Any event that results in sudden change of the normal functioning of a system may be thought of as a catastrophe. Stochastic processes involving catastrophes have very rich application in modeling of a dynamic population in areas of ecology, marketing, queueing theory, etc. When the size of the population reduces abruptly as a whole, due to a catastrophe, it is termed as the total catastrophe. However, in many real-life circumstances the catastrophes have a mild influence on the population and have a sequential effect on the individuals. This paper presents a discrete-time catastrophic model in which the catastrophes occur according to renewal process, and it eliminates each individual of the population in sequential order with probability p until the one individual survives or the entire population wipes out. The individuals arrive according to the discrete-time Markovian arrival process. Using the supplementary variable technique, we obtain the steady-state vector generating function (VGF) of the population size at various epochs. Further using the inversion method of VGF, the population size distribution is expressed in terms of the roots of the associated characteristic equation. We further give a detailed computational procedure by considering inter-catastrophe time distributions as discrete phase-type as well as arbitrary. Finally, a few numerical results in form of tables and graphs are presented. Moreover, the impact of the correlation of arrival process on the mean population size is also investigated.

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具有离散马尔可夫到达过程的更新生成的几何突变模型

导致系统正常功能突然改变的任何事件都可以视为灾难。涉及灾难的随机过程在生态，市场营销，排队论等领域的动态种群建模中具有非常广泛的应用。当种群的大小由于灾难而突然整体减少时，则称为总灾难。但是，在许多现实生活中，灾难对人口的影响不大，对个人有序贯的影响。本文提出了一个离散时间灾难性模型，其中灾难根据更新过程发生，并且它以概率p连续消除人口的每个个体。直到一个人幸存下来或整个人口全部消灭。个体根据离散时间马尔可夫到达过程到达。使用补充变量技术，我们获得了各个时期人口规模的稳态向量生成函数（VGF）。进一步使用VGF的反演方法，种群大小分布用相关特征方程的根表示。通过考虑突变间时间分布是离散相类型还是任意相，我们进一步给出了详细的计算过程。最后，以表格和图表的形式给出了一些数值结果。此外，还研究了到达过程的相关性对平均人口规模的影响。