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3. Queuing Theory, M/M/1/N/N Queuing Model (M/M/1/K/K) example
( Enter your problem )
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Algorithm and examples
- Formula
- Example-1: `lambda=8`, `mu=9`, `N=3`
- Example-2: `lambda=6`, `mu=7`, `N=3`
- Example-3: `lambda=1`, `mu=1.2`, `N=6`
- Example-4: `lambda=25`, `mu=40`, `N=12`
- Example-5: `lambda=1.5`, `mu=2.1`, `N=10`
- Example-6: `lambda=1/10`, `mu=1/4`, `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=25`, `mu=40`, `N=12`
Queuing Model = mm1nn, Arrival Rate `lambda=25` per 1 hr, Service Rate `mu=40` per 1 hr, Limited Customer `N=12`
Solution:
Arrival Rate `lambda=25` per 1 hr and Service Rate `mu=40` per 1 hr (given)
Queuing Model : M/M/1/N/N
Arrival rate `lambda=25,` Service rate `mu=40,` Machine `N=12` (given)
1. Traffic Intensity
`rho=lambda/mu`
`=(25)/(40)`
`=0.625`
2. Probability of no customers in the system
`P_0=[sum_{n=0}^(N) (N!)/((N-n)!)*rho^n]^(-1)`
`=[sum_{n=0}^(12) (12!)/((12-n)!)*(0.625)^n]^(-1)`
`=[1+(12!)/(11!)*(0.625)^1+(12!)/(10!)*(0.625)^2+(12!)/(9!)*(0.625)^3+(12!)/(8!)*(0.625)^4+(12!)/(7!)*(0.625)^5+(12!)/(6!)*(0.625)^6+(12!)/(5!)*(0.625)^7+(12!)/(4!)*(0.625)^8+(12!)/(3!)*(0.625)^9+(12!)/(2!)*(0.625)^10+(12!)/(1!)*(0.625)^11+(12!)/(0!)*(0.625)^12]^(-1)`
`=[1+(12)*(0.625)+(12xx11)*(0.390625)+(12xx11xx10)*(0.24414062)+(12xx11xx10xx9)*(0.15258789)+(12xx11xx10xx9xx8)*(0.09536743)+(12xx11xx10xx9xx8xx7)*(0.05960464)+(12xx11xx10xx9xx8xx7xx6)*(0.0372529)+(12xx11xx10xx9xx8xx7xx6xx5)*(0.02328306)+(12xx11xx10xx9xx8xx7xx6xx5xx4)*(0.01455192)+(12xx11xx10xx9xx8xx7xx6xx5xx4xx3)*(0.00909495)+(12xx11xx10xx9xx8xx7xx6xx5xx4xx3xx2)*(0.00568434)+(12xx11xx10xx9xx8xx7xx6xx5xx4xx3xx2xx1)*(0.00355271)]^(-1)`
`=[1+7.5+51.5625+322.265625+1812.74414062+9063.72070312+39653.77807617+148701.66778564+464692.71183014+1161731.77957535+2178247.08670378+2722808.85837972+1701755.53648733]^(-1)`
`=[8428850.21180688]^(-1)`
`=0.00000012` or `0.00000012xx100=0.000012%`
3. Probability that there are n customers in the system
`P_n=(N!)/((N-n)!)*rho^n*P_0`
`P_n=(12!)/((12-n)!)*(0.625)^n*P_0`
`P_1=(12!)/((12-1)!)*(0.625)^1*0.00000012=0.00000089`
`P_2=(12!)/((12-2)!)*(0.625)^2*0.00000012=0.00000612`
`P_3=(12!)/((12-3)!)*(0.625)^3*0.00000012=0.00003823`
`P_4=(12!)/((12-4)!)*(0.625)^4*0.00000012=0.00021506`
`P_5=(12!)/((12-5)!)*(0.625)^5*0.00000012=0.00107532`
`P_6=(12!)/((12-6)!)*(0.625)^6*0.00000012=0.00470453`
`P_7=(12!)/((12-7)!)*(0.625)^7*0.00000012=0.01764199`
`P_8=(12!)/((12-8)!)*(0.625)^8*0.00000012=0.05513121`
`P_9=(12!)/((12-9)!)*(0.625)^9*0.00000012=0.13782803`
`P_10=(12!)/((12-10)!)*(0.625)^10*0.00000012=0.25842755`
`P_11=(12!)/((12-11)!)*(0.625)^11*0.00000012=0.32303443`
`P_12=(12!)/((12-12)!)*(0.625)^12*0.00000012=0.20189652`
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*0.00000012+1*0.00000089+2*0.00000612+3*0.00003823+4*0.00021506+5*0.00107532+6*0.00470453+7*0.01764199+8*0.05513121+9*0.13782803+10*0.25842755+11*0.32303443+12*0.20189652`
`=10.40000019`
Or
`L_s=N-mu/lambda(1-P_0)`
`=12-40/25(1-0.00000012)`
`=12-1.59999981`
`=10.40000019`
5. Average number of customers in the queue
`L_q=sum_{n=1}^(N) (n-1)P_n`
`=sum_{n=1}^(12) (n-1)*P_n`
`=0*P_1+1*P_2+2*P_3+3*P_4+4*P_5+5*P_6+6*P_7+7*P_8+8*P_9+9*P_10+10*P_11+11*P_12`
`=0*0.00000089+1*0.00000612+2*0.00003823+3*0.00021506+4*0.00107532+5*0.00470453+6*0.01764199+7*0.05513121+8*0.13782803+9*0.25842755+10*0.32303443+11*0.20189652`
`=9.40000031`
Or
`L_q=N-((lambda+mu)/lambda)(1-P_0)`
`=12-((25+40)/25)*(1-0.00000012)`
`=12-(2.6)*(0.99999988)`
`=12-2.59999969`
`=9.40000031`
6. Effective Arrival rate
`lambda_e=lambda(N-L_s)`
`=25*(12-10.40000019)`
`=39.99999525`
7. Average time spent in the system
`W_s=(L_s)/(lambda_e)=(L_s)/(lambda(N-L_s))`
`=(10.40000019)/(39.99999525)`
`=0.26000004` hr or `0.26000004xx60=15.60000214` min
8. Average Time spent in the queue
`W_q=(L_q)/(lambda_e)=(L_q)/(lambda(N-L_s))`
`=(9.40000031)/(39.99999525)`
`=0.23500004` hr or `0.23500004xx60=14.10000214` min
9. Utilization factor
`U=L_s-L_q`
`=10.40000019-9.40000031`
`=0.99999988` or `0.99999988xx100=99.999988%`
This material is intended as a summary. Use your textbook for detail explanation.
Any bug, improvement, feedback then
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