Continuity Analysis of the Minimum Reachable Access Bandwidth in the Theory of Particle Access

To address persistent limitations in quality of service (QoS) mechanisms for large-scale networks, we previously proposed the idea and theory of particle access. In this theory, end-to-end traffic flows are modeled as discrete information particles, each characterized by attributes such as deadline and bearing capacity. The minimum reachable access bandwidth, a key performance metric directly determined by these attributes, represents the lowest bandwidth satisfying the delay constraints of the information particle group, with its inflection point marking the turning point in the minimum access bandwidth requirement. Earlier results demonstrated that increasing a particle’s deadline or decreasing its bearing capacity yields a non-increasing minimum reachable access bandwidth. Building on this foundation, we rigorously prove that slight variations in the deadlines of multiple particles within a group lead to continuous, rather than abrupt, changes in both the minimum reachable access bandwidth and its inflection-point time, with such changes negligible when deadline variations are slight. Furthermore, when deadline variations are bounded, the resulting bandwidth changes remain proportionally bounded. These continuity properties ensure that, even under deadline fluctuations, both the minimum reachable access bandwidth and its inflection point remain controllable, thereby supporting the reliable and scalable application of particle access theory in large-scale networks.