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

Queuing Model : M/M/s/N/N

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

1. Traffic Intensity
`rho=lambda/mu`

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)`

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):}`

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+1}^(N) (n-s)P_n`

6. Effective Arrival rate
`lambda_e=lambda(N-L_s)`

7. Average time spent in the system
`W_s=(L_s)/(lambda_e)=(L_s)/(lambda(N-L_s))`

8. Average Time spent in the queue
`W_q=(L_q)/(lambda_e)=(L_q)/(lambda(N-L_s))`

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