Gamma Distribution

The exponential distribution computes the probability of the interval time between a single event. The Gamma distribution is used for calculating the probability of the interval time for the \(r\)-th arrivals. The probability density function (PDF) of the Gamma distribution is:

\[ f(t, r, \lambda) = \frac{e^{-\lambda t} \lambda^r t^{r-1}}{\Gamma(r)} \]

where \(\Gamma(r)=(r-1)!\), and \(\lambda\) is rate. Denote \(T_r\) as the random time for the \(r\)-th arrival, and \(I\) as the single event interval time:

\[ T_r = I_1 + I_2 + I_3 + \cdots + I_r \\ T_r \sim \text{Gamma}(r, \lambda) \]

The cumulative distribution function (CDF) of the Gamma distribution for \(r > 0\) :

\[ P(T_r > t) = P(N_t \leq r-1) = \sum_{k=0}^{r-1} \frac{e^{-\lambda t}(\lambda t)^t}{k!} \]

where \(T_r\) is the continuous time interval for the \(r\)-th arrival, and \(N_t\) is the number of Poisson events that occur over the time interval \(t\).