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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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4. Example-3: `lambda=40`, `mu=1`, `s=10`, `N=10`
Queuing Model = mmsnn, Arrival Rate `lambda=40` per 1 hr, Service Rate `mu=1` per 1 hr, Number of servers `s=10`, Limited Customer `N=10`
Solution:
Arrival Rate `lambda=40` per 1 hr and Service Rate `mu=1` per 1 hr (given)
Queuing Model : M/M/s/N/N
Arrival rate `lambda=40,` Service rate `mu=1,` Number of servers `s=10,` Machine `N=10` (given)
1. Traffic Intensity
`rho=lambda/mu`
`=40`
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}^(9) (10!)/((10-n)!*n!)*(40)^n + sum_{n=10}^(10) (10!)/((10-n)!*10!*10^(n-10))*(40)^n]^(-1)`
`=[(1+(10!)/(9!*1!)*(40)^1+(10!)/(8!*2!)*(40)^2+(10!)/(7!*3!)*(40)^3+(10!)/(6!*4!)*(40)^4+(10!)/(5!*5!)*(40)^5+(10!)/(4!*6!)*(40)^6+(10!)/(3!*7!)*(40)^7+(10!)/(2!*8!)*(40)^8+(10!)/(1!*9!)*(40)^9) + ((10!)/(0!*10!*10^(0))*(40)^10)]^(-1)`
`=[(1+(10)/(1)*(40)^1+(10xx9)/(2)*(40)^2+(10xx9xx8)/(6)*(40)^3+(10xx9xx8xx7)/(24)*(40)^4+(10xx9xx8xx7xx6)/(120)*(40)^5+(10xx9xx8xx7xx6xx5)/(720)*(40)^6+(10xx9xx8xx7xx6xx5xx4)/(5040)*(40)^7+(10xx9xx8xx7xx6xx5xx4xx3)/(40320)*(40)^8+(10xx9xx8xx7xx6xx5xx4xx3xx2)/(362880)*(40)^9) + ((10xx9xx8xx7xx6xx5xx4xx3xx2xx1)/(3628800*1)*(40)^10)]^(-1)`
`=[(1+400+72000+7680000+537600000+25804800000+860160000000+19660800000000+294912000000000+2621440000000000) + (10485760000000000)]^(-1)`
`=[13422659310152400]^(-1)`
`=0` or `0xx100=0%`
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={((10!)/((10-n)!*n!)*(40)^n*P_0, "for "0<=n<10),((10!)/((10-n)!*10!*10^(n-10))*(40)^n*P_0, "for "10<=n<=10):}`
`P_1=0`
`P_2=0`
`P_3=0`
`P_4=0.00000004`
`P_5=0.00000192`
`P_6=0.00006408`
`P_7=0.00146475`
`P_8=0.02197121`
`P_9=0.1952996`
`P_10=0.7811984`
4. Average number of customers in the system
`L_s=sum_{n=0}^(N) nP_n`
`=sum_{n=0}^(10) n*P_n`
`=0*P_0+1*P_1+2*P_2+3*P_3+4*P_4+5*P_5+6*P_6+7*P_7+8*P_8+9*P_9+10*P_10`
`=0*0+1*0+2*0+3*0+4*0.00000004+5*0.00000192+6*0.00006408+7*0.00146475+8*0.02197121+9*0.1952996+10*0.7811984`
`=9.75609756`
5. Average number of customers in the queue
`L_q=sum_{n=s+1}^(N) (n-s)P_n`
`=sum_{n=11}^(10) (n-10)*P_n`
`=`
`=0`
6. Effective Arrival rate
`lambda_e=lambda(N-L_s)`
`=40*(10-9.75609756)`
`=9.75609756`
7. Average time spent in the system
`W_s=(L_s)/(lambda_e)=(L_s)/(lambda(N-L_s))`
`=(9.75609756)/(9.75609756)`
`=1` hr or `1xx60=60` min
8. Average Time spent in the queue
`W_q=(L_q)/(lambda_e)=(L_q)/(lambda(N-L_s))`
`=(0)/(9.75609756)`
`=0` hr or `0xx60=0` min
9. Utilization factor
`U=(L_s-L_q)/s`
`=(9.75609756-0)/(10)`
`=0.97560976` or `0.97560976xx100=97.560976%`
This material is intended as a summary. Use your textbook for detail explanation.
Any bug, improvement, feedback then
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