## 1 Introduction  
Many applications of wireless sensor networks (WSN) require timely actuation. For example, industrial process automation typically consists of a multitude of sensors and actuators that are required to interact in a very clearly-defined manner with respect to their timing. Traditionally, in this environment, real-time conditions have to be met. While using wireless communications and the increased complexity of modern factories makes hard real-time a more elusive goal, there is still a need for a predictable timing behavior of the system in order to avoid catastrophic behavior. Furthermore, WSNs have been proposed in the context of emergency response systems. Again, it is obvious that predictable timing behavior is key to the acceptance of such systems. More generally, many envisioned applications of cyber-physical systems (CPS) typically can be viewed as closed-loop systems [1], in terms of control theory, such that the sensing often becomes time-critical in order to ensure the stability of the system. As in many of the visions of CPS, using WSNs is an integral part for the sensing requirements, it can hardly be overemphasized that predictable timing is a necessity for WSNs in that context, as well. Consequently, a mathematical methodology to dimension, operate and control the timing behavior in WSNs is of outmost importance.  
There is a clear trend for WSNs to become of ever larger scale, and some examples can be found in the CitySee project [2], the GreenOrbs project [3] and more generally in the emergence of the Internet of Things (IoT), which partially can be seen as a world-wide sensor network (with many sensors using wireless communication). Hence, a second key criterion for WSNs is the scalability of its basic functions and, in particular, of the mathematical framework to analyze its timing behavior. To that end, there is a clear need for a fast mathematical methodology to predict the timing behavior in WSNs.  
In 2005, the sensor network calculus (SensorNC) was proposed as such a mathematical methodology [4], and in more than a decade, it was developed to meet the special requirementsof time-sensitive, large-scale WSNs. Mathematically, it can be viewed as a special instance of the general network calculus, which itself is based on a min-plus algebraic formulation for the performance analysis of queueing networks as typical in packet-switched networks [5; 6].  
The goal is to have a mathematical framework that allows one to compute useful performance characteristics of WSNs such as message transfer delays, required buffer sizes and link capacities, but also duty cycle durations, to name some of the most prominent ones. Sometimes, we may require absolute values for, e.g., delays, if a certain application has hard real-time requirements, but sometimes, it may also suffice to have a representative relative metric in order to compare different design alternatives of a WSN before its deployment, for instance.  
To that end, the SensorNC was customized in several dimensions and also extended over the general network calculus to capture the special requirements of WSNs, e.g., in-network processing. The goal of this article is to provide an overview of the efforts in over a decade of the development of the SensorNC framework, emphasizing the most important milestones in this.  
In the following section, some background on the general network calculus framework is provided, in order to alleviate the introduction of the basic SensorNC methodology in Section 3. Advanced concepts of the SensorNC, in particular, in-network processing, are presented and illustrated in Section 4. Section 5 provides an overview of the self-modeling capabilities of WSNs that employ SensorNC. In Section 6, the important aspect of tool support for the SensorNC is discussed. Clearly, a mathematical framework such as the sensor is only as useful as the problems it can solve, and thus, we present several examples of SensorNC applications in Section 7. Section 8 provides a discussion and some concluding remarks.