A loss network admits a call only when all its required resources have enough free capacity. Rejected calls do not queue. Under fixed routing and independent Poisson processes, its exact occupancy law has a product-form stationary distribution of a loss network.
Alternative routing lets a call try another resource path when its preferred route is blocked. Overflow onto longer paths can create feedback and multiple solutions of the Erlang fixed point approximation, even though an exact finite irreducible occupancy Markov chain has a unique stationary distribution.
A symmetric triangle with direct calls and two-link overflow routes has Erlang fixed point approximation equation . For and , this has at least three solutions, certified by the intermediate value theorem using signs at .
For capacity and offered traffic , the probability of a full single-resource loss system is . The stable recursion is , .
The Erlang fixed point approximation treats resource blocking as independent and applies the Erlang loss formula to traffic screened by other resources. Its self-consistency equations have a unique fixed point under fixed routing, but can have several under alternative routing.
For unit resource requirements in a loss network, the effective offered traffic to resource is , where are approximate resource blocking probabilities. The resource being modeled is excluded from its own screening product.
For calls arriving at rate with mean holding time , offered traffic is the dimensionless load , often measured in Erlangs. The Erlang loss formula distinguishes this offered traffic from the mean traffic actually carried.
Under fixed routing, each call type always requests the same collection of resources. The link-route incidence matrix records those requirements. This differs from alternative routing, in which a rejected direct request can try another route.
For a loss network with feasible occupancies , its stationary distribution is proportional to . It is the law of independent Poisson random variables conditioned on the resource constraints. With exponential holding times, detailed balance for a continuous-time Markov chain proves the formula directly.
For the standard fixed routing loss network with independent Poisson processes, the stationary occupancy law depends on independent holding-time distributions only through their means. The occupancy process alone need not be a Markov process when those holding times lack the memoryless property.
Articles by others on the same topic
A "Loss Network" generally refers to a type of network in telecommunications and network theory where packet loss occurs, often due to congestion or other adverse conditions. This can be in the context of data networks, where data packets may be dropped, leading to a loss of information. In such networks, performance analysis is crucial because packet loss can significantly affect the quality of service (QoS) and overall network reliability.