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
STAR NETWORK
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
BY INDEPENDENT POISSON PROCESSES
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