### Abstract

Using extension principle associated with a class of continuous Archimedean triangular norms, this paper studies a fuzzy random renewal process in which the interarrival times are assumed to be independent and identically distributed fuzzy random variables. Some limit theorems in chance measure and in expected value for the sum of fuzzy random variables are proved on the basis of the continuous Archimedean triangular norm based arithmetics. Furthermore, we discuss the fuzzy random renewal process based on the obtained limit theorems, and derive a fuzzy random elementary renewal theorem for the long-run expected renewal rate. The renewal theorem obtained in this paper can degenerate to the corresponding classical result in stochastic renewal process. Finally, two case studies of queueing systems are provided to illustrate the application of the fuzzy random elementary renewal theorem.

Original language | English |
---|---|

Pages (from-to) | 1232-1248 |

Number of pages | 17 |

Journal | Computers and Mathematics with Applications |

Volume | 57 |

Issue number | 7 |

DOIs | |

Publication status | Published - 2009 Apr |

### Fingerprint

### Keywords

- Archimedean triangular norm
- Elementary renewal theorem
- Extension principle
- Fuzzy random variables
- Interarrival times
- Queueing system
- Renewal process

### ASJC Scopus subject areas

- Computational Theory and Mathematics
- Modelling and Simulation
- Computational Mathematics

### Cite this

*Computers and Mathematics with Applications*,

*57*(7), 1232-1248. https://doi.org/10.1016/j.camwa.2009.01.030

**Fuzzy random renewal process with queueing applications.** / Wang, Shuming; Liu, Yan Kui; Watada, Junzo.

Research output: Contribution to journal › Article

*Computers and Mathematics with Applications*, vol. 57, no. 7, pp. 1232-1248. https://doi.org/10.1016/j.camwa.2009.01.030

}

TY - JOUR

T1 - Fuzzy random renewal process with queueing applications

AU - Wang, Shuming

AU - Liu, Yan Kui

AU - Watada, Junzo

PY - 2009/4

Y1 - 2009/4

N2 - Using extension principle associated with a class of continuous Archimedean triangular norms, this paper studies a fuzzy random renewal process in which the interarrival times are assumed to be independent and identically distributed fuzzy random variables. Some limit theorems in chance measure and in expected value for the sum of fuzzy random variables are proved on the basis of the continuous Archimedean triangular norm based arithmetics. Furthermore, we discuss the fuzzy random renewal process based on the obtained limit theorems, and derive a fuzzy random elementary renewal theorem for the long-run expected renewal rate. The renewal theorem obtained in this paper can degenerate to the corresponding classical result in stochastic renewal process. Finally, two case studies of queueing systems are provided to illustrate the application of the fuzzy random elementary renewal theorem.

AB - Using extension principle associated with a class of continuous Archimedean triangular norms, this paper studies a fuzzy random renewal process in which the interarrival times are assumed to be independent and identically distributed fuzzy random variables. Some limit theorems in chance measure and in expected value for the sum of fuzzy random variables are proved on the basis of the continuous Archimedean triangular norm based arithmetics. Furthermore, we discuss the fuzzy random renewal process based on the obtained limit theorems, and derive a fuzzy random elementary renewal theorem for the long-run expected renewal rate. The renewal theorem obtained in this paper can degenerate to the corresponding classical result in stochastic renewal process. Finally, two case studies of queueing systems are provided to illustrate the application of the fuzzy random elementary renewal theorem.

KW - Archimedean triangular norm

KW - Elementary renewal theorem

KW - Extension principle

KW - Fuzzy random variables

KW - Interarrival times

KW - Queueing system

KW - Renewal process

UR - http://www.scopus.com/inward/record.url?scp=61749094661&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=61749094661&partnerID=8YFLogxK

U2 - 10.1016/j.camwa.2009.01.030

DO - 10.1016/j.camwa.2009.01.030

M3 - Article

AN - SCOPUS:61749094661

VL - 57

SP - 1232

EP - 1248

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

SN - 0898-1221

IS - 7

ER -