Fixed Point Methods for Loss Networks

by

Mark Bebbington, Massey University

Phil Pollett, The University of Queensland

Ilze Ziedins, The University of Auckland

This consists of a group of nodes (locations),

*K* links (circuit groups),

circuits comprising link *j*, and

- a collection of routes.

Each route is a set of links. Calls using route *r*
are *offered* at rate as a Poisson stream,
and use circuits from link *j*.
indexes *independent* Poisson processes.

Calls requesting route *r* are *blocked* and lost if, on *
any* link *j*, there are fewer than free circuits. Otherwise,
the call is *connected* and simultaneously holds circuits
on each link *j* for the duration of the call.
For simplicity, we shall take .

Call durations are iid exponential random variables with unit mean,
and are *independent* of the arrival processes.

Let , where is the number of
calls in progress using route *r*, let , and let
.

The continuous-time Markov chain takes values in

and has a unique stationary distribution given by

where is given by

The stationary probability that a route-*r* call is blocked
is given by

where is the unit vector from
describing just one call in progress on route *r*

An explicit expression for !

However, the bad news is that
*can't usually be computed* in polynomial time.

For example, consider the trivial case of a fully-connected network with all possible single-link routes ( , ) and . Clearly, .

**Theorem**
*
(Kelly (1986)) There is a unique vector
satisfying
*

* *

*
and
*

* *

*
where*

is *Erlang's Formula* for the loss probability on
a *single link* with *C* circuits and Poisson traffic offered at
rate .

The celebrated *Erlang Fixed Point Approximation* is
obtained by using to estimate the probability that link *j*
is full, and using to estimate the route-*r* blocking probability.

(Independent blocking)

If links along route *r* were blocked *independently* (they are
clearly not) and if *were* the link-*j* blocking probability,
then would be the route-*r* blocking probability:

Carrying this rationale further, the traffic offered to
link *j* would be Poisson (at rate , say) and the
*carried traffic*
(that which is accepted) on link *j* would be

The Erlang Fixed Point Approximation stipulates that the blocking probabilities should be consistent with this level of carried traffic:

The (unique) *fixed point* of the system

is called the *Erlang Fixed Point*.

**Remarks** The existence of a fixed point is easy to prove using
the *Brouwer fixed point theorem*; these equations determine a continuous
mapping from a compact convex set into itself. The uniqueness
(Kelly) is difficult to prove.

For more complex systems, there may be more than one fixed point. This
may be associated with multiple stable states for the
*network*. For example, in networks with *Random Alternative
Routing* the system can fluctuate between a *low blocking*
state, where calls are accepted readily, and a *high blocking*
state, where the likelihood of a call being accepted can be quite low.

There are two limiting regimes under which the EFP is asymptotically valid:

**Moderate loading** (Kelly (1986)) Consider a sequence
of networks indexed by *N* (arbitrary), and index the capacities
and arrival rates accordingly:

**Theorem** *Suppose that as
*

* *

*
( Network topology fixed.)
If is the route-r loss probability, then, for each
, ,
where
*

* *

*
and is the Erlang Fixed Point determined by
and .*

**Diverse routing** (General formulation - Hunt (1990)) Consider a
sequence of networks indexed by *K* (the number of links), and index the
routing matrix and arrival rates accordingly:

Suppose that

and

(*Capacities fixed*.)
The traffic along link *j* is moderate ( ),
but that which is common to *any two* links becomes small.

There are no general results. Examples include *star networks*
(Ziedins and Kelly (1989)) and networks with *alternative routing*
(Gibbens, Hunt and Kelly (1990)); there are many more.

This consists of a collection of *K* outer nodes, which communicate
via a single central node. Take to be all
those routes consisting of a *pair* of links
(there are of them).
Fix and take for all ,
and fix for all links .

**Theorem** *
If is the common route loss probability, then
, given by
*

* *

*
and B, the Erlang Fixed Point, is the unique solution to*

where is Erlang's Formula.

The EFP can perform badly if the network exhibits any of the following features:

- highly linear structure (lines and rings)

- low capacities

- priority controls (eg, trunk reservation)

In these cases, the independent blocking assumption may not be valid.

Can we model the link dependencies, and thus obtain improved fixed point methods?

with two types of traffic

We shall examine a network consisting of *K* nodes with *K* links
which form a loop (see Figure 3). consist of all 1-link routes
(type-1 traffic), as well as all 2-link routes (type-2 traffic)
comprising pairs of adjacent links (2*K* routes in all). Type-*t*
traffic is offered at rate on all type-*t* routes, for , and for each link *j*.

Take links 1 and 2 as ``reference'' links. The subnetwork consisting
of just these two links has three routes: .
Let denote the number of calls on route *r*, for .
Then, is the number of calls occupying capacity on
link 1 *but not* on link 2; is the number
occupying capacity on link 2 *but not* on link 1; and is the number of calls occupying capacity on both links (see
Figure 5).

First examine the correlation between links 1 and 2, in particular,

If is the common loss probability for type-*t* calls,
for , then the EFP approximation is

where the Erlang Fixed Point *B* is the unique solution to

where is Erlang's Formula.

This follows the work of Pallant (1992). The network is decomposed into independent subnetworks and the stationary distribution for each is evaluated.

The state space for the subnetwork is

and the stationary distribution is

We estimate *B*, the probability that a
link adjacent to the two-link subnetwork is fully occupied, by

and use these expressions iteratively
to determine the ``correct'' value of , and hence *B*.

Our second approximation uses additional knowledge of the state of a
given link in estimating the probability that the adjacent link is full.
We use state-dependent arrival rates, ,
, where is the probability that link *K* is fully
occupied, conditional on ( is also the probability that
link 3 is fully occupied, conditional on ), so that

Once is estimated and determined, we set

Similarly, can be expressed in terms of .

An estimate of can be found by assuming (incorrectly) that does not depend on . For , set

where

The dependence of on is due to the cyclic nature
of the network, but is expected to be slight for large networks. The
approximation is *exact* for the infinite line network, as
Zachary (1985) shows for a network which is equivalent to the one
considered here, with (no single-link traffic). The
expression for given above is equivalent to that obtained in his
paper for the infinite line network, although written in a different
form.

State-dependent arrival rates such as we have here are also discussed by various authors (for example, Pallant and Taylor (1994)). Kelly (1985) gives an exact expression in a line network (including some asymmetric cases).