Queuing Model = mmsn, Arrival Rate `lambda=30` per 1 hr, Service Rate `mu=20` per 1 hr, Number of servers `s=2`, Capacity `N=3`

Solution:
Arrival Rate `lambda=30` per 1 hr and Service Rate `mu=20` per 1 hr (given)

Queuing Model : M/M/s/N

Arrival rate `lambda=30,` Service rate `mu=20,` Number of servers `s=2,` Capacity `N=3` (given)


1. Traffic Intensity
`rho=lambda/mu`

`=(30)/(20)`

`=1.5`


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

`=[sum_{n=0}^(1) (rho^n)/(n!) + sum_{n=2}^(3) (rho^n)/(2! * 2^(n-2))]^(-1)`

`=[(1+(1.5)^1/(1!)) + (((1.5)^2)/(2! * 2^(0))+((1.5)^3)/(2! * 2^(1)))]^(-1)`

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

`=[(1+1.5) + (1.125+0.84375)]^(-1)`

`=[4.46875]^(-1)`

`=0.22377622` or `0.22377622xx100=22.377622%`


3. 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 "s<=n<= N):}`

`P_n={(((1.5)^n)/(n!)*P_0, "for "0<=n<2),(((1.5)^n)/(2!*2^(n-2))*P_0, "for "2<=n<=3):}`

`P_1=((1.5)^1)/(1!)*P_0=1.5/1*0.22377622=0.33566434`

`P_2=((1.5)^2)/(2!*2^(2-2))*P_0=2.25/(2*2^(0))*0.22377622=0.25174825`

`P_3=((1.5)^3)/(2!*2^(3-2))*P_0=3.375/(2*2^(1))*0.22377622=0.18881119`


4. Average number of customers in the system
`L_s=sum_{n=0}^(N) nP_n`

`=sum_{n=0}^(3) n*P_n`

`=0*P_0+1*P_1+2*P_2+3*P_3`

`=0*0.22377622+1*0.33566434+2*0.25174825+3*0.18881119`

`=1.40559441`


5. Average number of customers in the queue
`L_q=sum_{n=s}^(N) (n-s)P_n`

`=sum_{n=2}^(3) (n-2)*P_n`

`=(2-2)*P_2+(3-2)*P_3`

`=0*0.25174825+1*0.18881119`

`=0.18881119`


6. Effective Arrival rate
`lambda_e=lambda*(1-P_N)`

`=30*(1-0.18881119)`

`=24.33566434`


7. Average Time spent in the queue
`W_q=(L_q)/(lambda_e)=L_q/(lambda*(1-P_N))`

`=(0.18881119)/(24.33566434)`

`=0.00775862` hr or `0.00775862xx60=0.46551724` min


8. Average Time spent in the queue
`W_s=(L_s)/(lambda_e)=L_s/(lambda*(1-P_N))`

`=(1.40559441)/(24.33566434)`

`=0.05775862` hr or `0.05775862xx60=3.46551724` min

Or
`W_s=W_q+1/mu`

`=0.00775862+1/20`

`=0.00775862+0.05`

`=0.05775862` hr or `0.05775862xx60=3.46551724` min


9. Utilization factor
`U=(L_s-L_q)/s`

`=(1.40559441-0.18881119)/(2)`

`=0.60839161` or `0.60839161xx100=60.839161%`

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