Channel-Based Multi-Queue SBC Process Algebra

• The paper introduces a unified metamodel that coalesces structural and behavioral system views using channel-based multi-queue process algebra.
• It formalizes system interactions with a labeled transition system integrating states, communications through channels, and multi-queue scheduling.
• The approach improves model consistency and verification in safety-critical, distributed systems and transactional applications.

The Channel-Based Multi-Queue Structure-Behavior Coalescence Process Algebra (C-M-SBC-PA) constitutes a formal metamodel framework for Model-Based Systems Engineering (MBSE), enabling the unified representation of both structural and behavioral facets of complex systems. Developed to address the shortcomings in traditional SysML metamodels—particularly their inability to provide an integrated semantic foundation for disparate user model diagrams—C-M-SBC-PA ensures that views such as state machines, activity flows, and internal block diagrams can all be formally projected from a single, channel- and queue-aware process algebraic structure (Chao, 2021). Its methodological innovations also connect with broader developments in process algebraic modeling of concurrency, coordination, and multi-queue systems.

1. Motivation and Theoretical Basis

C-M-SBC-PA was introduced to resolve a fundamental deficiency in prevailing MBSE frameworks, particularly SysML, where the lack of a unified semantic basis results in inconsistencies and fragmentation among model views (Chao, 2021). Traditional approaches (OCL, fUML, Alf) either focus on structure or behavior but fail to integrate them at the metamodeling level. C-M-SBC-PA leverages the Structure-Behavior Coalescence (SBC) paradigm, which posits that “Systems Architecture = Systems Structure + Systems Behavior,” formalizing this integration via a process algebra capable of representing both system components (structural states) and their dynamical interactions (behavioral transitions) in a single, coalesced artifact (Chao, 2021).

2. Formal Elements and Key Constructs

At the core is the SBC Interaction Transition Graph (ITG), a labeled transition system that systematically incorporates:

• States ($Y$): Each capturing the behavioral signature of a system component or process.
• Transitions: Triggered by interactions specified over communication channels.
• Channels and Signatures ($A$, $O$, $K \subseteq A \times O$): Defining the named mechanisms and parameter lists for process communication.
• Entity Sets (see Table I in (Chao, 2021)): Including actors ($B$), blocks ($\Gamma$), interaction sets ($G$, $V$, $\Delta$), states ($\Psi$), etc.

The formal expression for the ITG is:

$ITG = (Y, s_0, E, A, O, \Gamma, ITGR)$

where $ITGR$ is the set of interaction-driven transition tuples, e.g., $(s_i, ch, p, b, s_j) \in ITGR$ representing a transition from $s_i$ to $s_j$ via channel $ch$ with parameters $p$ and block $b$.

Channels are value-passing and may be defined as:

$<\text{channel name}>(<\text{parameter list}>)$

with signature relation $K \subseteq A \times O$.

3. Channel-Based Multi-Queue Modeling

C-M-SBC-PA extends the original SBC Process Algebra (SBC-PA), which primarily models interactions over single or centralized channels, to accommodate multiple concurrent, asynchronously scheduled interaction queues. In this extension:

• Each channel or group of channels can maintain its own queue $Q_i$, enabling fine-grained modeling of asynchronous or prioritized communication flows.
• Multi-queue architectures natively support heterogeneous concurrency patterns and resource sharing, capturing real-world deployment constraints better than single-queue abstractions (Chao, 2021).
• Behavioral prefixes in process expressions are enhanced to:

$PX_i = (C_i, \Delta_i, \Pi_i, R_i, Q_i)$

where $C_i$ is a set of guard conditions, $\Delta_i$ a set of interactions, $\Pi_i$ optional code snippets, $R_i$ the continuation relation, and $Q_i$ the dedicated queue.

This approach enables precise modeling of queueing behavior, interaction order, buffering, and scheduling—features central in distributed transaction processing, embedded systems, and asynchronous communication architectures.

4. Semantic Integration and Projection

The metamodel serves as the formal semantic backbone of SysML within MBSE, facilitating the algorithmic projection of conventional SysML views from the unified model:

• Internal Block Diagrams (IBD): Projected from the ITG by extracting the structural block interconnections via channel signatures and state relations (Algorithm 1, (Chao, 2021)).
• State Machine Diagrams (SMD): Derived via pattern recognition of state transitions and underlying guard conditions (Algorithm 2, (Chao, 2021)).
• Activity Diagrams (AD): Extracted by identifying activity-centric transition chains within the ITG (Algorithm 3, (Chao, 2021)).

Formally, channel signatures and state relationships provide a comprehensive mapping from ITG tuples to these user-level diagrams, eliminating ambiguities inherent in non-unified metamodeling techniques.

5. Applications and Case Studies

C-M-SBC-PA is applicable wherever formal consistency and integration between static and dynamic system views are required:

• Model-Based Design of Safety-Critical Systems: Consistency between structure (e.g., hardware/software partitioning) and behavior (e.g., safety protocols and state transitions) is guaranteed by the unified algebraic model (Chao, 2021).
• Distributed Transactional Systems (e.g., ATM networks): Multi-queue enhancements enable precise modeling of concurrency, queue scheduling, and transaction ordering across distributed nodes (Chao, 2021).
• Industrial MBSE: The approach generalizes to any sector relying on SysML for orchestrating complex systems (aerospace, automotive, automation) by reducing inter-diagram inconsistency and supporting system-wide verification.

6. Theoretical Context and Connections

C-M-SBC-PA’s formalism aligns with and extends a body of process algebra research on supervisory coordination, multi-queue stability, and concurrency:

• Supervisory Coordination Process Algebra: Focuses on control, synchronization, and partial bisimulation for managing controllable/uncontrollable events (Baeten et al., 2011). While both frameworks model system composition, C-M-SBC-PA centers on channel-based buffering and queue management, rather than direct control loop explicitness.
• Multi-Queue Multi-Server Stability Theory: Provides polytope-based characterizations of stability for systems with multiple asynchronous queues, informing how multi-queue structures can be managed and analyzed within process algebraic frameworks (Halabian et al., 2011).
• Logic-Based Process Algebra and Deep Inference: Enables logical encoding of process semantics, offering pathways for proof-theoretic verification of channel-based, multi-queue process algebra properties (Roversi, 2012).
• Process Scheduling and Mutual Exclusion: Incorporates strategic interleaving, control states, and semaphore-based mutual exclusion, concepts amenable to multi-queue semantical generalization in C-M-SBC-PA (Middelburg, 2020).

7. Implications and Prospects

The adoption of C-M-SBC-PA as the core metamodel in MBSE is poised to enhance system reliability, facilitate rigorous model checking, and streamline the transformation between diverse modeling views (Chao, 2021). The channel-based multi-queue construct not only bridges structure and behavior in a formal sense but also provides operational expressivity to realize emergent requirements in modern systems engineering. Ongoing challenges include the management of state-space explosion in large multi-queue systems, fairness, deadlock avoidance, and the extension of formal semantics to fully cover interaction-rich, distributed scenarios (Chao, 2021). A plausible implication is that refinement and tool-support for C-M-SBC-PA could drive the next generation of unified, semantics-rich systems modeling practices.