**M.S. Bebbington,
P.K. Pollett and
I. Ziedins
**

** **

**Extended abstract**

This paper is concerned with the performance evaluation of loss
networks. For the simplest networks, there are explicit analytical
formulae for the important measures of performance, but for networks
which involve some level of dynamic control, such as routing schemes
which make use of spare capacity, exact analytical
methods have had only limited success. Under several regimes, the *
Erlang Fixed Point* (EFP) method provides a good approximation for the
blocking probabilities, but when these regimes are not operative, the
method can perform badly because the key assumption of independent
blocking may not hold. We propose several new models which specifically
account for the dependencies between neighbouring links. We derive
approximations for the blocking probabilities with errors typically
of that found using the EFP approximation. These are extended
to cases where admission controls, such as trunk reservation, are used.

The assumption of independence between links which drives the EFP
approximation holds under two limiting regimes. The first is one in
which the topology of the network is held fixed, while capacities and
arrival rates at links become large [4]; this has become known
as the *Kelly limiting regime*, or (somewhat misleadingly) as the
*heavy traffic limit*. Under the second limiting regime, called
*diverse routing*, the number of links, and the number of routes
which use these links, become large, while the capacities are held fixed
and the arrival rate on any single route becomes small. Examples of
this are star networks and fully-connected networks with alternate
routing [2, 3, 5, 7, 8]. If neither
regime is operative, the EFP method can perform
badly: in particular, in highly linear networks and/or networks
with low capacities. Further, adding controls to the network may destroy
independence under otherwise favourable regimes. A particularly
useful control is *trunk reservation*. Here, traffic streams are
assigned priorities and calls are accepted only if the occupancies of
links along their route are not above a given threshold, the size of
which depends on the type of call. This widely used control mechanism
is typically very robust to fluctuations in arrival rate and has the
added advantage of eliminating pathological behaviour such as
bistability [1]. With such a control operating in a network of
reasonable size, links may be highly dependent and the equilibrium
distribution will no longer have a product form, as it does for the
corresponding uncontrolled network. Modelling dependencies
in *this* context is thus critical.

We will focus attention on simple, highly linear networks, since here
the EFP approximation is expected to perform poorly. To illustrate our
methods, consider a loss network with *K* links forming a loop and each
link has the same capacity, *C*. There are two types of traffic: 1-link
routes (type-1 traffic) and 2-link routes comprising pairs of adjacent
links (type-2 traffic). Type-*t* traffic is offered at rate on
each type-*t* route. If is the common loss probability for
type-*t* calls, then the EFP approximation is

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

where is Erlang's Formula.

A significant improvement in accuracy can be obtained by adapting methods
of Pallant [6]. The network is decomposed into independent
subnetworks and the stationary distribution is evaluated for each.
Consider a two-link subnetwork consisting of say links 1 and 2. There
are three routes: , and . If denotes the
number of calls on route *r*, 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.
The state space for the subnetwork is
and the stationary distribution is

where *G* is a normalizing constant;
we estimate *B*, the probability that a
link adjacent to the two-link subnetwork is fully occupied, by

These expressions are used iteratively to determine a fixed point *B*
and then, as before, we set and .

Our second, and most accurate,
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 is found by assuming that does not depend on . For , we set

where

The dependence of on is due to the cyclic nature of the network, but is expected to be slight for large networks.

Full details of all fixed point methods will be described in the paper, including extensions to networks with trunk reservation. Detailed comparisons and error analysis will be provided for all methods.