Problem 1: System Time to Failure (TTF) The TTF consists of two components that alternatively work as an active and cold spare: • The spare component becomes the active component when the (current) active component fails; • when a component fails it starts repair immediately; • The failed component becomes the spare component as soon as it finishes repair; • the system fails if both components have failed; • the system is operational as long as at least one component is operational; • Lifetime of a component is a discrete uniform {1, 2, . . . , 5, 6}; • The repair time is 2.5 days. Question 1: Describe the states and the events characterizing the dynamic of the failure of the component. Question 2: Draw the events graph for the example showing the relationship between the events and the triggering conditions for each arc to be executed. 2 IEE 475 GP© ASU Question 3: Write the pseudo-code, as seen in class, for each event in terms of: state update and events scheduled. 3 IEE 475 GP© ASU Problem 2: The M/G/1 queue The receptionist in a medium-sized hospital helps direct entering patients and visitors to the relevant floor or wing of the building: • It is under discussion to replace the human receptionist with an electronic kiosk; • Patients arrive according to a Poisson process with rate λ (time between arrivals is exponential); • Service times are i.i.d. with mean τ and deviation σ time units. Question 1: Describe the states and the events characterizing the dynamics of the queue of interest. Question 2: Draw the events graph for the example showing the relationship between the events and the triggering conditions for each arc to be executed. 4 IEE 475 GP© ASU Question 3: Write the pseudo-code, as seen in class, for each event in terms of: state update and events scheduled. 5 IEE 475 GP© ASU Problem 3: The Stochastic Activity Network A construction project consists of a large number of activities. Some can be completed in parallel, while others are subject to constraints on the sequence. • A project is completed when all the ` = 1, 2, . . . , L activities are completed; • Each activity has a duration X` which is randomly distributed; • Planners are interested in information about the distribution of Y , i.e., the project finishing time, for instance: θ = P r (Y > tp ) where tp is the quoted duration of the project. In order to answer the questions refer to the small activity network in Figure 1. Figure 1: Stochastic Activity Network According to Figure 1, the project starts with milestone a (i.e., there is no delay between the start and a nodes) and the sequence of events is determined by the arcs representing the 5 activities and the related duration Xi , i = 1, 2, . . . , 5. These durations are random with a specific distribution. Milestone d is the final of the project and it gets to the “end” milestone with no delay. Question 1: Describe the states and the events characterizing the dynamic of the activity network. 6 IEE 475 GP© ASU Question 2: Draw the events graph for the example showing the relationship between the events and the triggering conditions for each arc to be executed. Question 3: Write the pseudo-code, as seen in class, for each event in terms of: state update and events scheduled. 7 IEE 475 GP© ASU General Principles of Simulation G. Pedrielli SCHOOL OF COMPUTING INFORMATICS & DECISION SYSTEMS ENGINEERING Pedrielli School of CIDSE 1 / 20 Table of contents 1 Learning Objectives 2 A simple System 3 Simulation Questions 4 Simulation Model Components Pedrielli School of CIDSE 2 / 20 Learning Objectives Learning Objectives Underlying ideas, methods, and issues in simulation Centered around an example of a simple processing system Decompose the problem Terminology Simulation by hand Some basic statistical issues Overview of a simulation study Pedrielli School of CIDSE 3 / 20 A simple System A Simple Processing System General Objective: Estimate expected production; Estimate the waiting time in queue, queue length, proportion of time machine is busy; Time Units: Can use different units in different places . . . must declare; Be careful to check the units when specifying inputs; Declare base time units for internal calculations, outputs; Be reasonable (interpretation, roundoff error). Pedrielli School of CIDSE 4 / 20 A simple System An Example Initially (time 0) empty and idle Base time units: minutes Input data (assume given for now . . .), in minutes: Stop when 20 minutes of (simulated) time have passed. Pedrielli School of CIDSE 5 / 20 Simulation Questions Output Performance Measures Before we develop a simulation model we should know which quantities we are interested in tracking!!! Total Production of parts over a simulation run Φ; Average waiting time of parts in queue: Pn wq,i W̄q = i=1 n Maximum waiting time of parts in queue: Wqmax = max wq,i i=1,2,...,n Time Average Queue: Q̄(T ) = Pedrielli 1 T Z T q (t) dt 0 School of CIDSE 6 / 20 Simulation Questions Output Performance Measures Maximum number of parts in queue: Qmax (T ) = max q (t) 0≤t≤T Average Cycle Time: PΦ i=1 ci C̄ = Φ Maximum Cycle Time of parts in queue: C max = Machine Utilization: ( Z 1 1 T B̄(T ) = b (t) dt, b(t) = T 0 0 Pedrielli max ci i=1,2,...,Φ If the machine is busy at time t Otherwise School of CIDSE 7 / 20 Simulation Model Components What to model Individual operations (arrivals, service times) will occur exactly as in reality; Movements, changes occur at the right “time”, in the right order Different pieces interact; Install “observers” to get output performance measures; Adopt a concrete, “brute-force” analysis approach; Nothing mysterious or subtle. In order to achieve your modeling objectives you can use: Entities; Attributes; Global Variables; Resources; Queues; Statistical Accumulators. Pedrielli School of CIDSE 8 / 20 Simulation Model Components Entities “Players” that move around, change status, affect and are affected by other entities; Dynamic objects — get created, move around, leave (maybe); Usually represent “real” things (“fake” entities for modeling “tricks”); Usually have multiple realizations “floating” around; different types of entities exist concurrently. Pedrielli School of CIDSE 9 / 20 Simulation Model Components Attributes Characteristic of all entities: describe, differentiate All entities have same attribute “slots” but different values for different entities, for example: Time of arrival Due date Priority Color Attribute value tied to a specific entity; Like “local” (to entities) variables. Pedrielli School of CIDSE 10 / 20 Simulation Model Components Global Variables Reflects a characteristic of the whole model, not of specific entities; Used for many different kinds of things; Travel time between all station pairs; Number of parts in system; Simulation clock (TNOW); Name, value of which there’s only one copy for the whole model; Not tied to entities; Entities can access, change variables. Pedrielli School of CIDSE 11 / 20 Simulation Model Components Resources What entities compete for; People Equipment Space Entity seizes a resource, uses it, releases it; Think of a resource being assigned to an entity, rather than an entity “belonging to” a resource; A resource can have several units of capacity; Seats at a table in a restaurant Identical ticketing agents at an airline counter Number of units of resource can be changed during the simulation. Pedrielli School of CIDSE 12 / 20 Simulation Model Components Queues Place for entities to wait when they can’t move on (maybe since the resource they want to seize is not available); Often tied to a corresponding resource; Can have a finite capacity to model limited space — have to model what to do if an entity shows up to a queue that’s already full; Usually interested in the length of a queue, waiting time in it. Pedrielli School of CIDSE 13 / 20 Simulation Model Components Statistical Accumulators Variables that “watch” what is happening; Depend on output performance measures desired; “Passive” in model — don’t participate, just watch; Many are automatic, but ...; At end of simulation, used to compute final output performance measures. Examples: Number of parts produced so far; Total of the waiting times spent in queue so far; No. of parts that have gone through the queue; Max time in queue we’ve seen so far; Total of times spent in system; Max time in system we’ve seen so far; Area so far under queue-length curve Q(t); Max of Q(t) so far; Area so far under server-busy curve B(t). Pedrielli School of CIDSE 14 / 20 Simulation Model Components Events and State variables Event Calendar: it is a list of event records Entity No., Event Time, Event Type; Keep sorted in increasing order on Event Time; Next event always in top record; Initially, schedule first Arrival, schedule stopping event. State variables: describe current status Server status B(t) = 1 for busy, 0 for idle; Number of customers in queue Q(t); Times of arrival of each customer now in queue (a list of random length). Pedrielli School of CIDSE 15 / 20 Simulation Model Components Simulating with Arena vs. Simulating with a general purpose language Arena uses The Process View to model a system. The basics are: Identify characteristic entities in the system; Multiple copies of entities co-exist, interact, compete; “Code” is non-procedural; Tell a “story” about what happens to a “typical” entity; May have many types of entities, “fake” entities for things like machine breakdowns; Usually requires special simulation software; Underneath, still executed as event-scheduling (see next). Pedrielli School of CIDSE 16 / 20 Simulation Model Components Simulating with Arena vs. Simulating with a general purpose language A general purpose simulator implements the Event-Scheduling “World View”; Identify characteristic events; Decide on logic for each type of event to: Change the state for each event type Observe statistics Update times of future events (maybe of this type, other types) Keep a simulation clock, future event calendar; Jump from one event to the next, process, observe statistics, update event calendar; Must specify an appropriate stopping rule. Pedrielli School of CIDSE 17 / 20 Simulation Model Components What do you need to code a discrete event simulator Use “utility” libraries; List processing; Random-number generation; Random-variate generation; Statistics collection; Event-list and clock management; Summary and output; Main program ties it together, executes events in order. Pedrielli School of CIDSE 18 / 20 Simulation Model Components Event Simulation for the Simple Production System Pedrielli School of CIDSE 19 / 20 Simulation Model Components Simulation By Hand Manually track state variables, statistical accumulators; Use “given” inter-arrival, service times; Keep track of event calendar; “Lurch” clock from one event to the next. Pedrielli School of CIDSE 20 / 20 Rework In some cases, it may be necessary to rework a part that has just been processed. A random number, RND, that falls (strictly) between 0 and 1 is drawn. 𝑡𝑟 is the rework setup time Notice that if we model REWORK in this manner, LEAVE, REWORK, and ENTER might all schedule a START vertex at the same time. This model illustrates problems that may occur when two events are scheduled at the same time. Giulia Pedrielli Simulating Stochastic Systems (IEE 545) 2 Rework (cont’d) (Solver Compliant): Increase the state space when R_{i}=1 the server is reserved by the i-th trigger (Q>0 && R_1==0&& R_2 == 0) S (S==1 && R_2==0 && R_3 == 0) S Q=Q+1 R_1=1*(S==1)+0*(S==0) (Okay in manual code the solver may give you troubles!)Perform state update as follows: • “Local Update” • Check Conditions • “Second Update” & Schedule Strt Leav S=1 S=R_1=R_2=R_3=0 Q=Q-1 (S==1) R_3=1*(Q>=1)+0*(Q==0) Rew (RND0) S Ent Q=Q+1 S=0*(S==1) Giulia Pedrielli (RND=1)+1*(Q==0) 3 Limited Waiting Space The amount of waiting space is called the buffer size and called B in the example below. The self-scheduling edge used to create customer arrivals is moved from the ENTER vertex to the ARRIVE vertex. We also conditioned the edge from ARRIVE to ENTER to require that there be an empty space for the arriving customer to wait. Giulia Pedrielli Simulating Stochastic Systems (IEE 545) 4 Assembly operation In an assembly operation, several different types of parts are put together into a single unit. The collected parts for a finished assembly are sometimes called a "kit." Giulia Pedrielli Simulating Stochastic Systems (IEE 545) 5 Different Servers Working in Parallel Some systems have two servers with different characteristics operating in parallel. In this example, there is a new, faster machine working with last year's slower model. {S[0]=1, S[1]=1} Giulia Pedrielli Simulating Stochastic Systems (IEE 545) 6 Simulating based on counters N is the number of cars washed. The simulation run stops when N reaches the value a. (N>=a) COU NT {N=0} {N=N+1} Giulia Pedrielli Simulating Stochastic Systems (IEE 545) 7 A Semi-Random Walk Build a simulation model of a semi-random walk. The location of the walker on the line is given by the variable, X. Every step is in an opposite direction and has an expected step length equal to 4 feet. However, the steps to the left are uniformly distributed between 3 and 5 while steps to the right are exactly 4 feet long. Would you expect the location of the walker to change much over time? (𝑇𝐼𝑀𝐸 ≥ 𝑒𝑛𝑑_𝑡𝑖𝑚𝑒) 1 RUN {X=0} Giulia Pedrielli RIGHT {X=X+4} LEFT END 1 {X=X-UNIF(3,5)} Simulating Stochastic Systems (IEE 545) 8 Event Parameters and Edge Attributes I+1 ~ (I < 7) EVENT (I) ~ 2 (I = = 7) Do EVENT: FOR I = 2 to 7 Giulia Pedrielli Simulating Stochastic Systems (IEE 545) 9 Many Servers of Many Types A generalization of the model with two types of servers is a model of many types of servers with many servers of each type. As example, consider a production department with 𝑁 different types of machines of different models and ages. There may be any number of each type of machine. Giulia Pedrielli Simulating Stochastic Systems (IEE 545) 10 TTF System (X>0) State Variables Number of functional components 𝑋 ∈ 0,1,2 Events RUN  FAIL: One component fails  REPAIR: One component is repaired  END: System fails tf {X=2} tf FAIL tr {X=X-1} (X>0) REPAIR Giulia Pedrielli (X==0) END Simulating Stochastic Systems (IEE 545) {X=X+1} 11 Final considerations  ERGs can be used to derive a formal representation of the system dynamics. Modeling a system using ERGs helps to understand the system dynamics and the simulation model we want to develop.  They are general enough to represent any system dynamics (they are Turing complete)  They are more general than others (e.g., Petri nets, GSMPs, etc.)  ERG models can be (http://sigmawiki.com/) Giulia Pedrielli implemented in Simulating Stochastic Systems (IEE 545) SIGMA software 12 Simple production system: Event graph and pseudo code. 𝜏𝐴 Q>0 𝜏𝐴 𝜏𝐷 𝜏𝐷 R>0 R>0 A D R=[R-1] if R>0 R=[R+1] if Q=0 Q=Q+1 Q=Q-1 Notation: • • • • Q>0 𝜏𝐷 A S D Q=Q+1 R=R-1 R=R+1 Q=Q-1 𝜏𝐴 : Inter-arrival time; 𝑅(𝑡) : Number of free servers at time t; 𝑄(𝑡): Level of the queue (including the server) at time t; 𝜏𝐷 : Processing time. Pseudo-code: Pseudo-code: A(Arrival) :: Input (𝜏𝐴 , 𝑅(𝑡), 𝜏𝐷 ) { If (R>0) // There is a free server { R=R-1; // Dicrease the num of free stations Schedule Event D(𝑡 + 𝜏𝐷 ); } Q=Q+1; //Increase the queue level Schedule Event A(𝑡 + 𝜏𝐴 ); } A(Arrival) :: Input (𝜏𝐴 , 𝑅(𝑡)) { If (R>0) { Schedule Event S(t); } Q=Q+1; Schedule Event A(𝑡 + 𝜏𝐴 ); } D(Departure) :: Input (𝑅(𝑡), 𝑄(𝑡), 𝜏𝐷 ) { R(t)=R+1; Q(t)=Q-1; If (Q>0) { Schedule Event D(𝑡 + 𝜏𝐷 ); R=R+1; } } S(Start) :: Input ( 𝑅(𝑡), 𝜏𝐷 ) { R=R-1; //Occupy the machine Schedule Event D(𝑡 + 𝜏𝐷 ); } D(Departure) :: Input (𝑅(𝑡), 𝑄(𝑡)) { R(t)=R(t)+1; Q=Q-1; If (Q>0) { Schedule Event S(t); } }
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