Loss networks

In a loss network calls, or customers, of various types are accepted for service provided that this can commence immediately; otherwise they are rejected. An accepted call remains in the network for some holding time, which is generally independent of the state of the network, and throughout this time requires capacity simultaneously from various network resources. The loss model was first introduced by Erlang as a model for the behaviour of just a single telephone link (see Brockmeyer et al. (1948). The typical example remains that of a communications network, in which the resources correspond to the links in the network, and a call of any type requires, for the duration of its holding time, a fixed allocation of capacity from each link over which it is routed (Kelly, 1986). This is the case for a traditional circuit-switched telephone network, but the model is also appropriate to modern computer communications networks which support streaming applications with minimum bandwidth requirements (Kelly et al., 2000). There are also other examples: for instance, in a cellular mobile network similar capacity constraints arise from the need to avoid interference (Abdalla and Boucherie, 2002). The mathematics of such networks has been widely studied, with interest in both equilibrium and, more recently, dynamical behaviour. Of particular importance are questions of call acceptance and capacity allocation (for example, routing), with the aim of ensuring good network performance which is additionally robust with respect to variations in network parameters. Call arrival rates, in particular, may fluctuate greatly. An excellent review of the state-of-the-art at the time of its publication is given by Kelly (1991)—see also the many papers cited therein, and the later survey by Ross (1995). We take as our model the following. Let R denote the finite set of possible call, or customer, types. Calls of each type r ∈ R arrive at the network as a Poisson process with rate νr, and each such call, if accepted by the network (see below), remains in it for a holding time which is exponentially distributed with mean μ−1 r . We shall discuss later the extent to which these assumptions, in particular the latter, are necessary. Calls which are rejected do not retry and are simply considered lost. All arrival processes and holding times are independent of one another. We denote the state of the network at time t by n(t) = (nr(t), r ∈ R), where nr(t) is the number of calls of each type r in progress at