TRAFFIC PROCESSES
Traffic process : is the (cumulative)
number of arrivals in . Assume that and that no two
calls can arrive at exactly the same time.
If N were a Poisson process
(with rate ), then
or, more generally,
Here 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).
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 and with
as follows:
(recall that is a stopping time if
). Let be
the sequence of arrival times, and, for fixed i, define
Consider the three possible cases
and observe that
Thus,
Setting we find that
and hence that .
For i=0 we get
which gives . Since
we deduce that has a Poisson distribution with parameter t.
We also require to be independent of the
past at s. Fix s and , and let . Then,
(2) gives
where 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.
CONDITIONAL INTENSITY
Let be an arbitrary traffic process.
Under mild regularity conditions, the limit
exists and defines the conditional intensity (process),
, and, if we set
then, for all ,
For stopping times and such that ,
where means expectation conditional on the past at time :
formally, the -algebra of events, A,
such that for each t.
POISSON APPROXIMATIONS
Let N be a traffic process with conditional intensity . 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 idea is to construct a Poisson process M using N.
Define , 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
with and .
This gives
.
Thus, by the previous argument, 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:
Proposition For arbitrary A,
where is the probability that A would have if the traffic
process were Poisson with parameter .
If the traffic process is stationary
with , then
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, , (to the network)
fixed and let the number of
switching nodes, K, become large; what happens?
A POISSON APPROXIMATION FOR THE
Let 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
Thus and
Recall that if A is any event determined by on , then
ASYMPTOTICALLY INDEPENDENT BLOCKING
Proposition (Ziedins and Kelly (1989)):
By keeping the total offered traffic, , fixed and letting the
number of switching nodes, K, become large, the links are blocked
independently.
Specifically, as ,
We deduce that the traffic offered to any given link
is
approximately Poisson, since
.
A MARKOVIAN QUEUEING NETWORK
Label the links and
make the usual simplifying assumptions:
A total effort (or capacity) of is provided by link j when there
are n messages whose transmission is incomplete.
A POISSON APPROXIMATION
Let 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
where is the rate of externally offered traffic
and is the proportion of messages emanating from
link j which next use link k.
Let . Then (in equilibrium)
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
from a Poisson process with rate .
STATE-INDEPENDENT CAPACITY
Suppose
Then
where
(we must have that ).
Thus, if for each j, is
and either
then the Poisson approximation is sure to be good.
SIMULTANEOUS APPROXIMATIONS
Let be l traffic processes and let
be their conditional intensities.
If A is an event which is determined by these processes on the interval
and is the probability that A would have if
these processes were independent
Poisson processes with
rates , then
If the system is in equilibrium
and are chosen such that
, then a simpler bound is given by
STAR NETWORK
BY INDEPENDENT POISSON PROCESSES