Discrete Event Systems

Transcription

1 DI DIPARTIMENTO DI INGEGNERIA DELL INFORMAZIONE E SCIENZE MATEMATICHE Lecture notes of Discrete Event Systems Simone Paoletti Version 0.3 October 27, 2015

2 Indice Notation 1 Introduction 2 1 Basics of systems theory Systems and mathematical models Static vs dynamic systems The concept of state Time-driven vs event-driven systems Discrete event systems Untimed models of discrete event systems State automata (with outputs) Graphical representation Timed models of discrete event systems Introduction Bibliography 23

3 Notation The main symbols, operators, functions, acronyms, etc. used in these notes, are introduced next. Symbols, operators and functions R the field of real numbers C the field of complex numbers Z the set of integer numbers N the set of natural numbers 2 X power set of X (set of all subsets of X) belongs to / does not belong to equal by definition dx, ẋ first derivative of x dt d 2 x, ẍ second derivative of x dt 2 Acronyms DES Discrete Event System

4 Introduction Discrete event systems are systems whose dynamic behaviour is driven by asynchronous occurrences of events. Some examples are: a manufacturing plant with machines, workers, buffers, etc.; a bank with different types of customers and services(desks, ATMs, etc.); an airport with passengers in different states (check-in, security control, gate, boarding, etc.); a computer system with processes accessing to resources; a road system with cars, roads, crosses, traffic lights, etc.; a hospital with different types of patients and wards; a fast-food restaurant with a staff and different types of customers. Broadly speaking, discrete event systems can be found in a variety of fields, such as control, computer science, automated manufacturing, and communication, information and transportation networks. The objective of this course is to equip the students with several modeling, analysis and simulation tools for discrete event systems. In this respect, modeling, probability and programming are the main contents of the course. Both untimed and timed automata will be introduced as models of discrete event systems. Markov chains will be addressed as an important class of timed automata. The main application of the tools presented in the course, will be queueing theory, since queueing systems are probably the most important and widespread class of discrete event systems. These lecture notes follow the line of teaching held in the course, but do not substitute the textbook [1]. They should be rather considered as a complement to it.

5 1 Basics of systems theory 1.1 Systems and mathematical models In general terms, a system is what is of interest for a particular study. A system is not necessarily associated with physical objects and natural laws: consider, for instance, economic mechanisms or human behavior. From the input-output point of view, a system can be represented as in Figure 1.1, where: u denotes the system inputs, or independent variables; y denotes the system outputs, or dependent variables. u system y Figure 1.1: Input-output representation of a system. Inputs are those variables that can be varied independently of the system, and are fed to the system to modify its behavior. Outputs are those variables describing how the systems reacts to the inputs. Different systems may react differently to the same inputs. For instance, the same force may determine a different acceleration depending on the mass of the body to which it is applied. Inthissense, theoutputsdependontheinputsappliedandonthesystemitself. A (mathematical) model describes, implicitly or explicitly, a functional relationship between the inputs and the outputs of a system. It should be

6 Basics of systems theory 4 always clear that the model is not the system. For instance, a resistor is not the Ohm s Law. The Ohm s Law describes the voltage-current characteristic of a resistor in the operating zone where it is approximately linear. In general, a model differs from the system due to parameter uncertainties, unmodeled dynamics, approximations. The structure, the accuracy and the complexity of a model depend on the system to be described and on its intended use. 1.2 Static vs dynamic systems A first basic classification is between static and dynamic systems. Definition 1.1 A system is termed static when the outputs at any time instant depend only on the inputs applied at the same time instant. Models of static systems are just functions mapping the inputs at time t to the outputs at time t: y(t) = f(u(t)). (1.1) AnexampleistheOhm slaw, mappingthecurrentipassingthrougharesistor to the voltage v across the resistor: where R is the resistance. v(t) = Ri(t) (1.2) Definition 1.2 A system is termed dynamic when the outputs at any time instant depend on the whole past history of the inputs. Differential equations are models of dynamic systems. Consider for instance the current-voltage relation of a capacitor: i(t) = C dv(t) dt (1.3) where C is the capacitance. If we integrate both sides of (1.3) between t 0 (initial time) and t > t 0, we get v(t) = v(t 0 )+ 1 C t t 0 i(τ)dτ (1.4) where it is apparent that the voltage (system output) at time t depends on the past values of the current (system input) up to time t. In this course, we will study a class of dynamic systems (namely, discrete event systems) which is not described by differential equations.

7 Basics of systems theory The concept of state Consider the following problem. Problem 1.1 Given a system, is it sufficient to know the inputs u(t) for all t t 0 in order to determine uniquely the outputs y(t) for all t t 0? For static systems, the answer is yes. What about dynamic systems? Example 1.1 A constant braking force is applied to a car. We are interested in the space needed to stop the car. This implies that, in our problem, the car is the system, the braking force is the input, and the displacement is the output. We can write the second Newton s law of motion for the car: Ma(t) = f (1.5) where M is the mass of the car, a(t) is the car acceleration, and f is the modulus of the braking force. Recalling that the acceleration is the first derivative of the velocity, and integrating both sides of (1.5) between t 0 (initial time) and t > t 0, we get v(t) = v(t 0 ) f M (t t 0) (1.6) where v denotes the velocity of the car. Recalling that the velocity is the first derivative ofthe position, andintegrating bothsides of (1.6) between t 0 (initial time) and t > t 0, we further get x(t) = v(t 0 )(t t 0 ) f 2M (t t 0) 2 (1.7) where x denotes the displacement. It can be observed that the displacement does not depend only on the input (the braking force f), but also on the value of the velocity of the car at time t 0. This implies that the knowledge of the input only, is not sufficient to determine uniquely the output. Also the value v(t 0 ) is needed. Notice that the velocity is neither an input, nor an output of the system. It is rather an additional variable whose value at time t 0 is necessary in order to determine uniquely the output for a given input. From Example 1.1 we understand that, for dynamic systems, the answer to Problem 1.1 is no. This leads to the following definition. Definition 1.3 The state of a dynamic system is a set of variables whose values at time t 0 are necessary to determine uniquely the outputs y(t) for all t t 0, given the inputs u(t) for all t t 0.

8 Basics of systems theory 6 Typically, the state of a system will be denoted by x. Providing the correct definition of state for a system is crucial in any modeling task. Example 1.2 The queueing system of Figure 1.2 is formed by a single server preceded by a queue (or storage space). The total capacity of the queueing system is K, therefore the number of slots in the queue is K 1. queue server Figure 1.2: Queueing system of Example 1.2. Assume that one is interested in whether the server is idle or busy, and to this aim defines the following variable: { 0 if the server is idle x = 1 otherwise. (1.8) The question is whether (1.8) can be taken as a definition of state for the system. The answer is no, and the reasoning is simple. Assume that x = 1. This implies that the server is busy. What happens when the current service terminates? Using only the information contained in x = 1, it is not possible to determine uniquely whether the next value will be x = 0 or x = 1. On the basis of Definition 1.3, the variable in (1.8) is therefore not a state for the queueing system. Notice that the next value will be x = 0 if no customer is waiting in the queue, and x = 1 otherwise. It is apparent that full information about the number of customers in the system is needed for a proper definition of state. This example will be continued in Example 1.4. Notice that, for a given system, the definition of state is not unique. As will be clear later (see, e.g., Example 1.5), different definitions of state can be given for the same system, depending on the problem or application which is of interest. Definition 1.4 A system is termed with continuous state when its state takes values in a continuous set. It is termed with discrete state when its state takes values in a discrete/countable set. There exists a class of systems, called hybrid systems, for which some components of the state are continuous, and others are discrete.

9 Basics of systems theory C hour Figure 1.3: Plot of ambient temperature in a summer day. 1.4 Time-driven vs event-driven systems An important classification is between time-driven and event-driven systems. Definition 1.5 A system is termed time-driven when all its variables (inputs, state and outputs) may change at any time instant with regular/synchronous ticking. The above definition implies that time is an independent variable for timedriven systems. There is an external clock, and all system variables may change at any clock tick. Figure 1.3 shows an example of this type, where the observed variable is the ambient temperature in a summer day. Differential equations are typical models used to describe time-driven systems. Definition 1.6 A system is termed event-driven when its state changes only upon the (typically irregular/asynchronous) occurrence of events. It will be clear later that, for event-driven systems, time (meaning the time instants when the events occur) is a dependent variable of the system. An example of sample path(or state trajectory) of an event-driven system is shown in Figure 1.4. Its most evident characteristic is that it is piecewise constant, with the time instants of the state jumps determined by the asynchronous occurrence of events. Example 1.3 A typical example of event-driven system is the queueing system of Example 1.2. If the state is defined as the number of customers in the

10 Basics of systems theory 8 x(t) Figure 1.4: Example of sample path of an event-driven system. t system, it is clear that the state changes only upon the occurrence of events, in particular: the state is increased by one when the arrival of a new customer occurs (except when the system is full); the state is decreased by one when the service of a customer terminates, and the customer leaves the system. An event should be thought of as occurring instantaneously. Events can be of different types: specific actions (e.g., somebody presses a button, a customer arrives in a queue, a job is terminated, etc.); spontaneous occurrences(e.g., the failure of a component, an interruption of service, etc.); fulfilment of logic conditions (e.g., a warning threshold is exceeded, etc.). In the following, a generic event will be typically denoted by e. 1.5 Discrete event systems We are now ready for the definition of discrete event system. Definition 1.7 A discrete event system (DES) is a dynamic, event-driven system with discrete state.

11 Basics of systems theory 9 a a a a K-1 K a d d d d Figure 1.5: State transition diagram for the queueing system of Example 1.4. It follows that a DES is characterized by: a discrete set E of events; a discrete state space X; an event-driven dynamics. There exist real systems that can be naturally viewed as discrete event systems. A typical example is provided next. Example 1.4 Consider again the queueing system of Example 1.2, and define the state x as the number of customers in the system. It turns out that x may take values in the discrete state space X = {0,1,2,...,K 1,K}. (1.9) Moreover, the state x may change only upon the arrival of a new customer, or upon the termination of a service (assuming that customers depart from the system after service). We can thus define the event set E = {a,d} (1.10) where a denotes the arrival of a new customer, and d denotes the termination of a service in the server. The event-driven dynamics of the queueing system can be represented through the state transition diagram in Figure 1.5. In the state transition diagram, the nodes represent the states of X, whereas labeled arcs represent the state transitions: an arc from x to x with label e means that, when the current state is x and the next event is e, the next state will be x. Notice that in state x = K the queueing system is full, and therefore arrivals of new customers (that are still possible) are rejected due to lack of space (the state of the system does not change). Notice also that the state transition diagram contains the information that event d is not possible in

12 Basics of systems theory 10 state x = 0 (there is no arc labeled with d exiting from node 0). Indeed, when the queueing system is empty, no termination of service is possible. In other cases, real systems that are naturally time-driven, can be modeled as event-driven for particular applications. Example 1.5 A cart moves along a track. Sensors are located at three points of the track (they are denoted by A, B and C in Figure1.6). Each sensor sends an impulse when the cart crosses the corresponding point, in both directions. For the sake of simplicity, it is assumed that the cart never changes direction when it is across a sensor. x x x x x x x x x x x x x z A B C Figure 1.6: Cart moving along a track. The cart can be naturally seen as a time-driven system, whose motion is modeled by the second Newton s law: M z(t) = f(t) (1.11) where M is the mass of the cart, z is the position of the cart along the track, and f is the tractive force applied to the cart. In principle, by integrating twice (1.11), and given the initial position z(t 0 ), the initial velocity ż(t 0 ), and the applied force f(t) for all t t 0, it is possible to know from the model the exact position z(t) of the cart at any t t 0. In practice, uncertainty on the aforementioned quantities may lead to discrepancies between the real system and the model. However, in some applications, it is neither required nor feasible to know the exact position of the cart. It is sufficient to know the interval in which the cart is localized. To this aim, the input provided by the sensors can be used. Define the state of the system as follows: 0 if z < A 1 if A z < B x = 2 if B z < C 3 if z C. Thus, we have a discrete state space (1.12) X = {0,1,2,3}. (1.13)

13 Basics of systems theory 11 a b c a b c Figure 1.7: State transition diagram for the cart system of Example 1.5. Moreover, define the set of events E = {a,b,c} (1.14) where a means that an impulse is received from the sensor located at A, and so on. With these definitions, we can construct the state transition diagram in Figure 1.7. Notice that the dynamics of the system so described, turns out to be an event-driven dynamics. From Example 1.5 we understand that the distinction between timedriven and event-driven systems is not so strict. The same real system could be suitably modeled as time-driven or event-driven, depending on the application. We also learn that the definition of state is not unique for a given system. In the case of the cart of Example 1.5, the state is the pair (position,velocity) for the time-driven model, and the interval where the cart is localized for the event-driven model.

14 2 Untimed models of discrete event systems In this chapter, we introduce state automata as untimed models of discrete event systems. The term untimed refers to the fact that these models do not include time. Given a sequence of events, a state automaton returns the corresponding sequence of states. No information about time instants when events (and thus state transitions) occur, is available from the model. In this respect, untimed models only describe the logical behaviour of discrete event systems. An input-output representation of untimed models is given in Figure 2.1, where e k denotes the kth event, and x k denotes the state after the kth event, with x 0 the initial state. {e 1,e 2,...} untimed model {x 0,x 1,x 2,...} Figure 2.1: Input-output representation of untimed models. Notice that the event sequence {e 1,e 2,...} is the model input, and the corresponding state sequence {x 0,x 1,x 2,...} is the model output, provided that event e k is possible in state x k 1 for all k = 1,2,... To understand this, recall the queueing system of Example 1.4, where event d is not possible in state 0. An event sequence such that event d should occur in state 0, is infeasible for the queueing system.

15 Untimed models of discrete event systems State automata (with outputs) Based on Definition 1.7, a natural class of untimed models of discrete event systems is represented by state automata. Definition 2.1 A state automaton is a 5-tuple (E,X,Γ,f,x 0 ), where: E is a countable set of events; X is a countable set of states; Γ : X 2 E is the active event function, such that Γ(x) E is the set of all events that are possible in state x X; f : X E X is the state transition function, such that x = f(x,e) is the next state when event e Γ(x) occurs in the current state x X; x 0 X is the initial state. The following rules describe how a state automaton operates, i.e. it generates the state sequence corresponding to a given event sequence. Algorithm Set k = 1 2. If e k / Γ(x k 1 ), then goto step 5 3. Compute x k = f(x k 1,e k ) 4. Set k = k +1 and goto step 2 5. Error exit Notice that an error exit occurs in step 5 of Algorithm 2.1. This is because the next event is one which is actually not possible in the current state. This leads to the following definition of feasibility for an event sequence. Definition 2.2 The event sequence {e 1,e 2,...} is said to be feasible for the DES described by the state automaton (E,X,Γ,f,x 0 ) if e k Γ(x k 1 ) for all k = 1,2,..., where x 0 is the initial state and x k = f(x k 1,e k ). If we draw the following analogies: the event set E is the alphabet;

16 Untimed models of discrete event systems 14 feasibility check state update x 0 = 0 e 1 = a Γ(0) = Γ(x 0 ) x 1 = f(x 0,e 1 ) = f(0,a) = 1 e 2 = a Γ(1) = Γ(x 1 ) x 2 = f(x 1,e 2 ) = f(1,a) = 2 e 3 = d Γ(2) = Γ(x 2 ) x 3 = f(x 2,e 3 ) = f(2,d) = 1 e 4 = a Γ(1) = Γ(x 3 ) x 4 = f(x 3,e 4 ) = f(1,a) = 2 e 5 = d Γ(2) = Γ(x 4 ) x 5 = f(x 4,e 5 ) = f(2,d) = 1 e 6 = d Γ(1) = Γ(x 5 ) x 6 = f(x 5,e 6 ) = f(1,d) = 0 Table 2.1: Iterations of Algorithm 2.1 in Example 2.1. the event sequences are the words; the set of the feasible event sequences is the language, we can think of characterizing a DES through the language it recognizes (or accepts). This establishes interesting connections with formal language theory. Example 2.1 Consider the queueing system of Example 1.2. The definitions of E and X are given in Example 1.4. The other elements of the corresponding state automaton (E,X,Γ,f,x 0 ) are defined next. Active event function Γ Γ(0) = {a} Γ(x) = {a,d}, x = 1,2,...,K State transition function f { x+1 if x = 0,1,...,K 1 f(x,a) = K otherwise f(x,d) = x 1, x = 1,2,...,K. The initial state is x 0 = 0, if we assume that the system is initially empty. Algorithm 2.1 can be applied to determine the state sequence corresponding to the event sequence {e 1 = a,e 2 = a,e 3 = d,e 4 = a,e 5 = d,e 6 = d}. The iterations ofthealgorithmarereportedintable2.1. It turnsout that thestate sequence is {x 0 = 0,x 1 = 1,x 2 = 2,x 3 = 1,x 4 = 2,x 5 = 1,x 6 = 0}. On the other hand, given the event sequence {e 1 = a,e 2 = a,e 3 = d,e 4 = d,e 5 = d}, the algorithm returns an error exit, meaning that the event sequence is infeasible starting from x 0 = 0. Definition 2.1 can be extended to the case of systems with outputs.

17 Untimed models of discrete event systems 15 Definition 2.3 Astateautomatonwithoutputsisa7-tuple(E,X,Γ,f,x 0,Y,g), where: (E,X,Γ,f,x 0 ) is a state automaton; Y is a countable set of outputs; g : X Y is the output function, such that y = g(x) is the output corresponding to state x X. For a state automaton with outputs, Algorithm 2.1 returns also the output sequence {y 0,y 1,y 2,...}, where y k = g(x k ), k = 0,1,2,... It is worthwhile to note that state automata with outputs include Moore machines as special cases. In theory of computation, a Moore machine is a finite-state machine whose output values are determined solely by the current state. Moore machines are typically used in sequential logic implementation. Example 2.2 Consider again the queueing system of Examples 1.2, 1.4 and 2.1, and assume that one is only interested in whether the server is idle or busy. To this aim, one can extend the state automaton of Example 2.1 by adding the output set Y = {0,1} and the output function { 0 if x = 0 g(x) = 1 otherwise where outputs 0 and 1 mean that the server is idle and busy, respectively. 2.2 Graphical representation A state automaton (with outputs) can be represented through an oriented graph with labeled arcs (and nodes), according to the following rules. The nodes of the graph are the states of the automaton. The oriented and labeled arcs describe the state transition function f: an oriented arc with label e is added from node x to node x if and only if e Γ(x) and x = f(x,e). e.. x x

18 Untimed models of discrete event systems 16 a a a a K-1 K a d d d d Figure 2.2: State transition diagram with outputs for the queueing system of Example 2.2. Thelabelsonthenodesdescribe theoutput functiong: alabely isadded on node x if and only if y = g(x).... y. x A special case is when the output is binary, i.e. y {0,1}. If g(x) = 0: x If g(x) = 1: x The initial state x 0 is represented by an arrow pointing to the corresponding node. e.. x x 0 x.. e The resulting graph is also called state transition diagram. Example 2.3 For the state automaton of Example 2.1, the state transition diagramisshowninfigure1.5. WhenthevariantofExample2.2isconsidered, the state transition diagram with outputs is shown in Figure 2.2.

19 Timed models of discrete event systems 3 As we have seen in the previous chapter, state automata describe only the logical behaviour of discrete event systems. Given a sequence of events, a state automaton returns the corresponding sequence of states. In this chapter, we introduce timed automata as models of discrete event systems including information about time instants when events(and thus state transitions) occur. 3.1 Introduction One could argue that event times (i.e. time instants when events occur) could be provided as inputs to the timed model of a DES. If this were the case, there would be little to say. Recall that in discrete event systems state transitions occur only simultaneously with events. If event times were known, the same would be for time instants when state transitions occur, and one would only need to know the next state, given the current state and the next event. That is, timed models of discrete event systems would remain merely logical models. Now, recalling the definition of system input given in Section 1.1, the question is whether event times can be actually considered as inputs to discrete event systems. In other words, is it true that event times can be decided independently of the system? The next example tries to answer this question. Example 3.1 Consider the following two queueing disciplines: First In-First Out (FIFO) is a queueing discipline where the oldest task of the queue is processed first;

20 Timed models of discrete event systems 18 task arrival time execution time # # # Table 3.1: Task specification for Example 3.1. Round Robin (RR) is a queueing discipline where time slices are assigned to each task of the queue in equal portions and in circular order. In the case of FIFO, the ongoing task uses the resource until completion, without interruption. In the case of RR, a time slice is assigned to the first task of the queue. If processing of the task does not terminate before the time slice expires, the task is relegated to the last position of the queue, and all other tasks are shifted one position forward. On the other hand, if processing of the task terminates before the time slice expires, the processed task leaves the system, and all other tasks are shifted one position forward in the queue. In both cases, the task on top of the queue is then assigned a new time slice. Thanks to the cyclic and equitable assignment of the resource to all tasks of the queue, one advantage of RR versus FIFO is that it is starvation-free. Consider a resource, and tasks needing to use the resource. The system can be modeled as the queueing system of Figure 1.2, where the resource plays the role of the server, and the queue is the list of waiting tasks. Given the tasks with characteristics specified in Table 3.1, we want to determine when each task will be completed in case either FIFO or RR is applied. The resource is assumed to be initially idle. In the following, we denote by a i and d i the arrival and the completion of the ith task, respectively. FIFO The initial time instant is t 0 = 0. At t 1 = 1.0, task #1 arrives (event a 1 ) and accesses the resource. Execution of task #1 will terminate at t = 5.0. #1 At t 2 = 3.5, task #2 arrives (event a 2 ) and is added to the waiting list.

21 Timed models of discrete event systems 19 #2 #1 At t 3 = 4.5, task #3 arrives (event a 3 ) and is added to the waiting list. #3 #2 #1 At t 4 = 5.0, execution of task #1 terminates (event d 1 ). Task #1 leaves the system, and task #2 accesses the resource. Execution of task #2 will terminate at t = 7.0. #3 #2 At t 5 = 7.0, execution of task #2 terminates (event d 2 ). Task #2 leaves the system, and task #3 accesses the resource. Execution of task #3 will terminate at t = 9.5. #3 At t 6 = 9.5, execution of task #3 terminates (event d 3 ). Task #3 leaves the system. The system is now is empty. The sample path determined by the FIFO queueing discipline is shown in Figure 3.1. a 1 a 2 a 3 d 1 d 2 d time Figure 3.1: Sample path determined by the FIFO queueing discipline. RR Inthecase ofrr, expirationof thetimeslice isanother event tobeconsidered. Wedenotethiseventbys. ItisassumedthatthetimesliceamountstoT s = 1.5 time units.

22 Timed models of discrete event systems 20 The initial time instant is t 0 = 0. At t 1 = 1.0, task #1 arrives (event a 1 ) and is assigned a time slice. The execution time of task #1 amounts to 4.0 time units, which is greater than the time slice. The assigned time slice will expire at t = 2.5. #1 Att 2 = 2.5,thetimesliceexpires(events). Sincetask#1istheonlytask currently in the system, it is again assigned a time slice. The residual execution time of task #1 amounts to 2.5 time units, which is greater than the time slice. The assigned time slice will expire at t = 4.0. #1 At t 3 = 3.5, task #2 arrives (event a 2 ) and is added to the waiting list with execution time equal to 2.0 time units. #2 #1 At t 4 = 4.0, the time slice expires (event s). Task #1 is added to the waiting list with residual execution time equal to 1.0 time unit. Task #2 is assigned a time slice. The execution time of task #2 amounts to 2.0 time units, which is greater than the time slice. The assigned time slice will expire at t = 5.5. #1 #2 At t 5 = 4.5, task #3 arrives (event a 3 ) and is added to the waiting list with execution time equal to 2.5 time units. #3 #1 #2

23 Timed models of discrete event systems 21 At t 6 = 5.5, the time slice expires (event s). Task #2 is added to the waiting list with residual execution time equal to 0.5 time units. Task#1 is assigned a time slice. The residual execution time of task #1 amounts to 1.0 time unit, which is less than the time slice. Task #1 will be completed at t = 6.5. #2 #3 #1 At t 7 = 6.5, execution of task #1 terminates (event d 1 ). Task #1 leaves the system, and task #3 is assigned a time slice. The execution time of task #3 amounts to 2.5 time units, which is greater than the time slice. The assigned time slice will expire at t = 8.0. #2 #3 At t 8 = 8.0, the time slice expires (event s). Task #3 is added to the waiting list with residual execution time equal to 1.0 time unit. Task #2 is assigned a time slice. The residual execution time of task #2 amounts to 0.5 time units, which is less than the time slice. Task #2 will be completed at t = 8.5. #3 #2 At t 9 = 8.5, execution of task #2 terminates (event d 2 ). Task #2 leaves the system, and task #3 is assigned a time slice. The residual execution time of task #3 amounts to 1.0 time unit, which is less than the time slice. Task #3 will be completed at t = 9.5. #3 At t 10 = 9.5, execution of task #3 terminates (event d 3 ). Task #3 leaves the system. The system is now is empty.

24 Timed models of discrete event systems 22 a 1 s a 2 s a 3 s d 1 s d 2 d time Figure 3.2: Sample path determined by the RR queueing discipline. The sample path determined by the RR queueing discipline is shown in Figure 3.2. By comparing Figures 3.1 and 3.2, it can be observed that events d 1 and d 2 occur at times 5.0 and 7.0, when the FIFO queueing discipline is applied, and at times 6.5 and 8.5, when the RR queueing discipline is applied. Since the queueing discipline is part of the definition of the queueing system, we can conclude these event times cannot be decided independently of the system, i.e. they cannot be considered as system inputs. The lesson we learn from Example 3.1, is that event times may depend on the system, and therefore they cannot be taken as system inputs, in general. Event times are rather system outputs, for which suitable modeling mechanisms should be devised.

25 Bibliography [1] C. G. Cassandras and S. Lafortune. Introduction to discrete event systems. Springer, 2nd edition, 2008.