On the Spatial-temporal Reachability of DTNs - Frontiers in Internet Technologies - page 45

PU ( ==



n )

(1)

m N m − n m n −

  

CWindow CWindow   α α

C

m

* * 1 − * C

n

* * 1 −

     

   

N m

S S π π

 

   

map map

We go back for nodes that were originally located in circle Sr . These nodes can

contact S from time 0, so the probability for q of N nodes that communicate with S is

−

q N − q

   

2 2

π * r π * r

(2)

q
N

PV ( == q ) C * * 1 −

   

   

S S

   

map map

Step 3: Probability for n nodes having Spatial-Temporal links with S
Based on (1) and (2), we get the probability for n nodes having Spatial-Temporal links
with a certain node S,
Theorem 1:

 

n



N





PX ( == n ) PU ( = i ) * PV ( =− = n i )

   

 

 

i = 0 m = i

 

m N − m

  


 


   

CW indow CW indow

 

m
N



C *    * 1 − 

   

  S S 

N

   



map map

 



 

i m − i



n

   α α 

mi =



  ** C

i
m

*1 − 

   

 

π π

 

     

i = 0

 

ni − N ni − +

     

π * r

2

π * r

2

* C

ni
N

−

* * 1 −

     

   

  

S S

   



map map

From Theorem 1, we get the probability distribution of n nodes having Spa-
tial-Temporal link with a certain node S within Message Life Time T:

Fig. 3. Probability for all the nodes having n Spatial-Temporal links with a certain node S

1. It shows that in T=15s, probability for all the nodes having 18 Spatial-Temporal links with a certain

node S is 9% in analytical results;

2. When T=20s, analytical result shows probability for all the nodes having 26 Spatial-Temporal links

with a certain node S is 8%.