Queuing Theory, M/M/1/N/N Queuing Model (M/M/1/K/K) Formula

Queuing Model : M/M/1/N/N

Arrival rate `lambda,` Service rate `mu,` Machine `N`

1. Traffic Intensity
`rho=lambda/mu`

2. Probability of no customers in the system
`P_0=[sum_{n=0}^(N) (N!)/((N-n)!)*rho^n]^(-1)`

3. Probability that there are n customers in the system
`P_n=(N!)/((N-n)!)*rho^n*P_0`

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

Or
`L_s=N-mu/lambda(1-P_0)`

5. Average number of customers in the queue
`L_q=sum_{n=1}^(N) (n-1)P_n`

Or
`L_q=N-((lambda+mu)/lambda)(1-P_0)`

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`

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