Introduction - Simulation. Simulation of industrial processes and logistical systems - MION40

Transcription

1 Introduction - Simulation Simulation of industrial processes and logistical systems - MION40 1

2 What is a model? A model is an external and explicit representation of part of reality as seen by the people who wish to use that model to understand, to change, to manage and to control that part of reality Pidd Tools for thinking, 1996 Model is defined as a representation of a system for the purpose of studying the system Banks, Carson and Nelson Discrete-Event System Simulation,

3 Model types PRESCRIPTIVE - formulate and optimize the problem vs. (analytical) DESCRIPTIVE - describe the behavior of the system (numerical) DISCRETE - state variables change only at a discrete set of time points (assuming only a finite set of values) vs. CONTINUOUS - state variables change continuously and smoothly over time (assuming any real values) 3

4 Model types PROBABILISTIC - a response may take a range of values given the initial state and input (uses random variables - STOCHASTIC model) DETERMINISTIC - all variables stated with certainty, completely determined by its initial state and inputs STATIC - represent system at one time point vs. DYNAMIC - describe system in time 4

5 What is simulation? Definitions of simulation: Simulation is the process of designing a model of a real system and conducting experiments with this model with the purpose of either understanding the behavior of the system or of evaluating various strategies (within the limits imposed by a criterion or set of criteria) for the operation of the system From Shannon R.E., Systems simulation: The art and science,

6 What is simulation? (cont) The basic principles are simple enough: the analyst builds a model of the system of interest, writes computer programs which embody model and uses a computer to imitate the system s behavior when subject to a variety of operating policies From Pidd, M., Computer Simulation in Management Science, Third Edition, John Wiley, New York,

7 System model - reality We use the concept system when we refer to the part of reality that we would like to study. We use the concept model for the picture of reality that we create. 7

8 Simulation Generally, imitation of a system via computer Involves a model validity? Does not provide an analytic solution Do not get exact results (bad) Allow for complex and realistic models (good) Approximate answer to exact problem is better than exact answer to approximate problem (good) One cannot predict the behavior of a model by using simulation (bad) 8

9 Steps in a simulation project 1. Problem formulation 2. Set objectives and overall project plan Phase 1 Problem Definition 3. Model conceptualization 4. Data Collection 5. Model Translation Phase 2 Model Building No No 6. Verified Yes 7. Validated Yes 8. Experimental Design 9. Model runs and analysis No Phase 3 Experimentation Phase 4 Implementation Yes 10. More runs No 11. Documentation, reporting and implementation 9

10 Simulation vs. direct experimentation Cost Simulation may often be cheaper alternative Time Simulation allows compress or expand time Replication Simulations are precisely repeatable Safety Simulation offers risk-free environment Legality 10

11 Continuous simulation State variables change continuously over time. Typically involve differential equations In this course, we will only deal with discrete event simulation 11

12 Time aspect in discrete event simulation Two techniques for time increment - next event (move clocktime to next event-time) - time slice (move forward one time-step and check if something has happened) 12

13 Simulation software or general programming tools? Simlicity for model construction Simulators Simulation software General programming software Flexibility 13

14 Decision model 14

15 Applications New projects System developments Problem solving Staff education Operational control 15

16 Pitfalls (fallgropar) Too detailed model (very complicated model) Looking for solutions that does not exist (e.g. Very high fill rates and low inventory levels) Instable and chaotic systems (simulation of systems which are highly sensitive to parameter values) 16

17 You cannot simulate everything It is hard to simulate social activities in a computerized model operational technical social 17

18 Queueing theory - basics Important queuing models with FIFO discipline The M/M/1 model The M/M/c model The M/M/c/K model (limited queuing capacity) The M/M/c/ /N model (limited calling population) Priority-discipline queuing models Application of Queuing Theory to system design and decision making 18

19 What is queueing theory? Mathematical analysis of queues and waiting times in stochastic systems. Used extensively to analyze production and service processes exhibiting random variability in market demand (arrival times) and service times. Queues arise when the short term demand for service exceeds the capacity Most often caused by random variation in service times and the times between customer arrivals. If long term demand for service > capacity the queue will explode! 19

20 Why is queueing theory important? Capacity problems are very common in industry and one of the main drivers of process redesign Need to balance the cost of increased capacity against the gains of increased productivity and service Queuing and waiting time analysis is particularly important in service systems Large costs of waiting and of lost sales due to waiting Prototype Example ER at County Hospital Patients arrive by ambulance or by their own accord One doctor is always on duty More and more patients seeks help longer waiting times Question: Should another MD position be instated? 20

21 A cost/capacity tradeoff model Cost Total cost Cost of service Cost of waiting Process capacity 21

22 Components of basic a queueing process Input Source The Queuing System Calling Population Jobs Queue Service Mechanism Served Jobs leave the system Arrival Process Queue Configuration Queue Discipline Service Process 22

23 Components of a queueing process The calling population The population from which customers/jobs originate The size can be finite or infinite (the latter is most common) Can be homogeneous (only one type of customers/ jobs) or heterogeneous (several different kinds of customers/jobs) The Arrival Process Determines how, when and where customer/jobs arrive to the system Important characteristic is the customers /jobs inter-arrival times To correctly specify the arrival process requires data collection of interarrival times and statistical analysis. 23

24 Components of a queueing process The queue configuration Specifies the number of queues Single or multiple lines to a number of service stations Their location Their effect on customer behavior Balking and reneging Their maximum size (# of jobs the queue can hold) Distinction between infinite and finite capacity 24

25 Example Two Queue Configurations Multiple Queues Servers Single Queue Servers 25

26 Multiple v.s. Single Customer Queue Configuration Multiple Line Advantages Single Line Advantages 1. The service provided can be differentiated Ex. Supermarket express lanes 2. Labor specialization possible 3. Customer has more flexibility 4. Balking behavior may be deterred Several medium-length lines are less intimidating than one very long line 1. Guarantees fairness FIFO applied to all arrivals 2. No customer anxiety regarding choice of queue 3. Avoids cutting in problems 4. The most efficient set up for minimizing time in the queue 5. Jockeying (line switching) is avoided 26

27 Components of a Basic Queuing Process The Service Mechanism Can involve one or several service facilities with one or several parallel service channels (servers) - Specification is required The service provided by a server is characterized by its service time Specification is required and typically involves data gathering and statistical analysis. Most analytical queuing models are based on the assumption of exponentially distributed service times, with some generalizations. The queue discipline Specifies the order by which jobs in the queue are being served. Most commonly used principle is FIFO. Other rules are, for example, LIFO, SPT, EDD Can entail prioritization based on customer type. 27

28 Mitigating Effects of Long Queues 1. Concealing the queue from arriving customers Ex. Restaurants divert people to the bar or use pagers, amusement parks require people to buy tickets outside the park, banks broadcast news on TV at various stations along the queue, casinos snake night club queues through slot machine areas. 2. Use the customer as a resource Ex. Patient filling out medical history form while waiting for physician 3. Making the customer s wait comfortable and distracting their attention Ex. Complementary drinks at restaurants, computer games, internet stations, food courts, shops, etc. at airports 4. Explain reason for the wait 5. Provide pessimistic estimates of the remaining wait time Wait seems shorter if a time estimate is given. 6. Be fair and open about the queuing disciplines used 28

29 Queueing Modeling and System Design Two fundamental questions when designing (queuing) systems Which service level should we aim for? How much capacity should we acquire? The cost of increased capacity must be balanced against the cost reduction due to shorter waiting time Specify a waiting cost or a shortage cost accruing when customers have to wait for service or Specify an acceptable service level and minimize the capacity under this condition The shortage or waiting cost rate is situation dependent and often difficult to quantify Should reflect the monetary impact a delay has on the organization where the queuing system resides 29

30 Analyzing Design-Cost Tradeoffs Given a specified shortage or waiting cost function the analysis is straightforward Define WC = Expected Waiting Cost (shortage cost) per time unit SC = Expected Service Cost (capacity cost) per time unit TC = Expected Total system cost per time unit The objective is to minimize the total expected system cost Cost TC SC Min TC = WC + SC WC Process capacity 30

31 Analyzing Linear Waiting Costs Expected Waiting Costs as a function of the number of customers in the system C w = Waiting cost per customer and time unit C w N = Waiting cost per time unit when N customers in the system WC = C n= np w n = 0 Expected Waiting Costs as a function of the number of customers in the queue C w L WC = C w L q 31

32 Analyzing Service Costs The expected service costs per time unit, SC, depend on the number of servers and their speed Definitions c = Number of servers µ = Average server intensity (average time to serve one customer) C S (µ) = Expected cost per server and time unit as a function of µ SC = c*c S (µ) 32

33 A Decision Model for System Design Determining µ and c Both the number of servers and their speed can be varied Usually only a few alternatives are available Definitions A = The set of available µ - options Min µ A,c = 0,1,... TC = c C s ( µ ) + WC Optimization Enumerate all interesting combinations of µ and c, compute TC and choose the cheapest alternative 33

34 The Poisson Process The standard assumption in many queuing models is that the arrival process is Poisson Two equivalent definitions of the Poisson process 1. The times between arrivals are independent, identically distributed and exponential 2. X(t) is a Poisson process with arrival rate λ iff: a) X(t) have independent increments b) For a small time interval h it holds that P(exactly 1 event occurs in the interval [t, t+h]) = λh + o(h) P(more than 1 event occurs in the interval [t, t+h]) = o(h) 34

35 Properties of the Poisson Process 35

36 Stochastic Processes in Continuous Time X(t)=# Calls t 36