Propagation of chaos for a fully connected loss network with alternate routing (Q1208939)

Consider a fully connected network of \(n\) cities: \(a,b,\dots\). The \(N:=n(n-1)/2\) links between the cities are denoted by \(ab=ba\), \(ac=ca\) and so on. Each link has a capacity for \(C\) calls. Calls arrive independently on each link \(ab\) according to a Poisson process with rate \(v\) and occupy one of \(C\) channels if the capacity is not attained. If the link is full, then there is a protocol for alternate routing: A third city \(c\) is chosen at random uniformly among the \(n-2\) others; the call is routed through links \(ac\) and \(bc\) if both are not full, and is lost if either is full. Call durations follow independent exponential laws of rate 1, and alternately routed calls release both channels simultaneously. The process of the number of occupied channels between \(a\) and \(b\) is denoted by \(X^ N_{ab}\). The main purpose of the paper is to show that the law of \((X_{a_ 1b_ 1},\cdots,X_{a_ kb_ k})\) is asymptotically approached to an indpendent product law in weak convergence or in total variation. Note that the family \(\{X^ N_{ab}:a\neq b\}\) itself is not Markovian, the connection with the other city \(c\) has to be considered. The martingale formulation (may not be necessary) and some coupling argument are adopted.