以階段型機率分佈表示異質生成衝擊系統

Abstract

考慮一個衝擊系統，它的衝擊依據異質生成過程而產生。這個系統有兩\n種類型的損壞。類型一的損壞可以被修理消除。類型二的損壞可以被不定\n期置換消除。假設兩個連續衝擊之間的時間間隔服從階段型分佈。例如，\n在一個特殊的階段型分佈—亞指數分佈—之下，我們發現穩定機率存在的\n條件。在這個模型下探討年齡置換策略，我們導出置換週期內的期望成本\n率。為了找到最小化期望成本率的最佳定期置換年齡，我們提供一個有效\n率的演算法並開發一個 MATLAB 工具來實現。一系列數值範例促使我們發\n現新的定理，它比以前的定理更簡單，更實際，更直觀。該定理表明最佳定期置換年齡的存在性。

We consider a system subject to shocks which occur according to a non-homogeneous pure birth process. The system has two types of failures. Type-I failure can be removed by a repair. Type-II failure can be removed by an unplanned replacement. We assume that the inter-arrival time between consecutive shocks follows phase-type distributions. For example, under a special PH-distribution that is a hypo-exponential distribution, we find the conditions of the existence of stationary probability. Under this model we investigate the age replacement policy. We derive the expected cost rate of a replacement cycle. To find the optimal planned replacement age that minimizes the expected cost rate, we give an efficient algorithm and develop a MALAB tool for implementation. A series of numerical examples motivate us to write a new theorem. That is simpler, more practical, and more intuitive than a previous theorem. This theorem shows the existence of the optimal planned replacement age.