Queuing Theory, M/M/s/N Queuing Model (M/M/c/K) Formula

Queuing Model : M/M/s/N

Arrival rate `lambda,` Service rate `mu,` Number of servers `s,` Capacity `N`

1. Traffic Intensity
`rho=lambda/mu`

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)`

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):}`

4. Average number of customers in the system
`L_s=sum_{n=0}^(N) nP_n`

5. Average number of customers in the queue
`L_q=sum_{n=s}^(N) (n-s)P_n`

6. Effective Arrival rate
`lambda_e=lambda*(1-P_N)`

7. Average Time spent in the queue
`W_q=(L_q)/(lambda_e)=L_q/(lambda*(1-P_N))`

8. Average Time spent in the queue
`W_s=(L_s)/(lambda_e)=L_s/(lambda*(1-P_N))`

Or
`W_s=W_q+1/mu`

9. Utilization factor
`U=(L_s-L_q)/s`

This material is intended as a summary. Use your textbook for detail explanation.
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1. Formula