Extended time Petri nets

TL;DR

This paper addresses modeling time in complex systems where time data can be inconsistent or distributed across multiple forms. It introduces Extended time Petri nets (xTPN), a unified formalism that couples two transition timing intervals and a place timing interval with token lifetimes, enabling activation decisions via activating subsets and a rich multiset calculus on K and M. The authors define p-state and t-state, formalize time-elapse state changes, and describe production and activation dynamics, as well as transformations to classical nets and extended arcs. The framework supports modeling with heterogeneous time data and has practical implications for biological and other complex systems, with tooling compatibility noted.

Abstract

In many complex systems that can be modeled using Petri nets time can be a very important factor which should be taken into account during creation and analysis of the model. Time data can describe starting moments of some actions or their duration before their immediate effects start to influence some other areas of the modeled system. Places in a Petri net often describe static components of the system, but they can also describe states. Such a state can have time restrictions, for example, telling how long it can influence other elements in the model. Time values describing some system may be inconsistent or incomplete, which can cause problems during the creation of the model. In this paper, a new extension of time Petri nets is proposed, which allows the creation of models with different types of time data, which previously were possible to be properly used in separate types of well-known time Petri nets. The proposed new time Petri net solves this problem by integrating different aspects of already existing time Petri nets into one unified net.

Figures (4)

• Figure 1: Extended time Petri net example with time intervals $\alpha$ and $\beta$ assigned to transitions and time interval $\gamma$ assigned to place. Transition of the extended time Petri net are represented by rectangles with hourglass, place is represented by a circle. It should be noted that both input and output transition can have two intervals assigned, even though, e.g., $t_0$ will not consume any tokens when in fires, nor $t_1$ will produce any tokens. Production time for $t_1$ is deterministic, because both $\beta$ values for $t_1$ are equal. The same is true for an activation time of $t_0$, that is $\tau^{\alpha}_{t_0} = 2$.
• Figure 2: On the left side example transition $t_0$ is active, because in both $p_0$ and $p_1$ there is enough number of old enough tokens to form activating subsets. On the right side example transition $t_0$ is not active, because in $p_0$ there are only two old enough tokens in $K_{p_0}$ (represented by numbers 1 and 2). Therefore, multiset $M^{t_0}$ representing activating subsets for $t_0$ contains $\{\#\}$ instead of a proper activating subset for $p_0$. As a result, $M^{t_0}$ in right side example is not a subset of $M$.
• Figure 3: In both left and right cases it is assumed that time progress by $\tau = 0.5$ units. The only, yet very important difference between these cases is, that on the left the youngest token have a lifetime equal to 1, while on the right side its lifetime is equal to 1.5. This will result in deactivation of $t_0$ after time greater than $\tau$ by any measurable value, while in the example on the right $t_0$ will remain active.
• Figure 4: Part a) b) and c) represents three stages of a scenario in which tokens are returned by a read arc, but with their lifetimes increased. Part d) represents a simple example of the behavior of inhibitor arc painted in red.

Theorems & Definitions (23)

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