论文标题
交互式随机环境中的出生和死亡过程
Birth and Death Processes in Interactive Random Environments
论文作者
论文摘要
本文研究了交互式随机环境中的出生和死亡过程，在这些环境环境中，生日和死亡率和环境状态的动态相互依赖。考虑了两个随机环境的模型：连续时间马尔可夫链（有限或无数的无限）和反射（跳跃）扩散过程。背景是由带有特定交互作用机制的Markov工艺确定的，其结构与产品形式相似。我们讨论了许多排队和人口增长模型，并建立了可以得出上述不变度度量的条件。 接下来，对正在考虑的模型进行了对平稳性的收敛速率的分析。我们考虑两个设置导致指数或多项式收敛率。在这两种情况下，我们都假设基本的环境马尔可夫过程具有指数的收敛速率，但是关节马尔可夫过程的收敛速率取决于某些关于出生和死亡率的条件。为了证明这些结果，耦合方法证明是有用的。
This paper studies birth and death processes in interactive random environments where the birth and death rates and the dynamics of the state of the environment are dependent on each other. Two models of a random environment are considered: a continuous-time Markov chain (finite or countably infinite) and a reflected (jump) diffusion process. The background is determined by a joint Markov process carrying a specific interactive mechanism, with an explicit invariant measure whose structure is similar to a product form. We discuss a number of queueing and population-growth models and establish conditions under which the above-mentioned invariant measure can be derived. Next, an analysis of the rate of convergence to stationarity is performed for the models under consideration. We consider two settings leading to either an exponential or a polynomial convergence rate. In both cases we assume that the underlying environmental Markov process has an exponential rate of convergence, but the convergence rate of the joint Markov process is determined by certain conditions on the birth and death rates. To prove these results a coupling method turns out to be useful.
