Chapter 2: Simulation Examples

 This chapter presents several examples of simulations that can be performed by devising
a simulation table either manually or with a spreadsheet.
 The simulation table provides a systematic method for tracking system state over time.
 These examples provide insight into the methodology of discrete-system simulation and
the descriptive statistics used for predicting system performance.

The simulations in this chapter are carried out by following three steps;

1. Determine the characteristics of each of the inputs to the simulation. Quite often,
these are modeled as probability distributions, either continuous or discrete.
2. Construct a simulation table. Each simulation table is different, for each is developed
for the problem at hand. An example of a simulation table is shown in Table 2.1. In
this example, there are p inputs, xij,j= 1 , 2,. .., p, and one response, yi' for each of
repetitions (or, trials) i =1, 2,. . ., n. Initialize the table by filling in the data for
repetition 1 .
3. For each repetition i, generate a value for each of the p inputs, and evaluate the
function, calculating a value of the response yi. The input values may be computed by
sampling values from the distributions chosen in step l. A response typically depends
on the inputs and one or more previous responses.

Inputs Response
Repetitions xi1 xi2 ….. ……. yi
1
2
.
.
n
Table: 2.1

This chapter gives a number of simulation examples in queueing, inventory, reliability, and
network analysis.

SIMULATION OF QUEUEING SYSTEMS

A queueing system is described by its:

 Calling population
 The nature of the arrivals,
 The service mechanism,
 The system capacity and
 The queueing discipline.
A simple single-channel queueing system is portrayed in Figure 2.1. 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 one at a time in a random fashion; once they join the waiting
line, they are 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 (often called FlFO: first in, first out) by a single server or channel.

Arrivals and services are defined by the distribution of the time between arrivals and the
distribution of service times, respectively.

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 and the units in the queue. The simulation clock is used to track
simulated time.
In a single-channel queueing simulation, inter-arrival times and service times are generated
from the distributions of these random variables.

Example 2.1: Single, Channel Queue

A small grocery store has only one checkout counter. Customers arrive at this checkout
counter at random times that are from 1 to 8 minutes apart. Each possible value of
interarrival time has the same probability of occurrence. The service times vary from 1 to 6
minutes, with the probabilities shown in Table 2.6. The problem is to analyze the system by
simulating the arrival and service of 100 customers.

Service time distribution

Service time Probability

(minutes)
1 0.10
2 0.20
3 0.30
4 0.25
5 0.10
6 0.05

SIMULATION OF INVENTORY SYSTEMS

Example 2.3: The News Dealer's Problem

A classical inventory problem concerns the purchase and sale of newspapers. The newsstand
buys the papers for 33 cents each and sells them for 50 cents each. Newspapers not sold at
the end of the day are sold as scrap for 5 cents each. Newspapers can be purchased in bundles
of 10. Thus, the newsstand can buy 50, 60, and so on. There are three types’ of news days:
"good"; "fair"; and "poor''; they have the probabilities 0.35, 0.45, and 0.20, respectively. The
distribution of newspapers demanded on each of these days is given, in Table 2.15. The
problem 'Is to compute the optimal number of papers the newsstand should purchase.
Simulate the problems for 20 days.

Example 2.6: Random Normal Numbers

Consider a bomber attempting to destroy an ammunition depot, as shown in Figure 2.15.

(This bomber has conventional rather than laser-guided weapons.) If a bomb falls anywhere
on the target, a hit is scored; otherwise, the bomb is a miss. The bomber flies in the
horizontal direction and carries 10 bombs. The aiming point is (0, 0). The point of impact is
assumed to be normally distributed around the aiming point with a standard deviation of 400
meters in the direction of flight and 200 meters in the perpendicular direction. The problem is
to simulate the operation and make statements about the number of bombs on target.
Example 2.7: Lead-Time Demand

A firm sells bulk rolls of newsprint. The daily demand is given by the following probability
distribution:

Daily demand (Rolls) 3 4 5 6

Probability 0.20 0.35 0.30 0.15

The lead time is the number of days from placing an order until the firm receives the order
from the supplier. In this instance, lead time is a random variable given by the following
distribution:

Lead Times (Days) 1 2 3

Probability 0.36 0.42 0.22
Simulate the lead time demand.