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Perturbed renewal equations with applications to M/M queuing systems. I. (English) Zbl 0955.60080
Theory Probab. Math. Stat. 60, 35-42 (2000) and Teor. Jmovirn. Mat. Stat. 60, 31-37 (1999).
The author considers the renewal equation \[ x_{\varepsilon}(t)=q_{}(t)+\int_0^t x_{\varepsilon}(t-s) F_{\varepsilon}(ds),\quad t\geq 0, \tag{1} \] where \(q_{\varepsilon}(s)\) is a measurable real-valued function on \([0,\infty)\) bounded on every finite interval, \(F_{\varepsilon}(t)\) is a distribution function on \([0,\infty)\) which is not concentrated at \(0\), but can be improper, i. e. \(F_{\varepsilon}(t)\leq 1.\) The author studies the exponential asymptotical relation of the following type \[ {x_{\varepsilon}(t_{\varepsilon}) \over \exp\{ p_{\varepsilon}t_{\varepsilon}\}} \to x_0(\infty) \quad \text{as}\quad \varepsilon\to 0. \] For example, he finds conditions under which there exists a unique nonnegative solution of equation (1) for all \(\varepsilon\) that are small enough. He proves that for any \(0\leq t_{\varepsilon}\to\infty\) in such a way that \(\varepsilon^{z} \exp\{ -{a/\varepsilon}\} t_{\varepsilon} \to \lambda_{z} \in [0,\infty)\) for some \(k\leq z \leq w\) the following asymptotical relation holds true \[ {x_{\varepsilon}(t_{\varepsilon}) \over \exp\{ -(b_{k}\varepsilon^{k}+\cdots + b_{z-1}\varepsilon^{z-1})e^{-a/ \varepsilon} t_{\varepsilon}\} }\to e^{-\lambda_{z}b_{z}} x_0(\infty) \quad \text{as} \varepsilon \to 0. \] Here \(b_{k}\) are some constants.

60K05 Renewal theory
60K15 Markov renewal processes, semi-Markov processes
60K25 Queueing theory (aspects of probability theory)