A Smooth-turn Mobility Model for Airborne Networks Page: 18
View a full description of this thesis.
Extracted Text
The following text was automatically extracted from the image on this page using optical character recognition software:
As each individual distribution is uniform, we can conclude that the N vehicles' node locations and
heading angles remain uniformly distributed at any time t > 0. The proof is complete.
D
Theorem 3.2 informs that the uniform distribution at the initial time is reserved. In the
next Theorem, we present that the steady-state distribution of node location and heading angle are
uniform and independent from the initial distribution.
THEOREM 3.3. N airborne vehicles move independently in the space [0,L] x [0, W] according to
the ST mobility model associated with wrap-around boundary model. Assuming that , is finite and
S54 0, the distributions of node locations and heading angles are uniform in [0, L) x [0, W) and
[0, 27c), respectively, in the limit of large time, regardless of the distribution at the initial time.
PROOF. Let us f rst sketch the structure of the proof. We f rst construct a Markov process with
states S(t) = (lx(t), ly(t), f(t), l(t), z(t)) and fnd the probability transition kernel for the Markov
chain def ned at the time sequence Ti, namely S(T). After that, we f nd the invariant distribution
of the Markov chain S(Ti). The Palm Formula [24, 1] is then used to f nd the limiting probability
distribution of the Markov process S(t).
First, we note that S(t) is a Markov process, because S(t + At) is only dependent upon S(t),
but not on any states before time t. S(T) for i = 0, 1, ... form a discrete-time Markov chain. The
transition probability kernel for the Markov chain S(T) is
(22) f(S(Ti+l) S(Ti))
f(lx(Ti+) = Ax(lx(Ti), D (Ti), r(Ti), Ti+, Ti),ly (Ti+1) =n Ay(ly(Ti), D(Ti), r(Ti), Ti+) , Ti),
QI)(Ti+1 )- =I Y(I(Ti),rF(Ti), Ti+ 1, Ti),- (Ti+1) , T"(Ti+ 1)1/x(Ti) ,1y(Ti) , Q(Ti), ,-(Ti) , T"(Ti))
r r
= {l (T+ )= x l (T) (Ti r(Ti T+1, Ti ( l T i+ 1)=A ly( T) , 1 i ( Ti), Ti+1Ti)}
11
Because 7(Ti) and T(Ti) are independently and identically distributed (i.i.d.) normal and18
Upcoming Pages
Here’s what’s next.
Search Inside
This thesis can be searched. Note: Results may vary based on the legibility of text within the document.
Tools / Downloads
Get a copy of this page or view the extracted text.
Citing and Sharing
Basic information for referencing this web page. We also provide extended guidance on usage rights, references, copying or embedding.
Reference the current page of this Thesis.
He, Dayin. A Smooth-turn Mobility Model for Airborne Networks, thesis, August 2012; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc149603/m1/24/: accessed April 23, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .