Queuing Model = mm1nn, Arrival Rate `lambda=6` per 1 hr, Service Rate `mu=7` per 1 hr, Limited Customer `N=3`
Solution:
Arrival Rate `lambda=6` per 1 hr and Service Rate `mu=7` per 1 hr (given)
Queuing Model : M/M/1/N/N
Arrival rate `lambda=6,` Service rate `mu=7,` Machine `N=3` (given)
1. Traffic Intensity
`rho=lambda/mu`
`=(6)/(7)`
`=0.85714286`
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)!)*(0.85714286)^n]^(-1)`
`=[1+(3!)/(2!)*(0.85714286)^1+(3!)/(1!)*(0.85714286)^2+(3!)/(0!)*(0.85714286)^3]^(-1)`
`=[1+(3)*(0.85714286)+(3xx2)*(0.73469388)+(3xx2xx1)*(0.62973761)]^(-1)`
`=[1+2.57142857+4.40816327+3.77842566]^(-1)`
`=[11.75801749]^(-1)`
`=0.08504835` or `0.08504835xx100=8.504835%`
3. Probability that there are n customers in the system
`P_n=(N!)/((N-n)!)*rho^n*P_0`
`P_n=(3!)/((3-n)!)*(0.85714286)^n*P_0`
`P_1=(3!)/((3-1)!)*(0.85714286)^1*0.08504835=0.21869576`
`P_2=(3!)/((3-2)!)*(0.85714286)^2*0.08504835=0.37490702`
`P_3=(3!)/((3-3)!)*(0.85714286)^3*0.08504835=0.32134887`
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*0.08504835+1*0.21869576+2*0.37490702+3*0.32134887`
`=1.93255641`
Or
`L_s=N-mu/lambda(1-P_0)`
`=3-7/6(1-0.08504835)`
`=3-1.06744359`
`=1.93255641`
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*0.21869576+1*0.37490702+2*0.32134887`
`=1.01760476`
Or
`L_q=N-((lambda+mu)/lambda)(1-P_0)`
`=3-((6+7)/6)*(1-0.08504835)`
`=3-(2.16666667)*(0.91495165)`
`=3-1.98239524`
`=1.01760476`
6. Effective Arrival rate
`lambda_e=lambda(N-L_s)`
`=6*(3-1.93255641)`
`=6.40466154`
7. Average time spent in the system
`W_s=(L_s)/(lambda_e)=(L_s)/(lambda(N-L_s))`
`=(1.93255641)/(6.40466154)`
`=0.30174216` hr or `0.30174216xx60=18.10452962` min
8. Average Time spent in the queue
`W_q=(L_q)/(lambda_e)=(L_q)/(lambda(N-L_s))`
`=(1.01760476)/(6.40466154)`
`=0.15888502` hr or `0.15888502xx60=9.53310105` min
9. Utilization factor
`U=L_s-L_q`
`=1.93255641-1.01760476`
`=0.91495165` or `0.91495165xx100=91.495165%`
This material is intended as a summary. Use your textbook for detail explanation.
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