Question

Listed below is the observed sequence of arrival times in a Poisson Process (the times are measured in minutes from a fixed starting time). Using the parameter "w"=5, calculate the first available Erlang Random Variable contained in this data-set.
....7.8, 12.8, 21.5, 30.5, 31.6, 35.1, 39.5, 45.6 ....

Answer 1

k=shape parameter, $\lambda=rate\,\, parameter$

R-commands:

x=c(7.8, 12.8, 21.5, 30.5, 31.6, 35.1, 39.5, 45.6)
n=length(x)
n
sum(x)
#[1] 224.4

Si=cumsum(x)
Si
#[1] 7.8 20.6 42.1 72.6 104.2 139.3 178.8 224.4

#Let Tn = Sn − Sn−1 be the elapsed time between the (n − 1)–th and the n–th event.
DF=diff(x)
DF
#[1] 5.0 8.7 9.0 1.1 3.5 4.4 6.1
Si=cumsum(DF)
Si
#[1] 5.0 13.7 22.7 23.8 27.3 31.7 37.8

#Erlang density (PDF=f(s;k,lambda)) is given as:
#lambda=w=5 (rate parameter)
s=Si
k=1:7 #k=shape parameter
Nr=5*exp(-5*s)*(5*s)^(k-1)
Dr=factorial(k-1)
f=Nr/Dr
f
[1] 6.943972e-11 6.102230e-28 1.642510e-45 2.926957e-46 3.785233e-52 6.084988e-60 2.622716e-71

round(f,2)
[1] 0 0 0 0 0 0 0

#####
d=diff(x,5)
d
[1] 27.3 26.7 24.1
x[6]-x[1]
[1] 27.3
x[7]-x[2]
[1] 26.7
x[8]-x[3]
[1] 24.1
C=cumsum(d)
C
[1] 27.3 54.0 78.1