Fixed Point Methods for Loss Networks
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
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.
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:
(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?
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 (2K 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
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).