Systems of Differential Equations of Infinite Order with Small Parameter and Countable Markov Chains

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Tikhonov-type Cauchy problems are investigated for systems of ordinary differential equations of infinite order with a small parameter \(\mu \) and initial conditions. It is studying the singular perturbated systems of ordinary differential equations of infinite order of Tikhonov-type \(\mu \dot{x}=F(x(t,g_x),y(t,g_y),t)\), \(\dot{y}=f(x(t,g_x),y(t,g_y),t)\) with the initial conditions \(x(t_0)=g_x\), \(y(t_0)=g_y\), where \(x, \, g_x \, \in X\), \(X\subset l_1\) and \(y, \, g_y \, \in Y\), \(Y \, \in \mathbf {R}^n\), \(t \in \left[ t_0, t_1 \right] \) (\(t_0 0\) is a small real parameter. The results may be applied to the queueing networks, which arise from the modern telecommunications.