Mark Bebbington, Massey University
Phil Pollett, The University of Queensland
Ilze Ziedins, The University of Auckland

A circuit-switched network

This consists of a group of nodes (locations),

K links (circuit groups),

tex2html_wrap_inline594 circuits comprising link j, and

tex2html_wrap_inline598 - a collection of routes.

Each route tex2html_wrap_inline600 is a set of links. Calls using route r are offered at rate tex2html_wrap_inline604 as a Poisson stream, and use tex2html_wrap_inline606 circuits from link j. tex2html_wrap_inline598 indexes independent Poisson processes.

Calls requesting route r are blocked and lost if, on any link j, there are fewer than tex2html_wrap_inline616 free circuits. Otherwise, the call is connected and simultaneously holds tex2html_wrap_inline616 circuits on each link j for the duration of the call. For simplicity, we shall take tex2html_wrap_inline622 .

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

The usual Markovian model

Let tex2html_wrap_inline624 , where tex2html_wrap_inline626 is the number of calls in progress using route r, let tex2html_wrap_inline630 , and let tex2html_wrap_inline632 .

The continuous-time Markov chain tex2html_wrap_inline634 takes values in


and has a unique stationary distribution given by


where tex2html_wrap_inline640 is given by


Blocking probabilities

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


where tex2html_wrap_inline648 is the unit vector from tex2html_wrap_inline650 describing just one call in progress on route r


An explicit expression for tex2html_wrap_inline656 !

However, the bad news is that tex2html_wrap_inline658 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 ( tex2html_wrap_inline660 , tex2html_wrap_inline662 ) and tex2html_wrap_inline664 . Clearly, tex2html_wrap_inline666 .

Erlang Fixed Point Approximation (EFP)

Theorem (Kelly (1986)) There is a unique vector tex2html_wrap_inline668 satisfying






tex2html_wrap_inline674 is Erlang's Formula for the loss probability on a single link with C circuits and Poisson traffic offered at rate tex2html_wrap_inline678 .

The celebrated Erlang Fixed Point Approximation is obtained by using tex2html_wrap_inline680 to estimate the probability that link j is full, and using tex2html_wrap_inline656 to estimate the route-r blocking probability.

The rationale

(Independent blocking)

If links along route r were blocked independently (they are clearly not) and if tex2html_wrap_inline680 were the link-j blocking probability, then tex2html_wrap_inline656 would be the route-r blocking probability:


Carrying this rationale further, the traffic offered to link j would be Poisson (at rate tex2html_wrap_inline702 , say) and the carried traffic (that which is accepted) on link j would be


The Erlang Fixed Point Approximation stipulates that the blocking probabilities tex2html_wrap_inline708 should be consistent with this level of carried traffic:


The Erlang Fixed Point

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 tex2html_wrap_inline714 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.

When does the EFP perform well?

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 tex2html_wrap_inline718


(Network topology fixed.) If tex2html_wrap_inline722 is the route-r loss probability, then, for each tex2html_wrap_inline600 , tex2html_wrap_inline728 , where


and tex2html_wrap_inline708 is the Erlang Fixed Point determined by tex2html_wrap_inline734 and tex2html_wrap_inline736 .

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




(Capacities fixed.) The traffic along link j is moderate ( tex2html_wrap_inline746 ), 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.

The symmetric star network

This consists of a collection of K outer nodes, which communicate via a single central node. Take tex2html_wrap_inline598 to be all those routes consisting of a pair of links (there are tex2html_wrap_inline752 of them). Fix tex2html_wrap_inline754 and take tex2html_wrap_inline756 for all tex2html_wrap_inline758 , and fix tex2html_wrap_inline664 for all links tex2html_wrap_inline762 .

Theorem If tex2html_wrap_inline764 is the common route loss probability, then tex2html_wrap_inline766 , given by


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


where tex2html_wrap_inline774 is Erlang's Formula.

When can the EFP be expected to perform badly?

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?

A symmetric ring network
with two types of traffic

We shall examine a network consisting of K nodes with K links which form a loop (see Figure 3). tex2html_wrap_inline598 consist of all 1-link routes (type-1 traffic), as well as all 2-link routes (type-2 traffic) comprising pairs of adjacent links (2K routes in all). Type-t traffic is offered at rate tex2html_wrap_inline786 on all type-t routes, for tex2html_wrap_inline790 , and tex2html_wrap_inline664 for each link j.

A two-link subnetwork

Take links 1 and 2 as ``reference'' links. The subnetwork consisting of just these two links has three routes: tex2html_wrap_inline802 . Let tex2html_wrap_inline804 denote the number of calls on route r, for tex2html_wrap_inline808 . Then, tex2html_wrap_inline810 is the number of calls occupying capacity on link 1 but not on link 2; tex2html_wrap_inline812 is the number occupying capacity on link 2 but not on link 1; and tex2html_wrap_inline814 is the number of calls occupying capacity on both links (see Figure 5).

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


The EFP approximation

If tex2html_wrap_inline818 is the common loss probability for type-t calls, for tex2html_wrap_inline790 , then the EFP approximation is


where the Erlang Fixed Point B is the unique solution to


where tex2html_wrap_inline774 is Erlang's Formula.

Approximation I

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 tex2html_wrap_inline844 , and hence B.

Approximation II

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, tex2html_wrap_inline848 , tex2html_wrap_inline850 , where tex2html_wrap_inline852 is the probability that link K is fully occupied, conditional on tex2html_wrap_inline856 ( tex2html_wrap_inline852 is also the probability that link 3 is fully occupied, conditional on tex2html_wrap_inline862 ), so that


Once tex2html_wrap_inline852 is estimated and tex2html_wrap_inline844 determined, we set


Similarly, tex2html_wrap_inline872 can be expressed in terms of tex2html_wrap_inline844 .

An estimate of tex2html_wrap_inline852 can be found by assuming (incorrectly) that tex2html_wrap_inline852 does not depend on tex2html_wrap_inline800 . For tex2html_wrap_inline882 , set




Some remarks on Approximation II

The dependence of tex2html_wrap_inline908 on tex2html_wrap_inline910 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 tex2html_wrap_inline912 (no single-link traffic). The expression for tex2html_wrap_inline852 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).