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6. Queuing Theory, M/M/s/N/N Queuing Model (M/M/c/K/K) example
( Enter your problem )
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Algorithm and examples
- Formula
- Example-1: `lambda=30`, `mu=20`, `s=2`, `N=3`
- Example-2: `lambda=30`, `mu=20`, `s=2`, `N=3`
- Example-3: `lambda=40`, `mu=1`, `s=10`, `N=10`
- Example-4: `lambda=45`, `mu=15`, `s=2`, `N=12`
- Example-5: `lambda=1.5`, `mu=2.1`, `s=3`, `N=10`
- Example-6: `lambda=1/10`, `mu=1/4`, `s=2`, `N=5`
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Other related methods
- M/M/1 Model
- M/M/1/N Model (M/M/1/K Model)
- M/M/1/N/N Model (M/M/1/K/K Model)
- M/M/s Model (M/M/c Model)
- M/M/s/N Model (M/M/c/K Model)
- M/M/s/N/N Model (M/M/c/K/K Model)
- M/M/Infinity Model
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2. Example-1: `lambda=30`, `mu=20`, `s=2`, `N=3`
1. Queuing Model = mmsnn, Arrival Rate `lambda=30` per 1 hr, Service Rate `mu=20` per 1 hr, Number of servers `s=2`, Limited Customer `N=3`
Solution:
Arrival Rate `lambda=30` per 1 hr and Service Rate `mu=20` per 1 hr (given)
Queuing Model : M/M/s/N/N
Arrival rate `lambda=30,` Service rate `mu=20,` Number of servers `s=2,` Machine `N=3` (given)
1. Traffic Intensity
`rho=lambda/mu`
`=(30)/(20)`
`=1.5`
2. Probability of no customers in the system
`P_0=[sum_{n=0}^(s-1) (N!)/((N-n)!*n!)*rho^n + sum_{n=s}^(N) (N!)/((N-n)!*s!*s^(n-s))*rho^n]^(-1)`
`=[sum_{n=0}^(1) (3!)/((3-n)!*n!)*(1.5)^n + sum_{n=2}^(3) (3!)/((3-n)!*2!*2^(n-2))*(1.5)^n]^(-1)`
`=[(1+(3!)/(2!*1!)*(1.5)^1) + ((3!)/(1!*2!*2^(0))*(1.5)^2+(3!)/(0!*2!*2^(1))*(1.5)^3)]^(-1)`
`=[(1+(3)/(1)*(1.5)^1) + ((3xx2)/(2*1)*(1.5)^2+(3xx2xx1)/(2*2)*(1.5)^3)]^(-1)`
`=[(1+4.5) + (6.75+5.0625)]^(-1)`
`=[17.3125]^(-1)`
`=0.05776173` or `0.05776173xx100=5.776173%`
3. Probability that there are n customers in the system
`P_n={((N!)/((N-n)!*n!)*rho^n*P_0, "for "0<=n< s),((N!)/((N-n)!*s!* s^(n-s))*rho^n*P_0, "for "s<=n<= N):}`
`P_n={((3!)/((3-n)!*n!)*(1.5)^n*P_0, "for "0<=n<2),((3!)/((3-n)!*2!*2^(n-2))*(1.5)^n*P_0, "for "2<=n<=3):}`
`P_1=0.2599278`
`P_2=0.3898917`
`P_3=0.29241877`
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*0.05776173+1*0.2599278+2*0.3898917+3*0.29241877`
`=1.91696751`
5. Average number of customers in the queue
`L_q=sum_{n=s+1}^(N) (n-s)P_n`
`=sum_{n=3}^(3) (n-2)*P_n`
`=1*P_3`
`=1*0.29241877`
`=0.29241877`
6. Effective Arrival rate
`lambda_e=lambda(N-L_s)`
`=30*(3-1.91696751)`
`=32.49097473`
7. Average time spent in the system
`W_s=(L_s)/(lambda_e)=(L_s)/(lambda(N-L_s))`
`=(1.91696751)/(32.49097473)`
`=0.059` hr or `0.059xx60=3.54` min
8. Average Time spent in the queue
`W_q=(L_q)/(lambda_e)=(L_q)/(lambda(N-L_s))`
`=(0.29241877)/(32.49097473)`
`=0.009` hr or `0.009xx60=0.54` min
9. Utilization factor
`U=(L_s-L_q)/s`
`=(1.91696751-0.29241877)/(2)`
`=0.81227437` or `0.81227437xx100=81.227437%`
This material is intended as a summary. Use your textbook for detail explanation.
Any bug, improvement, feedback then
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