Averaging principle for second-order approximation of heterogeneous models with homogeneous models

Area of Science:

• Mathematical Modeling
• Applied Mathematics
• Theoretical Computer Science

Background:

• Analyzing heterogeneous models is challenging compared to homogeneous ones.
• Homogeneous models use an average value to represent heterogeneity.
• Simplifying complex models is crucial for broader applications.

Purpose of the Study:

• To demonstrate the equivalence between heterogeneous and homogeneous models.
• To establish a mathematical principle for model simplification.
• To apply this principle to diverse fields like queuing, game theory, and social networks.

Main Methods:

• Mathematical analysis of heterogeneous models.
• Establishing conditions for model equivalence (differentiability and symmetry).
• Deriving the error term O(ε(2)) for the approximation.

Main Results:

• Heterogeneous models satisfying differentiability and symmetry are O(ε(2)) equivalent to homogeneous models.
• The level of heterogeneity (ε) dictates the approximation error.
• New insights were gained in queuing theory, auctions, and social network marketing.

Conclusions:

• A general averaging principle simplifies the analysis of heterogeneous systems.
• This principle bridges theoretical complexity with practical applicability.
• The findings offer a powerful tool for researchers across multiple disciplines.