Queuing Model = mms, Arrival Rate `lambda=20` per 8 hr, Service Rate `mu=1` per 40 min, Number of servers `s=3`

Solution:
Arrival Rate `lambda=20` per 8 hr and Service Rate `mu=1` per 40 min (given)

So, Arrival Rate `lambda=2.5` per hr and Service Rate `mu=0.025xx60=1.5` per hr

Queuing Model : M/M/s

Arrival rate `lambda=2.5,` Service rate `mu=1.5,` Number of servers `s=3` (given)


1. Traffic Intensity
`rho=lambda/mu`

`=(2.5)/(1.5)`

`=1.66666667`


2. Probability of no customers in the system
`P_0=[sum_{n=0}^(s-1) (rho^n)/(n!) + (rho^s)/(s!)*(s mu)/(s mu-lambda)]^(-1)`

`=[1+(1.66666667)^1/(1!)+(1.66666667)^2/(2!) + (1.66666667)^3/(3!)*(3*1.5)/(3*1.5-2.5)]^(-1)`

`=[1+(1.66666667)/(1)+(2.77777778)/(2) + (4.62962963)/(6)*(4.5)/(2)]^(-1)`

`=[1+1.66666667+1.38888889 + 1.73611111]^(-1)`

`=[5.79166667]^(-1)`

`=0.17266187` or `0.17266187xx100=17.266187%`


3. Average number of customers in the queue
`L_q=((rho^s)/((s-1)!)*(lambda mu)/(s mu-lambda)^2)*P_0`

`=((1.66666667)^3/(2!)*(2.5*1.5)/(3*1.5-2.5)^2)*0.17266187`

`=((4.62962963)/2*(3.75)/(2)^2)*0.17266187`

`=(2.17013889)*0.17266187`

`=0.37470024`


4. Average Time spent in the queue
`W_q=L_q/lambda`

`=0.37470024/2.5`

`=0.1498801` hr or `0.1498801xx60=8.99280576` min

Or
`W_q=((rho^s)/((s-1)!)*(mu)/(s mu-lambda)^2) * P_0`

`=((1.66666667)^3/(2!)*(1.5)/(3*1.5-2.5)^2)*0.17266187`

`=((4.62962963)/2*(1.5)/(2)^2)*0.17266187`

`=(0.86805556)*0.17266187`

`=0.1498801` hr or `0.1498801xx60=8.99280576` min


5. Average number of customers in the system
`L_s=L_q+rho`

`=0.37470024+1.66666667`

`=2.04136691`


6. Average Time spent in the queue
`W_s=L_s/lambda`

`=2.04136691/2.5`

`=0.81654676` hr or `0.81654676xx60=48.99280576` min

Or
`W_s=W_q+1/mu`

`=0.1498801+1/1.5`

`=0.1498801+0.66666667`

`=0.81654676` hr or `0.81654676xx60=48.99280576` min


7. Utilization factor
`U=rho/s=(lambda)/(s mu)`

`=1.66666667/3`

`=0.55555556` or `0.55555556xx100=55.555556%`

Or
`U=(L_s-L_q)/s`

`=(2.04136691-0.37470024)/(3)`

`=0.55555556` or `0.55555556xx100=55.555556%`


8. Probability that there are n customers in the system
`P_n={((rho^n)/(n!)*P_0, "for "0<=n< s),((rho^n)/(s!*s^(n-s))*P_0, "for "n>=3):}`

`P_n={(((1.66666667)^n)/(n!)*P_0, "for "0<=n<3),(((1.66666667)^n)/(3!*3^(n-3))*P_0, "for "n>=3):}`

`P_1=((1.66666667)^1)/(1!)*P_0=1.66666667/1*0.17266187=0.28776978`

`P_2=((1.66666667)^2)/(2!)*P_0=2.77777778/2*0.17266187=0.23980815`

`P_3=((1.66666667)^3)/(3!*3^(3-3))*P_0=4.62962963/(6*3^(0))*0.17266187=0.13322675`

`P_4=((1.66666667)^4)/(3!*3^(4-3))*P_0=7.71604938/(6*3^(1))*0.17266187=0.07401486`

`P_5=((1.66666667)^5)/(3!*3^(5-3))*P_0=12.8600823/(6*3^(2))*0.17266187=0.04111937`

`P_6=((1.66666667)^6)/(3!*3^(6-3))*P_0=21.43347051/(6*3^(3))*0.17266187=0.02284409`

`P_7=((1.66666667)^7)/(3!*3^(7-3))*P_0=35.72245085/(6*3^(4))*0.17266187=0.01269116`

`P_8=((1.66666667)^8)/(3!*3^(8-3))*P_0=59.53741808/(6*3^(5))*0.17266187=0.00705065`

`P_9=((1.66666667)^9)/(3!*3^(9-3))*P_0=99.22903013/(6*3^(6))*0.17266187=0.00391703`

`P_10=((1.66666667)^10)/(3!*3^(10-3))*P_0=165.38171688/(6*3^(7))*0.17266187=0.00217613`

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