When modelling and analysing business processes, the main emphasis is usually put on model validity and accuracy, i.e., the model meets the formal specification and also models the relevant system. In recent years, a series of metrics has begun to develop, which allows the quantification of the specific properties of process models. These characteristics are, for instance, complexity, comprehensibility, cohesion, and uncertainty. This work is focused on defining a method that allows us to measure the uncertainty of a process model, which was modelled by using stochastic Petri nets (SPN). The principle of this method consists of mapping of all reachable marking of SPN into the continuous-time Markov chain and then calculating its stationary probabilities. The uncertainty is then measured as the entropy of the Markov chain (it is possible to calculate the uncertainty of the specific subset of places as well as of whole net). Alternatively, the uncertainty index is quantified as a percentage of the calculated entropy against maximum entropy (the resulting value is normalized to the interval <0,1>). The calculated entropy can also be used as a measure of the model complexity.

This paper deals with the analysis of employment data of the 2008 economic crises. The analyses are done by using entropy measures that can help with predicting regional employment dynamics. Our finding suggests that the Shannon entropy and Tsallis entropy are significant predictors for the size of the employment downturn. The Rényi entropy is also a useful predictor of the rate of employment downturn in recession phase. When the Shannon entropy was growing through the recovery phase before the crisis, regions experience a higher rate of employment decrease in the following recession period and high Shannon entropy inferred the smaller employment downturn. The results indicate the different role of the Tsallis entropy that plays a different role in comparison to Shannon entropy. The higher the Tsallis entropy, the more severely the region was affected. In conclusion, the use of entropic measures as resilience indicators in terms of regional policy is a significant predictor of regional resilience.

In modelling and analysis of concurrent systems is usually placed the main emphasis on their accuracy and validity, i.e. the model meets the formal specification and also modelled the relevant facts. Moreover, in recent years in various fields, the importance of the concept of fairness starts to develop, which covers such requirements on system/model, such as uniformity of resource usage, fair queuing policy, the problem of starvation and many others. The aim of this work is to define a method of calculating the uniformity of the distribution of specific system states that can be used to investigate the uniformity of the workload of individual entities / elements of the system. The method is based on modelling by classical Place/Transition Petri nets and measuring the entropy of subsets of specific markings in the reachability graph of the Petri net. The presented method is an alternative view on the fairness of the already defined fairness which in the theory of Petri nets deals with transitions.

Complexity analysis of dynamic systems provides a better understanding of the internal behaviours that are associated with tension and efficiency, which in the socio-technical systems may lead to innovation. One of the popular approaches for the assessment of complexity is associated with self-similarity. The dynamic component of dynamic systems represents the relationships and interactions among the inner elements (and its surroundings) and fully describes its behaviour. The approach used in this work addresses complexity analysis in terms of system behaviour, i.e., the so-called behavioural analysis of complexity. The self-similarity of a system (structural or behavioural) can be determined, for example, using fractal geometry, whose toolbox provides a number of methods for the measurement of the so-called fractal dimension. Other instruments for measuring the self-similarity in a system, include the Hurst exponent and the framework of complex system theory in general. The approach introduced in this work defines the complexity analysis in a social-technical system under tension. The proposed procedure consists of modelling the key dynamic components of a discrete event dynamic system by any definition of Petri nets. From the stationary probabilities, one can then decide whether the system is self-similar using the abovementioned tools. In addition, the proposed approach allows for finding the critical values (phase transitions) of the analysed systems.

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Abstract. In recent years, a series of metrics began to develop that allow the quantification of specific properties of process models. These characteristics are, for example, complexity, comprehensibility, maintainability, cohesion and uncertainty. This work is focused on defining a method that allows to measure the uncertainty of process models that was modelled by Stochastic Petri Nets (SPN). The principle of this method consists in mapping the set of all reachable marking of SPN into the continuous-time Markov chain and then calculating its steady-state probabilities. The uncertainty is then measured as the Shannon entropy of the Markov chain (it is possible to calculate the uncertainty of the specific subset of places as well as whole Petri net). Alternatively, the uncertainty is quantified as a percentage of the calculated entropy against maximum entropy.

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