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for nodes; Message Life Time is set to T; total number of nodes is N. The number of
neighbors (node degree) is a critical metric for a network. C. Bettstetter[12] gave a
detailed analysis on the relationship between node degree and overall connectivity for a
mobile network. He argues that a graph is said to be k-connected (k =1, 2, 3,...) if for
each node pair there exist at least k mutually independent paths connecting them. In this
paper, we follow this idea to calculate the paths each node has in order to get the overall
Spatial-temporal Reachability for a DTN.
We first calculate the area size of the Contact Window (
CWindow
). In this window,
nodes have the opportunity to move into the transmitting range of a certain node S
within Message Life Time T; secondly, the probability for n of m nodes moving into
S's transmitting range is acquired; while
nmN
, we get the probability for n nodes
having Spatial-Temporal links with S in Message Life Time T ,
≤≤
, and the
probability for all the nodes having at least n Spatial-Temporal links in Message Life
Time T,
PX
(
=
n
)
PX
(
≥
n
)
.
4.2
Spatial-temporal Reachability in a Two-Dimensional Mobility Model
The space distribution of a Random Direction Model in any time is uniform[13]. For a
square map with area of
ma
S
and a certain node S, we set pause time and reset time 0.
The area size of ring BA and ring AC is
and
.
S
S
BA
AC
As shown by Fig. 2, we follow the analysis steps,
Step 1:
Calculating the area of the Contact Window.
For any node A, it has to be located in the circle
to get the opportunity
of contacting S within time T. We firstly put nodes within circle
Sr
SVTr
−
(*
+
)
−
out of consid-
eration, and get the area of contact window,
(
)
2
CWindow
=
π
V
*
T
+
r
−
π
r
2
Step 2:
Probability of n of m nodes moving into S's transmitting range.
As Random Direction Model fits normal distribution, the probability for m nodes
located in the Contact Window is
m
N
−
m
CWindow
CWindow
PU
(
*
==
m
)
C
m
N
*
* 1
−
S
S
map
map
For node A, the moving direction has to be within
∠
MAN
in order to get a contact
with S (AM, AN are tangents to circle
Sr
). We derived the average angle for all the
nodes that can contact S in the Contact Window,
−
S
S
rVT
*
α
+
*
α
'
.
α
=
BA
AC
2
2
2*
π
*(
+
*
)
−
π
*
r
Through the average angle, we get the probability for n of m nodes moving into S's
transmitting range in the Contact Window.
