HYPE with stochastic events

The process algebra HYPE was recently proposed as a fine-grained modelling approach for capturing the behaviour of hybrid systems. In the original proposal, each flow or influence affecting a variable is modelled separately and the overall behaviour of the system then emerges as the composition of these flows. The discrete behaviour of the system is captured by instantaneous actions which might be urgent, taking effect as soon as some activation condition is satisfied, or non-urgent meaning that they can tolerate some (unknown) delay before happening. In this paper we refine the notion of non-urgent actions, to make such actions governed by a probability distribution. As a consequence of this we now give HYPE a semantics in terms of Transition-Driven Stochastic Hybrid Automata, which are a subset of a general class of stochastic processes termed Piecewise Deterministic Markov Processes.

💡 Research Summary

The paper extends the HYPE process algebra, originally designed for fine‑grained hybrid system modelling, by turning its previously underspecified non‑urgent events into fully quantified stochastic events. In the original HYPE, system dynamics are expressed as a composition of flows (continuous influences on variables) and instantaneous events. Urgent events fire immediately when their guard becomes true, while non‑urgent events have an undefined guard (⊥) and may occur after an arbitrary, nondeterministic delay, providing no information about timing. This lack of quantitative information hampers probabilistic analysis.

To address this, the authors introduce stochastic events whose activation is governed by a probability distribution, most commonly an exponential distribution with a rate that may be constant or a function of the current continuous state. Each stochastic event is associated with a rate function f : ℝ^|V| → ℝ_{≥0} and a reset, analogous to the guard‑reset pair of instantaneous events. Consequently, the HYPE model now distinguishes two disjoint event sets: Ed for instantaneous (urgent) events and Es for stochastic events.

The semantics of stochastic HYPE is given via a mapping to Transition‑Driven Stochastic Hybrid Automata (TDSHA). A TDSHA consists of discrete modes, continuous variables, deterministic flows (ODEs) defined per mode, and two kinds of transitions: (i) instantaneous transitions triggered by logical guards, and (ii) stochastic transitions occurring after a random delay determined by the associated rate function. This structure is a subclass of Piecewise Deterministic Markov Processes (PDMP), a well‑studied class of stochastic processes that combine deterministic evolution with random jumps. By establishing this mapping, every stochastic HYPE model can be interpreted as a PDMP, opening the door to existing analytical and simulation techniques for PDMPs.

The paper illustrates the approach with a running example: an orbital temperature controller. The original HYPE model includes four flows affecting temperature K (cooling, heating, solar radiation, shading) and a controller that switches the heater and shade based on temperature thresholds. The stochastic extension adds a data‑download subsystem. A continuous variable D records the amount of stored data; two stochastic events, request and completed, model the initiation and termination of a download. The request event occurs with a constant exponential rate λ, while the completed event’s rate is λ·μ/(μ + D), i.e., it slows down as more data accumulates. The corresponding flows for data accumulation and download are expressed as influences (dw, r, const) and (dw, 0, const), switched by the stochastic events. A dedicated controller synchronises request and completed, preserving the compositional nature of HYPE.

Formally, a stochastic HYPE model is defined as a tuple (ConSys, V, IN, IT, Ed, Es, A, ec, iv, EC, ID). Here V is the set of continuous variables, IN the set of influence names, IT the set of influence types, A the set of activities (influence name, rate, type), iv maps influences to variables, and ID provides the semantic interpretation of each influence type (e.g., const → 1). Event conditions ec map each event to a pair (activation, reset); for instantaneous events activation is a logical guard, for stochastic events it is a rate function. Controlled systems are built from subcomponents (sequential agents with prefixed events), components, uncontrolled systems (parallel composition), and controllers (pure event sequences). The well‑formedness condition requires each subcomponent to be a self‑looping agent with an explicit init activity.

Compared with related work, stochastic process algebras such as PEPA focus on purely discrete state spaces and CTMC semantics, while hybrid process algebras (Hybrid CSP, Hybrid π‑calculus) require explicit ODE specifications for each component. HYPE’s distinctive feature is that flows are declared as modular influences; the overall ODE system emerges automatically from the composition, and stochastic events now provide a principled way to model non‑deterministic delays. This yields a richer modelling language that can capture both deterministic hybrid dynamics and probabilistic timing without sacrificing compositionality.

The authors conclude that stochastic HYPE, together with its TDSHA/PDMP semantics, offers a powerful framework for quantitative analysis of hybrid systems. Future work includes tool support for automatic translation to TDSHA, efficient simulation algorithms for the resulting PDMPs, and the development of verification techniques (e.g., model checking) tailored to stochastic HYPE models.