Queuing Model = mm1n, Arrival Rate `lambda=1.5` per 1 hr, Service Rate `mu=2.1` per 1 hr, Capacity `N=10`

Solution:
Arrival Rate `lambda=1.5` per 1 hr and Service Rate `mu=2.1` per 1 hr (given)

Queuing Model : M/M/1/N

Arrival rate `lambda=1.5,` Service rate `mu=2.1,` Capacity `N=10` (given)


1. Traffic Intensity
`rho=lambda/mu`

`=(1.5)/(2.1)`

`=0.71428571`


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

`=(1-0.71428571)/(1-(0.71428571)^(10+1))`

`=(0.28571429)/(1-(0.71428571)^11)`

`=(0.28571429)/(0.97530599)`

`=0.29294836` or `0.29294836xx100=29.294836%`


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

`=(0.71428571)^10*0.29294836`

`=0.03457161*0.29294836`

`=0.0101277`


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

`=0.71428571/(1-0.71428571) - ((10+1)*(0.71428571)^(10+1))/(1-(0.71428571)^(10+1))`

`=0.71428571/0.28571429 - (11*(0.71428571)^11)/(1-(0.71428571)^11)`

`=2.5 - (11*(0.02469401))/(1-(0.02469401))`

`=2.5 - (0.2716341)/(0.97530599069028012)`

`=2.5 - 0.27851167`

`=2.22148833`


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

`=1.5*(1-0.0101277)`

`=1.48480845`


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

`=2.22148833-1.48480845/2.1`

`=1.51443668`


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

`=(2.22148833)/(1.48480845)`

`=1.49614472` hr or `1.49614472xx60=89.76868315` min


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

`=(1.51443668)/(1.48480845)`

`=1.01995424` hr or `1.01995424xx60=61.19725458` min


9. Utilization factor
`U=L_s-L_q`

`=2.22148833-1.51443668`

`=0.70705164` or `0.70705164xx100=70.705164%`



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

`P_n=(0.71428571)^n*P_0`

`P_1=(0.71428571)^1*P_0=0.71428571*0.29294836=0.20924883`

`P_2=(0.71428571)^2*P_0=0.51020408*0.29294836=0.14946345`

`P_3=(0.71428571)^3*P_0=0.36443149*0.29294836=0.1067596`

`P_4=(0.71428571)^4*P_0=0.2603082*0.29294836=0.07625686`

`P_5=(0.71428571)^5*P_0=0.18593443*0.29294836=0.05446919`

`P_6=(0.71428571)^6*P_0=0.13281031*0.29294836=0.03890656`

`P_7=(0.71428571)^7*P_0=0.09486451*0.29294836=0.0277904`

`P_8=(0.71428571)^8*P_0=0.06776036*0.29294836=0.01985029`

`P_9=(0.71428571)^9*P_0=0.04840026*0.29294836=0.01417878`

`P_10=(0.71428571)^10*P_0=0.03457161*0.29294836=0.0101277`

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