Response:
(A)0.6600 (B) 1.325 (C) 0.997 or 1 minute
Clarification:
Resolution
It's provided that:
The fixed production time = 30 seconds
The customer arrival time determined by the Poisson distribution is = 45 seconds
Now,
(A) We calculate the typical line length of vehicles
The formula is as follows:
Lq = λ²/ 2μ ( μ -λ)
In this equation,
λ = signifies the average arrival rate
μ = denotes the average service rate
We first determine the average arrival rate as shown below:
The average arrival rate λ = arrival rate/ 60 seconds
= 60/45
= 1.33 customers per minute
Next, we find the average service rate which is detailed below
The average service rate μ = 60 seconds/ average rate
= 60/30 = 2 customers per minute
Now we will calculate the average line length in vehicles as shown here:
Lq = λ²/ 2μ ( μ -λ)
Lq = 1.33²/2*2 (2-1.33)
Lq = 1.7689/4 (0.67)
Lq = 1.7689/2.68
Lq = 0.6600
Thus, the average line length for vehicles is 0.6600 cars
(B) We calculate the average number of vehicles in the entire system
Ls = Lq + λ /μ
Ls = 0.600 + 1.33/2
Ls = 0.6600 + 0.665
Ls = 1.325
(C) Lastly, we need to ascertain the expected average duration in the system outlined here:
Ws = Ls/λ
Ws= 1.325/1.33 = 0.997 or 1.00
The predicted average duration in the system is 0.997 or 1.00 minutes.
