1. Queuing Model = mmsnn, Arrival Rate `lambda=30` per 1 hr, Service Rate `mu=20` per 1 hr, Number of servers `s=2`, Limited Customer `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/N

Arrival rate `lambda=30,` Service rate `mu=20,` Number of servers `s=2,` Machine `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) (N!)/((N-n)!*n!)*rho^n + sum_{n=s}^(N) (N!)/((N-n)!*s!*s^(n-s))*rho^n]^(-1)`

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

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

`=[(1+(3)/(1)*(1.5)^1) + ((3xx2)/(2*1)*(1.5)^2+(3xx2xx1)/(2*2)*(1.5)^3)]^(-1)`

`=[(1+4.5) + (6.75+5.0625)]^(-1)`

`=[17.3125]^(-1)`

`=0.05776173` or `0.05776173xx100=5.776173%`


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

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

`P_1=0.2599278`

`P_2=0.3898917`

`P_3=0.29241877`


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*0.05776173+1*0.2599278+2*0.3898917+3*0.29241877`

`=1.91696751`


5. Average number of customers in the queue
`L_q=sum_{n=s+1}^(N) (n-s)P_n`

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

`=1*P_3`

`=1*0.29241877`

`=0.29241877`


6. Effective Arrival rate
`lambda_e=lambda(N-L_s)`

`=30*(3-1.91696751)`

`=32.49097473`


7. Average time spent in the system
`W_s=(L_s)/(lambda_e)=(L_s)/(lambda(N-L_s))`

`=(1.91696751)/(32.49097473)`

`=0.059` hr or `0.059xx60=3.54` min


8. Average Time spent in the queue
`W_q=(L_q)/(lambda_e)=(L_q)/(lambda(N-L_s))`

`=(0.29241877)/(32.49097473)`

`=0.009` hr or `0.009xx60=0.54` min


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

`=(1.91696751-0.29241877)/(2)`

`=0.81227437` or `0.81227437xx100=81.227437%`