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,
.