Discrete Event System Specification (DEVS)

1 The Random Package

package Random import Modelica.Math; constant Real NV_MAGICCONST = 4 * exp( -0.5 ) / sqrt( 2.0 ); type Seed=Integer[3]; function random "input random number generator with external storage of the seed" input Seed si "input random seed"; output Real x "uniform random variate between 0 and 1"; output Seed so "output random seed"; algorithm so[1] := rem((171 * si[1]),30269); so[2] := rem((172 * si[2]),30307); so[3] := rem((170 * si[3]),30323); // zero is a poor Seed, therfore substitute 1; if so[1] == 0 then so[1] := 1; end if; if so[2] == 0 then so[2] := 1; end if; if so[3] == 0 then so[3] := 1; end if; x := rem((so[1]/30269.0 +so[2]/30307.0 + so[3]/30323.0),1.0); end random; function normalvariate "normally distributed random variable" input Real mu "mean value"; input Real sigma "standard deviation"; input Seed si "input random seed"; output Real x "gaussian random variate"; output Seed so "output random seed"; protected Seed s1, s2; Real z, zz, u1, u2; algorithm s1 := si; u2 := 1; while true loop (u1,s2) := random(s1); (u2,s1) := random(s2); z := NV_MAGICCONST*(u1-0.5)/u2; zz := z*z/4.0; if zz <= (- Math.log(u2)) then break; end if; end while; x := mu + z*sigma; so := s1; end normalvariate; connector discreteConnector discrete Boolean dcon; end discreteConnector; end Random;

2 DEVS Model

We will now describe a fairly comprehensive example on how DEVS models can be represented in Modelica. One difference in Modelica is that we have a built-in time variable for absolute time rather than a local timer tlocal that is reset at the most recent event. This local timer is not really needed for modeling purposes since time is available, but is included to show the relation to traditional DEVS models. The facilities for model structuring and reuse are in general much greater in Modelica due to its object-oriented properties compared to traditional DEVS. In order to simplify comparisons this example connects model components such as servers or generators in a way that is analogous to traditional DEVS.

connector EventPort Boolean signal; end EventPort;
connector EventPortInput = input Boolean;
connector EventPortOutput = output Boolean;
partial model BasicDEVSTwoPort import Modelica.Constants.inf; input EventPortInput inp; // Input port for external events output EventPortOutput outp; // Output port for emitted events Boolean DEVSactive(start=false); // Start in non-active mode (phase) Real tResidual(start=inf); // Time until internal event (sigma) Real tlocal(start=0); // Local timer reset at events (e) discrete Real tNextEvent(start=0); // Time of next internal event discrete Real tLastEvent(start=0); // Time of most recent event equation tResidual = if tNextEvent<time then inf else tNextEvent-time; tlocal = if inp or tNextEvent<time then 0 else time - tLastEvent; when {inp, pre(tNextEvent)<time} then tLastEvent = time; // Save the time of this event end when; end BasicDEVSTwoPort;

A simple job shop model illustrates DEVS based modeling. A generator for arriving jobs is connected to a simple DEVS based server.

The testDEVSServer model connects a job generator to a simple DEVS-based server.

2.1 testDEVSServer Model

The testDEVSServer model contains two main items: a generateJobEvents generator component that generates arriving jobs at regular intervals, and a simpleDEVSServer server component being a very simple model of a job shop system that services arriving jobs with a normally distributed service time. The SignalView class instances are needed to widen sharp peaks for plotting.

model testDEVSServer GenerateJobEvents generateJobEvents(period=1.5,startTime=3) "generator"; SimpleDEVSServer simpleDEVSServer "server"; SignalView viewSignal1; // Just for plotting SignalView viewSignal2; // Just for plotting equation connect(generateJobEvents.outp, simpleDEVSServer.inp); connect(simpleDEVSServer.outp, viewSignal2.inp); // Just for plotting connect(generateJobEvents.outp, viewSignal1.inp); // Just for plotting end testDEVSServer;
model GenerateJobEvents "Generates Job events periodically" EventPortOutput outp; // Output port parameter Real startTime = 0; parameter Real period = 2; equation outp = sample(startTime,period); end GenerateJobEvents;

2.2 SimpleDEVSServer

The SimpleDEVSServer class is a fairly straightforward extension of the basic DEVS class BasicDEVSTwoPort. A randomly distributed service time variable servTime has been added with a normal distribution determined by parameters mean and stdev. The discrete randomSeed variable is needed to store the seed between calls to normalvariate, which as all Modelica functions is a mathematical function whose value is completely determined by its inputs. The absolute value of servTime is used to guarantee positive service times.

model SimpleDEVSServer extends BasicDEVSTwoPort; Boolean internalEvent; Boolean externalEvent; parameter Real mean = 2.0 "mean of job service time"; parameter Real stdev = 0.5; discrete Real servTime "time needed to service current job"; discrete Random.Seed randomSeed(start={23,87,187}); equation internalEvent = pre(tNextEvent)<time; externalEvent = inp; outp = edge(internalEvent) and pre(DEVSactive); when {externalEvent,internalEvent} then (servTime,randomSeed) = Random.normalvariate(mean,stdev, pre(randomSeed)); // * External event – job arrives, start service if edge(externalEvent) and not pre(DEVSactive) then DEVSactive = true; tNextEvent = time + abs(servTime);//abs of normal distr servTime // * External event – job arrives but is not serviced since busy elseif edge(externalEvent) and pre(DEVSactive) then DEVSactive = true; tNextEvent = pre(tNextEvent); // Next internal event unchanged // * Internal event – servicing current job finished else // Here effectively: edge(internalEvent) DEVSactive = false; tNextEvent = pre(tNextEvent); // time now >= tNextEvent end if; end when; end SimpleDEVSServer;
model SignalView "Increase width of sample trigger signals" EventPortInput inp; EventPortOutput outp; parameter Real width=0.001; discrete Real T0; equation when inp then T0 = time; end when; outp = (time>=T0) and (time<T0+width); end SignalView;

Remember the following two connect equations from the testDEVSServer model. When using the currently available Modelica tools it is necessary to employ some kind of SignalView peak widening facility in order to plot sharp peaks that have zero width, i.e. have no time duration. Otherwise these peaks would not be visible in the diagram.

class Conn1 equation connect(simpleDEVSServer.outp, viewSignal2.inp); end Conn1;
class Conn2 equation connect(generateJobEvents.outp, viewSignal1.inp); end Conn2;

2.3 Simulation of testDEVSServer

We simulate the testDEVSServer model to be able to view some of the input and output signals. The plot agrees with the specification of the generateJobEvents instance, where the first event is generated at time = 3 and subsequent events are generated with a periodicity 1.5 seconds.

loadModel(Modelica)
simulate( testDEVSServer, stopTime=10 )

2.3.1 Plot

plot( { viewSignal1.outp, viewSignal2.outp, simpleDEVSServer.DEVSactive } )

Diagram 1: viewSignal1.outp.signal, viewSignal2.outp.signal and DEVSactive

2.3.2 Plot

plot( { viewSignal1.outp, viewSignal2.outp } )

Diagram 2: - viewSignal1.outp.signal and viewSignal2.outp.signal

2.3.3 Plot

plot( viewSignal1.outp )

Diagram 3: - The output signal outp of the generateJobEvents generator component in testDEVSServer.

2.3.4 Plot

plot( simpleDEVSServer.tNextEvent )

Diagram 4: - The output signal simpleDEVSServer.tNextEvent.

2.3.5 Plot

plot( viewSignal2.outp)

Diagram 5: - A signal is emitted through simpleDEVSServer.outp when completing \\ service of a job, here Job 1 and Job 3.

2.3.6 Plot

plot( simpleDEVSServer.DEVSactive )

Diagram 6: - The variable simpleDEVSServer.DEVSactive is true while a job is being serviced.

2.3.7 Plot

plot( simpleDEVSServer.servTime )

Diagram 7: - The service time variable simpleDEVSServer.servTime obtains new values at events.

2.3.8 Plot

plot( simpleDEVSServer.internalEvent )

Diagram 8: - The output signal simpleDEVSServer.internalEvent.

3 A Coupled Pipelined DEVS Model

Several basic DEVS model components can be connected together to form a coupled model, which itself is

a DEVS model. This is illustrated by the Modelica realization of a simple DEVS coupled model by placing

three simple servers in series to form a three-server pipeline.

model ThreeServerPipeline EventPortInput extInSignal; EventPortOutput outSignal; SimpleDEVSServer S1(mean=2.0); SimpleDEVSServer S2(mean=2.5); SimpleDEVSServer S3(mean=1.5); equation connect(extInSignal,S1.inp); connect(S1.outp, S2.inp); connect(S2.outp, S3.inp); connect(S3.outp, outSignal); end ThreeServerPipeline;

The three-server coupled DEVS pipeline model can be tested using the following model

model testPipeline ThreeServerPipeline pipeLine; GenerateJobEvents jobgenerator(period=4); SignalView viewJobGen; // Just for plotting SignalView viewS1; // Just for plotting SignalView viewS2; // Just for plotting SignalView viewS3; // Just for plotting equation connect(jobgenerator.outp, pipeLine.extInSignal); connect(jobgenerator.outp, viewJobGen.inp); // Just for plotting connect(pipeLine.S1.outp, viewS1.inp); // Just for plotting connect(pipeLine.S2.outp, viewS2.inp); // Just for plotting connect(pipeLine.S3.outp, viewS3.inp); // Just for plotting end testPipeline;
simulate(testPipeline,stopTime=10)
plot({viewS3.outp,pipeLine.S3.DEVSactive,viewS1.outp})