[Paper Review] Optimal Rate Scheduling via Utility-Maximization for J-User MIMO Markov Fading Wireless Channels with Cooperation

Wanyang Dai|arXiv (Cornell University)|Jun 22, 2011

Advanced Wireless Network Optimization3 citations

TL;DR

This paper proposes a utility-maximizing dynamic rate scheduling policy for J-user MIMO wireless systems with Markov fading and cooperation, using a Nash equilibrium solution to optimize throughput under random channel conditions. It establishes asymptotic optimality via a reflecting diffusion with regime-switching (RDRS) model, proving the policy minimizes workload in heavy traffic and applies to both MIMO MAC and BC with convex, time-varying capacity regions.

ABSTRACT

We design a dynamic rate scheduling policy of Markov type via the solution (a social optimal Nash equilibrium point) to a utility-maximization problem over a randomly evolving capacity set for a class of generalized processor-sharing queues living in a random environment, whose job arrivals to each queue follow a doubly stochastic renewal process (DSRP). Both the random environment and the random arrival rate of each DSRP are driven by a finite state continuous time Markov chain (FS-CTMC). Whereas the scheduling policy optimizes in a greedy fashion with respect to each queue and environmental state and since the closed-form solution for the performance of such a queueing system under the policy is difficult to obtain, we establish a reflecting diffusion with regime-switching (RDRS) model for its measures of performance and justify its asymptotic optimality through deriving the stochastic fluid and diffusion limits for the corresponding system under heavy traffic and identifying a cost function related to the utility function, which is minimized through minimizing the workload process in the diffusion limit. More importantly, our queueing model includes both J-user multi-input multi-output (MIMO) multiple access channel (MAC) and broadcast channel (BC) with cooperation and admission control as special cases. In these wireless systems, data from the J users in the MAC or data to the J users in the BC is transmitted over a common channel that is fading according to the FS-CTMC. The J-user capacity region for the MAC or the BC is a set-valued stochastic process that switches with the FS-CTMC fading. In any particular channel state, we show that each of the J-user capacity regions is a convex set bounded by a number of linear or smooth curved facets. Therefore our queueing model can perfectly match the dynamics of these wireless systems.

Motivation & Objective

• To design an optimal dynamic rate scheduling policy for J-user MIMO wireless systems operating over Markov-modulated fading channels with user cooperation.
• To address the intractability of performance evaluation under greedy scheduling by developing fluid and diffusion limit models for heavy-traffic analysis.
• To unify the modeling of MIMO multiple access (MAC) and broadcast (BC) channels under a common queueing framework with random environment and doubly stochastic arrivals.
• To establish asymptotic optimality of the proposed policy by linking utility maximization to workload minimization in the diffusion limit.
• To prove that the J-user capacity region in each fading state is a convex set with linear or smooth curved facets, enabling tractable optimization.

Proposed method

• Formulates a generalized processor-sharing queueing model with job arrivals following a doubly stochastic renewal process (DSRP) driven by a finite-state continuous-time Markov chain (FS-CTMC).
• Solves a utility-maximization problem over the time-varying capacity region using Karush-Kuhn-Tucker (KKT) conditions to derive a social optimal Nash equilibrium point as the scheduling policy.
• Develops a stochastic fluid limit and a diffusion limit for the system under heavy traffic, leading to a reflecting diffusion with regime-switching (RDRS) process for workload.
• Identifies a cost function in the diffusion limit that corresponds to the original utility function and is minimized when workload is minimized, proving asymptotic optimality.
• Applies convex optimization, implicit function theorem, and duality theory to show that the J-user MIMO MAC and BC capacity regions are convex sets bounded by linear or smooth curved facets in each fading state.
• Uses cross-layer design to implement the DSRP by synchronizing user arrival rates with the FS-CTMC fading process.

Experimental results

Research questions

• RQ1 How can a dynamic rate scheduling policy be designed to maximize system utility in a J-user MIMO wireless system with Markov fading and cooperation?
• RQ2 What is the performance limit of such a scheduling policy under heavy traffic, and can it be characterized analytically despite intractable exact analysis?
• RQ3 How do the time-varying capacity regions of MIMO MAC and BC behave in each fading state, and are they convex?
• RQ4 Can a fluid and diffusion limit model be rigorously derived for the workload process under the proposed policy?
• RQ5 Is the proposed scheduling policy asymptotically optimal, and how is this linked to minimizing workload in the diffusion limit?

Key findings

• The proposed rate scheduling policy, derived as a social optimal Nash equilibrium via utility maximization, is asymptotically optimal under heavy traffic conditions.
• The fluid and diffusion limits of the workload process are characterized by a reflecting diffusion with regime-switching (RDRS), which captures the system dynamics under Markov-modulated fading.
• The cost function minimized in the diffusion limit corresponds directly to the original utility function, validating the policy's optimality.
• The J-user MIMO MAC and BC capacity regions are proven to be convex sets in each fading state, bounded by linear or smooth curved facets, enabling efficient optimization.
• The optimal solution to the utility maximization problem is continuous and differentiable with respect to the channel state, ensuring stable policy adaptation.
• The framework successfully unifies the analysis of MIMO MAC and BC under a single queueing model with doubly stochastic arrivals and random environment.