On the Spatial-temporal Reachability of DTNs - Frontiers in Internet Technologies - page 46

Step 4: Probability for the network being K-Reachable
Our goal is to get the overall reachability for the whole network. In Theorem 1 we
acquired the probability for n nodes having Spatial-Temporal links with a certain node
S. In a large scale network, events can be seen as independent from each other[13, 14].
As a result, when , we finally get the probability for all the nodes having at least

n Spatial-Temporal links, which means the network being K-Reachable:

N >> 1

N



n





PX ( ≥=− n ) 1 PX ( = j ) =

 

(3)

 

j = 0

N

 

 

m N − m

 

   

CWindow CWindow

   

 

C

m
N

* * 1 −

   

   

   

 S S 

N

   



map map

 

   



i m − i



   

n j

 

mi =

α α



 1 −  ** C

i
m

*1 −  

 

 

 

 π π 

 

 

j i == 00

 

 

 

ji − N ji − +

 

     

π * r

2

π * r

2

* C

ji
N

−

* * 1 −

        

   

  

S S

     



map map



Fig. 4. Probability for the network being K-Reachable

With Message Life Time T=15s, analytical result shows probability for the network being 3-Reachable is
92%, while simulation shows 100%; when T=20s, analytical result shows probability for the network being
3-Reachable is 95%, while simulation indicates 100%.

5 Simulations and Discussions

In our simulation, we use C++ to build a Random Direction Model as an example for
DTN simulation environment. For Random Direction Model, we set node number
1000, map size 2000m*2000m, node moving speed 20m/s, transmitting range 100m.
We run the simulation 1000 times so that we can get the statistical distribution of n
nodes having Spatial-Temporal links with a certain node S within Message Life
Time T: