# Discrete-Event Simulation

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1 Discrete-Event Simulation Introduction to Simulation WS01/02 - L 04 1/40 Graham Horton

2 Contents Models and some modelling terminology How a discrete-event simulation works The classic example - the queue in the bank Example for a discrete-event simulation Introduction to Simulation WS01/02 - L 04 2/40 Graham Horton

3 Simulation A definition of simulation: 1. Imitation of the operation of a real-world system 2. Generation of an artificial history of a system 3. Observation of the artificial history Simulation is performed using a model Introduction to Simulation WS01/02 - L 04 3/40 Graham Horton

4 Advantages of Simulation Simulation has many advantages: study new designs without interrupting real system study new designs without needing extra resources Improve understanding of system Manipulate time Less dangerous / expensive / intrusive Introduction to Simulation WS01/02 - L 04 4/40 Graham Horton

5 Model A model: is a set of assumptions about the operation of the system These assumptions can be: algorithmic (sequence of steps) mathematical (equations) logical (conditions) This model can be "run" in order to study the real system Introduction to Simulation WS01/02 - L 04 5/40 Graham Horton

6 Abstraction & Idealisation There are two main techniques for building models: -Abstraction - Idealisation Abstraction means leaving out unnecessary details - Represent only selected attributes of a customer Idealisation means replacing real things by concepts - Replace a set of measurements by a function Introduction to Simulation WS01/02 - L 04 6/40 Graham Horton

7 Systems System System environment A group of objects that interact System boundary Placing the system boundary is the first difficult task in modelling Introduction to Simulation WS01/02 - L 04 7/40 Graham Horton

8 Systems Example: Modelling a bank: System environment System Customers Cashiers System boundary Time of day Special events Introduction to Simulation WS01/02 - L 04 8/40 Graham Horton

9 Systems Example: Modelling the Earth's orbit: System environment Earth System Sun Pluto? System Moon boundary Asteroids? Introduction to Simulation WS01/02 - L 04 9/40 Graham Horton

10 Entities, Attributes & Activities An entity is an object of interest in the system customer manager cashier An attribute is a (relevant) property of an entity account balance gender skills Attributes are state variables Introduction to Simulation WS01/02 - L 04 10/40 Graham Horton

11 Activities & Delays An activity is a duration of known length check balance drink coffee serve customer Activities form part of the model specification A delay is a duration of unknown length Waiting time in queue Delays form part of the simulation result Introduction to Simulation WS01/02 - L 04 11/40 Graham Horton

12 State, State Variables The (system) state is a description which is - complete and - minimal at any point in time A state variable... length of queue (0, 1, 2,...)... is a variable needed to describe the state current activity of manager (sleeping, drinking coffee,... Introduction to Simulation WS01/02 - L 04 12/40 Graham Horton

13 Events An event is an occurrence which is instantaneous may change the state of the system Customer arrives from outside Manager wakes up Service is completed Introduction to Simulation WS01/02 - L 04 13/40 Graham Horton

14 Steps in a Simulation Study Problem formulation Set objectives and plan Conceptual model Collect data Validation Documentation & report Production runs & analysis Experimental design Create simulation model Verification Introduction to Simulation WS01/02 - L 04 14/40 Graham Horton

15 Validation & Verification Validation means creating a correct model ("building the right model") Conceptual model Validation Collect data Verification means writing a correct program ("building the model right") Create simulation model Verification Introduction to Simulation WS01/02 - L 04 15/40 Graham Horton

16 Input-Output Transformation View the (real or simulated) system as a "black box" Input variables System Output variables Black box transforms input variables into output variables Input variables are obtained by observing the real system Output variables are obtained by observing the real system and from the simulation experiment Introduction to Simulation WS01/02 - L 04 16/40 Graham Horton

17 Model Specification Discrete-event modelling raises the following questions: How does each event affect the system state and attributes? How are activities defined? What events mark the beginning and the end? What conditions (if any) must hold? How are delays defined? How must the simulation be initialised? Introduction to Simulation WS01/02 - L 04 17/40 Graham Horton

18 A Simulation Classic The single-server queue: Bank Service Queue Introduction to Simulation WS01/02 - L 04 18/40 Graham Horton

19 A Simulation Classic One possible problem formulation: "Customers have to wait too long in my bank" A typical objective: Determine the effect of an additional cashier on the mean queue length Data needed: Inter-arrival times of customers Service times Introduction to Simulation WS01/02 - L 04 19/40 Graham Horton

20 A Simulation Classic A typical simulation result: N Q T Introduction to Simulation WS01/02 - L 04 20/40 Graham Horton

21 Event Notice, Event List Event notice A data record specifying an event The event notice must contain all the information necessary to execute the event (in particular the time it is scheduled to occur) (Future) event list... A list of event notices for future events The event list is the main data structure in a discrete-event simulator Introduction to Simulation WS01/02 - L 04 21/40 Graham Horton

22 The Event List The (future) event list (FEL) controls the simulation The FEL contains all future events that are scheduled The FEL is ordered by increasing time of event notice Example FEL (at some simulation time t 1 ): (t 1, Event 1 ) (t 2, Event 2 ) (t 3, Event 3 ) (t 4, Event 4 ) t 1 t 2 t 3 t 4 Introduction to Simulation WS01/02 - L 04 22/40 Graham Horton

23 Conditional and Primary Events A primary event An event whose occurrence is scheduled at a certain time Arrivals of customers A conditional event An event which is triggered by a certain condition becoming true Customer moving from queue to service Introduction to Simulation WS01/02 - L 04 23/40 Graham Horton

24 Conditional and Primary Events Primary event: WHENEVER T >= TArrive DO NQueue^ := NQueue + 1; TArrive^ := T + IArrive; END Conditional event: WHENEVER (NServer = 0) AND (NQueue > 0) DO NQueue^ := NQueue - 1; NServer^ := NServer + 1; TService^ := T + IService; END Introduction to Simulation WS01/02 - L 04 24/40 Graham Horton

25 The Event List Example: Simulation of the Mensa: Some state variables: # people in line 1 # people at meal line 1 & 2 # people at cashier 1 & 2 # people eating at tables (t 1, Arrival) (t 2, Service completion at cashier 2) (t 3, Service completion at meal 1) (t 4, Finish eating) (t 5, Finish eating) Introduction to Simulation WS01/02 - L 04 25/40 Graham Horton

26 The Event List Operations on the FEL: Insert an event into FEL (at appropriate position!) Remove first event from FEL for processing Delete an event from the FEL The FEL is thus usually stored as a linked list Introduction to Simulation WS01/02 - L 04 26/40 Graham Horton

27 The Event List The simulator spends a lot of time processing the FEL In some cases, up to 40% of overall simulation time! Efficiency is thus very important Sequential search when inserting must be avoided: O(n) Solution: more advanced data structures (trees): O(log n) Introduction to Simulation WS01/02 - L 04 27/40 Graham Horton

28 The Event List Example: Event "Customer arrives" in the bank model: # Customer arrives WHENEVER T >= TArrive DO NQueue^ := NQueue + 1; TArrive^ := T + IArrive; DISPLAY ("T= %f Customer arrives\n",t); END Executing this event at time T causes the event (T + IArrive, Customer arrives) to be inserted into the FEL Introduction to Simulation WS01/02 - L 04 28/40 Graham Horton

29 The Simulation Algorithm yes FEL empty? no Remove and process 1st primary event Remove and process conditional event yes Conditional event enabled? no Introduction to Simulation WS01/02 - L 04 29/40 Graham Horton

30 The Simulation Algorithm Remove and process 1st primary event: Remove 1st primary event from FEL Advance simulation time Update state variables Enter new future events into FEL Compute statistics Every discrete-event simulator works like this (even if the programming model looks different!) Introduction to Simulation WS01/02 - L 04 30/40 Graham Horton

31 The Simulation Algorithm (t 1, Arrival) (t 1, Move up) (t 2, Arrival) (t 2, Arrival) (t 4, Arrival) (t 3, Service complete) (t 3, Service complete) Introduction to Simulation WS01/02 - L 04 31/40 Graham Horton

32 Timing Sometimes, activities last for an exact amount of time: The clock cycle in a computer Each phase of a traffic light A certain operation in a manufacturing line Such exact times are deterministic Introduction to Simulation WS01/02 - L 04 32/40 Graham Horton

33 Timing Usually, activities last for varying amounts of time: Inter-arrival times at bank Service times at bank Time to failure for a machine Time that a user program runs Such times are random or stochastic Introduction to Simulation WS01/02 - L 04 33/40 Graham Horton

34 Random Variables If any quantity in a simulation is random, then the simulation result must also be random This can make things complicated: - The simulator will need to use random variables - We will need to do some statistics Introduction to Simulation WS01/02 - L 04 34/40 Graham Horton

35 Random Variables The best-known random variable: Discrete and uniformly distributed on [1,6] Discrete and uniformly distributed on [1,4] Introduction to Simulation WS01/02 - L 04 35/40 Graham Horton

36 Simulating the Bank by Hand Simulation clock: Arrival interval Customer arrives Begin service Service duration Service complete Introduction to Simulation WS01/02 - L 04 36/40 Graham Horton

37 Computing Statistics Average waiting time for a customer: 1.25 Arrival interval Customer arrives Begin service Service duration Service complete Introduction to Simulation WS01/02 - L 04 37/40 Graham Horton

38 Computing Statistics P(customer has to wait): 0.75 Arrival interval Customer arrives Begin service Service duration Service complete Introduction to Simulation WS01/02 - L 04 38/40 Graham Horton

39 Computing Statistics P(Server busy): 0.66 Arrival interval Customer arrives Begin service Service duration Service complete Introduction to Simulation WS01/02 - L 04 39/40 Graham Horton

40 Computing Statistics Average queue length: 0.33 Arrival interval Customer arrives Begin service Service duration Service complete Introduction to Simulation WS01/02 - L 04 40/40 Graham Horton

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