MODELLING RANDOM FLUCTUATIONS

IN A BISTABLE TELECOMMUNICATIONS NETWORK

by

Phil Pollett

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

- links

*C* - circuits on each link

Poisson traffic offered at rate

Exponentially distributed holding times

(mean 1)

*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

2-D ``SUMMARY'':

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 ,

,

,

where .

**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

where solves

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 [1]) 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

and

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
,

and

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,
.