MODELLING BISTABILITY IN TELECOMMUNICATIONS NETWORKS
Department of Mathematics
The University of Queensland
A SYMMETRIC FULLY-CONNECTED NETWORK
N - nodes
C - circuits on each link
Poisson traffic offered at rate
Exponentially distributed holding times (mean 1)
THE GHK MODEL
Let be the number of links with j circuits in use at time t (for a network with K links).
Let . Then, is a continuous-time Markov chain which takes values in
The transition rates, , , of the process are given by
where is the unit vector with 1 as its entry.
THE BEHAVIOUR AS THE NETWORK BECOMES LARGE
Let , where is the proportion of links with j circuits in use at time t.
If, as , , then , where is a deterministic process with initial point and which satisfies ,
THE EQUILIBRIUM POINTS
If is an equilibrium point it must be of the form given by
The quantity , given by
is Erlang's formula for the loss probability of a single link with C circuits and with Poisson traffic offered at rate . It is usually more convenient to calculate the equilibrium points by setting and solving the equation
MODELLING RANDOM FLUCTUATIONS
The central limit law shows that the random fluctuations about any given equilibrium point, , are Gaussian. Moreover, it shows that these fluctuations can be approximated by an Ornstein-Uhlenbeck (OU) process.
Let be an equilibrium point. Then, if
the family of processes , defined by
converges weakly to an Ornstein-Uhlenbeck process with initial value and with a local drift matrix, B, and a local covariance matrix, G, which can be determined from the parameters of the model.
In particular, is normally distributed with mean
and covariance matrix
where , the stationary covariance matrix, satisfies
We can conclude that, for K large, has an approximate normal distribution for each s, and an approximation for the mean and the covariance matrix of is given by