Gugrajah_Yuvaan_ Ramesh_2003.pdf

Adaptation ofthe Fixed Point Approximation to Ad Hoc Networks Chapter 5
pair in an N node network, where the maximum length of a route is H intermediate
hops, is given by
M =1+f eN -2)! ; H 5, N-l
hoo2(N -h-l)!
(5-2)
Equation (5-2) is proved in the Appendix. If the number of hops in a 50 node
network is limited to 10 hops, then the number of possible routes between each node
pair is 6.24x10 14 routes, if all possible topologies are considered. Even in a moderate
sized network of 10 nodes with a maximum route length of 5 hops, the number of
possible routes between each of the 45 node pairs is 2081 routes.
Due to the above-mentioned problems the analytical model is limited to the
evaluation of a fixed topology multihop wireless ad hoc network. The fixed topology
allows only routes that exist to be evaluated.
5.3. Analytical Model
The fixed-point approximation described here is based on that proposed by [LiuOO]..
The approximation method involves solving simultaneously for four fixed-point
variables using three mappings. Once solved iteratively, the fixed-point variables are
used to calculate the blocking probability for each node pair.
5.3.1. Notation
The network consists of N nodes, indexed n and has a total of R =N(N-l)/2 node
pairs, each indexed r. Between each node pair r there are a total ofM r routes listed in
order from least cost (best route) to highest cost (worst route). The evaluation of the
routes depends on the particular routing protocol that is under evaluation. The mth
route between node pair r is referred to as (r, m) and consists of h rm intermediate
nodes. Calls arrive to a node pair with a Poisson arrival rate of A r and have an
exponential mean holding time of lip.. The offered traffic between a node pair
consisting of nodes a and b is the sum of the traffic from a to b and of traffic from b
to a.
5-5