Hybrid Petri Nets

• Hybrid Petri Nets are a unified modeling framework combining discrete events and continuous flows to represent hybrid dynamic systems.
• They enable phase-wise simulation where ODE integration and discrete transitions alternate for performance evaluation and control synthesis.
• Their modular and compositional design supports scalable analysis in applications like communication networks, industrial process control, and energy systems.

Hybrid Petri nets (HPNs) are mathematical modeling frameworks that extend classical Petri nets by incorporating both discrete and continuous dynamics within a unified formalism. HPNs are particularly suited for representing and analyzing hybrid dynamic systems (HDS) where event-driven and flow-based behaviors coexist, as in communication networks, transport systems, industrial process control, and energy systems. The formal integration of discrete tokens and continuous “fluid” quantities enables modular, scalable analysis and supports advanced methodologies in performance evaluation, control, and stochastic modeling (Caramihai et al., 2017, Culita et al., 2017, 0706.1716, Hüls et al., 2020).

1. Formalism and Structural Elements

A Hybrid Petri Net is typically defined as a tuple

$$\text{HPN} = (P_D, P_C, T_D, T_C, \mathrm{Pre}, \mathrm{Post}, h, m_0)$$

where:

• $P_D$, $P_C$ are disjoint finite sets of discrete and continuous places, respectively.
• $T_D$, $T_C$ are disjoint finite sets of discrete and continuous transitions.
• $\mathrm{Pre}, \mathrm{Post}$: incidence functions assigning non-negative real arc weights.
• $h$: node-typing function indicating discrete/continuous status.
• $m_0$: initial marking, assigning integer values to $P_D$ (tokens) and non-negative reals to $P_C$ (fluid volumes).

Discrete transitions are enabled under classical rules: $P_D$0 for all discrete input places $P_D$1. Continuous transitions are “$P_D$2-enabled” with degree

$P_D$3

enforcing positive quantities for continuous arcs and ensuring consistency between the fluid and event-driven semantics. Firing in the discrete part generates instantaneous marking changes, while continuous transitions operate at real-valued, potentially time-varying firing rates.

HPNs generalize both classical discrete-event Petri nets and continuous (fluidified) Petri nets (0706.1716). The formalism supports further extensions, including stochastic transitions with general firing-time distributions (Hüls et al., 2020).

2. Continuous Dynamics and Hybrid Evolution

The continuous sector of an HPN is described by a marking vector $P_D$4 and a continuous incidence matrix $P_D$5. The temporal evolution follows a linear ordinary differential equation (ODE): $P_D$6 where $P_D$7 is the instantaneous vector of firing rates. The rates are constrained by maximal speeds $P_D$8 for each $P_D$9, the local enabling degree, and possible priority/sharing rules that resolve conflicts among transitions sharing input places. In typical implementations, continuous transitions fire at rates $P_C$0 with $P_C$1 determined by the net’s priority or sharing policies (Caramihai et al., 2017).

Simulation proceeds in discrete–continuous phases. Each phase is characterized by a fixed discrete marking and priority configuration, with the ODE solved exactly (since $P_C$2 is constant) until a discrete transition fires or a continuous place exhausts its marking. This hybrid evolution can be captured by a phase graph or, for verification purposes, translated into a hybrid automaton (0706.1716).

3. Modular and Compositional Modeling

HPNs support modular construction of large-scale hybrid systems by composing smaller subnets. Typical compositional strategies involve assigning subnets to logical components (nodes, links, buffers) and coupling them via interface places and transitions. For communication networks:

• Each node is represented by a continuous place (buffer or in-flight packet fluid), with outgoing continuous transitions modeling forwarding/processing.
• Link availability or reconfiguration is described by small discrete subnets (places and transitions controlling up/down).
• Subnets for connected nodes are merged by tying their continuous and discrete interface elements to reflect shared resources and connectivity (Caramihai et al., 2017, Culita et al., 2017).

This modularity facilitates reuse of subnet models, scalable simulation, and adaptation to scenario variations (e.g., link failures, subsystem upgrades) (Culita et al., 2017).

4. Performance Evaluation and Decision Support

HPN-based simulation engines implement phase-wise execution: alternating between discrete events and continuous flows. High-level pseudocode comprises initialization, repeated enabling and speed computation, event prediction, continuous ODE integration, and discrete transition firings, iterating until a termination condition (e.g., all packets delivered or time bound exceeded).

Common performance metrics:

• Throughput: rate of fluid delivered to destination.
• Latency: end-to-end delay per packet or fluid unit.
• Reliability: fraction of deliveries completed without disruption.

Optimization objectives include minimizing total delivery time, maximizing throughput, or ensuring reliability under constraints. Decision support systems (DSS) built atop HPNs use scenario analysis and simple priority-sharing heuristics to search for feasible or near-optimal configurations, and may suggest added path diversity or altered routing when no scenario meets performance targets (Caramihai et al., 2017, Culita et al., 2017).

5. Stochastic and Nondeterministic Extensions

Hybrid Petri nets are extended to include stochastic timing by introducing general (randomly delayed) transitions (HPnGs). The state-space semantics require encoding of both evolution ordering (using the “Parametric Location Tree,” PLT) and joint probability densities over random firing times.

In the PLT, each location corresponds to a unique ordering and interval configuration of random variables associated with stochastic transitions. Algorithms generate and restrict multidimensional integration domains using symbolic arithmetic (Fourier–Motzkin elimination) or geometric convex-polytopes. Probabilities for transient states are computed via multidimensional integration—by Monte Carlo methods, triangulation, or direct interval arithmetic. Each approach offers tradeoffs in terms of scalability, domain expressiveness, and support for property checking (Hüls et al., 2020).

A case study in battery-backup systems demonstrates tractable construction and probability evaluation for high-dimensional PLTs, with various integration strategies compared quantitatively.

6. Analytical Methods and Control Synthesis

Analysis of HPNs includes:

• Reachability: Determining whether certain hybrid states are attainable, typically after encoding the HPN as a hybrid automaton.
• Safety and Liveness: Verification that constraints (e.g., buffer overflows, deadlocks) are never breached along any trajectory.
• Controller Synthesis: Computing maximally-permissive guards for a subset of controllable transitions to ensure invariance or exclusion of forbidden states.

Translation is performed through a hierarchical approach that first maps the discrete net to a timed automaton (tracking clocks for timed or stochastic transitions), then embeds the continuous (fluid) part within each location, finally flattening to a hybrid automaton. The latter supports symbolic reachability analysis, safety verification, and supervisory control synthesis using well-established hybrid systems tools (e.g., PHAVer, HyTech, UPPAAL) (0706.1716).

7. Applications and Case Studies

HPNs are extensively applied in communication network modeling, process industries, and safety-critical infrastructure. Representative models include:

• Communication subnet with alternative routing and link failures: precise scenario-based performance optimization exposes the influence of routing-priority allocation on end-to-end delay and buffer occupancy. Experiments with 1000 packets and varied forwarding priority yield significant latency reductions (from 857 to 286 units) by eliminating buffer bottlenecks (Caramihai et al., 2017).
• Battery-backup systems under grid failure and load-shift, with both deterministic and stochastic transitions, integrated into the PLT analysis for transient-state probabilities and cost/risk assessment (Hüls et al., 2020).
• Process engineering (e.g., multi-tank mixing, valve–actuator scheduling) modeled as D-elementary hybrid PNs, supporting controller synthesis to honor invariance constraints on fluid levels (0706.1716).

A plausible implication is that the generality and expressiveness of HPNs—combining discrete, continuous, and stochastic phenomena—make them uniquely positioned for complex hybrid systems where modularity, scalability, and rigorous verification are critical.

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References:

(Caramihai et al., 2017, Culita et al., 2017, 0706.1716, Hüls et al., 2020)