Markov Chain-Based Models in Queuing Theory: An Operations Research Perspective
DOI:
https://doi.org/10.53573/rhimrj.2025.v12n7.006Keywords:
Markov Chains, Queuing Theory, Operations Research, M/M/1 Model, Jackson Networks, System Utilization, Waiting Time, Simulation, Capacity Planning, Stochastic ProcessesAbstract
Queuing theory is a fundamental discipline within operations research that focuses on analyzing systems characterized by congestion and waiting lines. Markov chain-based models offer a mathematically rigorous and probabilistically sound framework for modeling such systems, particularly those governed by random arrival and service processes. This paper examines key queuing configurations—including M/M/1, M/M/c, M/M/1/K, and Jackson Networks—using continuous-time Markov chains (CTMCs) to derive performance metrics such as system utilization, waiting time, queue length, and blocking probability. Analytical solutions were complemented by simulation models to validate findings under varied operational scenarios. Results reveal that Markovian models accurately predict system behavior under stable conditions and provide actionable insights for resource allocation and capacity planning. However, the study also identifies limitations in addressing non-exponential input distributions and complex system dynamics. Consequently, it advocates for hybrid approaches that combine Markov models with simulation and machine learning for enhanced scalability and adaptability. The findings reinforce the enduring relevance of Markov chain-based queuing models while pointing toward future directions for intelligent operations research.
References
Banks, J., Carson, J. S., Nelson, B. L., & Nicol, D. M. (2010). Discrete-event system simulation (5th ed.). Pearson.
Bolch, G., Greiner, S., de Meer, H., & Trivedi, K. S. (2006). Queueing networks and Markov chains: Modeling and performance evaluation with computer science applications (2nd ed.). Wiley-Interscience.
Erlang, A. K. (1909). The theory of probabilities and telephone conversations. Nyt Tidsskrift for Matematik, B (20), 33–39.
Green, L. V., Soares, J., Giglio, J. F., & Green, R. A. (2006). Using queuing theory to increase the effectiveness of emergency department provider staffing. Academic Emergency Medicine, 13(1), 61–68.
Gross, D., Shortle, J. F., Thompson, J. M., & Harris, C. M. (2018). Fundamentals of queueing theory (5th ed.). Wiley.
Harchol-Balter, M. (2013). Performance modeling and design of computer systems: Queueing theory in action. Cambridge University Press.
Jackson, J. R. (1957). Networks of waiting lines. Operations Research, 5(4), 518–521.
Jun, J. B., Jacobson, S. H., & Swisher, J. R. (1999). Application of discrete-event simulation in health care clinics: A survey. Journal of the Operational Research Society, 50(2), 109–123.
Kelly, F. P. (2011). Reversibility and stochastic networks. Cambridge University Press.
Kleinrock, L. (1975). Queueing systems, volume I: Theory. Wiley-Interscience.
Law, A. M. (2015). Simulation modeling and analysis (5th ed.). McGraw-Hill Education.
Neuts, M. F. (1981). Matrix-geometric solutions in stochastic models: An algorithmic approach. The Johns Hopkins University Press.
Puterman, M. L. (2005). Markov decision processes: Discrete stochastic dynamic programming. Wiley.
Wolff, R. W. (1989). Stochastic modeling and the theory of queues. Prentice Hall.
