Welcome to an introduction to Petri nets. Petri nets are a mathematical modeling language and graphical tool used to describe distributed systems. They were developed by Carl Adam Petri in 1962 for his doctoral dissertation. Petri nets consist of places, represented by circles, transitions shown as rectangles, and arcs connecting them. Places may contain tokens, shown as dots, which represent resources or states in the system. This simple example shows a Petri net with three places and one transition. The arc from place P1 to transition T1 has a weight of 2, meaning the transition requires two tokens to fire.
Let's examine the basic components of Petri nets in more detail. Places, represented by circles, model states or conditions in the system. Transitions, shown as rectangles, represent events or actions that can occur. Arcs connect places to transitions and transitions to places, showing the flow of tokens. Tokens, depicted as dots inside places, represent resources, objects, or control flow. In this example, we have a simple workflow with idle, processing, and done states, plus a resource pool. The start transition requires both an idle process and a resource to fire, while the finish transition returns the resource to the pool.
Now let's explore the dynamic behavior of Petri nets through firing rules. A transition is enabled when all its input places have at least as many tokens as the weight of the corresponding arc. In this example, transition T1 has two input places: P1 requiring 2 tokens and P2 requiring 1 token. Since P1 has 3 tokens and P2 has 1 token, transition T1 is enabled. When T1 fires, it consumes 2 tokens from P1 and 1 token from P2, then produces 1 token in output place P3. This firing happens atomically, meaning all token movements occur simultaneously as a single indivisible action. This mechanism allows Petri nets to model concurrent and asynchronous behavior in distributed systems.
Petri nets have a wide range of applications across various domains. In manufacturing systems, they model production workflows, resource allocation, and synchronization between different processes. This example shows a manufacturing system with raw materials, two processing machines, and quality control. Business process modeling uses Petri nets to represent workflows, document management, and approval processes. In computer science, they model distributed algorithms, communication protocols, and concurrent systems. Petri nets are also applied in biology to model metabolic pathways, in logistics for supply chain management, and in software engineering for system design and verification. Their ability to represent concurrency, resource constraints, and synchronization makes them valuable in many complex system domains.
To summarize what we've learned about Petri nets: They are powerful mathematical modeling tools specifically designed for distributed systems. The basic components include places represented by circles, transitions shown as rectangles, arcs connecting them, tokens that flow through the network, and weights that determine firing conditions. The dynamic behavior of Petri nets is governed by well-defined firing rules, where transitions consume tokens from input places and produce tokens in output places. This makes Petri nets particularly effective at modeling concurrency, synchronization, and resource allocation in complex systems. Their applications span numerous fields including manufacturing systems, business process modeling, computer science, and even biological systems. The visual nature of Petri nets, combined with their mathematical foundation, makes them both intuitive to understand and rigorous for analysis.