law, which states that the average number of customers in a system is equal to the average arrival rate (λ) multiplied by the average time a customer spends in the system (W).

Average arrival rate (λ) = 120 per hour
Average service rate (μ) = 180 per hour

Utilization (ρ) = λ / μ = 120 / 180 = 0.6667 (or 66.67%)

Using Little's Law:
L = λ * W

We need to determine the average time a customer spends in the system (W).

2. Average time a customer spends in the system (W):
The average time a customer spends in the system can be calculated using Little's Law:

W = L / λ
= (λ / μ) / λ
= 1 / μ

W = 1 / 180 = 0.0056 hours (or 20 seconds)

Now that we have the value for W, we can calculate L using Little's Law:

L = λ * W
= 120 * 0.0056
= 0.672 customers (or approximately 0.67 customers)

3. Average waiting time for a customer:
The average waiting time can be calculated by subtracting the average service time from the average time a customer spends in the system:

Average waiting time = Average time a customer spends in the system - Average service time

Average service time (1 / μ) = 1 / 180 = 0.0056 hours (or 20 seconds)

Average waiting time = 0.0056 - 0.0056 = 0 hours (or 0 seconds)

4. Utilization of the service facility:
Utilization is a measure of how busy the service facility is. It is calculated by dividing the average arrival rate by the average service rate:

Utilization (ρ) = λ / μ = 120 / 180 = 0.6667 (or 66.67%)

In summary, the operating characteristics of this system are as follows:
- Average number of customers in the system (L) = 0.67 customers
- Average waiting time for a customer = 0 seconds
- Utilization of the service facility = 66.67%