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

• the links have C circuits,

• calls arrive as independent Poisson processes,

• these have the same rate, ,

• call lengths are exponential with mean 1,

• these are mutually independent.

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:

• the links are perfectly reliable and are not subject to noise, so that message transmission times are determined by their length.

• the time taken by the nodes to switch, and, if necessary, buffer, reassemble and acknowledge messages, is negligible in comparison with their transmission times.

• traffic entering the network from external sources is Poisson and message lengths are mutually independent and exponentially distributed with mean 1.

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

(1)
is ,

(2)
is , or

(3)
is ,

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