Introduction to Manufacturing Systems

Derivation of Exponential Distribution

The graph of the exponential distribution is shown in Figure 1.

Figure 1: Graph of Markov Process for Exponential Distribution. (Figure by MIT OpenCourseWare.)

The transition equations are

$$\begin{array}{}\text{(1)}& p(0,t+\delta t)=\mu \delta tp(1,t)+p(0,t)+o(\delta t)\end{array}$$

$$\begin{array}{}\text{(2)}& p(1,t+\delta t)=(1-\mu \delta t)p(1,t)+(0)p(0,t)+o(\delta t)\end{array}$$

or,

$$\begin{array}{}\text{(3)}& \frac{dp(0,t)}{dt}=\mu p(1,t)\end{array}$$

$$\begin{array}{}\text{(4)}& \frac{dp(1,t)}{dt}=-\mu p(1,t)\end{array}$$

Solve differential equations (3), (4) and we get

$$\begin{array}{}\text{(5)}& p(0,t)=1-e^{-\mu t}\end{array}$$

$$\begin{array}{}\text{(6)}& p(1,t)=e^{-\mu t}\end{array}$$

Function $p(0,t)$ is actually the cumulative density function of the exponential distribution

$$\begin{array}{}\text{(7)}& F(t)=p(0,t)\end{array}$$

Then the density function of the exponential distribution is

$$\begin{array}{}\text{(8)}& f(t)=\frac{dF(t)}{dt}=\mu e^{-\mu t}\end{array}$$