Asymptotic optimal disciplines in a model of a complex repairable system (Q1406366)

Let \(\varepsilon\lambda_k(e(t))\) be the failure of the \(k\)-th device refusal at the tmoment\(t\) and \(\epsilon\) be a small parameter. \(\,\lambda_k(e) = 0\,\) if \(\,e_k = 1\), \(\,G_k(x) = P\{\eta_k< x\}\,\) is the function of the repair time distribution for the \(k\)-th device, \(\,E_+ = \{e\: \phi(e) = 0\}\,\) is the set of the working orders. The author gives assertions of the following type. If \(\,\lambda_k(e)\), \(G_k(x)\), \(n\) and \(E_+\) are fixed and there exists the average \(\,\int\limits_0^\infty t\,dG_k(t)<\infty\), then \[ \lim\limits_{\varepsilon\to 0}\, P\bigg\{\frac{\tau(u)}{E\tau(u)} > x\bigg\} = e^{-x} \] uniformly with respect to the class of service disciplines.