Abstract
Stochastic processes describe the dynamic behavior of systems modeled by formalisms such as queuing networks, network planning models, semi-Markov processes, and some other. Flowgraph models provide an analytical approach to the problem of stochastic process duration distribution law (SPDDL) finding. The problem is solved in two stages. Initially, the moment generating function (MGF) of the process duration is to be obtained using the graphical evaluation and review technique (GERT) or the flowgraph algebra. This stage is straightforward in contrast to the second one—the analytical transition from the MGF to the process duration distribution law in terms of probability distribution function (cumulative or non-cumulative). The transition is nontrivial and is implemented in this study using MATLAB Symbolic Math Toolbox along with various examples of finding SPDDLs when the processes are represented as ordered activity sets. Also, the capabilities of the statistical flowgraph methodology can be extended over the case when flowgraphs have parallel branches. The results of symbolic calculations are validated via simulation using GPSS World software. This study opens up real possibilities of replacing simulation with symbolic mathematics when searching for duration distribution laws of stochastic processes spawned by flowgraph models.
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Appendix
Appendix
Results for the Uniform Distribution. The Padé approximant orders: a = 6, b = 7. Expectation and variance are presented in Table 4. For comparative charts, see Fig. 11. As a result of the goodness-of-fit test, we failed to reject H0 hypothesis with p = 0.99.
Results for the Triangular Distribution. The Padé approximant orders: a = 6, b = 7. Moment comparison is given in Table 5. For charts of PDFs and CDFs, see Fig. 12. Here, we failed to reject the H0 hypothesis with p = 0.10.
Results for the Erlang-3 Distribution. The Padé approximant orders: a = 6, b = 7. Moment comparison is presented in Table 6. For comparative charts of PDFs and CDFs, see Fig. 13. We failed to reject the H0 hypothesis with p = 0.16.
Results for Concurrent Activities Distributed According to the Uniform Law. Let each of the parallel arcs of activity “34” from Sect. 3.2 be uniformly distributed with the same parameters, and all other distributions of activity durations are also uniform. When simulating in the GPSS World, 5000 iterations were carried out for greater accuracy, and we took a = 12, b = 13 for the Padé approximation.
As a result, we obtained correct values of expectation and variance (see Table 7), and Fig. 14 shows the analytical PDF and CDF compared with the empirical ones constructed from the results of simulation. We failed to reject the H0 hypothesis (p = 0.64).
Analytical approach versus simulation for the uniform distribution
Analytical approach versus simulation for the triangular distribution
Analytical approach versus simulation for the Erlang-3 distribution
Results for Concurrent Activities Distributed According to the Exponential Law. The Padé approximant orders: a = 12, b = 13. Moment comparison is presented in Table 8. For comparative charts of PDFs and CDFs, see Figs. 13 and 15. We failed to reject the H0 hypothesis with p = 0.53.
Analytical approach versus simulation for the uniform distribution (two concurrent twin activities)
Analytical approach versus simulation for the exponential distribution (two concurrent twin activities)
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Zhemelev, G., Sidnev, A. (2021). Using Symbolic Computing to Find Stochastic Process Duration Distribution Laws. In: Voinov, N., Schreck, T., Khan, S. (eds) Proceedings of International Scientific Conference on Telecommunications, Computing and Control. Smart Innovation, Systems and Technologies, vol 220. Springer, Singapore. https://doi.org/10.1007/978-981-33-6632-9_7
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