MMAPs to Model Complex Multi-State Systems with Vacation Policies in the Repair Facility


  • Juan Eloy Ruiz-Castro Department of Statistics and Operations Research and Mathematics Institute IMAG. University of Granada, Granada, Spain
  • Christian Acal Department of Statistics and Operations Research and Mathematics Institute IMAG. University of Granada, Granada, Spain



Phase-type distribution (PH), Marked Markovian arrival process (MMAP), vacation policy, preventive maintenance


Two complex multi-state systems subject to multiple events are built in an algorithmic and computational way by considering phase-type distributions and Markovian arrival processes with marked arrivals. The internal performance of the system is composed of different degradation levels and internal repairable and non-repairable failures can occur. Also, the system is subject to external shocks that may provoke repairable or non-repairable failure. A multiple vacation policy is introduced in the system for the repairperson. Preventive maintenance is included in the system to improve the behaviour. Two types of task may be performed by the repairperson; corrective repair and preventive maintenance. The systems are modelled, the transient and stationary distributions are built and different performance measures are calculated in a matrix-algorithmic form. Cost and rewards are included in the model in a vector matrix way. Several economic measures are worked out and the net reward per unit of time is used to optimize the system. A numerical example shows that the system can be optimized according to the existence of preventive maintenance and the distribution of vacation time. The results have been implemented computationally with Matlab and R (packages: expm, optim).


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Author Biographies

Juan Eloy Ruiz-Castro, Department of Statistics and Operations Research and Mathematics Institute IMAG. University of Granada, Granada, Spain

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-Markov models, Phase Type distributions and Markovian arrival processes using matrix analysis methods. An interesting aspect of his research is the construction of theoretical models in medicine to analyse 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, Department of Statistics and Operations Research and Mathematics Institute IMAG. University of Granada, Granada, Spain

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 Modelling 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.


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How to Cite

Ruiz-Castro, J. E. ., & Acal, C. . (2022). MMAPs to Model Complex Multi-State Systems with Vacation Policies in the Repair Facility. Journal of Reliability and Statistical Studies, 15(02), 473–504.



Reliability and Stochastic Processes