Operations Management: Processes and Supply Chains, 12th Edition

Key Equations

Structure of Waiting-Line Problems

1. 1. Customer arrival Poisson distribution:

   $$P_n=\frac{{\left(\lambda T\right)}^n}{n!}{e}^{-\lambda T}$$

2. 2. Service time exponential distribution:

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

Using Waiting-Line Models to Analyze Operations

1. 3. Average utilization of the system:

   $$\rho =\frac{\lambda }{\mu }$$

2. 4. Probability that n customers are in the system:

   $$P_n=\left(1-\rho \right){\rho }^n$$

3. 5. Probability that zero customers are in the system:

   $$P_0=1-\rho$$

4. 6. Average number of customers in the service system:

   $$L=\frac{\lambda }{\mu -\lambda }$$

5. 7. Average number of customers in the waiting line:

   $$L_q=\rho L$$

6. 8. Average time spent in the system, including service:

   $$W=\frac{1}{\mu -\lambda }$$

7. 9. Average waiting time in line:

   $$W_q=\rho W$$

8. 10. Little’s law: