#### Cite this article as:

Kocheganov V. M. Markov Chain States Classification in a Tandem Model with a Cyclic Service Algorithm with Prolongation. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2020, vol. 20, iss. 2, pp. 257-265. DOI: https://doi.org/10.18500/1816-9791-2020-20-2-257-265

# Markov Chain States Classification in a Tandem Model with a Cyclic Service Algorithm with Prolongation

There is a limited list of papers about crossroads tandems. Usually the following service algorithms are under consideration: a cyclic algorithm with fixed duration, a cyclic algorithm with a loop a cyclic algorithm with regime changes etc. To construct a formal mathematical model of queuing systems nets and crossroads tandems in particular a descriptive approach is usually used. Using this approach input flows and service algorithms are set at the level of content, service duration distribution is known and set via a particular customer service distribution function. However with this approach one can not find nodes output flows distribution, as well as investigate customers’ noninstanteneous transfering between systems and with dependent, different service time distributions. In this paper a new approach is utilized to construct probability models of tandems for conflict queuing systems with different service algorithms in subsystems. Within this approach one can solve a problem of choosing the description for ω elementary outcomes of the stochastic experiment and mathematically correctly define the stochastic process, which describes the entire system, as well as solve the above mentioned problems. Based on a constructively given probabilistic space one can strictly justify the reachability of one state from another the other which in turn gives a full description of the entire essential state space.

- Haight F. A. Mathematical Theories of Traffic Flow. New York, Academic, 1963. 241 p.
- Inose H., Hamada T. Road Traffic Control. Tokyo, Univ. of Tokyo Press, 1975. 331 p.
- Drew D. R. Traffic Stream Theory and Control. New York, McGraw-Hill, 1968. 467 p.
- Fedotkin M. A. On a class of stable algorithms for control of conflicting flows or arriving airplanes. Problems of Control and Information Theory, 1977, vol. 6, no. 1, pp. 17–27.
- Fedotkin M. A. Construction of a model and investigation of nonlinear algorithms for control of intense conflict flows in a system with variable structure of servicing demands. I. Lithuanian mathematical journal, 1977, vol. 7, no. 1, pp. 129–137. DOI: https://doi.org/10.1007/BF00968503
- Litvak N. V., Fedotkin M. A. A probabilistic model for the adaptive control of conflict flows. Automation and Remote Control, 2000, vol. 61, no. 5, pp. 777–784.
- Proidakova E. V., Fedotkin M. A. Control of output flows in the system with cyclic servicing and readjustments. Automation and Remote Control, 2008, vol. 69, no. 6, pp. 993– 1002.
- Yamada K., Lam T. N. Simulation analysis of two adjacent traffic signals. Proceedings of the 17th Winter Simulation Conference. New York, ACM, 1985, pp. 454–464.
- Kocheganov V. M., Zorine A. V. Sufficient condition of low-priority queue stationary distribution existence in a tandem of queuing systems. Vestnik Volzhskoi gosudarstvennoiakademii vodnogo transporta [Bulletin of the Volga State Academy of Water Transport], 2017, vol. 50, pp. 47–55 (in Russian).
- Kocheganov V., Zorine A. V. Sufficient condition for primary queues stationary distribution existence in a tandem of queuing systems. Vestnik TVGU. Ser. Prikl. Matem. [Herald of Tver State University. Ser. Appl. Math.], 2018, no. 2, pp. 49–74 (in Russian). DOI: https://doi.org/10.26456/vtpmk193

- 60 reads