Gugrajah_Yuvaan_ Ramesh_2003.pdf

EvaLuation ofNetwork Blocking ProbabiLity Chapter 4
In 1917 the Danish mathematician A. K. Erlang published his loss formula for the
loss probability of a telephone system [Kelly95]. Erlang's formula refers to a single
link with calls at a single rate. Although a loss network can be used to model an
entire network of links, it needs to be modelled as a multidimensional Markov
process when there are multiple links, multiple call rates with different bandwidth
requirements and a fixed route associated with each source-destination pair. Since
the number of calls of each class on each feasible route uniquely defines a state of
the network, the dimension of the state space of the process is the product of the
number of the service bandwidth requirements and the number of routes allowed in
the network.
The difficulty arises when alternate routes are present in addition to fixed routes. The
Markov process no longer has a product form, and the whole set of detailed balance
equations need to be written out and solved to obtain the equilibrium state
probabilities. Since the computational complexity of the detailed balance equations is .
both exponential in the number of routes and exponential in the number of service
classes, this approach is not practical when there are many routes. To provide
accurate estimates of blocking probabilities in reasonable time frames, other
computational techniques are required.
One approach is to model wired networks as loss networks and analyse them using
the reduced load approximation. The reduced load approximation is based on two
assumptions:
i) Link independence: blocking occurs independently from link to link.
ii) Poisson assumption: traffic flow to each individual link is Poisson
[LiuOO] proposes a fixed-point approximation that is one type of reduced load
approximation for use with multihop, multirate, loss networks with state-dependent
routing. An iterative process is used to solve simultaneously for fixed-point variables
that are then used to evaluate the blocking probability.
In trying to develop an analytical model for ad hoc networks, the wireless
transmission environment needs to be considered. In wireless cellular networks
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