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%`
This material is intended as a summary. Use your textbook for detail explanation.
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