Question

Suppose customers in a certain queue are served one at a time sequentially, that the time for each individual customer is Exponential, and the average service time per customer equals 12 minutes. (Assume service times are independent of each other.) The time, in minutes, until the next 3 services are completed is Gamma (more specifically, Erlang) with what shape and rate parameters?

Answer 1

Answer:

The shape and rate parameters are $\frac{1}{12}$ and $3$.

Step-by-step explanation:

Let X = service time for each individual.

The average service time is, β = 12 minutes.

The random variable follows an Exponential distribution with parameter, $\lambda=\frac{1}{\beta}=\frac{1}{12}$.

The service time for the next 3 customers is,

Z = X₁ + X₂ + X₃

All the X$_{i}$'s are independent Exponential random variable.

The sum of independent Exponential random variables is known as a Gamma or Erlang random variable.

The random variable Z follows a Gamma distribution with parameters (α, n).

The parameters are:

$\alpha =\lambda=\frac{1}{12}\\n=3$

Thus, the shape and rate parameters are $\frac{1}{12}$ and $3$.