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5. Queuing Theory, M/M/s/N Queuing Model (M/M/c/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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5. Example-4: `lambda=45`, `mu=15`, `s=2`, `N=12`
Queuing Model = mmsn, Arrival Rate `lambda=45` per 1 hr, Service Rate `mu=15` per 1 hr, Number of servers `s=2`, Capacity `N=12`
Solution:
Arrival Rate `lambda=45` per 1 hr and Service Rate `mu=15` per 1 hr (given)
Queuing Model : M/M/s/N
Arrival rate `lambda=45,` Service rate `mu=15,` Number of servers `s=2,` Capacity `N=12` (given)
1. Traffic Intensity
`rho=lambda/mu`
`=(45)/(15)`
`=3`
2. Probability of no customers in the system
`P_0=[sum_{n=0}^(s-1) (rho^n)/(n!) + sum_{n=s}^(N) (rho^n)/(s! *s^(n-s))]^(-1)`
`=[sum_{n=0}^(1) (rho^n)/(n!) + sum_{n=2}^(12) (rho^n)/(2! * 2^(n-2))]^(-1)`
`=[(1+(3)^1/(1!)) + (((3)^2)/(2! * 2^(0))+((3)^3)/(2! * 2^(1))+((3)^4)/(2! * 2^(2))+((3)^5)/(2! * 2^(3))+((3)^6)/(2! * 2^(4))+((3)^7)/(2! * 2^(5))+((3)^8)/(2! * 2^(6))+((3)^9)/(2! * 2^(7))+((3)^10)/(2! * 2^(8))+((3)^11)/(2! * 2^(9))+((3)^12)/(2! * 2^(10)))]^(-1)`
`=[(1+(3)^1/(1)) + (((3)^2)/(2*1)+((3)^3)/(2*2)+((3)^4)/(2*4)+((3)^5)/(2*8)+((3)^6)/(2*16)+((3)^7)/(2*32)+((3)^8)/(2*64)+((3)^9)/(2*128)+((3)^10)/(2*256)+((3)^11)/(2*512)+((3)^12)/(2*1024))]^(-1)`
`=[(1+3) + (4.5+6.75+10.125+15.1875+22.78125+34.171875+51.2578125+76.88671875+115.33007812+172.99511719+259.49267578)]^(-1)`
`=[773.47802734]^(-1)`
`=0.00129286` or `0.00129286xx100=0.129286%`
3. Probability that there are n customers in the system
`P_n={((rho^n)/(n!)*P_0, "for "0<=n< s),((rho^n)/(s!*s^(n-s))*P_0, "for "s<=n<= N):}`
`P_n={(((3)^n)/(n!)*P_0, "for "0<=n<2),(((3)^n)/(2!*2^(n-2))*P_0, "for "2<=n<=12):}`
`P_1=((3)^1)/(1!)*P_0=3/1*0.00129286=0.00387858`
`P_2=((3)^2)/(2!*2^(2-2))*P_0=9/(2*2^(0))*0.00129286=0.00581788`
`P_3=((3)^3)/(2!*2^(3-2))*P_0=27/(2*2^(1))*0.00129286=0.00872682`
`P_4=((3)^4)/(2!*2^(4-2))*P_0=81/(2*2^(2))*0.00129286=0.01309022`
`P_5=((3)^5)/(2!*2^(5-2))*P_0=243/(2*2^(3))*0.00129286=0.01963533`
`P_6=((3)^6)/(2!*2^(6-2))*P_0=729/(2*2^(4))*0.00129286=0.029453`
`P_7=((3)^7)/(2!*2^(7-2))*P_0=2187/(2*2^(5))*0.00129286=0.0441795`
`P_8=((3)^8)/(2!*2^(8-2))*P_0=6561/(2*2^(6))*0.00129286=0.06626925`
`P_9=((3)^9)/(2!*2^(9-2))*P_0=19683/(2*2^(7))*0.00129286=0.09940388`
`P_10=((3)^10)/(2!*2^(10-2))*P_0=59049/(2*2^(8))*0.00129286=0.14910582`
`P_11=((3)^11)/(2!*2^(11-2))*P_0=177147/(2*2^(9))*0.00129286=0.22365874`
`P_12=((3)^12)/(2!*2^(12-2))*P_0=531441/(2*2^(10))*0.00129286=0.3354881`
4. Average number of customers in the system
`L_s=sum_{n=0}^(N) nP_n`
`=sum_{n=0}^(12) 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+11*P_11+12*P_12`
`=0*0.00129286+1*0.00387858+2*0.00581788+3*0.00872682+4*0.01309022+5*0.01963533+6*0.029453+7*0.0441795+8*0.06626925+9*0.09940388+10*0.14910582+11*0.22365874+12*0.3354881`
`=10.08015742`
5. Average number of customers in the queue
`L_q=sum_{n=s}^(N) (n-s)P_n`
`=sum_{n=2}^(12) (n-2)*P_n`
`=(2-2)*P_2+(3-2)*P_3+(4-2)*P_4+(5-2)*P_5+(6-2)*P_6+(7-2)*P_7+(8-2)*P_8+(9-2)*P_9+(10-2)*P_10+(11-2)*P_11+(12-2)*P_12`
`=0*0.00581788+1*0.00872682+2*0.01309022+3*0.01963533+4*0.029453+5*0.0441795+6*0.06626925+7*0.09940388+8*0.14910582+9*0.22365874+10*0.3354881`
`=8.08662172`
6. Effective Arrival rate
`lambda_e=lambda*(1-P_N)`
`=45*(1-0.3354881)`
`=29.90303538`
7. Average Time spent in the queue
`W_q=(L_q)/(lambda_e)=L_q/(lambda*(1-P_N))`
`=(8.08662172)/(29.90303538)`
`=0.27042812` hr or `0.27042812xx60=16.22568737` min
8. Average Time spent in the queue
`W_s=(L_s)/(lambda_e)=L_s/(lambda*(1-P_N))`
`=(10.08015742)/(29.90303538)`
`=0.33709479` hr or `0.33709479xx60=20.22568737` min
Or
`W_s=W_q+1/mu`
`=0.27042812+1/15`
`=0.27042812+0.06666667`
`=0.33709479` hr or `0.33709479xx60=20.22568737` min
9. Utilization factor
`U=(L_s-L_q)/s`
`=(10.08015742-8.08662172)/(2)`
`=0.99676785` or `0.99676785xx100=99.676785%`
This material is intended as a summary. Use your textbook for detail explanation.
Any bug, improvement, feedback then
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