# Key Equations

# Structure of Waiting-Line Problems

1. Customer arrival Poisson distribution:

$${P}_{n}=\frac{{\left(\lambda T\right)}^{n}}{n!}\text{\hspace{0.17em}}{e}^{\text{\hspace{0.17em}}-\lambda T}$$2. Service time exponential distribution:

$$P(t\text{\hspace{0.17em}}\le T)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\text{\hspace{0.17em}}{e}^{-\mu T}$$

# Using Waiting-Line Models to Analyze Operations

3. Average utilization of the system:

$$\rho \text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{\lambda}{\mu}$$4. Probability that

`n`customers are in the system:$${P}_{n}=\left(1-\rho \right){\rho}^{n}$$5. Probability that zero customers are in the system:

$${P}_{0}=1-\rho $$6. Average number of customers in the service system:

$$L=\frac{\lambda}{\mu \text{}-\text{}\lambda}$$7. Average number of customers in the waiting line:

$${L}_{q}=\rho L$$8. Average time spent in the system, including service:

$$W=\frac{1}{\mu \text{}-\text{}\lambda}$$9. Average waiting time in line:

$${W}_{q}=\rho W$$10. Little’s law:

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