
240 Data Analysis and Statistics for Geography, Environmental Science, and Engineering
Using a similar argument, we can calculate the pdf of arrival times as RV T
k
taking values τ.
Say, the density
P( )(,) (,)
k
pk Pk
1 (7.12)
Now use Equation 7.11
pk
k
(,)
() exp( )
()!
exp( )
()!
τ
λτ λτ
λ
λτ λτ
=
−
−
=
−
−
(7.13)
This is the pdf of an Erlang RV of order k. The cdf is
Fk
k
k
k
k
(,)exp( )
!
τ
λτ
=− −
=
−
∑
1
0
1
(7.14)
Figure 7.19 shows both pdf and cdf. As you can see, the curve looks more and more as a normal as
we increase the order k (Acevedo, 1980; Acevedo et al. 1996; Hennessey, 1980; Lewis, 1977).
When you look carefully at the cu