Quantitative Analysis for Management, 13/e

Key Equations

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

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

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

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

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

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

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

   Equations 12-8 through ...