MODELLING RANDOM FLUCTUATIONS
IN A BISTABLE TELECOMMUNICATIONS NETWORK
Department of Mathematics
The University of Queensland
1. Pollett, P.K., ``Modelling random fluctuations in a bistable telecommunications network'', Ed. W. Henderson, Proc. 7th Austral. Teletraffic Res. Seminar, (1992), 335-345.
2. Pollett, P.K., ``Diffusion approximations for a circuit switching network with random alternative routing'', Austral. Telecomm. Res. 25, (1992), 45-52.
3. Pollett, P.K., ``On a model for interference between searching insect parasites'' J. Austral. Math. Soc., Ser. B 31, (1990), 133-150.
4. Pollett, P.K. and Vassallo, A., ``Diffusion approximations for some simple chemical reaction schemes Adv. Appl. Probab. 24, (1992), 875-893.
N - nodes
C - circuits on each link
Poisson traffic offered at rate
Exponentially distributed holding times
r retries (initially take r=1)
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.
Let , where is the proportion of links with j circuits in use at time t. The process is itself a Markov chain, but one which takes values in a lattice contained in the simplex
is proportion of links full; this estimates link blocking probability.
And, ; this estimates the mean number of circuits in use on any given link.
If, as , , then , where is a deterministic process with initial point and which satisfies ,
Theorem. If, as , , then, for all and ,
where is the unique solution to the above system of differential equations with .
So, for example, converges in probability to and, since for each s, is uniformly bounded, dominated convergence implies that
on all finite time intervals.
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
The central limit law (Theorem 4.2 of ) shows that the random fluctuations about any given equilibrium point, , are approximately 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 Z(0)=z 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, Z(s) 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
If C=1 (one circuit on each link), set . Then, F has a unique zero, , on (0,1), for all (stable equilibrium point of dx/dt=F(x)). Further, has an approximate normal distribution with
where , and
The magnitude of B, and hence the stability of , increases as becomes large, but the stationary variance, , increases from 0 to a maximum around and then decreases to 0.
When C>1, change coordinates:
where the rows of A are the left-eigenvectors of . Since the column sums of B are all equal to 0, B has a zero eigenvalue, and so one of the components of , say , is identically zero, since because , we have that .
The sequence , where (for convenience) , converges weakly to an OU process, , whose individual components are, themselves, OU processes. Its local drift matrix is , where are the non-zero eigenvalues of B, and its local covariance matrix, S, is obtained from the matrix by deleting the zeroth row and column.
In particular, W(s) has a properly C-dimensional normal distribution with ,
for , where w=Az.
The change of coordinates allows us to use some powerful results of Andrew Barbour, which establish asymptotic results on the time of first exit of from a region containing .
For example, suppose that is a stable and let be the time when first crosses the contour
where T, the stationary covariance matrix of , has elements and is a sequence which tends to ; to order , the contour delimits the rectangle
Then, if , the random variable
where , converges weakly to a unit-mean exponential random variable as . Thus, provided , the time at which first crosses the contour is of order .
The corresponding result for the C=1 case is more straightforward. One can show that the time that first leaves the interval
is of order
whenever . Hence, it is asymptotically larger than any power of K if, for example, .