An approach to non-homogenous phase-type distributions through multiple cut-points

Abstract

A new class of distributions based on phase-type distributions is introduced in the current paper to model lifetime data in the field of reliability analysis. This one is the natural extension of the distribution proposed by Acal et al. (One cut-point phase-type distributions in reliability. An application to resistive random access memories. Mathematics 9(21):2734, 2021) for more than one cut-point. Multiple interesting measures such as density function, hazard rate or moments, among others, were worked out both for the continuous and discrete case. Besides, a new EM-algorithm is provided to estimate the parameters by maximum likelihood. The results have been implemented computationally in R and simulation studies reveal that this new distribution reduces the number of parameters to be estimated in the optimization process and, in addition, it improves the fitting accuracy in comparison with the classical phase-type distributions, especially in heavy tailed distributions. An application is presented in the context of resistive memories with a new set of electron devices for nonvolatile memory circuits. In particular, the voltage associated with the resistive switching processes that control the internal behavior of resistive memories has been modeled with this new distribution to shed light on the physical mechanisms behind the operation of these memories.

KEYWORDS:

• Markov processes
• phase-type distributions
• multiple cut-points
• modeling
• EM-algorithm
• simulation
• resistive random access memory (RRAM)

Additional information

Funding

This paper is partially supported by the project FQM-307 of the Government of Andalusia (Spain) and by the project PID2020-113961GB-I00 of the Spanish Ministry of Science and Innovation (also supported by the FEDER program). Authors also acknowledge the financial support of the Consejería de Conocimiento, Investigación y Universidad, Junta de Andalucía10.13039/501100018943 (Spain) and the FEDER program for projects A-TIC-117-UGR18, A-FQM-66-UGR20 and B-TIC-624-UGR20. Additionally, the authors would like to express their gratitude for financial support by the IMAG–María de Maeztu grant CEX2020-001105-M/AEI/10.13039/501100011033.

Notes on contributors

Juan Eloy Ruiz-Castro

Juan Eloy Ruiz Castro is a Full Professor in the Department of Statistics and Operations Research at the University of Granada (Spain). He works mainly, since 1993, on survival and reliability analysis considering Markovian and semi-Markovian models, Phase Type distributions and Markovian arrival processes using matrix analytic methods. An interesting aspect of his research is the construction of theoretical models in medicine to analyze the behavior of various diseases. For example, he has applied these models to study the evolution of breast cancer subject to multiple treatments. His research in the field of reliability focuses on the analysis of repairable systems with and without loss of units. As a result of his research, he has more than thirty publications in a large number of high-impact scientific journals and has been invited to participate in multiple conferences to show his contributions. His editorial activity is extensive, and he belongs to the editorial board of several journals recognized as prestigious in the JCR. Currently, he is interested in incorporating phase distributions and Makovian arrival processes in various fields, such as electronics and physics, to analyze the behavior of complex devices.

Christian Acal

Christian Acal is Substitute Teaching Tutor in the Department of Statistics and Operation Research at University of Granada (Spain). He received his International Ph.D. Degree in Mathematical and Applied Statistics from University of Granada in 2021. His areas of interest is focused on the Stochastic Modeling and Forecasting of high dimension data. In particular, his main research line is the Functional Data Analysis and its applications in different areas of knowledge, although he also works in Survival and Reliability Analysis. He has participated in numerous national e international congresses, many of them as invited, and he has multiple publications in high-impact scientific journals indexed in the Journal Citation Reports. Currently, he belongs to the work team of several research projects awarded by the Spanish Ministry of Science and Innovation and by the Government of Andalusia (Spain) and before, he was director of a project for young researchers promoted by the University of Granada. Finally, he is researcher attached to the Institute of Mathematics at University of Granada, which has the Seal of National Excellence ‘María de Maeztu’ and he is also member of the Spanish Society of Statistics and Operation Research.

Juan B. Roldán

Juan B Roldán received a degree in physics and a Ph.D. degree from the University of Granada, Granada, Spain, in 1993 and 1997, respectively. His current research interests include physical simulation and compact modeling of resistive switching memories. He is currently a Full Professor with the Universidad de Granada.