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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.

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