Delayed Renewal Process with Uncertain Random Inter-Arrival Times

Xiaoli Wang,Yuhong Sheng,Gang Shi

doi:10.3390/sym13101943

Abstract

An uncertain random variable is a tool used to research indeterminacy quantities involving randomness and uncertainty. The concepts of an ’uncertain random process’ and an ’uncertain random renewal process’ have been proposed in order to model the evolution of an uncertain random phenomena. This paper designs a new uncertain random process, called the uncertain random delayed renewal process. It is a special type of uncertain random renewal process, in which the first arrival interval is different from the subsequent arrival interval. We discuss the chance distribution of the uncertain random delayed renewal process. Furthermore, an uncertain random delay renewal theorem is derived, and the chance distribution limit of long-term expected renewal rate of the uncertain random delay renewal system is proved. Then its average uncertain random delay renewal rate is obtained, and it is proved that it is convergent in the chance distribution. Finally, we provide several examples to illustrate the consistency with the existing conclusions.

Highlights

The delayed renewal process is a variation of the normal renewal process, which allows the first arrival time to be different from other processes
Under the uncertainty theory framework, Liu [4] proposed a definition of uncertain process, and an uncertain renewal process was proposed to simulate the sudden jump in uncertain systems
The concept of the uncertain random delay renewal process is proposed, regarding the arrival interval as uncertain random variables, and allowing the chance distribution of the first arrival interval to be different from other times

Summary

Introduction

The delayed renewal process is a variation of the normal renewal process, which allows the first arrival time to be different from other processes. There are both uncertainty and randomness in complex systems To describe such a system, Liu [8] founded the chance theory and proposed chance measure, defined uncertain random variables, gave their chance distribution, and defined the expected value and variance of uncertain random variables. The concept of the uncertain random delay renewal process is proposed, regarding the arrival interval as uncertain random variables, and allowing the chance distribution of the first arrival interval to be different from other times.

Uncertain Random Variable

Uncertain Random Delayed Renewal Process

Elementary Uncertain Random Delayed Renewal Theorem

Conclusions