A hybrid quantitative framework for performance evaluation of cyber-physical systems with explicit structural information

Cyber-Physical Systems (CPS) combine discrete control, continuous physical evolution, stochastic uncertainty, and resource constraints. Existing formal methods typically focus either on verification of global system behaviors or on compositional modeling of interacting processes, but provide limited support for quantitative evaluation explicitly connected to system structure. As a result, it is difficult to analyze how particular sub-systems, parallel branches, or component interactions affect overall system performance. To address this problem, this paper presents a hybrid quantitative framework consisting of Hybrid Quantitative Process Algebra (HyQPA) and Hybrid Structural Metric Language (HSML). HyQPA extends process algebra with continuous evolution described by ordinary differential equations, probabilistic branching, timing, and cost annotations in a compositional form. HSML is introduced to specify quantitative properties, including probability, response time, and resource consumption, directly over explicit process structure. On this basis, we define the operational semantics of HyQPA and develop recursive evaluation procedures for HSML formulas using finite symbolic regions. Two representative CPS case studies are used to illustrate the proposed framework within bounded-time finite symbolic abstractions. The results indicate that HyQPA can capture hybrid system behaviors involving continuous dynamics, discrete interactions, probabilistic branching, timing constraints, and cost annotations. HSML further enables component-specific and interaction-specific quantitative evaluation while retaining process-level structural information. These findings suggest the potential of the proposed framework for CPS scenarios that require both hybrid dynamical modeling and structural quantitative reasoning.