1.2.1. Service-Desk-0: Modeling queueLength as a global variable

As discussed above, the simplest model for the service desk problem with maximum queue length statistics (available in the Sim4edu library as Service-Desk-0) has only one global variable: queueLength, which is a non-negative integer, and a global function for computing the random service time, but no object type.

An information model for Service-Desk-0 consists of a special class for defining model variables and functions, and two classes for defining the event types Arrival and Departure, as shown in Figure 1-1. An information design model for Service-Desk-0.

In addition to an information design model for defining the simulation system's state structure, we also need to make a process design model for defining the dynamics of the simulation system. The dynamics of a system consists of events triggering state changes and follow-up events. A process model can be expressed with the help of event rules, which define what happens when an event (of a certain type) occurs, or, more specifically, which state changes and which follow-up events are caused by an event of that type.

Event rules can be expressed with the help of a process model diagram or in pseudo-code, or in a simulation or programming language. The following Event Graph provides a process design model for the Service-Desk-0 simulation scenario. Circles represent events (or, more precisely, event types) and arrows, which may be annotated with a delay expression, such as +serviceTime(), represent event scheduling relationships. An arrow with a mini-diamond at its source end represents a conditional event scheduling relationship where the condition is expressed in brackets below or above the arrow.

Event Graphs have originally been proposed by L. Schruben (1983). Their visual syntax has been improved and harmonized with the business process modeling language BPMN in the Discrete Event Process Modeling Notation (DPMN) proposed by Wagner (2018) and more thoroughly described in the book Discrete Event Simulation Engineering.

The following table shows the two event rules defined by the above Event Graph, expressed in pseudo-code.

INCREMENT queueLength
IF queueLength = 1 THEN
  sTime := serviceTime()
  SCHEDULE Departure @ (t + sTime)

DECREMENT queueLength
IF queueLength > 0 THEN
  sTime := serviceTime()
  SCHEDULE Departure @ (t + sTime)

1.2.2. Service-Desk-1: Modeling queueLength as an attribute

In our extended model (Service-Desk-1) we represent the state variable queueLength as an attribute of an object type ServiceDesk. This results in a model with three classes, the object class ServiceDesk with an attribute queueLength, and the event classes Arrival and Departure, both with a reference property serviceDesk for referencing the service desk at which an event occurs. When we also want to compute the service utilization statistics, we need to add an attribute serviceTime to the Departure class for being able to update the service utilization statistics when a customer departs.

Both event types, Arrival and Departure, now have a many-to-one association with the object type ServiceDesk. This expresses the fact that any such event occurs at a particular service desk, which participates in the event. This association is implemented in the form of a reference property serviceDesk in each of the two event types, as shown in Figure 1-3. An information design model for Service-Desk-1.

In addition to an information model, we need to make a process model, which captures the dynamics of the service desk system consisting of arrival and departure events triggering state changes and follow-up events. The following DPMN Process Diagram provides a process design model for the Service-Desk-1 simulation scenario. As in Event Graphs, circles represent event types and arrows represent event scheduling relationships. DPMN extends Event Graphs by adding object rectangles, attached to event circles, representing state change patterns for objects that are affected by events of that type.

The following table shows the two event rules defined by the DPMN diagram, which now account for the fact that both types of events occur at a particular service desk that is referenced by the event expression parameter sd.

INCREMENT sd.queueLength
IF sd.queueLength = 1 THEN
  sTime := ServiceDesk.serviceTime()
  SCHEDULE Departure( sTime, sd) @(t + sTime)

DECREMENT sd.queueLength
IF sd.queueLength > 0 THEN
  sTime := ServiceDesk.serviceTime()
  SCHEDULE Departure( sTime, sd) @(t + sTime)

2.1.1. Time Granularity

A model's time granularity is the time delay until the next moment, such that the model does not allow considering an earlier next moment. This is captured by the simulation parameter nextMomentDeltaT used by the simulator for scheduling immediate events with a minimal delay. When a simulation model is based on discrete time, nextMomentDeltaT is set to 1, referring to the next time point. When a simulation model is based on continuous time, nextMomentDeltaT is set to the default value 0.001, unless the model parameter sim.model.nextMomentDeltaT is explicitly assigned in the simulation.js file.

2.1.2. Time Progression

An important issue in simulation is the question how the simulation time is advanced by the simulator. The OES paradigm supports next-event time progression and fixed-increment time progression, as well as their combination.

An OESjs-Core1 model with fixed-increment time progression has to define a suitable periodic time event type, like EachSecond or EachDay in the form of an exogenous event type with a recurrence function returning the value 1. Such a model can be used for

1. modeling continuous state changes (e.g., objects moving in a continuous space), or
2. making a discrete model that abstracts away from explicit events and uses only implicit periodic time events ("ticks"), which is a popular approach in social science simulation.

Examples of discrete event simulation models with fixed-increment time progression and no explicit events are the Schelling Segregation Model and the Susceptible-Infected-Recovered (SIR) Disease Model.

2.2.1. Model Variables and Functions

In the simple model of a service desk discussed in the previous section, we define one (global) model variable, queueLength, one model function, serviceTime(), and two event types, as shown in the following class diagram:

Notice that this model does not define any object type, which implies that the system state is not composed of the states of objects, but of the states of model variables, here it consists of the state of the model variable queueLength. The discrete random variable for modeling the random variation of service durations is implemented as a model function serviceTime shown in the Global Variables and Functions class. It samples integers between 2 and 4 from the empirical probability distribution Frequency{ 2:0.3, 3:0.5, 4:0.2}. The model can be coded with OESjs-Core1 in the following way:

// (global) model variable
sim.model.v.queueLength = 0;
// (global) model function
sim.model.f.serviceTime = function () {
  var r = math.getUniformRandomInteger( 0, 99);
  if ( r < 30) return 2;         // probability 0.30
  else if ( r < 80) return 3;    // probability 0.50
  else return 4;                 // probability 0.20
};

You can run this Service-Desk-0 model from the project's GitHub website. An example of a run of this model is shown in the following simulation log:

2.2.2. Object Types

Object types are defined in the form of classes. Consider the object type ServiceDesk defined in the following Service-Desk-1 model:

While queueLength was defined as a global variable in the Service-Desk-0 model, it is now defined as an attribute of the object type ServiceDesk:

class ServiceDesk extends oBJECT {
  constructor({ id, name, queueLength}) {
    super( id, name);
    this.queueLength = queueLength;
  }
  static serviceTime() {
    var r = math.getUniformRandomInteger( 0, 99);
    if ( r < 30) return 2;         // probability 0.3
    else if ( r < 80) return 3;    // probability 0.5
    else return 4;                 // probability 0.2
  }
}
ServiceDesk.labels = {"queueLength":"qLen"};  // for the log

Notice that, in OESjs, object types are defined as subtypes of the pre-defined class oBJECT, from which they inherit an integer-valued id attribute and an optional name attribute. When a property has a label (defined by the class-level (map-valued) property labels), it is shown in the simulation log.

You can run this simulation model from the project's GitHub website.

2.2.3. Event Types

In OES, there is a distinction between two kinds of events:

1. events that are caused by other event occurrences during a simulation run;
2. exogenous events that seem to happen spontaneously, but may be caused by factors, which are external to the simulation model.

Here is an example of an exogenous event type definition in OESjs-Core1:

class CustomerArrival extends eVENT {
  constructor({ occTime, serviceDesk}) {
    super( occTime);
    this.serviceDesk = serviceDesk;
  }
  onEvent() {
    ...
  }
  ...
}

The definition of the CustomerArrival event type includes a reference property serviceDesk, which is used for referencing the service desk object at which a customer arrival event occurs. In OESjs, event types are defined as subtypes of the pre-defined class eVENT, from which they inherit an attribute occTime, which holds the occurrence time of an event. As opposed to objects, events do normally not have an ID, nor a name.

Each event type needs to define an onEvent method that implements the event rule for events of the defined type. Event rules are discussed below.

Exogenous events occur periodically. They are therefore defined with a recurrence function, which provides the time in-between two events (often in the form of a random variable). The recurrence function is defined as a class-level ("static") method:

class CustomerArrival extends eVENT {
  ...
  static recurrence() {
    return math.getUniformRandomInteger( 1, 6);
  }
}

Notice that the recurrence function of CustomerArrival is coded with the library method math.getUniformRandomInteger, which allows sampling from discrete uniform probability distribution functions.

In the case of an exogenous event type definition, a createNextEvent method has to be defined for assigning event properties and returning the next event of that type, which is scheduled by invoking the recurrence function for setting its ocurrenceTime and by copying all participant references (such as the serviceDesk reference).

class CustomerArrival extends eVENT {
  ...
  createNextEvent() {
    return new CustomerArrival({
      occTime: this.occTime + CustomerArrival.recurrence(),
      serviceDesk: this.serviceDesk
    });
  }
  static recurrence() {...}
}

When an OE simulator processes an exogenous event e of type E, it automatically schedules the next event of type E by invoking the createNextEvent method on e, if it is defined, or, otherwise by duplicating e and resetting its occurrence time by invoking E.recurrence().

For an exogenous event type, it is an option to define a maximum number of event occurrences by setting the static attribute maxNmrOfEvents, as in the following example:

CustomerArrival.maxNmrOfEvents = 3;

The second event type of the Service-Desk-1 model, Departure, is an example of a type of caused events:

class CustomerDeparture extends eVENT {
  constructor({ occTime, serviceDesk}) {
    super( occTime);
    this.serviceDesk = serviceDesk;
  }
  onEvent() {
    ...
  }
}

A caused event type does neither define a recurrence function nor a createNextEvent method.

2.2.4. Event Rules

An event rule for an event type defines what happens when an event of that type occurs, by specifying the caused state changes and follow-up events. In OESjs, event rules are coded as onEvent methods of the class that implements the event type. These methods return a set of events (more precisely, a set of JS objects representing events).

Notice that in the DES literature, event rule methods are called event routines.

For instance, in the CustomerArrival class, the following event rule method is defined:

class CustomerArrival extends eVENT {
  ...
  onEvent() {
    var followupEvents=[];
    // increment queue length due to newly arrived customer
    this.serviceDesk.queueLength++;
    // update statistics
    sim.stat.arrivedCustomers++;
    if (this.serviceDesk.queueLength > sim.stat.maxQueueLength) {
      sim.stat.maxQueueLength = this.serviceDesk.queueLength;
    }
    // if the service desk is not busy
    if (this.serviceDesk.queueLength === 1) {
      followupEvents.push( new CustomerDeparture({
        occTime: this.occTime + ServiceDesk.serviceTime(),
        serviceDesk: this.serviceDesk
      }));
    }
    return followupEvents;
  }
}

The context of this event rule method is the event that triggers the rule, that is, the variable this references a JS object that represents the triggering event. Thus, the expression this.serviceDesk refers to the service desk object associated with the current customer arrival event, and the statement this.serviceDesk.queueLength++ increments the queueLength attribute of this service desk object (as an immediate state change).

The following event rule method is defined in the CustomerDeparture class.

class CustomerDeparture extends eVENT {
  ...
  onEvent() {
    var followupEvents=[];
    // decrement queue length due to departure
    this.serviceDesk.queueLength--;
    // update statistics
    sim.stat.departedCustomers++;
    // if there are still customers waiting
    if (this.serviceDesk.queueLength > 0) {
      // start next service and schedule its end/departure
      followupEvents.push( new CustomerDeparture({
        occTime: this.occTime + ServiceDesk.serviceTime(),
        serviceDesk: this.serviceDesk
      }));
    }
    return followupEvents;
  }
}

2.2.5. Event Priorities

An OES model may imply the possibility of several events occurring at the same time. Consequently, a simulator (like OESjs) must be able to process simultaneous events. In particular, simulation models based on discrete time may create simulation states where two or more events occur at the same time, but the model's logic requires them to be processed in a certain order. Defining priorities for events of a certain type helps to control the processing order of simultaneous events.

Consider an example model based on discrete time with three exogenous event types StartOfMonth, EachDay and EndOfMonth, where the recurrence of StartOfMonth and EndOfMonth is 21, and the recurrence of EachDay is 1. In this example we want to control that on simulation time 1 + i * 21 both a StartOfMonth and an EachDay event occur simultaneously, but StartOfMonth should be processed before EachDay, and on simulation time 21 + i * 21 both an EndOfMonth and an EachDay event occur simultaneously, but EndOfMonth should be processed after EachDay. This can be achieved by defining a high priority, say 2, to StartOfMonth, a middle priority, say 1, to StartOfMonth, and a low priority, say 0, to EndOfMonth.

Event priorities are defined as class-level properties of event classes in the event type definition file. Thus, we would define in StartOfMonth.js:

StartOfMonth.priority = 2;

and in EachDay.js:

EachDay.priority = 1;

and finally in EndOfMonth.js:

EndOfMonth.priority = 0;

2.2.6. Library Methods for Sampling Probability Distribution Functions

Random variables are implemented as methods that sample specific probability distribution functions (PDFs). Simulation frameworks typically provide a library of predefined parametrized PDF sampling methods, which can be used with one or several (possibly seeded) streams of pseudo-random numbers.

The OESjs simulator provides the following predefined parametrized PDF sampling methods:

The OESjs library rand.js supports both unseeded and seeded random number streams. By default, its PDF sampling methods are based on an unseeded stream, using Marsaglia’s high-performance random number generator xorshift that is built into the Math.random function of modern JavaScript engines.

A seeded random number stream, based on David Bau's seedable random number generator seedrandom, can be obtained by setting the scenario parameter sim.scenario.randomSeed to a positive integer value.

Additional streams can be defined and used in the following way:

var stream1 = new Random( 1234);
var stream2 = new Random( 6789);
var service1Duration = stream1.exponential( 0.5);
var service2Duration = stream2.exponential( 1.5);

Avoid using JavaScript's built-in Math.random in simulation code. Always use rand.uniform, or one of the other sampling functions from the rand.js library described above, for generating random numbers.

Otherwise, using a random seed does not guarantee reproducible simulation runs!

2.3.1. Model Parameters

While model variables are state variables whose values are changed as an effect of an event occurrence, model parameters are not part of the dynamic state of the simulated system, but are rather used for providing values that can only be read during a simulation run. The main purpose of model parameters is to allow parameter variation experiments.

2.3.2. Simulation Scenario Parameters

A few simulation parameters are predefined as attributes of the simulation scenario. The most important ones are:

• durationInSimTime - this attribute allows defining the duration of a simulation run; which runs forever when this attribute s not set;
• randomSeed: Setting this optional parameter to a positive integer allows to obtain a specific fixed random number sequence (generated by a random number generator). This can be used for performing simulation runs with the same (repeated) random number sequence, e.g., for testing a simulation model by checking if expected results are obtained.

2.3.3. Initial State

Defining an initial state means:

1. assigning initial values to global model variables, if there are any;
2. defining which objects exist initially, and assigning initial values to their properties;
3. defining which events are scheduled initially.

A setupInitialState procedure takes care of these initial state definitions. A global model variable is initialized in the following way:

sim.scenario.setupInitialState = function () {
  // Initialize model variables
  sim.model.v.queueLength = 0;
  // Create initial objects
  ...
  // Schedule initial events
  ...
};

An initial state object is created by instantiating an object type of the simulation model with suitable initial property values, as shown in the following example:

sim.scenario.setupInitialState = function () {
  // Initialize model variables
  ...
  // Create initial objects
  const serviceDesk1 = new ServiceDesk({id: 1, queueLength: 0});
  // Schedule initial events
  ...
};

Notice that object IDs are positive integers.

Instead of assigning a fixed value to a property like queueLength for defining an object's initial state, as in queueLength: 0, we can also assign it an expression, as in queueLength: Math.round(12/30).

An initial event is scheduled (or added to the Future Events List), as shown in the following example:

sim.scenario.setupInitialState = function () {
  // Initialize model variables
  ...
  // Create initial objects
  const desk1 = new ServiceDesk({id: 1, queueLength: 0});
  // Schedule initial events
  sim.schedule( new CustomerArrival({occTime:1, serviceDesk: desk1}));
};

Initial objects or events can be parametrized with the help of model parameters.

2.3.4. Defining Alternative Scenarios with Different Initial States

For running a model on top of different initial states, one can define a list of scenarios, each with its own setupInitialState procedure:

sim.scenarios[1] = {
  scenarioNo: 1,
  title: "Scenario with two service desks",
  setupInitialState: function () {
    // Create initial objects
    var sD1 = new ServiceDesk({id: 1, queueLength: 0}),
        sD2 = new ServiceDesk({id: 2, queueLength: 0});
    // Schedule initial events
    sim.FEL.add( new CustomerArrival({occTime: 1, serviceDesk: sD1}));
    sim.FEL.add( new CustomerArrival({occTime: 2, serviceDesk: sD2}));
  }
};
sim.scenarios[2] = {...}

Before running a simulation, a specific scenario can be chosen in the user interface.

Do not set model parameters in the setupInitialState procedure! This would interfere with parameter variation experiments in which the same parameter(s) are used.

Simple Experiments

A simple experiment type is defined with a sim.experimentType record on top of a model by defining (1) the number of replications and (2) possibly a list of seed values, one for each replication. The following code shows an example of a simple experiment type definition:

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sim.experimentType = {
  title: "Simple Experiment with 10 replications, each running for 1000 time units (days)",
  nmrOfReplications: 10,
  seeds: [123, 234, 345, 456, 567, 678, 789, 890, 901, 1012]
};

Running this simple experiment means running the underlying scenario 10 times, each time with another random seed, as specified by the list of seeds. The resulting statistics are composed of the statistics for each replication complemented with summary statistics listing averages, standard deviations, min/max values and 95% confidence intervals, as shown in the following example:

When no seeds are defined, the experiment is run with implicit random seeds using JavaScript's built-in random number generator, which implies that experiment runs are not reproducible.

Parameter Variation Experiments

A parameter variation experiment is defined with (1) a number of replications, (2) a list of seed values (one for each replication), and (3) one or more experiment parameters.

An experiment parameter must have the same name as the model parameter to which it refers. It defines a set of values for this model parameter, either using a values field or a combination of a startValue and endValue field (and stepSize for a non-default increment value) as in the following example.

The following code shows an example of a parameter variation experiment definition (on top of the Inventory-Management simulation model):

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sim.experimentTypes[1] = {
  id: 1,
  title: "Parameter variation experiment for exploring reorderInterval and targetInventory",
  nmrOfReplications: 10,
  seeds: [123, 234, 345, 456, 567, 678, 789, 890, 901, 1012],
  parameterDefs: [
    {name:"reviewPolicy", values:["periodic"]},
    {name:"reorderInterval", values:[2,3,4]},
    {name:"targetInventory", startValue:80, endValue:100, stepSize:10},
  ]
};

Notice that this experiment definition defines 9 experiment scenarios resulting from the combinations of the values 2/3/4 and 80/90/100 for the parameters reorderInterval and targetInventory. Running this parameter variation experiment means running each of the 9 experiment scenarios 10 times (each time with another random seed, as specified by the list of seeds). The resulting statistics, as shown in the following table, is computed by averaging all statistics variables defined for the given model.

Storage and Export of Experiment Results

In OESjs-Core1, an experiment's output statistics data is stored in a browser-managed database using JavaScript's IndexedDB technology. The name of this database is the same as the name of the simulation model. It can be inspected with the help of the browser's developer tools, which are typically activated with the key combination [Shift]+[Ctrl]+[I]. For instance, in Google's Chrome browser, one has to go to Application/Storage/IndexedDB.

The experiment statistics database consists of three tables containing data about (1) experiment runs, (2) experiment scenarios, and (3) experiment scenario runs, which can be exported to a CSV file.

Visualizing Objects

For being able to observe objects in a simulation run, they have to be visualized in some form. OESjs supports both the visualization of objects in space in spatial models and of objects in non-spatial models.

In a visualization of a non-spatial model, such as the ServiceDesk-1 model, all object views have to be explicitly positioned in an observation canvas.

In the case of our ServiceDesk-1 model, we may, for instance, visualize the service desk using either an image or simply a fixed-size rectangle, and its queue in the form of a growing and shrinking bar.

Two-dimensional visualizations can be obtained by using the web technology of Scalable Vector Graphics (SVG) in the definition of the observation UI. For defining an observation UI with SVG-based visualization, the following settings have to be made:

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sim.config.obs.ui.type = "SVG";
sim.config.obs.ui.canvas.width = 600;
sim.config.obs.ui.canvas.height = 300;

In addition, one can define a CSS style for the canvas element in the following way, e.g., for setting a background color or background image:

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sim.config.obs.ui.canvas.style = "background-color: azure";

Then we can define fixed elements of a visualization, giving each one a name (here: "desk") and defining an SVG shape with attributes and a CSS style:

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sim.config.obs.ui.fixedElements = {
  "desk": {
    shapeName: "rect",
    shapeAttributes: { x: 350, y: 200, width: 50, height: 30},
    style: "fill:brown; stroke-width:0"
  }
};

The main issue in visualization is to visualize simulation objects by defining suitable views for them and then map some of their attributes to suitable visual parameters such as color, shape width and height, etc. A view can be defined either for all instances of an object type or for specific instances only.

For instance, we may want to visualize the waiting line of the object "serviceDesk1" in the form of a rectangle and map the service desk's queueLength attribute to the width of that rectangle, as in the following object view definition:

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sim.config.obs.ui.objectViews = {
  "serviceDesk1": {  // the name of the object
    visualizationAttributes: ["queueLength"],
    attributesViewItemsRecords: [
      { attributes:["queueLength"],
        viewItems: [  // a list of 2 view elements for the object "serviceDesk1"
          {shapeName: "rect",  // a rectangle
           shapeAttributes: {  // left-upper corner (x,y) as well as width and height
             x: sd => Math.max( 0, 330 - sd.queueLength * 20),
             width: sd => Math.min( 300, sd.queueLength * 20),
             y: 150, height: 80 },
           style:"fill:yellow; stroke-width:0"},
          {shapeName: "text",
           shapeAttributes: {x: 325, y: 250, textContent: sd => sd.queueLength},
           style: "font-size:14px; text-anchor:middle"}
    ]
  }
};

Notice that the view consists of two elements: a rectangle and an attached text displaying the queue length. In the view definition, certain attributes are assigned a fixed value, while others are assigned a JS function expression, which codes the mapping of object attributes to visual parameters.

Alternatively, instead of defining a view for a specific service desk object, we can also define a view for all service desk objects, like so

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sim.config.obs.ui.objectViews = {
  "ServiceDesk": [  // the name of the object type/class
    { shapeName: "rect",  // a rectangle defined by 
      shapeAttributes: {  // left-upper corner (x,y) as well as width and height
        x: function (sd) {return Math.max( 0, 330 - sd.queueLength * 20);},
        width: function (sd) {return Math.min( 300, sd.queueLength * 20);},
        y: 150, height: 80
      },
      style:"fill:yellow; stroke-width:0"
    }
};

Visualizing Events