Queuing Model = mmsnn, Arrival Rate `lambda=10` per 1 hr, Service Rate `mu=3` per 1 hr, Number of servers `s=2`, Limited Customer `N=3`

Solution:
Arrival Rate `lambda=10` per 1 hr and Service Rate `mu=3` per 1 hr (given)

Queuing Model : M/M/s/N/N

Arrival rate `lambda=10,` Service rate `mu=3,` Number of servers `s=2,` Machine `N=3` (given)


1. Traffic Intensity
`rho=lambda/mu`

`=(10)/(3)`

`=3.33333333`


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!)*(3.33333333)^n + sum_{n=2}^(3) (3!)/((3-n)!*2!*2^(n-2))*(3.33333333)^n]^(-1)`

`=[(1+(3!)/(2!*1!)*(3.33333333)^1) + ((3!)/(1!*2!*2^(0))*(3.33333333)^2+(3!)/(0!*2!*2^(1))*(3.33333333)^3)]^(-1)`

`=[(1+(3)/(1)*(3.33333333)^1) + ((3xx2)/(2*1)*(3.33333333)^2+(3xx2xx1)/(2*2)*(3.33333333)^3)]^(-1)`

`=[(1+10) + (33.33333333+55.55555556)]^(-1)`

`=[99.88888889]^(-1)`

`=0.01001112` or `0.01001112xx100=1.001112%`


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!)*(3.33333333)^n*P_0, "for "0<=n<2),((3!)/((3-n)!*2!*2^(n-2))*(3.33333333)^n*P_0, "for "2<=n<=3):}`

`P_1=0.10011123`

`P_2=0.33370412`

`P_3=0.55617353`


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*0.01001112+1*0.10011123+2*0.33370412+3*0.55617353`

`=2.43604004`


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*0.55617353`

`=0.55617353`


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

`=10*(3-2.43604004)`

`=5.63959956`


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

`=(2.43604004)/(5.63959956)`

`=0.43195266` hr or `0.43195266xx60=25.91715976` min


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

`=(0.55617353)/(5.63959956)`

`=0.09861933` hr or `0.09861933xx60=5.91715976` min


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

`=(2.43604004-0.55617353)/(2)`

`=0.93993326` or `0.93993326xx100=93.993326%`

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