Queuing Model = mm1, Arrival Rate `lambda=30` per 1 day, Service Rate `mu=1` per 36 min

Solution:
Arrival Rate `lambda=30` per 1 day and Service Rate `mu=1` per 36 min (given)

So, Arrival Rate `lambda=30/(24xx60)=0.02083333` per min and Service Rate `mu=0.02777778` per min

Queuing Model : M/M/1

Arrival Rate `lambda=0.02083333,` Service Rate `mu=0.02777778` (given)


1. Traffic Intensity
`rho=lambda/mu`

`=(0.02083333)/(0.02777778)`

`=0.74999982`


2. Probability of no customers in the system
`P_0=1-rho`

`=1-0.74999982`

`=0.25000018` or `0.25000018xx100=25.000018%`


3. Average number of customers in the system
`L_s=lambda/(mu-lambda)`

`=(0.02083333)/(0.02777778-0.02083333)`

`=(0.02083333)/(0.00694445)`

`=2.99999712`


4. Average number of customers in the queue
`L_q=L_s-rho`

`=2.99999712-0.74999982`

`=2.2499973`

Or
`L_q=(lambda^2)/(mu(mu-lambda))`

`=((0.02083333)^2)/(0.02777778*(0.02777778-0.02083333))`

`=(0.00043403)/(0.02777778*(0.00694445))`

`=(0.00043403)/(0.0001929)`

`=2.2499973`


5. Average time spent in the system
`W_s=L_s/lambda`

`=(2.99999712)/(0.02083333)`

`=143.9998848` min

Or
`W_s=1/(mu-lambda)`

`=1/(0.02777778-0.02083333)`

`=1/(0.00694445)`

`=143.9998848` min


6. Average Time spent in the queue
`W_q=L_q/lambda`

`=(2.2499973)/(0.02083333)`

`=107.99988768` min

Or
`W_q=(lambda)/(mu(mu-lambda))`

`=(0.02083333)/(0.02777778*(0.02777778-0.02083333))`

`=(0.02083333)/(0.02777778*(0.00694445))`

`=(0.02083333)/(0.0001929)`

`=107.99988768` min


7. Utilization factor
`U=L_s-L_q`

`=2.99999712-2.2499973`

`=0.74999982` or `0.74999982xx100=74.999982%`



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

`P_n=(0.74999982)^n*P_0`

`P_1=(0.74999982)^1*P_0=0.74999982*0.25000018=0.18750009`

`P_2=(0.74999982)^2*P_0=0.56249973*0.25000018=0.14062503`

`P_3=(0.74999982)^3*P_0=0.4218747*0.25000018=0.10546875`

`P_4=(0.74999982)^4*P_0=0.31640595*0.25000018=0.07910154`

`P_5=(0.74999982)^5*P_0=0.2373044*0.25000018=0.05932614`

`P_6=(0.74999982)^6*P_0=0.17797826*0.25000018=0.0444946`

`P_7=(0.74999982)^7*P_0=0.13348366*0.25000018=0.03337094`

`P_8=(0.74999982)^8*P_0=0.10011272*0.25000018=0.0250282`

`P_9=(0.74999982)^9*P_0=0.07508452*0.25000018=0.01877114`

`P_10=(0.74999982)^10*P_0=0.05631338*0.25000018=0.01407836`

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