A New Method for Markovian Adaptation of the Non-Markovian Queueing System Using the Hidden Markov Model
Abstract
:1. Introduction
2. Analysis of the Current Solution Queueing Systems M/Er/1/∞ and Ek/Er/1/∞
- Customers do not arrive at Poisson’s arrival process—a group of “k” customers arrive at Poisson’s arrival process. For this reason, the ratio of the number of clients in the queueing systems is (3);
- Customers are not served by Erlang’s distribution—customers are served by the exponential distribution. A group of “k” customers are served by Erlang’s distribution.
3. Analytical Solution of the Queueing System M(λ)/Ek(μ)/1/∞
3.1. Elementary Case of Poison’s Birth and Erlang’s Death
3.2. The Probability of State of the Queueing System M(λ)/Ek(μ)/∞
3.3. Comparison of the System M(λ)/M(μe)/1/∞ and M(λ)/Ek(μk)/1/∞
4. Queueing System M(λ)/N(ω, σ)/1/∞
4.1. An Approximate Solution
4.2. Calibration of the Solution for a Normal Service Time
- The first Erlang system of order k′, i.e., queueing system M(λ)/E(k′, ω2/σ2)/1/∞. Let us denote its probability as si′;
- The second Erlang system of order k″, i.e., queueing system M(λ)/E(k″, ω2/σ2)/1/∞. Let us identify its probabilities with si″.
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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Tanackov, I.; Prentkovskis, O.; Jevtić, Ž.; Stojić, G.; Ercegovac, P. A New Method for Markovian Adaptation of the Non-Markovian Queueing System Using the Hidden Markov Model. Algorithms 2019, 12, 133. https://doi.org/10.3390/a12070133
Tanackov I, Prentkovskis O, Jevtić Ž, Stojić G, Ercegovac P. A New Method for Markovian Adaptation of the Non-Markovian Queueing System Using the Hidden Markov Model. Algorithms. 2019; 12(7):133. https://doi.org/10.3390/a12070133
Chicago/Turabian StyleTanackov, Ilija, Olegas Prentkovskis, Žarko Jevtić, Gordan Stojić, and Pamela Ercegovac. 2019. "A New Method for Markovian Adaptation of the Non-Markovian Queueing System Using the Hidden Markov Model" Algorithms 12, no. 7: 133. https://doi.org/10.3390/a12070133