1. Queuing Model = mm1nn, Arrival Rate `lambda=8` per 1 hr, Service Rate `mu=9` per 1 hr, Limited Customer `N=3`

Solution:
Arrival Rate `lambda=8` per 1 hr and Service Rate `mu=9` per 1 hr (given)

Queuing Model : M/M/1/N/N

Arrival rate `lambda=8,` Service rate `mu=9,` Machine `N=3` (given)


1. Traffic Intensity
`rho=lambda/mu`

`=(8)/(9)`

`=0.88888889`


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

`=[sum_{n=0}^(3) (3!)/((3-n)!)*(0.88888889)^n]^(-1)`

`=[1+(3!)/(2!)*(0.88888889)^1+(3!)/(1!)*(0.88888889)^2+(3!)/(0!)*(0.88888889)^3]^(-1)`

`=[1+(3)*(0.88888889)+(3xx2)*(0.79012346)+(3xx2xx1)*(0.70233196)]^(-1)`

`=[1+2.66666667+4.74074074+4.21399177]^(-1)`

`=[12.62139918]^(-1)`

`=0.07923052` or `0.07923052xx100=7.923052%`


3. Probability that there are n customers in the system
`P_n=(N!)/((N-n)!)*rho^n*P_0`

`P_n=(3!)/((3-n)!)*(0.88888889)^n*P_0`

`P_1=(3!)/((3-1)!)*(0.88888889)^1*0.07923052=0.21128138`

`P_2=(3!)/((3-2)!)*(0.88888889)^2*0.07923052=0.37561135`

`P_3=(3!)/((3-3)!)*(0.88888889)^3*0.07923052=0.33387675`


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*0.07923052+1*0.21128138+2*0.37561135+3*0.33387675`

`=1.96413433`

Or
`L_s=N-mu/lambda(1-P_0)`

`=3-9/8(1-0.07923052)`

`=3-1.03586567`

`=1.96413433`


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

`=sum_{n=1}^(3) (n-1)*P_n`

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

`=0*0.21128138+1*0.37561135+2*0.33387675`

`=1.04336485`

Or
`L_q=N-((lambda+mu)/lambda)(1-P_0)`

`=3-((8+9)/8)*(1-0.07923052)`

`=3-(2.125)*(0.92076948)`

`=3-1.95663515`

`=1.04336485`


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

`=8*(3-1.96413433)`

`=8.28692533`


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

`=(1.96413433)/(8.28692533)`

`=0.23701605` hr or `0.23701605xx60=14.22096317` min


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

`=(1.04336485)/(8.28692533)`

`=0.12590494` hr or `0.12590494xx60=7.55429651` min


9. Utilization factor
`U=L_s-L_q`

`=1.96413433-1.04336485`

`=0.92076948` or `0.92076948xx100=92.076948%`