Limit theorems for nonsymmetric transportation networks. (Russian. English summary) Zbl 1099.60509

A model of an asymmetric transportation network with \(N\) nodes and \(n\) regions described by a Markov process \(U_N(t)\) is considered, here \(n\) is fixed and \(N\) goes to the infinity. This process has values in a compact subset of the finite-dimensional real vector space \({\mathbb R}^{\alpha}\). It is proved that for any finite interval of time the evolution satisfies the large number law, and the system of nonlinear differential equations for the limit dynamics is provided. It is proved that the corresponding Cauchy problem has a unique solution \(u(t,g)\). Under natural conditions on the convergence of initial distribution, the convergence in distribution of \(U_N(t)\) to the dynamical system \(g\to u(t,g)\) is obtained. The author shows that the dynamic system has a unique invariant measure to which the invariant measures of the processes \(U_N(t)\) converges as \(N\to \infty\).


60J05 Discrete-time Markov processes on general state spaces