@article {Longo20151540,
	title = {Two-layer symbolic representation for stochastic models with phase-type distributed events},
	journal = {International Journal of Systems Science},
	volume = {46},
	number = {9},
	year = {2015},
	note = {cited By 2},
	pages = {1540-1571},
	publisher = {Taylor and Francis Ltd.},
	abstract = {<p>Among the techniques that have been proposed for the analysis of non-Markovian models, the state space expansion approach showed great flexibility in terms of modelling capacities.The principal drawback is the explosion of the state space. This paper proposes a two-layer symbolic method for efficiently storing the expanded reachability graph of a non-Markovian model in the case in which continuous phase-type distributions are associated with the firing times of system events, and different memory policies are considered. At the lower layer, the reachability graph is symbolically represented in the form of a set of Kronecker matrices, while, at the higher layer, all the information needed to correctly manage event memory is stored in a multi-terminal multi-valued decision diagram. Such an information is collected by applying a symbolic algorithm, which is based on a couple of theorems. The efficiency of the proposed approach, in terms of memory occupation and execution time, is shown by applying it to a set of non-Markovian stochastic Petri nets and comparing it with a classical explicit expansion algorithm. Moreover, a comparison with a classical symbolic approach is performed whenever possible. {\textcopyright} 2013 Taylor \&amp; Francis.</p>
},
	keywords = {Decision diagram, Decision theory, efficient memory occupation, Many valued logics, Markov processes, Non-Markovian, Petri nets, phase type distributions, Stochastic models, Stochastic systems, symbolic representation},
	issn = {00207721},
	doi = {10.1080/00207721.2013.822940},
	url = {http://www.scopus.com/inward/record.url?eid=2-s2.0-84928644988\&partnerID=40\&md5=3f853e6fcb8bddeebf2e890076ff798d},
	author = {Francesco Longo and Marco Scarpa}
}
@proceedings {Distefano201045,
	title = {Symbolic representation techniques in dynamic reliability evaluation},
	journal = {Proceedings of the 12th IEEE International Symposium on High Assurance Systems Engineering (HASE)},
	year = {2010},
	note = {cited By 9; Conference of 2010 IEEE 12th International Symposium on High Assurance Systems Engineering, HASE 2010 ; Conference Date: 3 November 2010 Through 4 November 2010; Conference Code:83929},
	pages = {45-53},
	publisher = {IEEE Computer Society},
	address = {San Jose, CA, United States, 3-4 November 2010},
	abstract = {<p>The increasing demand of quality presses towards more specific requirements, tighter constraints, and higher standards. It is thus necessary to provide new paradigms, techniques, and tools to adequately model and evaluate complex systems. This paper mainly focuses on reliability aspects, also taking into account dynamic-dependent interactions among components. Starting from the conservation of reliability principle, we characterize the time to failure of the system components through continuous phase type distributions. The system reliability is thus modeled by an expanded Markov chain expressed in terms of Kronecker algebra in order to face the state space explosion and to represent the memory policies related to the aging process. A two-component system is taken as example to demonstrate the effectiveness of the technique and to validate it. {\textcopyright} 2010 IEEE.</p>
},
	keywords = {Aging process, Algebra, Complex systems, Continuous phase, Dynamic reliability, Kronecker algebra, Markov Chain, Markov processes, phase type distributions, Quality assurance, Reliability, State-space explosion, symbolic representation, System components, System reliability, Systems engineering, Time to failure, Two component systems},
	isbn = {9780769542928},
	issn = {15302059},
	doi = {10.1109/HASE.2010.28},
	url = {http://www.scopus.com/inward/record.url?eid=2-s2.0-79951924261\&partnerID=40\&md5=8ed2be2a9f6a7e9d891ed3fb58789d6f},
	author = {Salvatore Distefano and Francesco Longo and Marco Scarpa}
}
