1. Queuing Model = mm1n, Arrival Rate `lambda=8` per 1 hr, Service Rate `mu=9` per 1 hr, Capacity `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

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


1. Traffic Intensity
`rho=lambda/mu`

`=(8)/(9)`

`=0.88888889`


2. Probability of no customers in the system
`P_0=(1-rho)/(1-rho^(N+1))`

`=(1-0.88888889)/(1-(0.88888889)^(3+1))`

`=(0.11111111)/(1-(0.88888889)^4)`

`=(0.11111111)/(0.37570492)`

`=0.29574037` or `0.29574037xx100=29.574037%`


3. Probability of N customers in the system
`P_N=rho^N*P_0`

`=(0.88888889)^3*0.29574037`

`=0.70233196*0.29574037`

`=0.20770791`


4. Average number of customers in the system
`L_s=rho/(1-rho) - ((N+1)*rho^(N+1))/(1-rho^(N+1))`

`=0.88888889/(1-0.88888889) - ((3+1)*(0.88888889)^(3+1))/(1-(0.88888889)^(3+1))`

`=0.88888889/0.11111111 - (4*(0.88888889)^4)/(1-(0.88888889)^4)`

`=8 - (4*(0.62429508))/(1-(0.62429508))`

`=8 - (2.49718031)/(0.375704923030026)`

`=8 - 6.64665314`

`=1.35334686`


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

`=8*(1-0.20770791)`

`=6.33833671`


6. Average number of customers in the queue
`L_q=L_s-(lambda_e)/(mu)=L_s-(lambda(1-P_N))/(mu)`

`=1.35334686-6.33833671/9`

`=0.64908722`


7. Average time spent in the system
`W_s=(L_s)/(lambda_e)=(L_s)/(lambda(1-P_N))`

`=(1.35334686)/(6.33833671)`

`=0.21351767` hr or `0.21351767xx60=12.81105991` min


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

`=(0.64908722)/(6.33833671)`

`=0.10240655` hr or `0.10240655xx60=6.14439324` min


9. Utilization factor
`U=L_s-L_q`

`=1.35334686-0.64908722`

`=0.70425963` or `0.70425963xx100=70.425963%`



10. Probability that there are n customers in the system
`P_n=rho^n*P_0`

`P_n=(0.88888889)^n*P_0`

`P_1=(0.88888889)^1*P_0=0.88888889*0.29574037=0.26288032`

`P_2=(0.88888889)^2*P_0=0.79012346*0.29574037=0.2336714`

`P_3=(0.88888889)^3*P_0=0.70233196*0.29574037=0.20770791`

`P_4=(0.88888889)^4*P_0=0.62429508*0.29574037=0.18462925`

`P_5=(0.88888889)^5*P_0=0.55492896*0.29574037=0.16411489`

`P_6=(0.88888889)^6*P_0=0.49327018*0.29574037=0.1458799`

`P_7=(0.88888889)^7*P_0=0.43846239*0.29574037=0.12967103`

`P_8=(0.88888889)^8*P_0=0.38974434*0.29574037=0.11526313`

`P_9=(0.88888889)^9*P_0=0.34643942*0.29574037=0.10245612`

`P_10=(0.88888889)^10*P_0=0.30794615*0.29574037=0.09107211`