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?

The shape and rate parameters are and .

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, .

The service time for the next 3 customers is,

Z = X₁ + X₂ + X₃

All the X'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:

Thus, the shape and rate parameters are and .

the smallest possible measurement of the width is 14 feet

step-by-step explanation:

let's assume length of parallelogram is 'l'

width of parallelogram is 'w'

the perimeter of a parallelogram must be no less than 40 feet

so, we get

we are given

length of parallelogram is 6 feet

so,

l=6

we can plug it

now, we can solve for w

subtract both sides by 12

divide both sides by 2

and we get

so,

the smallest possible measurement of the width is 14 feet