## Section: New Results

### Stability Properties of Loss Networks

Participants : Nelson Antunes, Christine Fricker, Philippe Robert, Danielle Tibi.

A new class of stochastic networks has been introduced and analyzed. Their dynamics combine the key characteristics of the two main classes of queueing networks : loss networks and Jackson type networks.

1. Each node of the network has finite capacity so that a request entering a saturated node is rejected as in a loss network.

2. Requests visit a subset of nodes along some (possibly) random route as in Jackson or Kelly's networks.

This class of networks is motivated by the mathematical representation of cellular wireless networks. Such a network is a group of base stations covering some geographical area. The area where mobile users communicate with a base station is referred to as a cell . A base station is responsible for the bandwidth management concerning mobiles in its cell. New calls are initiated in cells and calls are handed over (transfered) to the corresponding neighboring cell when mobiles move through the network. A new or a handoff call is accepted if there is available bandwidth in the cell, otherwise, it is rejected.

Figure 1. The time evolution of these networks has been analyzed by considering two limiting regimes

• Heavy traffic limits.

The arrival rates and capacities at nodes are proportional to some factor Nwhich gets large.

• Thermodynamic limits.

The number of nodes of the network goes to infinity.

The time evolution of the network can be (roughly) described as follows. A stochastic process associated with the state of the network for the parameter Nis introduced : is the vector describing the number of requests of different classes at the nodes of the network. As Ngoes to infinity, it is proved that converges to some function ( x( t)) , satisfying the deterministic equation (1)

The equilibrium points of the limiting process are contained in the set of solutions xof the equation F( x) = 0 .

It is shown in Antunes et al.    that for the heavy traffic limit, there is a unique equilibrium point. The proof uses a dual method approach to study the fixed point equations together with some convenient inequalities.

For the thermodynamic limit, it is shown in Antunes et al.    that there are situations where several equilibrium points coexist. This result has practical important implications for communication networks : It implies that, in some cases, the network will stay a long time in a set of states where a class of calls will be rejected and after this long time, it will switch to a set of states where this class of calls has a higher acceptance rate and, again after a long time, it switches back to the first set of states and so on. At the mathematical level, this is the situation where the function Fhas at least two stable points and a saddle point. The proof uses an interesting correspondence between two energy functions defined in different state spaces.

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