$\lambda =$ mean number of arrivals per time period

$\mu =$ mean number of people or items served per time period

Equations 12-1 through 12-7 describe operating characteristics in single-channel models that have Poisson arrival and exponential service rates.

1. (12-1) $\begin{array}{lll}L\hfill & =\hfill & \text{Average number of units (customers) in}\hfill \\ \hfill & \hfill & \text{the system}\hfill \\ \hfill & =\hfill & \frac{\lambda }{\mu -\lambda }\hfill \end{array}$

2. (12-2) $\begin{array}{lll}W\hfill & =\hfill & \text{Average time a unit spends in the system}\hfill \\ \hfill & \hfill & \text{(Waiting time + Service time)}\hfill \\ \hfill & =\hfill & \frac{1}{\mu -\lambda }\hfill \end{array}$

3. (12-3) $\begin{array}{lll}{L}_q\hfill & =\hfill & \text{Average number of units in the queue}\hfill \\ \hfill & =\hfill & \frac{{\lambda }^2}{\mu \left(\mu -\lambda \right)}\hfill \end{array}$

4. (12-4) $\begin{array}{lll}{W}_q\hfill & =\hfill & \text{Average time a unit spends waiting in the queue}\hfill \\ \hfill & =\hfill & \frac{\lambda }{\mu \left(\mu -\lambda \right)}\hfill \end{array}$

5. (12-5) $\rho =\text{Utilization} \text{factor} \text{for} \text{the} \text{system}=\frac{\lambda }{\mu }$

6. (12-6) $\begin{array}{lll}{P}_0\hfill & =\hfill & \text{Probability of 0 units in the system}\hfill \\ \hfill & \hfill & \text{(i}\text{.e}\text{., the service unit is idle)}\hfill \\ \hfill & =\hfill & 1-\frac{\lambda }{\mu }\hfill \end{array}$

7. (12-7) $\begin{array}{lll}{P}_{n>k}\hfill & =\hfill & \text{Probability of more than }k\text{ units in the system}\hfill \\ \hfill & =\hfill & {\left(\frac{\lambda }{\mu }\right)}^{k+1}\hfill \end{array}$

   Equations 12-8 through 12-12 are used for finding the costs of a queuing system.

8. (12-8) Total service cost = mC_s

   where

   • m = number of channels

   • C_s = service cost (labor cost) of each channel

9. (12-9) Total waiting cost $=\left(\lambda W\right)C_w$ where $C_w=\text{cost} \text{of} \text{waiting}$

   Waiting time cost based on time in the system.

10. (12-10) $\text{Total} \text{waiting} \text{cost} =\left(\lambda W_q\right)C_w$

    Waiting time cost based on time in the queue.

11. (12-11) $\text{Total} \text{cost}=m{C}_{s }+\lambda W{C}_w$

    Waiting time cost based on time in the system.

12. (12-12) $\text{Total} \text{cost}=m{C}_s+\lambda W_q{C}_w$

    Waiting time cost based on time in the queue.

    Equations 12-13 through 12-18 describe operating characteristics in multichannel models that have Poisson arrival and exponential service rates, where $\text{m}=$the number of open channels.

13. (12-13) $P_0=\frac{1}{\left[{\displaystyle \sum _{n=0}^{n=m\mathrm{-}1}}\frac{1}{n!}{\left(\frac{\lambda }{\mu }\right)}^n\right]+\frac{1}{m!}{\left(\frac{\lambda }{\mu }\right)}^m\frac{m\mu }{m\mu -\lambda }}$

    $\text{for}m\mu >\lambda$

    Probability that there are no people or units in the system.

14. (12-14) $L=\frac{\lambda \mu {\left(\lambda /\mu \right)}^m}{\left(m-1\right)!{\left(m\mu -\lambda \right)}^2}{P}_0+\frac{\lambda }{\mu }$

    Average number of people or units in the system.

15. (12-15) $W=\frac{\mu {\left(\lambda /\mu \right)}^m}{\left(m-1\right)!{\left(m\mu -\lambda \right)}^2}{P}_0+\frac{1}{\mu }=\frac{L}{\lambda }$

    Average time a unit spends in the waiting line or being serviced (that is, in the system).

16. (12-16) $L_q=L-\frac{\lambda }{\mu }$

    Average number of people or units in line waiting for service.

17. (12-17) $W_q=W-\frac{1}{\mu }=\frac{L_q}{\lambda }$

    Average time a person or unit spends in the queue waiting for service.

18. (12-18) $\rho =\frac{\lambda }{m\mu }$

    Utilization rate.

    Equations 12-19 through 12-22 describe operating characteristics in single-channel models that have Poisson arrival and constant service rates.

19. (12-19) $L_q=\frac{{\lambda }^2}{2\mu \left(\mu -\lambda \right)}$

    Average length of the queue.

20. (12-20) $W_q=\frac{\lambda }{2\mu \left(\mu -\lambda \right)}$

    Average waiting time in the queue.

21. (12-21) $L=L_q+\frac{\lambda }{\mu }$

    Average number of units in the system.

22. (12-22) $W=W_q+\frac{1}{\mu }$

    Average waiting time in the system.

    Equations 12-23 through 12-28 describe operating characteristics in single-channel models that have Poisson arrival and exponential service rates and a finite calling population.

23. (12-23) $P_0=\frac{1}{{\displaystyle \sum _{n=0}^N}\frac{N!}{\left(N-n\right)!}{\left(\frac{\lambda }{\mu }\right)}^n}$

    Probability that the system is empty.

24. (12-24) $L_q=N-\left(\frac{\lambda +\mu }{\lambda }\right)\left(1-P_0\right)$

    Average length of the queue.

25. (12-25) $L=L_q+\left(1-P_0\right)$

    Average number of units in the system.

26. (12-26) $W_q=\frac{L_q}{\left(N-L\right)\lambda }$

    Average waiting time in the queue.

27. (12-27) $W=W_q+\frac{1}{\mu }$

    Average time in the system.

28. (12-28) $P_n=\frac{N!}{\left(N-n\right)!}{\left(\frac{\lambda }{\mu }\right)}^n{P}_0\text{for} n=0, 1,\dots , N$

    Probability of units in the system.

    Equations 12-29 to 12-30 are Little’s Flow Equations, which can be used when a steady state condition exists. Equation 12-31 is an assumption that must be met in order for steady state conditions to exist.

29. (12-29) $L=\lambda W$

30. (12-30) $L_q=\lambda W_q$

31. (12-31) $W=W_q+1/\mu$