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Home > Operation Research > Queuing Theory > Queuing Theory M/M/s Queuing Model (M/M/c) example
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4. Queuing Theory, M/M/s Queuing Model (M/M/c) example
( Enter your problem )
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Algorithm and examples
- Formula
- Example-1: `lambda=30`, `mu=20`, `s=2`
- Example-2: `lambda=10`, `mu=3`, `s=4`
- Example-3: `lambda=1` per 6 min, `mu=1` per 4 min, `s=2`
- Example-4: `lambda=15` per 1 hr, `mu=1` per 5 min, `s=2`
- Example-5: `lambda=20` per 8 hr, `mu=1` per 40 min, `s=3`
- Example-6: `lambda=1` per 5 hr, `mu=1` per 1 hr, `s=4`
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1. Formula
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Queuing Model : M/M/s
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Arrival rate `lambda,` Service rate `mu,` Number of servers `s`
1. Traffic Intensity
`rho=lambda/mu`
2. Probability of no customers in the system
`P_0=[sum_{n=0}^(s-1) (rho^n)/(n!) + (rho^s)/(s!)*(s mu)/(s mu-lambda)]^(-1)`
3. Average number of customers in the queue
`L_q=((rho^s)/((s-1)!)*(lambda mu)/(s mu-lambda)^2)*P_0`
4. Average Time spent in the queue
`W_q=L_q/lambda`
Or
`W_q=((rho^s)/((s-1)!)*(mu)/(s mu-lambda)^2) * P_0`
5. Average number of customers in the system
`L_s=L_q+rho`
6. Average Time spent in the queue
`W_s=L_s/lambda`
Or
`W_s=W_q+1/mu`
7. Utilization factor
`U=rho/s=(lambda)/(s mu)`
Or
`U=(L_s-L_q)/s`
8. 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 "n>=2):}`
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