MASSACHUSETTS
OF
TECHNOLOGY
INSTITUTE
1.203J/6.281J/13.665J/15.073J/16.76J/ESD.216J
Logistical and Transportation
P
lanning Methods
Fall 2002
Consider link
1.
Define
X2
to be the distance from the right most point of link
1.
Then the
conditional travel distance, given that
X1
is defined to be the distance on link
7
from its left
most point and that
X2
is on link
1,
is
(Dl,
7)
=
X1
+
X2
+
1.
Let's define
V
=
X1
+
X2.
Note that
XI, X2

U(0,l)
i.i.d. Either by convolution, or by the "never fail" sample space
method using cumulative distribution functions, or by recalling problem
2(e)(i)
of HW1, we
find
d
t
[O,
11
fv(d)
=
otherwise
Now the conditional pdf we want for link
1
is
fv(d)
"shifted to the right" by one unit of
distance. Call this conditional pdf
f (Dl,7)(d).
Then we have for link
1, f (Dl,7)(d)
=
fv(d

1).
By inspection we also have
f (D3,7)(d)
=
f (D8,7)(d)
=
f (Dll,
7)(d)
=
f
(Dl,
7)(d)
=
fv(d

1).
For the remaining links in Set
1,
links
4, 5, 6, 9
and
10
"touch"
link
7,
so there is no shifting of the pdf by one. That is, there is no intermediate link between
them that would add
1.0
km to the travel distance. Hence,
f (D4,7)(d)
=
f (D5,7)(d)
=
f(D6,7)(d)
=
f(D9,7)(d)
=
f(Dl0,7)(d)
=