Customers arrive at service window every 3.6 minutes(exponentially distributed) and service time is 2.4 minutes )exponentially distributes) What is the average lenght of waiting line and waiting time?
2+2=2
To find the average length of the waiting line, we need to calculate the average number of customers in the system, including those being serviced and those waiting in line.
In an M/M/1 queuing system, where arrivals are exponentially distributed, and service times are exponentially distributed, the average number of customers in the system (L) can be calculated using the following formula:
L = λ / (μ - λ)
Where:
λ = Arrival rate (customers per minute)
μ = Service rate (customers per minute)
In this case, the arrival rate (λ) is 1 customer every 3.6 minutes, and the service rate (μ) is 1 customer every 2.4 minutes.
λ = 1 / 3.6 = 0.2778 customers per minute
μ = 1 / 2.4 = 0.4167 customers per minute
Now we can substitute these values into the formula and calculate the average number of customers in the system:
L = 0.2778 / (0.4167 - 0.2778) = 0.2778 / 0.1389 = 1.9998
Therefore, the average length of the waiting line is approximately 2 customers.
To find the average waiting time, we use Little's Law, which states that the average number of customers in the system (L) multiplied by the average time a customer spends in the system (W) is equal to the average arrival rate (λ). Mathematically:
L * W = λ
Substituting the values we have:
2 * W = 0.2778
Solving for W:
W = 0.2778 / 2 = 0.1389 minutes
Therefore, the average waiting time is approximately 0.1389 minutes (or 8.33 seconds).