DETERMINATION OF RECURRENT ALGORITHMS AND OPTIMAL STRATEGIES FOR CONTROLLED MARKOV CHAINS

Аннотация

Article controlled Markov chains for recurrent algorithms and optimal strategies determination to the issues dedicated . Managed random processes , especially industrial​ of enterprises work activity planning such as practical in the fields application​ seeing The states of the system are Markov random . process with modeled , each one in case various to strategies suitable to pass probabilities and income analysis The goal is to​ one in step maximum average​ income optimal strategy that provides find . Recurring algorithms using expected income and optimal strategies consecutively is considered . From this except , asymptotic formulas optimal strategies using determination methods Example​ ​ as two stately and two strategic Markov process for calculations and tables presented is done , this optimal strategies through is determined . Article practical and theoretical in terms of controlled Markov processes study​ for important source isconsidered . Key Keywords: Supervised Markov chains , recurrent algorithms , optimal strategies , random processes ,transitions probabilities ,expected income,asymptoticformulas ,strategies package , Markov process ,industry planning . Introduction: Supervised Markov Chains random of processes important class are , they are various in sectors including industry​ enterprises planning , economic analysis , logistics decision acceptance to do and other many practical issues in solution wide is used in these processes . system Markov properties of states has to be , every one in case acceptance to be done strategies transition probabilities and expected income Determines optimal strategies . to determine , that is every one in step maximum average​ income providing decisions set to find is in the field main from issues This is one of them . article controlled Markov chains for recurrent algorithms and optimal strategies to find asymptotic methods learns . In the article theoretical approaches practical examples with strengthened , two stately and two strategic Markov process in the example of calculations and optimal strategies determination process This is brought . research controllable random processes deep understanding and them practical in https://ijmri.de/index.php/jmsi volume 4,issue3,2025 751 matters application​ for importantbasis creates . Literature analysis Supervised Markov chains and they for recurrent algorithms field many fundamental and​ practical of research topic become Markov processes​ foundations AA Markov formed in the works (1906) Although the concept of Controlled Markov Chains was first proposed in the mid- 20th century , in particular by R. Bellman (1957). dynamic programming methods within Bellman developed optimal strategies in his work " Dynamic Programming" find for recurrent equations system offer did this​ later controlled Markov processes in learning main from approaches to one H. Howard (1960) in his book “Dynamic Programming and Markov Processes” controlled Markov chains for iterative algorithms , in particular , value iteration and strategy iteration methods improved . His optimal strategies for work in determining practical calculations for important basis Later , D. Bertsekas (1995) in his book “ Dynamic Programming and Optimal Control” this algorithms modern optimization methods with together , wide extensive to applications adapted.Asymptotic formulas and their in controlled Markov processes application according to research and by J. Filar and K. Vrieze (1997) in their work “Competitive Markov Decision Processes” deep illuminated . Their work , especially long​ term average​ income maximize in matters asymptotic solutions to find aimed at was . With that together , E. Altman (1999) in his work “ Constrained Markov Decision Processes” limited conditions optimal strategies under to determine circle new approaches offer did. Local research point of view from the point of view of Uzbekistan scientists , for example , A. Ikramov and by T. Shirinov (2015) announcement made in articles controllable random of processes industry and economic in systems application studied . Their in their work local industry in enterprises optimal allocation of resources to the issues attention However , local​ ​ in literature asymptotic methods and recurrent algorithms deep analysis according to research relatively limited . Research methodology Research controlled Markov chains for recurrent algorithms and optimal strategies Methodology​ ​ ​ following from stages consists of: Theoretical analysis : Markov processes , optimal strategies and recurrent algorithms literature studied.Mathematics Modeling : System states Markov process with modeled , transition probabilities and income based on recurrent equations was compiled.Asymptotic Methods : Long term income maximize for asymptotic equations system solved.Practical calculations : Two situation and to the strategy Markov process​ in the example of transition probabilities and income matrices optimal strategies u