{"title":"Joint Probability Functions for Scenarios Arising From Multi-State Series and Multi-State Parallel Systems","authors":"Leena Kulkarni, Sanjeev Sabnis, Sujit K. Ghosh","doi":"10.1002/asmb.2922","DOIUrl":null,"url":null,"abstract":"<div>\n \n <p>Consider multi-state series and multi-state parallel systems consisting of <span></span><math>\n <semantics>\n <mrow>\n <mi>N</mi>\n </mrow>\n <annotation>$$ N $$</annotation>\n </semantics></math> independent components each. It is assumed that (i) each component and both the systems take values in the set <span></span><math>\n <semantics>\n <mrow>\n <mo>{</mo>\n <mn>0</mn>\n <mo>,</mo>\n <mn>1</mn>\n <mo>,</mo>\n <mn>2</mn>\n <mo>}</mo>\n </mrow>\n <annotation>$$ \\left\\{0,1,2\\right\\} $$</annotation>\n </semantics></math>, (ii) each system and each component start out in state 2, the perfect state, and they make the transition to state 1, depending upon system configuration, and, eventually, each system enters state 0, the failed state. This multi-state nature of components and systems leads to <span></span><math>\n <semantics>\n <mrow>\n <mi>N</mi>\n </mrow>\n <annotation>$$ N $$</annotation>\n </semantics></math> scenarios under which each of the systems makes the transition from state 2 to state 1, and eventually to state 0. The joint probability function for times spent in state 2 and state 1 is obtained based on these <span></span><math>\n <semantics>\n <mrow>\n <mi>N</mi>\n </mrow>\n <annotation>$$ N $$</annotation>\n </semantics></math> scenarios for each of the systems. It is interesting to note that by merely changing set <span></span><math>\n <semantics>\n <mrow>\n <mo>{</mo>\n <mn>0</mn>\n <mo>,</mo>\n <mn>1</mn>\n <mo>}</mo>\n </mrow>\n <annotation>$$ \\left\\{0,1\\right\\} $$</annotation>\n </semantics></math> of a standard binary series (parallel) system to a set <span></span><math>\n <semantics>\n <mrow>\n <mo>{</mo>\n <mn>0</mn>\n <mo>,</mo>\n <mn>1</mn>\n <mo>,</mo>\n <mn>2</mn>\n <mo>}</mo>\n </mrow>\n <annotation>$$ \\left\\{0,1,2\\right\\} $$</annotation>\n </semantics></math> of a multi-state series (multi-state parallel) system, renders expression of the joint probability function of system spending times in state 2 and state 1 of a multi-state series (multi-state parallel) system is quite complex as compared to the univariate survival probability of the binary series (parallel) system being in the functioning state. As a proof of concept, graphical comparison between these analytical joint probability functions and joint empirical probability functions for each of the multi-state series and multi-state parallel systems based on the Farlie-Gumbel-Morgernsten distribution is made, and it is found that they compare very well. The graphical comparison between corresponding univariate cumulative distribution functions is also found to be very good. The theoretical results mentioned above for multi-state series and multi-state parallel systems have been partially generalized for multi-state systems made up of <span></span><math>\n <semantics>\n <mrow>\n <mi>k</mi>\n </mrow>\n <annotation>$$ k $$</annotation>\n </semantics></math> subsystems that are themselves either multi-state series or multi-state parallel systems. Finally, an attempt has been made to demonstrate the result for the multi-state series system with a real-life scenario involving head and neck cancer data using copula models such as Gumbel, Clayton, FGM, and Frank.</p>\n </div>","PeriodicalId":55495,"journal":{"name":"Applied Stochastic Models in Business and Industry","volume":"41 1","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2025-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Stochastic Models in Business and Industry","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/asmb.2922","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 0
Abstract
Consider multi-state series and multi-state parallel systems consisting of independent components each. It is assumed that (i) each component and both the systems take values in the set , (ii) each system and each component start out in state 2, the perfect state, and they make the transition to state 1, depending upon system configuration, and, eventually, each system enters state 0, the failed state. This multi-state nature of components and systems leads to scenarios under which each of the systems makes the transition from state 2 to state 1, and eventually to state 0. The joint probability function for times spent in state 2 and state 1 is obtained based on these scenarios for each of the systems. It is interesting to note that by merely changing set of a standard binary series (parallel) system to a set of a multi-state series (multi-state parallel) system, renders expression of the joint probability function of system spending times in state 2 and state 1 of a multi-state series (multi-state parallel) system is quite complex as compared to the univariate survival probability of the binary series (parallel) system being in the functioning state. As a proof of concept, graphical comparison between these analytical joint probability functions and joint empirical probability functions for each of the multi-state series and multi-state parallel systems based on the Farlie-Gumbel-Morgernsten distribution is made, and it is found that they compare very well. The graphical comparison between corresponding univariate cumulative distribution functions is also found to be very good. The theoretical results mentioned above for multi-state series and multi-state parallel systems have been partially generalized for multi-state systems made up of subsystems that are themselves either multi-state series or multi-state parallel systems. Finally, an attempt has been made to demonstrate the result for the multi-state series system with a real-life scenario involving head and neck cancer data using copula models such as Gumbel, Clayton, FGM, and Frank.
期刊介绍：
ASMBI - Applied Stochastic Models in Business and Industry (formerly Applied Stochastic Models and Data Analysis) was first published in 1985, publishing contributions in the interface between stochastic modelling, data analysis and their applications in business, finance, insurance, management and production. In 2007 ASMBI became the official journal of the International Society for Business and Industrial Statistics (www.isbis.org). The main objective is to publish papers, both technical and practical, presenting new results which solve real-life problems or have great potential in doing so. Mathematical rigour, innovative stochastic modelling and sound applications are the key ingredients of papers to be published, after a very selective review process.
The journal is very open to new ideas, like Data Science and Big Data stemming from problems in business and industry or uncertainty quantification in engineering, as well as more traditional ones, like reliability, quality control, design of experiments, managerial processes, supply chains and inventories, insurance, econometrics, financial modelling (provided the papers are related to real problems). The journal is interested also in papers addressing the effects of business and industrial decisions on the environment, healthcare, social life. State-of-the art computational methods are very welcome as well, when combined with sound applications and innovative models.
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