Answer:
(A)0.6600 (B) 1.325 (C) 0.997 or 1 minute
Explanation:
Solution
Given that:
The constant rate = 30 seconds
The arrival rate according to Poisson distribution is = 45 seconds
Now,
(A) We solve for the average length line of cars
The formula is given below:
Lq = λ²/ 2μ ( μ -λ)
Here,
λ = this is the mean time of arrival rate
μ = This is the mean service rate
Thus we compute for the mean time arrival rate which is given below:
The mean arrival rate λ = arrival rate/ 60 seconds
= 60/45
= 1.33 customer per minute
Then we solve for the means service rate which is given below
The mean service rate μ = 60 seconds/ mean rate
= 60/30 = 2 customer per minute
We will now solve for the average line length in cars which is shown below:
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
Therefore the average length in line for cars is 0.6600 cars
(B) We solve for the average number of cars in the system
Ls =Lq + λ /μ
Ls =0.600 + 1.33/2
Ls =0.6600 + 0.665
Ls = 1.325
(C) Finally we need to find the expected average time in the system which is shown below:
Ws = Ls/λ
Ws= 1.325/1.33 = 0.997 or 1.00
The expected time average in the system is 0.997 or 1.00 minutes.
