PUBLIC SERVICE SYSTEMS: PROBABILITIES AND OPTIMIZATION

Аннотация

Queueing theory is a field of probabilistic systems that studies random demand flows and service processes. This theory was developed based on A. Erlang’s research on telephone exchanges and is applied in various fields such as telecommunication, commerce, transportation, maintenance, and others. The main elements of the system include: incoming flow, queue, service node, and outgoing flow. The incoming flow represents the process of random demand arrivals, while the queue reflects the order in which services are awaited. The service node may consist of one or multiple channels, and the outgoing flow represents the served demands. The Poisson process is the most common type of incoming flow, characterized by stationarity, ordinariness, and lack of aftereffects. Service timeoften follows an exponential distribution. The efficiency of the system is assessed using indicators such as the utilization coefficient, waiting time in queue, time spent in the system by requests, and both relative and absolute throughput. Queueing systems can be open or closed, with or without waiting, and single-channel or multi-channel. The theory plays an important role in managing queues, optimizing resources, and improvingservice qualityin practice. Keywords:Queueing system, demand, arrival flow, queue, service node, departure flow, probabilistic system, stationary flow, ordinary flow, Poisson process, service time, transition probability, utilization coefficient, average number ofrequests, average waiting timeinqueue, average timea request spends in thesystem, relative throughput, absolutethroughput. Introduction. Operations research and optimal control are fields of study focused on decision-making and system optimization. 1. Operations research uses mathematical models, linear programming, game theory, andnetwork optimization methods toefficiently allocate resources. 2. Optimal control deals with determining the best control strategies for systems. Its main methods include Pontryagin's Maximum Principle and Bellman's dynamic programming. Literature Review:Queuing Theory page1120 INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE ISSN:2692-5206,ImpactFactor:12,23 AmericanAcademicpublishers,volume05,issue04,2025 Journal: https://www.academicpublishers.org/journals/index.php/ijai A literature review on Queuing Theory is essential for understanding the fundamental principles, historical development, and practical applications of this field. Below, key literary sources and their contents are briefly analyzed, along with an overview of the general directions in current research. A.K.Erlang’s Works Danish mathematician Agner Krarup Erlang (1878–1929) is considered the founder of queuing theory. His work on modeling traffic in telephone exchanges, particularly the 1917 paper “Solution of Some Problems in the Theory of Probabilities of Significance in Automatic Telephone Exchanges”, significantly contributed to the field. Erlang's studies developed methods for determining system congestion and queue length based on models using Poisson arrivals and exponential service distributions. These works laid the foundation for classical queuing system models such as M/M/1 and M/M/c, and they remain relevant today in telecommunications and network analysis. Analysis: Erlang’s contributions are fundamental in shaping the theoretical basis of queuing systems. However, his research primarily focused on telephone networks, and thus requires extension for application to modern complex systems. L. Kleinrock’s “Queueing Systems”(1975–1976) Leonard Kleinrock’s two-volumework “Queueing Systems” plays acrucial role in the modern development of queuing theory. The first volume (Theory) covers the mathematical foundations of queuing systems, including Poisson processes, Markov chains, and service time distributions. The second volume (Computer Applications) focuses on the application of the theory in computer networks and information systems. Kleinrock's work popularized important analytical tools such as Little’s Law and Kendall’s Notation. Analysis: Kleinrock’s book provides a balanced approach between theory and practical application, greatly influencing the field of computer networks. However, its mathematical rigor may be challenging for beginners. D. Gross and C.M. Harris – “Fundamentals of Queueing Theory” (1974, later editions up to 2013) This book is considered one of the most important introductory texts on queuing theory. The authors explain models such as M/M/1, M/M/c, M/G/1, and also cover the statistical characteristics of queuing systems, simulation methods, and practical examples. Later editions of the book expand on modern issues, including the analysis of network systems and call centers. Analysis: The book is reader-friendly and enriched with examples and exercises. However, it mainly focuses on classical models and does not deeply explore complex dynamicsystems. Research Methodology Theresearch methodology in queuing systems focuses on anal