For many military and civilian large-scale, real-world systems of interest, data are first acquired asynchronously, i. e. at irregular intervals of time, at geographically-dispersed sites, processed utilizing decision-making algorithms, and the processed data then disseminated to other appropriate sites. The term real-world refers to systems under computer control that relate to everyday life and are beneficial to the society in the large. The traditional approach to such problems consists of designing a central entity which collects all data, executes a decision making algorithm sequentially to yield the decisions, and propagates the decisions to the respective sites. Centralized decision making algorithms are slow and highly vulnerable to natural and artificial catastrophes. Recent literature includes successful asynchronous, distributed, decision making algorithm designs wherein the local decision making at every site replaces the centralized decision making to achieve faster response, higher reliability, and greater accuracy of the decisions. Two key issues include the lack of an approach to synthesize asynchronous, distributed, decision making algorithms, for any given problem, and the absence of a comparative analysis of the quality of their decisions. This paper proposes MFAD, a Mathematical Framework for Asynchronous, Distributed Systems, that permits the description of centralized decision-making algorithms and facilities the synthesis of distributed decision-making algorithms. MFAD is based on the Kohn-Nerode distributed hybrid control paradigm. It has been a belief that since the centralized control gathers every necessary data from all entities in the system and utilizes them to compute the decisions, the decisions may be "globally" optimal. In truth, however, as the frequency of the sensor data increases and the environment gets larger, dynamic, and more complex, the decisions are called into question. In the distributed decision-making system, the centralized decision-making is replaced by those of the constituent entities that aim at minimizing a Lagrangian, i. e. a local, non-negative cost criterion, subject to the constraints imposed by the global goal. Thus, computations are carried out locally, utilizing locally obtained dataand appropriate information that is propagated from other sites. It is hypothesized that with each entity engaged in optimizing its individual behavior, asynchronously, concurrently, and independent of other entities, the distributed system will approach "global" optimal behavior. While it does not claim that such algorithms may be synthesized for all centralized real-world systems, this paper implements both the centralized and distributed paradigms for a representative military battlefield command, control, and communication (C3) problem. It also simulates them on a testbed of a network of workstations for a comparative performance evaluation of the centralized and decentralized paradigms in the MFAD framework. While the performance results indicate that the decentralized approach consistently outperforms the centralized scheme, this paper aims at developing a quantitative evaluation of the quality of decisions under the decentralized paradigm. To achieve this goal, it introduces a fundamental concept, embodied through a hypothetical entity termed "Perfect Global Optimization Device (PGOD)," that generates perfect or ideal decisions. PGOD possesses perfect knowledge, i. e. the exact state information of every entity of the entire system, at all times, unaffected by delay. PGOD utilizes the same decision-making algorithm as the centralized paradigm and generates perfect globally-optimal decisions which, though unattainable, provide a fundamental and absolute basis for comparing the quality of decisions. Simulation results reveal that the quality of decisions in the decentralized paradigm are superior to those of the centralized approach and that they approach PGOD's decisions.