Generalized phase-type distributions based on multi-state systems

Abstract

It is a well-known definition that the time until entering an absorbing state in a finite state Markov process follows a phase-type distribution. In this article we extend this distribution through adding two new events: one is that the number of transitions among states reaches a specified threshold; the other is that the sojourn time in a specified subset of states exceeds a given threshold. The system fails when it enters the absorbing state or two new events happen, whichever occurs first. We develop three models in terms of three circumstances: (i) the two thresholds are constants; (ii) the number of transitions is random while the sojourn time in the specified states is constant; and (iii) the sojourn time in the specified states is random while the number of transitions is constant. To the performance of such systems, we employ the theory of aggregated stochastic processes and obtain closed-form expressions for all reliability indexes, such as point-wise availabilities, various interval availabilities, and distributions of lifetimes. We select special distributions of two thresholds for models 2 and 3, which are the exponential and geometric distributions. The corresponding formulas are presented. Finally, some numerical examples are given to demonstrate the proposed formulas.

Keywords:

• Phase-type distribution
• aggregated stochastic process
• reliability index
• availability
• lifetime

Acknowledgments

The authors thank the anonymous referees for their valuable suggestions that significantly improved this paper.

Additional information

Funding

This work was supported by the NSF of China under grant 71631001.

Notes on contributors

Bei Wu

Bei Wu received a B.S. degree in mathematics from Beijing Institute of Technology, Beijing, China, in 2012. She is currently working toward a Ph.D. degree in management science and engineering in the School of Management and Economics, Beijing Institute of Technology. Her research interests include system reliability, stochastic modeling and applications of probability.

Lirong Cui

Lirong Cui is a professor in the School of Management & Economics at Beijing Institute of Technology. He received his Ph.D. degree in probability and statistics from the University of Wales, UK, in 1994. He has worked on quality and reliability-related problems since 1986 and published more than 100 papers and technical reports. In 2000, he co-authored a book on reliability published by Kluwer Academic Publishers. He served as an associate editor of IEEE Transactions on Reliability from 2005 to 2015. He currently serves as an associate editor for Quality Technology & Quantitative Management and Communications in Statistics: Theory and Methods, Simulation and Computation. In 2005, he was awarded the New Century Excellent Talents in China. His recent research interests are in stochastic modeling, Hawkes processes, quality and reliability engineering, simulation and optimization, operations research, and applications of probability and statistics in various fields.

Chen Fang

Chen Fang is a lecturer in the School of Management & Economics at Zhejiang Ocean University. She received her master’s degree in transportation planning and management from Dalian Maritime University, Dalian, China, in 2011. She is currently working toward a Ph.D. degree in management at the School of Management & Economics, Beijing Institute of Technology, Beijing, China. Her research interests are in system reliability and applications of probability.