Chapter 22 Simulation with Process Model to accompany Operations Research: Applications and Algorithms 4th edition by Wayne L. Winston Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

2 Description In Chapter 9 we learned how to build simulation models of many different situations. In this chapter we will explain how the powerful, user- friendly simulation package Process Model can be used to simulate queuing systems.

Simulating an M/M/1 Queuing System After installing Process Model you can start it by selecting Start – Programs – Process Model. You will see the process model screen appear. The book contains a labeled diagram showing key icons. Process Model is an easy to use software that allows you to simulate queuing systems. To simulate a M/M/1 queuing systems having λ=10 arrivals/hour and µ=15 customers/hour. (See the file MM1.igx on the cd from the book.)

4 The book walks you through the steps of creating the model. During the simulation an on-screen scoreboard tracks the following quantities:  Quality Processed-Total Number of units to leave system  Cycle Time-Average time a unit spends in system  Value Added Time-Time which a unit spends in service  Cost Per Unit-If costs are associated with the resources, the cost incurred per unit serviced is computed. After completing the simulation you are asked if you want to view the output. Sample output can be seen in the book.

Simulating a M/M/2 System Let us suppose that we had two telephone operators who can handle calls. To modify the previous example we simply need to change the number of operators to 2 and ensure that up to 2 operators can be working on calls at the same time. See the file MM2.igx. The book contains the output of running this file for 1000 hours.

A Series System This section uses Process Model to simulate a series queuing system. The auto assembly line example used previously will be used.

7 Example The last two things that are done to a car before its manufacture is complete are installing the engine and putting on tires. An average of 54 cars per hour arrive requiring these two tasks. One worker is available to install the engine and can service an average of 60 cars per hour. After the engine is installed, the car goes to the tire station and waits for its tires to be attached Three workers server at the tire station.

8 Each works on one car at a time and can put tires on a car in an average of 3 minutes. Assume interarrival times and service times are exponential. Simulate this system for 400 hours.

9 Solution See the file Carassembly.igx. The key to creating a queuing network with Process Model is to build the diagram one service center at a time. Begin by creating arrivals in Example 1 and then create the engine production center and the tire center. The tire center will need to have the number of servers changed. The system is flexible and can be modeled to accommodate changes in Move Time and other factors.

10 The Effect of a Finite Buffer The output form this model can be found in the book. Suppose we only have enough space for two cars to wait for Tire installation. This is called a buffer of size 2. To model this change the Input Capacity in the Tire Activity dialog box needs to be changed to 2. Running the simulation shows the average wait is 6 hours. Clearly we need more storage space.

Simulating Open Queuing Networks - Example An open queuing network consists of two servers: Server 1 and Server 2. An average of 8 customers per hour arrive from outside at Server 1. An average of 17 customers per hour arrive from outside at Server 2. Interarrival times are exponential. Server 1 can server at an exponential rate of 20 customers per hour and Server 2 can serve at an exponential rate of 30 customers per hour.

12 After completing service at Server 1, half the customers leave the system and half go to Server 2. After completing Service at Server 2 75% of the customers complete service and 25% return to Server 1. Simulate this system for 400 hours.

13 Solution See the file Open.igx. To begin we create two arrival entities: one arrival entity representing external arrivals to Server 1 and one arrival entity representing external arrivals to Server 2. After creating the Servers we use the connector tool to create a link from Server 1 to Server 2, a link Server 2 to Server 1, and a link from Server 2 to Server 1, and a link from Server 1 and 2 to the exiting system.

14 We input the arrival and service rates, move times along with the routing information into the software. The project diagram can be seen in the book. Note as the simulation runs some calls move between the servers and some exit the system! Seeing this movement really makes the concept of an open queuing network come alive!

Simulating Erlang Service Times Service times often do not follow exponential distribution. Usually the Erlang distribution is used to model nonexponential service times. An Erlang distribution can be defined by a mean and a shape parameter k. The shape parameter must be an integer. It can be shown that Standard Deviation of Erlang =

16 Therefore is we know the mean and the standard deviation of the service times we may determine an appropriate value of k. The syntax for generating Erlang service times in Process Model is ER(Mean, k).

What Else Can Process Model Do? A brief description of other modeling features included in Process Model follows:  Bulk Arrivals and Services Often at a restaurant people arrive in groups. This arrival pattern is called bulk arrivals.  Reneging Perhaps people hang up when calling an 800 number if they are put on hold more than 5 minutes. Process Model can model such balking or reneging behavior.  Variation in Arrival pattern At a restaurant or bank arrival rate varies substantially over the course of the day. Variable arrival rate patterns can easily be modeled with Process Model.

18  Variation of Number of Servers During the day workers take breaks and go to lunch. Also many companies vary the number of servers during the day, Process Model can easily model variation in server capacity.  Priorities In an emergency room more seriously ill patients are given priority over earlier arriving less ill patients. Process Model can handle complex priority mechanisms.