TRAFFIC PROCESSES
Traffic process
If N were a Poisson process
(with rate
or, more generally,
Here
We say that N has a deterministic past-conditional arrival rate.
WATANABE'S THEOREM
Now let N be an arbitrary traffic process.
Theorem If N has a deterministic past-conditional
arrival rate, then N is a Poisson process.
Outline Proof For simplicity, suppose that
Now extend (1) to stopping times
(recall that
Consider the three possible cases
and observe that
Thus,
Setting
and hence that
For i=0 we get
which gives
We also require
where
CONDITIONAL INTENSITY
Let
Under mild regularity conditions, the limit
exists and defines the conditional intensity (process),
then, for all
For stopping times
where
POISSON APPROXIMATIONS
Let N be a traffic process with conditional intensity
The idea is to construct a Poisson process M using N.
Define
with
This gives
Thus, by the previous argument,
Proposition For arbitrary A,
where
A STAR NETWORK
There are K outer nodes which communicate via a single central node.
Thus, there are K links (circuit groups),
and each route consists of a pair of links
Assume that
Keep the total offered traffic,
A POISSON APPROXIMATION FOR THE
Let
Thus
Recall that if A is any event determined by
ASYMPTOTICALLY INDEPENDENT BLOCKING
Proposition (Ziedins and Kelly (1989)):
By keeping the total offered traffic,
We deduce that the traffic offered to any given link
is
approximately Poisson, since
A MARKOVIAN QUEUEING NETWORK
Label the links
A total effort (or capacity) of
A POISSON APPROXIMATION
Let
where
Let
An equilibrium exists if and only if, for each j,
in which case, the states of the individual links are
independent and the probability that there are n messages at link j
is given by
We can then show that
This leads to a bound on the degree of deviation of
STATE-INDEPENDENT CAPACITY
Suppose
Then
where
then the Poisson approximation is sure to be good.
SIMULTANEOUS APPROXIMATIONS
Let
If A is an event which is determined by these processes on the interval
If the system is in equilibrium
and :
is the (cumulative)
number of arrivals in
. Assume that
and that no two
calls can arrive at exactly the same time.
), then
represents expectation conditional on the
past at time s
(formally,
, where
is an
increasing, right-continuous family of
-algebras, with
being
measurable and
being complete).
and
with
as follows:
is a stopping time if
). Let
be
the sequence of arrival times, and, for fixed i, define
we find that
.
. Since
we deduce that
has a Poisson distribution with parameter t.
to be independent of the
past at s. Fix s and
, and let
. Then,
(2) gives
refers to expectation conditional on the past at time t,
including the fixed event A. But the previous analysis then
gives a Poisson
distribution for
, that is, a Poisson
distribution with parameter t for
, conditional on A.
This establishes the required independence.
be an arbitrary traffic process.
, and, if we set
,
and
such that
,
means expectation conditional on the past at time
:
formally, the
-algebra of events, A,
such that
for each t.
. Fix
and let A be an event determined by N on
. It is
proposed to approximate
by
, where
is the
probability that A would have if N were Poisson with given rate
.
, the (pseudo) inverse
of the increasing (random) function
. In general
, but
, with inequality
occurring at only those t for which
is constant around an interval
containing t. Certainly
for
, and so we may apply
the above (stopping time) result, namely
and
.
.
is a unit-rate Poisson
process and so M, where
, is a Poisson
process with rate
.
A familiar coupling argument then leads to the following result:
is the probability that A would have if the traffic
process were Poisson with parameter
.
If the traffic process is stationary
with
, then
.
,
, (to the network)
fixed and let the number of
switching nodes, K, become large; what happens?
STAR NETWORK
be the number of calls offered to link k in the
time-interval
and let
be the number of circuits in use on link j at time s.
Clearly
, the conditional intensity of
, is given by
and
on
, then
, fixed and letting the
number of switching nodes, K, become large, the links are blocked
independently.
Specifically, as
,
.
and
make the usual simplifying assumptions:
is provided by link j when there
are n messages whose transmission is incomplete.
be the net traffic offered to a given link,
k, on the interval
.
Since all messages have unit mean length, the rate at which messages are
transmitted by link j is
when there are n messages present.
Thus, if
is the number of messages present at link j at time s,
then
, the conditional intensity of
, is given by
is the rate of externally offered traffic
and
is the proportion of messages emanating from
link j which next use link k.
. Then (in equilibrium)
from a Poisson process with rate
.
(we must have that
).
Thus, if for each j,
is
and either
is
,
is
, or
is
,
BY INDEPENDENT POISSON PROCESSES
be l traffic processes and let
be their conditional intensities.
and
is the probability that A would have if
these processes were independent
Poisson processes with
rates
, then
are chosen such that
, then a simpler bound is given by