Introduction to Event-based Modeling & Simulation Chapter 4 “Little people discuss other people. Average people discuss events. Big people discuss ideas.”

Introduction to Event-based Modeling & Simulation

Chapter 4

“Little people discuss other people. Average people discuss events. Big people discuss ideas.”

– quoted by R. E. Kalman

Last revision July 14, 2013

For more information on this book and materials, visit the official website, http://www.vms-technology.com/book/

Modeling and Simulation of Discrete-Event Systems

Chapter 4 – Introduction to Event-based M&S

1. Introduction (1/1)

1. Introduction

2. Modeling and Simulation of Single Server System

3. Execution Rules and Specifications of Event Graph

4. Event Graph Modeling Templates

5. Event Graph Modeling Examples

6. Execution of Event Graph Models with SIGMA®

7. Developing Your Own Event Graph Simulator

If something that happened results in some meaningful changes it is called an event. If we can identify the logical and temporal relationships among those events, we understand our present situation better and may even be able to predict the future.

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2. Modeling & Simulation of Single Server System (0/2)

Table of Contents

2-1. Event-based Modeling of Single Server System

2-2. Simulation of Single Server System

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2-1. Event-based Modeling of Single Server System

Reference Model

Formal Model (Event Graph)

Event Transition table

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2-2. Simulation of Single Server System

Event Graph Model

SIGMA/EGS Your own simulator

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3. Execution Rules and Specifications of Event Graph (0/3)

Table of Contents

3-1. Event Graph Execution Rules

3-2. Tabular Specification of Event Graph Model

3-3. Algebraic Specification of Event Graph Model

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3. Execution Rules and Specifications of Event Graph (1/3)3-1. Event Graph Execution Rules

1) Execution of an Event having a Scheduling Edge

“Whenever the originating event E1 occurs, the state s changes according to fE1(s). Then, if the edge condition c1 is true, the destination event E3 is scheduled to occur after a time delay of t1.”.

2) Execution of an Event having a Canceling Edge

“Whenever the originating event E2 occurs, the state‘s’ changes to fE2(s). Then, if the edge condition c2 is true, the scheduled event E3 is canceled immediately.”

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3. Execution Rules and Specifications of Event Graph (2/3)3-2. Tabular Specification of Event Graph Model

“Whenever E1 occurs, the state‘s’ changes to fE1(s). Then, 1) if the edge condition c1 is true, E2 is scheduled to occur after t1; 2) if the edge condition c2 is true, E3 is canceled immediately.”

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3. Execution Rules and Specifications of Event Graph (3/3)3-3. Algebraic Specification of Event Graph Model

An event graph is a directed graph consisting of a set of event vertices V, a set of directed edges E, and a set of state variables S. The edge set E represents the temporal and logical relationships between a pair of vertices. Associated with each vertex v V is ∈ state transition function fv; associated with each edge e E are an ∈ edge condition ce C, ∈ time delay de D, and ∈ action type ae A (schedule or cancel). Thus, an ∈ event graph model M can be defined as a 7-tuple structure.

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1. M = <V, E, S, F, C, D, A>, where

2. V = {v}: set of event vertices

3. E = {eod= (vo, vd)}: set of edges // vo: originating event; vd: destination event

4. S = {s}: set of state variables

5. F = {fv: S → S v V}: set of state transition functions associated with V∀ ∈

6. C = {ce: S→[0,1] e E}: set of conditions associated with E∀ ∈

7. D = {de R0 e E}: set of time delays associated with E∀ ∈

8. A = {ae [scheduling, canceling] e E}: action type set∈ ∀ ∈

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H/W-1(Exercise 4.1): Specify the algebraic components of the event graph in Figure 4.5.



4. Event Graph Modeling Templates (1/6)

Table of Contents

4-1. Single Queue Models (1) Multiple Server Models (Flexible multi-server Template) (2) Limited Waiting Space Model (Balking Template) (3) Impatient Customer Model (Reneging with balking) (4) Non-stationary Arrival Rate Model (Fluctuating arrival Template) (5) Batched Service Models (6) Assembly Operation Model (7) Resource Priority Model (8) A Resource Failure Models

4-2. Tandem Queue Models (1) Limited-buffer Tandem Queue Model (2) Time-constrained Processing Tandem Queue Model)

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4-1. Single Queue Models

Baseline Model

1) Multiple Server Models (Flexible multi-server Template)- The baseline model becomes a (fixed) multi-server model if M= n in the initialize-event box.

2) Limited Waiting Space Model (Balking Template)

If the limited-capacity waiting space is full (Q=c), an arriving job may not be able to enter the system and leave the system for good, which is referred to as balking.

- Flexible multi-server model if the number of servers M = n(t) varies over time.

H/W-2(Exercise 4.2): Simplify the baseline model Figure 4.1(b) by removing the ‘Load’ event.

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4-1. Single Queue Models - Continued

3) Impatient Customer Model (Reneging with balking)The entered customers may choose to leave it if they have waited too long, which is often called reneging in queuing theory. A Leave event is introduced to manage the reneging situation: (1) every Enter event schedules a Leave event to occur after tr minutes; (2) every Unload event cancels the oldest Leave event that has been scheduled, if there is one; and (3) every Leave event decreases Q by one (denoting the reneging customer).

4) Non-stationary Arrival Rate Model (Fluctuating arrival Template)

The next Arrive event is scheduled to occur after an inter-arrival time ta with a bounding arrival rate λ*, and the Arrive event will schedule an Enter event only when the thinning test is passed (i.e. U < λ(t)/λ*).

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Exercise 4.3. Construct a single server system event graph for reneging without balking.




4-1. Single Queue Models - Continued 5) Batched Service Models

Batched service occurs when a batch of jobs is processed at the same time. In general, there are maximum number (b) and minimum number (a) of jobs that can be processed at a time, which is denoted as ‘a ≤ J ≤ b’ where J is the actual number of jobs in a batched service. If a = b, we have a full-batched service; if a < b, we have a partial-batched service.

6) Assembly Operation Model

Parts of type j for j=1, 2 arrive at the system every t j minutes, and a pair of parts, one from each type, are assembled together by a machine. Let Q denote number of part pairs and P j denote number of parts of type j, then the assembly operation is modeled as follows:

H/W-3(Exercise 4.4): Revise the event graph presented in Figure 4.10 to make it a partial batched service model

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4-1. Single Queue Models - Continued7) Resource Priority ModelWhen there are two servers having different priorities, a customer is served by the high priority server (M1) if it is free. The customer is directed to the low priority server only when M1 is busy

8) A Resource Failure Models

Failures may occur only when the server is busy and the job is discarded. Assume that service time S = 10, repair time R = 50, and inter-failure time F = 1000. The remaining time to failure (tf) is updated every time the server completes a cycle.

Failures may occur even when the server is idle and the job is discarded. A Fail event with an inter-failure time tf and a Repair event with a repair time tr are introduced. The Fail event will cancel the scheduled Unload event and schedule a Repair event to occur after tr minutes.

H/W-4(Exercise 4.6): Modify the event graph Figure 4.12 to make it a queue length balanced line.

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4-2. Tandem Queue Models

Baseline Model

1) Limited-buffer Tandem Queue Model

If the buffer after a machine has a limited capacity, a finished job may not be unloaded from the machine when the buffer is full. This situation is referred to as blocking.

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4-2. Tandem Queue Models - Continued

2) Time-constrained Processing Tandem Queue Model

A processing situation where a job that had been processed on a machine (M1) has to start the next processing step on the next machine (M2) within a time-out limit (to) is called a time constrained processing. The time-out job is discarded (or sent back to M1 for reprocessing).

H/W-5(Exercise 4.8): Modify the event graph Figure 4.18 so that the time out jobs are reprocessed.

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5. Event Graph Modeling Examples (0/11)

Table of Contents

5-1. Simple Service Shop (Flexible Multi-Server System) Modeling

5-2. Car Repair Modeling

5-3. Project Management Modeling

5-4. Conveyor-driven Serial Line Modeling

5-5. Inline-type Manufacturing Cell Modeling

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5-1. Simple Service Shop (Flexible Multi-Server System) ModelingIn a flexible multi-server system, the resource levels change over time in order to cope with the changes in arrival rates. Let λ(t) and n(t) denote arrival rates and number of servers at time t, respectively; then, by combining the flexible multi-server template and the fluctuating arrival template, an event graph model of a flexible multi-server system with fluctuating arrival rates can be constructed as follows.

Flexible multi-server template

Fluctuating arrival template

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5-2. Car Repair ModelingThe entities are the cars coming in for repair and the resources are technicians and repairmen. There are three types of activities: Fasten, Inspect, and Repair, with processing times t1, t2 and t3, respectively. The fasten operation is performed by a technician; an inspection operation needs both a technician and a repairman; a repair operation is handled by a repairman..

under a same operator policy

without the same operator policy?

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5-3. Project Management ModelingShown below is an AON (activity-on-node) PERT diagram involving 9 activities (A1~A9) that serves as a reference model of the project management problem. A node denotes an activity, and an edge represents the finish-start precedence relation between the two nodes (i.e., the first activity has to be finished before starting the second activity). Indicated in each node are the activity ID (A j), activity time (tj), and the critical resource (Rk) needed for each activity (non-critical resources are not explicitly identified).

Event Graph ModelWithoutResourceConstraints

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5-4. Conveyor-driven Serial Line ModelingConsider a 3-stage serial production line shown in Fig.4.30 consisting of three work stations (WS) connected by conveyors (CV). Base-parts (jobs) stored in the input buffer (Buffer-I) are moved along the line, and sub-parts are assembled to the base-part at each work station. The assembled base-parts (i.e., products) are stored in the output buffer (Buffer-O). Entities in the system are the base-parts, and resources are the WSs, CVs and Buffers. Activities are the production operations at the WSs and the transport operations by the conveyors.

Mj = workstation-j status (1: free, 0: busy); Bj = blocking of workstation-j (1: blocked), Qj = total number of jobs at CV-j, and Rj = number of jobs ready at CV-j (that have been transported).

pj = processing time at WSj; tj = transport time of the conveyor-j; cj = capacity of CVj (c1 = cn+1 = ∞).

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5-5. Inline-type Manufacturing Cell Modeling

In the photolithography cell found in a modern TFT-LCD (thin film transistor-liquid crystal display) panel Fab, the jobs are glasses that go through photoresist (PR) coating, exposure, and development processes in the cell. The glasses are handled in batches with each batch stored in a cassette. The arriving cassettes that are stored in the inline stocker are moved into the I/O port, which is called the cassette loading (CL) operation. The glasses are loaded inline using a loading robot, with one glass being loaded at every takt time (τ). It takes a flow time (π) for a loaded glass to reach the end of the cell where it is unloaded into the unloading cassette located at the I/O port. The unloading cassette departs when it is filled with finished glasses. In unloading the glasses, only one unloading cassette is unloaded at a time.

(0) Source System Description

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Reference modeling starts with identifying the physical modeling components and then proceeds with defining logical modeling components. • The physical modeling components are the inline Stocker, I/O port, loading Robot, and Inline. • The state variables are Q (number of arriving cassettes in the Stocker), B (number of arriving cassettes

in the I/O port), E (number of empty shelves in the I/O port), and R (status of Robot).• The activities are (1) cassette arrival, (2) cassette loading, (3) glass loading, (4) glass unloading, and

(5) cassette departure.

(1) Overall Reference

5-5. Inline-type Manufacturing Cell Modeling - continued

Formal Modeling is carried out in three parts: The loading, processing, and unloading regions

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The time required for processing all glasses in an arriving cassette is t1 = ga * τ, where τ is the takt time of a glass loading. The events involved in the loading region model are CA (end of cassette arrival), CL (end of cassette loading), FGL (start of first glass loading), and LGL (end of last glass loading).

(2) Formal Modeling of the Loading Region


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The events at the start of the Inline are the FGL and LGL that were defined in the Loading region, whereas the events at the end of the Inline are FGU (start of the first glass unloading) and LGU (end of the last glass unloading). The FGU event is scheduled by the FGL event to occur after the flow time (π), and the LGU is scheduled by the LGL after π.

(3) Formal Modeling of the Processing Region (Inline)


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The events at the unloading region are the FGU, LGU, and CD (end of cassette departure). An arriving cassette is identified by job type (Ja) and number of glasses (ga), while the unloading cassette is specified by job type (Ju) and the number of empty slots (eu). An unloading cassette departs either when it is full or there is a job type change in the unloaded glasses.

(4) Formal Modeling of the Unloading Region


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In practice, the restriction that all glasses in an unloading cassette must have the same job type (requiring partially filled unloading cassettes to be removed when there is a job type change) may be relaxed in order to reduce the model complexity. Then, the cassettes are fully loaded during handling (ga = λ).

(4) Formal Modeling of the Unloading Region - continued


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By combining the three event graphs, the event graph for the entire inline cell is obtained. There are six events in the model: CA (cassette arrival), CL (cassette loading), FGL (first glass loading), LGL (last glass loading), FGU (first glass unloading), and CD (cassette departure).

(5) Event Graph Model of Entire Cell


Loading Region Model Inline Region Model Unloading Region Model

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6. Execution of Event Graph Models with SIGMA®(0/7)

Table of Contents

6-1. The overall procedure for simulation with SIGMA®

6-2. Simulation of Conveyor Driven Serial Line with SIGMA

No.30




1. Create a SIGMA-generated event graph consisting of vertices and edges.

6-1. The overall procedure for simulation with SIGMA®

Event Graph Model

No.31




2. Declare state variables Q and M in a State Variable Editor dialog box.

6-1. The overall procedure for simulation with SIGMA® - Continued

No.32

3. Define the Run vertex: state variables are listed as parameters in an Edit Vertex dialog box.




4. Define the Event vertices Arrive, Load and Unload in Edit Vertex dialog boxes.


No.33

5. Define the Edges ‘from Arrive to Load’, etc. in Edit Edge dialog boxes.




6. Specify run options in the Run Options dialog box and run the simulation.


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0. Convert to a SIGMA-compatible event graph

1. Build a SIGMA-generated event graph

4. Define Event Vertices 5. Define Edges

2. Declare State Variables

6. Define Run Options

Event Graph

3. Define Run Vertex


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(1) Modifying the Event Graph to Collect the Waiting Time Statistics.

6-2. Simulation of Conveyor Driven Serial Line with SIGMA®

Event Graph Model

Waiting time at the conveyor?

The waiting time (WT) of a job is computed by subtracting the job’s entering time from its leaving time, for which a queue (ranked-list) is defined for the job-entering times. In SIGMA, (1) the current simulation clock time is obtained from the function CLK, (2) the function PUT{FIF;L} is used for en-queuing a record into the FIFO queue L and GET{FST;L} for de-queuing, and (3) the record for en-queue/de-queue is stored in the built-in array ENT[].

Modified Event Graph Model

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(2) Simulating the Modified Event Graph Model with SIGMA®.

6-2. Simulation of Conveyor Driven Serial Line with SIGMA® - Continued

1. Create a SIGMA-generated event graph consisting of vertices and edges.

2. Declare State variables; 3. Define the Run vertex;

4. Define Events vertices; 5. Define Edges; 6. Define Run Options

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7. Developing Your Own Event Graph Simulator (0/6)

Table of Contents

7-1. Functions for Handling Events and Managing Queues

7-2. Functions for Generating Random Variates

7-3. Event Routines

7-4. Next-Event Scheduling Algorithm

7-5. Single Server System Simulator

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(1) Event Handling Functions (FEL: future event list)

7-1. Functions for Handling Events and Managing Queues

Schedule-event (event-name, time); Retrieve-event (event-name, time); Cancel-event (event-name)

(2) Queue Handling Functions

(j, x) Q; Q (j, x)

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7-2. Functions for Generating Random Variates

Input Variable Types Distributions Parameter Estimation Generation Methods

Inter-arrival time Exponential (θ) Maximum likelihood method Inverse-transform

Erlang (k, θ) Method of moment Convolution of exponential

Service time (Repair time)

Uniform (a, b) - -

Triangular (a, b, c) - Composition method

Beta (α, β) Method of moment Acceptance-rejection

Normal (μ, σ) Maximum likelihood method Box & Muller method

Lognormal (μ, σ) Maximum likelihood method Conversion of normal variate

Inter-failure time Weibull (α, β) Rank regression method Inverse-transform

Exp (a): { If (a <= 0) then return False; u = Math.random (); Return (– a * Math.log (u)); }Uni (a, b): { If (a >= b) then return False; u = Math.random (); Return (a + (b –a) * u); }

There are available ‘free’ Java functions for generating random variates from major standard distributions. One of them is a library, named randvar, which is a part of “Stochastic Simulations in Java” (SSJ), developed in Department of Operational Research and Information, at the University of Montreal, Canada. More details can be found in the “SSJ User’s Guide” document.

No.40

http://www.iro.umontreal.ca/~simardr/ssj/doc/pdf/guiderandvar.pdf

http://www.iro.umontreal.ca/~simardr/ssj/doc/pdf/guiderandvar.pdf




7-3. Event Routines

The above event graph indicates that “whenever E0 occurs, the state variable s changes to fE0(s). Then, if edge condition C1 is true, E1 is scheduled to occur after t1; if edge condition C2 is true, E2 is scheduled to occur after t2; and if edge condition C3 is true, E3 is canceled immediately”.

The event routine for the E0 event in the above event graph can be expressed as follows:

Originating Event State Change Edge Condition Action Delay Destination Event

E0 s = fE0(s)1 C1 schedule t1 E1

2 C2 schedule t2 E2

3 C3 cancel 0 E3

E0

{s = fE0(s)}

t1E1 E3

(C3)(C1)E2

t2(C2)

Execute-E0-event-routine (Now) { s = fE0(s); // state change If (C1) Schedule-event (E1, Now+ t1); If (C2) Schedule-event (E2, Now+ t2); If (C3) Cancel-event (E3);}.

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7-4. Next-Event Scheduling Algorithm

No.42




7-5. Single Server System Simulator

(1) SSS Simulator Main Program for Computing AQL (average queue length)

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7-5. Single Server System Simulator - Continued

(2) Event Transition Table and Event Routines

Execute-Initialize-routine (Now) { Q = 0; M = 1; Before = 0; SumQ = 0; Schedule-event (Arrive, Now); }Execute-Arrive-event-routine (Now) { SumQ = SumQ + Q*(Now – Before); Before = Now; Q = Q + 1; Schedule-event (Arrive, Now+ Exp (5)); If (M>0) Schedule-event (Load, Now); }Execute-statistics-routine (Now) { SumQ = SumQ + Q*(Now – Before); AQL = SumQ / CLK; }.

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Introduction to Event-based Modeling & Simulation Chapter 4 “Little people discuss other people. Average people discuss events. Big people discuss ideas.”