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Special Digraph Models
Published in Jonathan L. Gross, Jay Yellen, Mark Anderson, Graph Theory and Its Applications, 2018
Jonathan L. Gross, Jay Yellen, Mark Anderson
definition: A (discrete-time) Markov chain is a phenomenon whose behavior can be modeled by a sequence {Xt}, t = 0, 1, 2,…, such that the (one-step) transition probability pij that Xt+1 = sj, given that Xt = si, does not depend on any earlier terms in the sequence {Xt} and does not depend on t. Thus, pij is the conditional probability given by pij=prob(Xt+1=sj|Xt=si),fort=1,2,…
Queueing Theory
Published in Erchin Serpedin, Thomas Chen, Dinesh Rajan, Mathematical Foundations for SIGNAL PROCESSING, COMMUNICATIONS, AND NETWORKING, 2012
A discrete-time Markov chain can be characterized by a (one-step) transition probability matrix () P=[p00p01p02⋯p10p11p12⋯::::]
Markov Processes
Published in Michael W. Carter, Camille C. Price, Ghaith Rabadi, Operations Research, 2018
Michael W. Carter, Camille C. Price, Ghaith Rabadi
A Markov analysis has been used to identify the steady-state production capability of manufacturing systems. For specified reliability and maintainability characteristics, the model tracks failures in a multi-unit manufacturing system. Certain assumptions are necessary. First, although the manufacturing system is a continuous-time process, in this case, it is assumed that it can be accurately modeled as a discrete-time Markov chain if the time increments are small enough. Second, the issue of stationarity must be addressed. It is assumed that the probability that a functioning element in the system becomes non-operational in a given interval is independent of the length of time it has been functioning.
Application of a novel approach of production system modelling, analysis and improvement for small and medium-sized manufacturers: a case study
Published in International Journal of Production Research, 2023
The commonly used mathematical models for characterising such random behaviour of production operations include the Bernoulli reliability model, geometric reliability model, and exponential reliability model. Under the Bernoulli reliability model, the machine status (up or down) is modelled as Bernoulli random variables, while the geometric reliability model formulates the up- and downtime of a machine as geometric random variables. Production system models with Bernoulli and/or geometric reliability machines are characterised by discrete-time Markov chains. Similarly, the exponential reliability model formulates the up- and downtime of a machine as exponential random variables and production system models with exponential reliability machines are characterised by continuous-time Markov chains.
Developing capabilities for supply chain resilience in a post-COVID world: A machine learning-based thematic analysis
Published in IISE Transactions, 2023
Dun Li, Bangdong Zhi, Tobias Schoenherr, Xiaojun Wang
Bilevel optimization is a special optimization where one problem is embedded (nested) within another, and is also frequently used to design supply chain networks (e.g., Ghomi-Avili et al., 2021). Although the models mentioned above have been widely used in the network design literature, other models have been emerging with which supply chain networks under disruption risks can be designed, such as Discrete-Time Markov chains and Dynamic Bayesian networks (Hosseini et al., 2020).
Documenting occupant models for building performance simulation: a state-of-the-art
Published in Journal of Building Performance Simulation, 2022
Marika Vellei, Elie Azar, Karol Bandurski, Christiane Berger, Salvatore Carlucci, Bing Dong, Matteo Favero, Ardeshir Mahdavi, Marcel Schweiker
Regarding the Modelling Formalism, occupant behaviour is often modelled by assuming it is a stochastic process. Following the definition of Coleman (1974), a stochastic process can be defined as ‘a system which evolves in time while undergoing chance fluctuations’, which means that there is a certain probability of getting a certain outcome for each observation at a specific time. Therefore, to implement occupant behaviour in BPS, two fundamental aspects need to be considered: (i) the probability of a certain outcome; and (ii) the evolution of this probability over time. For modelling the probability of a certain outcome, we can use analytical or statistical modelling approaches, which include generalized linear models, i.e. a broad class of models where the response relates to the linear predictor through a link function (denoted by ). This is not to be confused with general linear models, which refers to conventional linear regression models. However, if the ‘identify link’ () is selected, general linear models can be viewed as a particular case of generalized linear models. For modelling the evolution of the probability over time, we identify the following three main approaches: Bernoulli process is used to model the probability of having a certain state (or event) independently from the previous state (or event).A discrete-time Markov-chain technique is used to model the probability of changing state or event (i.e. transitions probabilities) depending on the previous state (i.e. the conditions just before the occupants undertake the action).Survival model is used to model the time until a certain state or event occurs.