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Stability of multi-dimensional Markov chains with monotone drifts
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In this work, we investigate the stability of a class of n-dimensional discrete-time Markov chains with state space Nn and monotonic drift conditions. While exact stability criteria are well understood for one- and two-dimensional cases, extending these results to higher dimensions remains an open problem. To address this, we present an analytical approach to establish upper and lower bounds for the stability region for n ≥ 2. A key feature of the considered Markov chain is the monotonicity of the drift vector, which not only reduces the analytical complexity but also reflects realistic dynamics observed in practical applications, such as communication networks. Our analysis uses the Foster–Lyapunov criterion and generalizes the existing stability results to higher-dimensional settings. To illustrate the bounds obtained, we also provide numerical examples. 

Keywords [en]
Markov chains, multi-dimensional Markov chains, Foster theorem, Lyapunov function, dependent rates, ergodicity, quarter plane, stability analysis
National Category
Probability Theory and Statistics
Research subject
Applied Mathematics and Statistics
Identifiers
URN: urn:nbn:se:uu:diva-571401OAI: oai:DiVA.org:uu-571401DiVA, id: diva2:2013006
Available from: 2025-11-11 Created: 2025-11-11 Last updated: 2025-11-13
In thesis
1. Modelling and Performance of Cellular Networks: Stochastic Geometry, Queuing, and Learning Approaches
Open this publication in new window or tab >>Modelling and Performance of Cellular Networks: Stochastic Geometry, Queuing, and Learning Approaches
2025 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis is based on seven papers concerning mathematical models for wireless cellular networks with retransmissions, buffering, and interference. The analysis combines stochastic geometry with queuing theory to capture complex stochastic aspects of the physical model. Paper I introduces a downlink model with transmitter buffers, providing performance measures such as coverage probability, delay, and loss probability. Paper II extends the modeling approach to quantify Shannon capacity under finite and infinite buffer regimes. Paper III studies multi-tier networks, extending the previous approach. The paper introduces biased load balancing and discusses the increase in capacity compared with single-tier systems. Pa-per IV derives a stability condition for buffered uplink traffic, for a special case of no noise and unbounded attenuation. The paper further refines the analytical stability bound through simulations. Paper V considers the network with heterogeneous users with different arrival rates and powers, and establishes user-specific stability bounds. Paper VI uses the well-known Foster criteria for two-dimensional Markov chains and extends them to derive both stability and transience criteria for Markov chains in higher dimensions with monotone drifts. Finally, Paper VII studies a model of a buffered cellular network in terms of reinforcement learning (RL) methodology. It introduces a decentralized mean-field RL method, where base stations act as agents who aim to maximize their channel capacity via dynamically adjusting the transmission intensity.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2025. p. 64
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 2615
Keywords
Cellular networks, performance evaluation, stochastic geometry, stochastic modelling, Shannon capacity, coverage probability, Markov chains, reinforcement learning.
National Category
Communication Systems Mathematical sciences
Research subject
Applied Mathematics and Statistics
Identifiers
urn:nbn:se:uu:diva-571533 (URN)978-91-513-2675-7 (ISBN)
Public defence
2026-01-14, Polhemsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 17:29 (English)
Opponent
Supervisors
Available from: 2025-12-18 Created: 2025-11-13 Last updated: 2025-12-18

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Language
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