# Applications of Discrete Event Systems

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**DOI:**https://doi.org/10.1007/978-1-4471-5102-9_59-2

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## Abstract

This entry provides an overview of the problems addressed by DES theory, with an emphasis on their connection to various application contexts. The primary intentions are to reveal the caliber and the strengths of this theory and to direct the interested reader, through the listed citations, to the corresponding literature. The concluding part of the article also identifies some remaining challenges and further opportunities for the area.

## Keywords

Discrete event systems Applications## Introduction

Discrete event system (DES) theory Models for Discrete Event Systems: An Overview Christos Cassandras, (Cassandras and Lafortune 2008; Seatzu et al. 2013) emerged in the late 1970s/early 1980s from the effort of the controls community to address the control needs of applications concerning some complex production and service operations, like those taking place in manufacturing and other workflow systems, telecommunication and data processing systems, and transportation systems. These operations were seeking the ability to support higher levels of efficiency and productivity and more demanding notions of quality of product and service. At the same time, the thriving computing technologies of the era, and in particular the emergence of the microprocessor, were cultivating, and to a significant extent supporting, visions of ever-increasing automation and autonomy for the aforementioned operations. The DES community set out to provide a systematic and rigorous understanding of the dynamics that drive the aforementioned operations and their complexity and to develop a control paradigm that would define and enforce the target behaviors for those environments in an effective and robust manner.

In order to address the aforementioned objectives, the controls community had to extend its methodological base, borrowing concepts, models, and tools from other disciplines. Among these disciplines, the following two played a particularly central role in the development of the DES theory: (i) theoretical computer science (TCS) and (ii) operations research (OR). As a new research area, DES thrived on the analytical strength and the synergies that resulted from the rigorous integration of the modeling frameworks that were borrowed from TCS and OR. Furthermore, the DES community substantially extended those borrowed frameworks, bringing in them many of its control-theoretic perspectives and concepts.

In general, DES-based approaches are characterized by (i) their emphasis on a rigorous and formal representation of the investigated systems and the underlying dynamics; (ii) a double focus on time-related aspects and metrics that define traditional/standard notions of performance for the considered systems but also on a more behaviorally oriented analysis that is necessary for ensuring fundamental notions of “correctness,” “stability,” and “safety” of the system operation, especially in the context of the aspired levels of autonomy; (iii) the interplay between the two lines of analysis mentioned in item (ii) above and the further connection of this analysis to structural attributes of the underlying system; and (iv) an effort to complement the analytical characterizations and developments with design procedures and tools that will provide solutions provably consistent with the posed specifications and effectively implementable within the time and other resource constraints imposed by the “real-time” nature of the target applications.

The rest of this entry overviews the current achievements of DES theory with respect to (w.r.t.) the different classes of problems that have been addressed by it and highlights the potential that is defined by these achievements for a range of motivating applications. On the other hand, the constricted nature of this entry does not allow an expansive treatment of the aforementioned themes. Hence, the provided coverage is further supported and supplemented by an extensive list of references that will connect the interested reader to the relevant literature.

## A Tour of DES Problems and Applications

### DES-Based Behavioral Modeling, Analysis, and Control

The basic characterization of behavior in the DES theoretic framework is through the various event sequences that can be generated by the underlying system. Collectively, these sequences are known as the (formal) language generated by the plant system, and the primary intention is to restrict the plant behavior within a subset of the generated event strings. The investigation of this problem is further facilitated by the introduction of certain mechanisms that act as formal representations of the studied systems, in the sense that they generate the same strings of events (i.e., the same formal language). Since these models are concerned with the representation of the event sequences that are generated by DES, and not by the exact timing of these events, they are frequently characterized as *untimed* DES models. In the practical applications of DES theory, the most popular such models are the finite state automaton (FSA) (Hopcroft and Ullman 1979; Cassandras and Lafortune 2008), Supervisory Control of Discrete Event Systems W. Murray Wonham, Diagnosis of Discrete Event Systems Stephane Lafortune and the Petri net (PN) (Murata 1989; Cassandras and Lafortune 2008), Modeling, Analysis and Control of Discrete Event Systems as Petri Nets Manuel Silva.

In the context of DES applications, these modeling frameworks have been used to provide succinct characterizations of the underlying event-driven dynamics and to design controllers, in the form of supervisors, that will restrict these dynamics so that they abide to safety, consistency, fairness, and other similar considerations Supervisory Control of Discrete Event Systems W. Murray Wonham. As a more concrete example, in the context of contemporary manufacturing, DES-based behavioral control – frequently referred to as supervisory control (SC) – has been promoted as a systematic methodology for the synthesis and verification of the control logic that is necessary for the support of the so-called SCADA (supervisory control and data acquisition) function. This control function is typically implemented through the programmable logic controllers (PLCs) that have been employed in contemporary manufacturing shop floors, and DES SC theory can support it (i) by providing more rigor and specificity to the models that are employed for the underlying plant behavior and the imposed specifications and (ii) by offering the ability to synthesize control policies that are provably correct by construction. Some example works that have pursued the application of DES SC along these lines can be found in Balemi et al. (1993), Brandin (1996), Park et al. (1999), Chandra et al. (2003), Endsley et al. (2006), and Andersson et al. (2010).

On the other hand, the aforementioned activity has also defined a further need for pertinent interfaces that will translate (a) the plant structure and the target behavior to the necessary DES-theoretic models and (b) the obtained policies to PLC executables. This need has led to a line of research, in terms of representational models and computational tools, which is complementary to the core DES developments described in the previous paragraphs. Indicatively we mention the development of GRAFCET (David and Alla 1992) and of the sequential function charts (SFCs) (Lewis 1998) from the earlier times, while some more recent endeavor along these lines is reported in Wightkin et al. (2011) and Alenljung et al. (2012) and the references cited therein.

Besides its employment in the manufacturing domain, DES SC theory has also been considered for the coordination of the communicating processes that take place in various embedded systems (Feng et al. 2007); the systematic validation of the embedded software that is employed in various control applications, ranging from power systems and nuclear plants to aircraft and automotive electronics (Li and Kumar 2012); the synthesis of the control logic in the electronic switches that are utilized in telecom and data networks; and for the modeling, analysis, and control of the operations that take place in health-care systems (Sampath et al. 2008). Wassyng et al. (2011) gives a very interesting account of the gains but also the extensive challenges, experienced by a team of researchers who have tried to apply formal methods, similar to those that have been promoted by the behavioral DES theory, to the development and certification of the software that manages some safety critical operations for Canadian nuclear plants.

Apart from control, untimed DES models have also been employed for the diagnosis of critical events, like certain failures, that cannot be observed explicitly, but their occurrence can be inferred from some resultant behavioral patterns (Sampath et al. 1996) Diagnosis of Discrete Event Systems Stephane Lafortune. More recently, the relevant methodology has been extended with prognostic capability (Kumar and Takai 2010), while an interesting variation of it addresses the “dual” problem that concerns the design of systems where certain events or behavioral patterns must remain undetectable by an external observer who has only partial observation of the system behavior; this last requirement has been formally characterized by the notion of “opacity” in the relevant literature, and it finds application in the design and operation of secure systems (Dubreil et al. 2010; Saboori and Hadjicostis 2012, 2014; Wu and Lafortune 2013) Opacity for Discrete Event Systems Christoforos Hadjicostis.

### Dealing with the Underlying Computational Complexity

As revealed from the discussion of the previous paragraphs, many of the applications of DES SC theory concern the integration and coordination of behavior that is generated by a number of interacting components. In these cases, the formal models that are necessary for the description of the underlying plant behavior may grow their size very fast, and the algorithms that are involved in the behavioral analysis and control synthesis may become practically intractable. Nevertheless, the rigorous methodological base that underlies DES theory provides also a framework for addressing these computational challenges in an effective and structured manner.

More specifically, DES SC theory provides conditions under which the control specifications can be decomposable to the constituent plant components while maintaining the integrity and correctness of the overall plant behavior Supervisory Control of Discrete Event Systems W. Murray Wonham, (2006). The aforementioned works of Brandin (1996) and Endsley et al. (2006) provide some concrete examples for the application of modular control synthesis. But there are also fundamental problems addressed by SC theory and practice that require a holistic view of the underlying plant and its operation, and thus, they are not amenable to the aforementioned decomposing solutions. DES SC theory can provide effective and tractable solutions for many of these cases as well, by, e.g., (i) helping identify special plant structure, of practical relevance, for which the target supervisors are implementable in a computationally efficient manner, or (ii) developing customized structured approaches that can systematically trade-off the original specifications for computational tractability. Additional substantial leverage in such endeavors is provided by the availability of more than one formal framework for tackling these problems, with complementary modeling and analytical capabilities. In particular, Petri nets can be an especially useful tool in the context of these endeavors, since they (a) provide very compact representations of the underlying plant dynamics, (b) capture effectively the connection of these dynamics to the structural properties of the plant, and also (c) admit analytical techniques of a more algebraic nature that do not require an explicit enumeration of the underlying state space (Murata 1989; Holloway et al. 1997; Cabasino et al. 2013) Modeling, Analysis and Control of Discrete Event Systems as Petri Nets Manuel Silva.

A particular application that has benefited from, and, at the same time, has significantly promoted the capabilities of DES SC theory to deal in an effective and structured manner with the high inherent complexity of the targeted behaviors, is that concerning the deadlock-free operation of many systems where a set of processes that execute concurrently and in a staged manner, are competing, at each of their processing stages, for the allocation of a finite set of reusable resources. In DES theory, this problem is known as the liveness-enforcing supervision of sequential resource allocation systems (RAS) (Reveliotis 2005; Zhou and Fanti 2004), and it underlies the operation of many contemporary applications: from the resource allocation taking place in contemporary manufacturing shop floors (Ezpeleta et al. 1995; Reveliotis and Ferreira 1996; Jeng et al. 2002) to the traveling and/or work-space negotiation in robotic systems (Reveliotis and Roszkowska 2011), automated railway (Giua et al. 2006), and other guidepath-based traffic systems (Reveliotis 2000), to Internet-based workflow management systems like those envisioned for e-commerce and certain banking and insurance claim processing applications (Van der Aalst 1997), and to the allocation of the semaphores that control the accessibility of shared resources by concurrently executing threads in parallel computer programs (Liao et al. 2013). A comprehensive and systematic introduction to the DES-based modeling of RAS and the problem of their liveness-enforcing supervision can be found in Reveliotis (2017).

Closing the above discussion on the ability of DES theory to address effectively the complexity that underlies the DES SC problem, we should point out that the same merits of the theory have also enabled the effective management of the complexity that underlies problems related to the performance modeling and control of the various DES applications. We shall return to this capability in the next subsection that discusses the achievements of DES theory in this domain.

### DES Performance Control and the Interplay Among Structure, Behavior, and Performance

DES theory is also interested in the performance modeling, analysis, and control of its target applications w.r.t. time-related aspects like throughput, resource utilization, experienced latencies, and congestion patterns. To support this type of analysis, the untimed DES behavioral models are extended to their *timed* versions. This extension takes place by endowing the original untimed models with additional attributes that characterize the experienced delays between the activation of an event and its execution (provided that it is not preempted by some other conflicting event). Timed models are further classified by the extent and the nature of the randomness that is captured by them. A basic such categorization is between deterministic models, where the aforementioned delays take fixed values for every event, and stochastic models which admit more general distributions. From an application standpoint, timed DES models connect DES theory to the multitude of applications that have been addressed by dynamic programming, stochastic control, and scheduling theory (Bertsekas 1995; Meyn 2008; Pinedo 2002), Control and Optimization of Stochastic Discrete Event Systems Xiren Cao. Also, in their most general definition, stochastic DES models provide the theoretical foundation of discrete event simulation (Banks et al. 2009).

Similar to the case of behavioral DES theory, a practical concern that challenges the application of timed DES models for performance modeling, analysis, and control is the very large size of these models, even for fairly small systems. DES theory has tried to circumvent these computational challenges through the development of methodology that enables the assessment of the system performance, over a set of possible configurations, from the observation of its behavior and the resultant performance at a single configuration. The required observations can be obtained through simulation, and in many cases, they can be collected from a single realization – or sample path – of the observed behavior; but then, the considered methods can also be applied on the actual system, and thus, they become a tool for real-time optimization, adaptation, and learning.

Collectively, the aforementioned methods define a “sensitivity”-based approach to DES performance modeling, analysis, and control (Cassandras and Lafortune 2008), Perturbation Analysis of Discrete Event Systems Yorai Wardi. Historically, DES sensitivity analysis originated in the early 1980s in an effort to address the performance analysis and optimization of queueing systems w.r.t. certain structural parameters like the arrival and processing rates (Ho and Cao 1991). But the current theory addresses more general stochastic DES models that bring it closer to broader endeavors to support incremental optimization, approximation, and learning in the context of stochastic optimal control (Cao 2007; Wardi et al. 2018). Some particular applications of DES sensitivity analysis for the performance optimization of production, telecom, and computing systems can be found in Cassandras and Strickland (1988), Cassandras (1994), Panayiotou and Cassandras (1999), Homem-de Mello et al. (1999), Fu and Xie (2002), Santoso et al. (2005), and Li and Reveliotis (2015).

Another interesting development in time-based DES theory is the theory of (max,+) algebra (Baccelli et al. 1992; Hardouin et al. 2018). In its practical applications, this theory addresses the timed dynamics of systems that involve the synchronization of a number of concurrently executing processes with no conflicts among them, and it provides important structural results on the factors that determine the behavior of these systems in terms of the occurrence rates of various critical events and the experienced latencies among them. Motivational applications of (max,+) algebra can be traced in the design and control of telecommunication and data networks, manufacturing, and railway systems, and more recently, the theory has found considerable practical application in the computation of repetitive/cyclical schedules that seek to optimize the throughput rate of automated robotic cells and of the cluster tools that are used in semiconductor manufacturing (Park et al. 1999; Lee 2008; Kim and Lee 2012).

Both, sensitivity-based methods and the theory of (max,+) algebra, that were discussed in the previous paragraphs, are enabled by the explicit, formal modeling of the DES structure and behavior in the pursued performance analysis and control. This integrative modeling capability that is supported by DES theory also enables a profound analysis of the impact of the imposed behavioral-control policies upon the system performance and, thus, the pursuance of a more integrative approach to the synthesis of the behavioral and the performance-oriented control policies that are necessary for any particular DES instantiation. This is a rather novel topic in the relevant DES literature, and some recent works in this direction can be found in Cao (2005), Markovski and Su (2013), David-Henriet et al. (2013), Li and Reveliotis (2015), and Li and Reveliotis (2016).

### The Roles of Abstraction and Fluidification

The notions of “abstraction” and “fluidification” play a significant role in mastering the complexity that arises in many DES applications. Furthermore, both of these concepts have an important role in defining the essence and the boundaries of DES-based modeling.

In general systems theory, abstraction can be broadly defined as the effort to develop simplified models for the considered dynamics that retain, however, adequate information to resolve the posed questions in an effective manner. In DES theory, abstraction has been pursued w.r.t. the modeling of, both, the timed and untimed behavior, giving rise to hierarchical structures and models. A theory for hierarchical SC is presented in Wonham (2006), while some applications of hierarchical SC in the manufacturing domain are presented in Hill et al. (2010) and Schmidt (2012). In general, hierarchical SC relies on a “spatial” decomposition that tries to localize/encapsulate the plant behavior into a number of modules that interact through the communication structure that is defined by the hierarchy. On the other hand, when it comes to timed DES behavior and models, a popular approach seeks to define a hierarchical structure for the underlying decision-making process by taking advantage of the different time scales that correspond to the occurrence of the various event types. Some particular works that formalize and systematize this idea in the application context of production systems can be found in Gershwin (1994) and Sethi and Zhang (1994) and the references cited therein.

In fact, the DES models that have been employed in many application areas can be perceived themselves as abstractions of dynamics of a more continuous, time-driven nature, where the underlying plant undergoes some fundamental (possibly structural) transition upon the occurrence of certain events that are defined either endogenously or exogenously w.r.t. these dynamics. The combined consideration of the discrete event dynamics that are generated in the manner described above, with the continuous, time-driven dynamics that characterize the modalities of the underlying plant, has led to the extension of the original DES theory to the so-called hybrid systems theory. Hybrid systems theory is itself very rich, and it is covered in another section of this encyclopedia (see also Connections Between Discrete Event Systems and Hybrid Systems Alessandro Giua). From an applications standpoint, it increases substantially the relevance of the DES modeling framework and brings this framework to some new and exciting applications. Some of the most prominent such applications concern the coordination of autonomous vehicles and robotic systems, and a nice anthology of works concerning the application of hybrid systems theory in this particular application area can be found in the IEEE Robotics and Automation magazine of September 2011. These works also reveal the strong affinity that exists between hybrid systems theory and the DES modeling paradigm. Along similar lines, hybrid systems theory underlies also the endeavors for the development of the automated highway systems that have been explored for the support of the future urban traffic needs (Horowitz and Varaiya 2000; Fleck et al. 2016). Finally, hybrid systems theory and its DES component have been explored more recently as potential tools for the formal modeling and analysis of the molecular dynamics that are studied by systems biology (Curry 2012).

Fluidification, on the other hand, is the effort to represent as continuous flows, dynamics that are essentially of discrete event type, in order to alleviate the computational challenges that typically result from discreteness and its combinatorial nature. The resulting models serve as approximations of the original dynamics; frequently, they have the formal structure of hybrid systems, and they define a basis for developing “relaxations” for the originally addressed problems. Usually, their justification is of an ad hoc nature, and the quality of the established approximations is empirically assessed on the basis of the delivered results (by comparing these results to some “baseline” performance). There are, however, a number of cases where the relaxed fluid model has been shown to retain important behavioral attributes of its original counterpart (Dai 1995). Furthermore, some recent works have investigated more analytically the impact of the approximation that is introduced by these models on the quality of the delivered results (Wardi and Cassandras 2013), while an additional important extension of the fluid modeling framework for DES is through the notion of “stochastic flow models (SFMs),” which allow the flow rates themselves to be random processes. Some works introducing the SFM framework and providing a first set of results for it can be found in Cassandras et al. (2002) and Sun et al. (2004), while a brief but more tutorial exposition of this framework is provided in Wardi et al. (2018). On the other hand, some works exemplifying the application of fluidification in the DES-theoretic modeling frameworks, and the potential advantages that this approach brings in various application contexts, can be found in Srikant (2004), Meyn (2008), David and Alla (2005), Cassandras and Yao (2013), and Ibrahim and Reveliotis (2019). Finally, the work of Vázquez et al. (2013) provides a nice introduction on the pursuance of the “fluidification” concept in the Petri net modeling framework, while the recent work of Ibrahim and Reveliotis (2018) demonstrates very vividly how the corresponding results enable a synergistic employment of all the different representations of DES behavior that have been discussed in this document, in a totally integrative and seamless manner.

## Summary and Future Directions

The discussion of the previous section has revealed the extensive application range and potential of DES theory and its ability to provide structured and rigorous solutions to complex and sometimes ill-defined problems. On the other hand, the same discussion has revealed the challenges that underlie many of the DES applications. The complexity that arises from the intricate and integrating nature of most DES models is perhaps the most prominent of these challenges. This complexity manifests itself in the involved computations but also in the need for further infrastructure, in terms of modeling interfaces and computational tools, that will render DES theory more accessible to the practitioner.

The DES community is aware of this need, and the last few years have seen the development of a number of computational platforms that seek to implement and leverage the existing theory by connecting it to various application settings; indicatively, we mention DESUMA (Ricker et al. 2006), SUPREMICA (Akesson et al. 2006), and TCT (Feng and Wonham 2006), that support DES behavioral modeling, analysis, and control along the lines of DES SC theory, while the website entitled “The Petri Nets World” has an extensive database of tools that support modeling and analysis through untimed and timed variations of the Petri net model. Model checking tools, like SMV and NuSPIN, which are used for verification purposes, are also important enablers for the practical application of DES theory, and, of course, there are a number of programming languages and platforms, like Arena, AutoMod, Simio, and SimEvents that support discrete event simulation (SimEvents also supports simulation of hybrid systems). However, with the exception of the discrete event simulation software, which is a pretty mature industry, the rest of the aforementioned endeavors currently evolve primarily within the academic and the broader research community. Hence, a remaining challenge for the DES community is the strengthening and expansion of the aforementioned computational platforms to robust and user-friendly computational tools. The availability of such industrial strength computational tools, combined with the development of a body of control engineers well-trained in DES theory, will be catalytic for bringing all the developments that were described in the earlier parts of this document even closer to the industrial practice.

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