Queuing Model = mm1, Arrival Rate `lambda=6` per 12 hr, Service Rate `mu=7` per 1 hr
Solution:
Arrival Rate `lambda=6` per 12 hr and Service Rate `mu=7` per 1 hr (given)
Queuing Model : M/M/1
Arrival Rate `lambda=0.5,` Service Rate `mu=7` (given)
1. Traffic Intensity
`rho=lambda/mu`
`=(0.5)/(7)`
`=0.07142857`
2. Probability of no customers in the system
`P_0=1-rho`
`=1-0.07142857`
`=0.92857143` or `0.92857143xx100=92.857143%`
3. Average number of customers in the system
`L_s=lambda/(mu-lambda)`
`=(0.5)/(7-0.5)`
`=(0.5)/(6.5)`
`=0.07692308`
4. Average number of customers in the queue
`L_q=L_s-rho`
`=0.07692308-0.07142857`
`=0.00549451`
Or
`L_q=(lambda^2)/(mu(mu-lambda))`
`=((0.5)^2)/(7*(7-0.5))`
`=(0.25)/(7*(6.5))`
`=(0.25)/(45.5)`
`=0.00549451`
5. Average time spent in the system
`W_s=L_s/lambda`
`=(0.07692308)/(0.5)`
`=0.15384615` hr or `0.15384615xx60=9.23076923` min
Or
`W_s=1/(mu-lambda)`
`=1/(7-0.5)`
`=1/(6.5)`
`=0.15384615` hr or `0.15384615xx60=9.23076923` min
6. Average Time spent in the queue
`W_q=L_q/lambda`
`=(0.00549451)/(0.5)`
`=0.01098901` hr or `0.01098901xx60=0.65934066` min
Or
`W_q=(lambda)/(mu(mu-lambda))`
`=(0.5)/(7*(7-0.5))`
`=(0.5)/(7*(6.5))`
`=(0.5)/(45.5)`
`=0.01098901` hr or `0.01098901xx60=0.65934066` min
7. Utilization factor
`U=L_s-L_q`
`=0.07692308-0.00549451`
`=0.07142857` or `0.07142857xx100=7.142857%`
8. Probability that there are n customers in the system
`P_n=rho^n*P_0`
`P_n=(0.07142857)^n*P_0`
`P_1=(0.07142857)^1*P_0=0.07142857*0.92857143=0.06632653`
`P_2=(0.07142857)^2*P_0=0.00510204*0.92857143=0.00473761`
`P_3=(0.07142857)^3*P_0=0.00036443*0.92857143=0.0003384`
`P_4=(0.07142857)^4*P_0=0.00002603*0.92857143=0.00002417`
This material is intended as a summary. Use your textbook for detail explanation.
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