Lecture 3 APPLICATION OF SIMULATION IN SERVICE OPERATIONS MANAGEMENT


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1 Lecture 3 APPLICATION OF SIMULATION IN SERVICE OPERATIONS MANAGEMENT Learning Objective To discuss application of simulation in services
2 1. SIMULATION Simulation is a powerful technique for solving a wide variety of problems. Simulation is an imitation of the operation of a real world processor system over time. Simulation involves the generation of an artificial history of a system and the observation of that artificial history to draw inferences concerning of the operating characteristics of the real system. Simulation is done either manually or using computers. Simulation is basically an experimental technique. It is a fast and relatively inexpensive method of doing an experiment on the computer. The basic idea behind simulation include Model the given system by means of some equations. Determine its time dependent behavior. The simplicity of approach when combined with the computational power of the high speed digital computer makes simulation a power full tool. Simulation is mostly used when An exact analytic expression for the behavior of the system under investigation is not available The analytic solution is too time consuming or expensive. Simulation modeling can be used as An analysis tool: To predict the effect of changes to the effect of changes to existing system. A design tool: to predict the performance of new systems under varying set of circumstances. Examples of simulation applications in services Simulating aircraft delay absorption Runway schedule determination by simulation optimization
3 Modeling ship arrivals in ports Modeling and simulation of telephonic call centre Baggage screening at airports Telecommunication billing system Modeling front office and patient care in ambulatory health care practices Project management Scheduling of police patrols Hazardous waste handling 2. DEFINITION OF SYSTEM AND STATE OF A SYSTEM System A system is defined as a group of objects that are joined together in some regular interaction or interdependence toward the accomplishment of some purpose. System can be categorized as discrete or continuous. State of a system The state of a system is defined to be a collection of variables necessary to describe the system at any time relative to the objective of the study. In case of banking the possible state variables are a number of busy tellers, the number of customers waiting in line or being served, and the arrival time of the next customer. Discrete system A discrete system is one in which the state variables change only at a discrete set of points in time. The number of customer waiting in line in a bank is an example of discrete system. Continuous systems A continuous system is one in which the state variables change continuously over time. An example is arrival time of the next customer in a bank.
4 Model of a system A model of a system is defined as a representation of a system for the purpose of studying the system. Types of model Models can be classified as being mathematical model or physical as categorized in Figure Models Mathematical model Physical model Simulation models Static or dynamic Deterministic or Stochastic Discrete or continuous Figure 11.8: Classification of models Static simulation models It represents a system at a particular point in time. It is otherwise called as Monte Carlo simulation. Dynamic simulation models It represents systems as they change over time. Example: The simulation of a bank from 9 A.M. to 4 P.M for the time of arrival of customers. Deterministic simulation models Simulation models that contain no random variables are classified as deterministic. Deterministic models have a known set of inputs which will result in unique set of
5 outputs. Example: deterministic arrivals would occur at a dentist s office if all patients arrived at the scheduled appointment time. Stochastic simulation models A stochastic simulation model has one or more random variables as inputs. Random inputs lead to random outputs. Since the outputs are random they can be considered only as estimates of the true characteristics of the model. Example: the simulation of a bank would usually involve random interval times and random service times. Thus in a stochastic simulation, the output measures like the average number of people waiting, the average waiting time of a customer, must be treated as statistical estimates of the true characteristics of the system. Discrete event system simulation It is the modeling of systems in which the state variables change only at a discrete set of points in time. The simulation models are analyzed by numerical methods rather than by analytical methods. Analytical methods employ the deductive reasoning of mathematics to solve the model. For example, differential calculus can be used to compute the minimum cost policy for some inventory models. Numerical methods employ computational procedure to solve mathematical models. The process of system simulation is presented in Figure 2.
6 Problem formulation Setting of objectives and overall project plan Model conceptualization Data collection Model translation No Verified Yes No Validated No Yes Experimental design Production runs and analysis Yes More runs? Yes No Documentation and reporting Implementation Figure 2: Steps in system simulation process
7 MONTE CARLO METHOD A Monte Carlo method is a stochastic technique that involves use of random numbers and probability statistics to solve the problems. The term Monte Carlo Method was coined by S. Ulam and Nicholas Metropolis in reference to games of chance, a popular attraction in Monte Carlo, Monaco (Hoffman, 1998; Metropolis and Ulam, 1949). This method can be used in many areas from economics, nuclear physics to regulating the flow of traffic. To call something a "Monte Carlo" experiment, all you need to do is use random numbers to examine some problem. The Monte Carlo method is just one of many methods for analyzing uncertainty propagation, where the goal is to determine how random variation, lack of knowledge, or error affects the sensitivity, performance, or reliability of the system that is being modeled. Monte Carlo simulation is a method for iteratively evaluating a deterministic model using sets of random numbers as inputs. This method is often used when the model is complex, nonlinear, or involves more than just a couple of uncertain parameters. A simulation can typically involve over 10,000 evaluations of the model, a task which in the past was only practical using super computers. 3. SIMULATION OF QUEUING SYSTEMS A queuing system is described by its population, the nature of arrivals, the service mechanism, the system capacity, and the queuing discipline as discussed in Module 9. Many real world queuing applications are too complex to be modeled analytically and hence computer simulation helps to analyze the model. A simple single channel queuing system is given in the following Figure
8 Server Calling population Waiting line Figure 11.10: Queuing system In the single channel queue the calling population is infinite; that is, if a unit leaves the calling population and joins the waiting line or enters service, there is no change in the arrival rate of other units that could need service. Arrivals for service occur at a time in a random fashion; once they join the waiting line, they eventually served. In addition, service times are of some random length according to a probability distribution which does not change over time. The system capacity has no limit, meaning that any number of units can wait in line. Finally units are served in the order of their arrival by a server or channel (often called FIFO: first in first out). Arrivals and number of customer served are defined by the distribution of time between arrivals and the distribution of service times respectively. For any simple single or multi channel queue, the overall effective arrival rater must be less than the total service rate, or the waiting line will grow without bound. When queue grow without bound, they are termed Explosive or unstable. The state of the system is the number of units in the system and the status of the server, busy or idle. An event is a set of circumstances that causes an instantaneous change in the state of the system. In a single channel queuing system, there are only two possible events that can affect the state of the system. They are the entry of a unit into the system (the arrival event) and the completion of service on a unit (the departure event). The queuing system includes the server, the unit being served (if one is being serviced), and the units in the queue (if any are waiting). The simulation clock is used to track simulated time.
9 If a unit has just completed service, the simulation proceeds in the manner shown in the following Figure Departure event Begin server idle time No Another unit Yes Remove the waiting unit from the queue Begin servicing the unit Figure 11.1: Flow diagram presenting a service just completed The arrival event occurs when a unit enters the system. The flow diagram for the arrival event is shown in Figure Arrival event No Server busy? Yes Unit enters service Unit enters queue for service Figure 11.12: Unit entering system flow diagram The unit will find the server either idle or busy; therefore, either the unit begins to be served immediately, or it enters the queue for the server. The unit follows the course of action shown in Table 11.8.
10 Table 11.8: Potential unit actions upon arrival Queue status Non empty Empty Server Busy Enter queue Enter queue status Idle Impossible Enter service If the server is busy, the units enter the queue. If the server is idle and the queue is empty, the unit begins service. It is not possible for the server to be idle while the queue is nonempty. After the completion of a service, the server either will become idle or will remain busy with the next unit. The relationships of these two out comes to the status of the queue is shown in Table Table 11.9: server outcomes after the completion of service Queue status Non empty Empty Server busy impossible outcomes idle Impossible If the queue is not empty, another unit will enter the server and it will be busy. If the queue is empty, the server will be idle after a service is completed. These two possibilities are shaded portion of Table It is impossible for the server to be idle after a service is completed when the queue is not empty. Simulation clock times for arrivals and departures are computed in a simulation table customized for each problem. In simulation, events usually occur at random times, the
11 randomness imitating uncertainty in real life. For example, it is not known with certainty when the next customer will arrive at a grocery checkout counter, or how long the bank teller will take to complete a transaction. In these cases, a statistical model of the data is developed either from data collected and analyzed or from subjective estimates and assumptions. The randomness needed to imitate real life is made possible through the use of random numbers. Random numbers are distributed uniformly and independently on the interval (0, 1). Random digits are uniformly distributed on the set {0,1,..,9}. Random digits can be used to form random numbers by selecting the proper number of digits for each random number and placing a decimal point to the left of the value selected. When numbers are generated by using a procedure, they are often referred to as pseudo random numbers. Because the procedure is fully known, it is always possible to predict the sequence of numbers that will be generated prior to the simulation. In a single channel queuing simulation, inter arrival times and service times are generated from the distributions of these random variables. Illustration The following table contains a set of five inter arrival times that were generated by rolling a die five times and recording the up face. These five inter arrival times are used to compute the arrival times of six customers at the queuing system. The first customer is assumed to arrive at clock time 0. This starts the clock in operation. The second arrives two units later at clock time 2. The third customer arrives four time units later, at clock time 6; and so on.
12 Table 11.10: Inter arrival and clock times Customer Inter arrival time Arrival time on clock The second time of interest is the service time. The following table contains service times generated at random from a distribution of service times. The only possible time units are one, two, three and four. Assuming that all four values are equally likely to occur, these values could have been generated by placing the numbers one through four on chips and drawing the chips from a hat with replacement, being sure to record the numbers selected. Now the inter arrival times and service times must be meshed to simulate the single channel queuing system as shown in Table Table 11.11: Service times Customer Service time
13 A Customer Number Table 11.12: Simulation table emphasizing clock times B Arrival time (Clock) C Time service Begins (Clock) D Service time (duration) E Service time ends (clock) As in table 11.12, the first customer arrives at clock time 0 and immediately begins service which requires two minutes. Service is completed at clock time 2. The second customer arrives at clock time 2 and is finished at clock time 3. Note that the fourth customer arrived at clock time 7, but service could not begin until clock time 9.this occurred because customer 3 did not finish service until clock time 9. This simulation table is designed specifically for a single channel queue that serves customers on a first in first out (FIFO) basis. It keeps track of the clock at which time unit an event occurs. The occurrence of the two types of events in chronological order is shown in table and figure Table 11.13: Chronological ordering of events Event type Customer number Clock time Arrival 1 0 Departure 1 2 Arrival 2 2 Departure 2 3
14 Arrival 3 6 Arrival 4 7 Departure 3 9 Arrival 5 9 Departure 4 11 departure 5 12 Arrival 6 15 Departure 6 19 Number of customers in the system Clock time Figure 11.13: Number of customers in the system Example: Single channel Queue A small grocery store has only one checkout counter. Customers arrives at this checkout counter at random times that are from 1 to 8 minutes apart. Each possible value of inter arrival time has the same probability of occurrence as shown in table
15 Table 11.14: Distribution of time between arrivals at grocery stores Time between arrivals (minutes) probability Cumulative probability Random digit assignments The service times vary from 1 to 6 minutes, with the probability shown in table The problem is to analyze the system by simulating the arrival and service of 100 customers. Service time (minutes) Table 11.15: Service time distribution at grocery store probability Cumulative probability Random digit assignments
16 A set of uniformly distributed random numbers is needed to generate the arrivals at the checkout counter. Such random numbers have the following properties. The set of random numbers is uniformly distributed between 0 and 1. Successive random numbers are independent. The time between arrivals is presented in table The service times are presented in table Table 11.16: Time between arrival Determinations Customer Random digits Time between arrivals (Minutes) Customer Random digits Time between arrivals (Minutes)
17 Customer Table 11.17: Time between arrival Determinations Random digits Service time (minutes) Customer Random digits Service Time (Minutes) Simulation table for first 10 customers is shown in table
18 Table 11.18: Simulation table for single channel queuing problem in a grocery store Customer Inter arrival Time (minutes) Arrival time Service time (Minutes) Time service begins Waiting time in queue (Minutes) Time service ends Time consumer spends in system (Minutes) Idle time of server Total The following findings can be determined from the simulation table. Total time customers wait in queue (minutes) Average waiting time = Total number of customers
19 Probability of customers waiting = Number of customers who wait Total number of customers Probability of getting server idle Total idle time of server (minutes) = Total run time of simulation (minutes) Average service time Total service time (minutes) = Total number of customers Similarly, expected service time, average time between arrivals, average waiting time of those who wait, average time consumer spends in the system, average time customer spends in the system can also be calculated. Questions 1. Consider the following continuously operating maintenance job shop. Inter arrival times of jobs are distributed as follows: Time between arrivals (Hours) Probability Processing times for jobs are normally distributed with mean 50 minutes and standard deviation 8 minutes. Construct a simulation table and perform a simulation for 10 new jobs. Assume that, when simulation begins, there is one job
20 being processed (scheduled to be completed in 25 minutes) and there is one job with a 50 minutes processing time in the queue. (i) What was the average time in the queue for the 10 new jobs? (ii) What was the average processing time of the 10 new jobs? (iii) What was the maximum time in the system for the 10 new jobs? 2. Small town taxi operates one vehicle during the 9.AM to 5 PM period. Currently, consideration is being given to the addition of a second vehicle to the fleet. The demand for the taxis follows the distribution shown: Time between calls (minutes) Probability The distribution of time to complete a service as follows: Service time (Minutes) Probability Simulate 5 individual days of operation of the current system and of the system with additional taxi cab. Compare the two systems with respect to the waiting times of the customers and any other measures that might shed light on the situation
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