NBS

PUBLICATIONS

NAT'L INST OF STAND S TECH

U.S. Depar

of Commerce

National Bureau

of Standards

Computer Science

and Technology

NBS Special Publication 500-123

Guide on

Workload Forecasting

Helen Letmanyi

QC—

100

U57

#500-123

1984
rwi

M he National Bureau of Standards' was established by an act of Congress on March 3, 1901. The

IP Bureau's overall goal is to strengthen and advance the nation's science and technology and facilitate their effective application for public benefit. To this end, the Bureau conducts research and provides: (1) a

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The Bureau's technical work is performed by the National Measurement Laboratory, the National

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Conducts research and provides scientific and technical services to aid Federal agencies in the selection, acquisition, application, and use of computer technology to improve effectiveness and economy in Government

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Gaithersburg, MD 20899. 'Some divisions within the center are located at Boulder, CO 80303. 'Located at Boulder, CO, with some elements at Gaithersburg, MD.
Computer Science

and Technology

NBS

NBS Special Publication 500-123

Guide on

Workload Forecasting

Helen Letmanyi

Center for Programming Science and Technology

Institute for Computer Sciences and Technology

National Bureau of Standards

Gaithersburg, MD 20899

U.S. DEPARTMENT OF COMMERCE

Malcolm Baldrige, Secretary

National Bureau of Standards

Ernest Ambler, Director

Issued March 1985
Reports on Computer Science and Technology

The National Bureau of Standards has a special responsibility within the Federal

Government for computer science and technology activities. The programs of the

NBS Institute for Computer Sciences and Technology are designed to provide ADP

standards, guidelines, and technical advisory services to improve the effectiveness

of computer utilization in the Federal sector, and to perform appropriate research

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at the end of this publication.

Library of Congress Catalog Card Number: 85-600504

National Bureau of Standards Special Publication 500-123

Natl. Bur. Stand. (U.S.), Spec. Publ. 500-123, 68 pages (Mar. 1985)

CODEN: XNBSAV

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1985

For sale by the Superintendent of Documentsi, U.S. Government Printing Office, Washington, DC 20402
CONTENTS

Page

I. INTRODUCTION 2 A. Purpose 2 B. Background 3

C. Overview 3

II. OVERVIEW OF WORKLOAD FORECASTING TECHNIQUES 7

A. Forecasting in General 7 1. Forecast Spans 8

2. Data Patterns 8

2.1 Trend Data Pattern 8

2.2 Cyclical Data Pattern 10

2.3 Seasonal Data Pattern 10

2.4 Stationary Data Pattern 10

B. Criteria for Selecting Forecasting Techniques.... 10

C. Forecasting Techniques 12

1. Simple Moving Averages 13

2. Exponential Smoothing ' 13

3. Classical Decomposition Method 14

4. Simple Linear Regression Analysis 15

5. Multiple Linear Regression Analysis 16

6. Box-Jenkins Method 16

III. WORKLOAD FORECASTING STEPS 18

STEP 1. Analyze Historical Data and Collect

Future Requirements 18

1.1 Identify Forecasting Objectives 18

1.2 Organize Forecasting Team 18

1.3 Collect Data by Workload Levels 18

1.3.1 Level 1. Agency Mission 20

1.3.2 Level 2. Agency Functions 20

1.3.3 Level 3. Quantifiable Events.... 21

1.3.4 Level 4. ADP Operations

Performed 22

1.3.5 Level 5. Resources Consumed 22

1.4 Analyze Historical Data 22

1.4.1 Analyze Historical Data by

Application Systems 23

1.4.2 Analyze Historical Data for the

Total Workload 23

STEP 2. Select Forecasting Technique 24

STEP 3. Perform Workload Forecasting 25

3.1 Organizational Approach to Forecasting. 25

3.1.1 Forecast Level 3 Workload

Requirements 29

3.1.2 Forecast Level 4 Workload

Requirements 29

3.1.3 Forecast Level 5 Workload

Requirements 29

3.2 Apply Forecasting Techniques 30

3.2.1 Forecasting with Simple Moving

Averages 30

i i i
3.2.2 Forecasting with Exponential

Smoothing 34

3.2.3 Forecasting with Classical

Decomposition Method 35

3.2.4 Forecasting with Simple Linear...

Regression 43

3.2.5 Forecasting with Multiple Linear

Regression 49

3.2.6 Forecasting with Box-Jenkins

Method 53

STEP 4. Analyze Forecast Results 58

SUMMARY 60

APPENDIX A: SUGGESTED READINGS AND REFERENCES 61

IV
GUIDE ON WORKLOAD FORECASTING

Helen Letmanyi

ABSTRACT

The purpose of this guide is to provide ADP managers and

technical personnel with useful quantitative techniques for

forecasting future workload requirements. It additionally

provides a step-by-step approach to the forecasting process.

Readers can then, in a timely manner, provide the computing

resources needed to perform the user's workload at required

service levels throughout the life-cycle of an ADP system.

These techniques are described so that readers with little

or no training in statistics should find them useful.

However, this guide does not intend to give an exhaustive

treatment of the techniques discussed. Readers requiring

more information are referred to Appendix A ("Suggested

Readings and References").

Key words: causal models; forecast spans; time-series

models; workload forecasting techniques; workload levels;

workload transition.

1
I. INTRODUCTION

A. Purpose

As a general planning tool, forecasting is a means by

which the impact of today's decisions concerning future

activities can be evaluated. As an ADP planning tool, the

forecasting and evaluation of future workload requirements

of an organization provides a basis for informed management

decisions.

The purpose of this guide is to provide ADP managers

and technical personnel with useful quantitative techniques

for forecasting future workload requirements. It

additionally provides a step-by-step approach to the

forecasting process. Readers can then, in a timely manner,

provide the computing resources needed to perform the user's

workload at required service levels throughout the

life-cycle of an ADP system. These techniques are described

so that readers with little or no training in statistics

should find them useful. However, this guide does not

intend to give an exhaustive treatment of the techniques

discussed. Readers requiring more information are referred

to Appendix A ("Suggested Readings and References").

There are circumstances when the quantitative methods

presented in this guide are either inappropriate or need to

be applied with extreme caution. This is true in those

situations where either historical data does not exist or

where a trend is expected not to continue. In such an

environment, qualitative, or non-quantitative, methods in

lieu of or as a complement to those discussed in this guide

may be appropriate.

No general guide of this kind can address every

contingency; thus, specific decisions and actions in

support of the workload forecasting process will vary from

agency to agency. However, the workload forecast procedure

described in this guide is applicable throughout the

life-cycle of an ADP system, and should be an on-going

process within an organization.

Portions of this guide are directed to technical staff,

ADP management, and functional management. The

Introduction, Overview, and the Overview of Workload

Forecasting Techniques sections, as well as workload

forecasting STEP 4 ("Analyze Forecast Results"), will be

useful to functional management. Technical staff and ADP

managers, who generally are responsible for forecasting the

workload, should find the entire document useful.

2
B. Background

Forecasting the user's workload requirements plays an

essential role in the life-cycle management of an ADP

system. It significantly impacts the success that an

organization has in providing the necessary computing

resources in a timely manner.

A detailed description of the life-cycle of an ADP

system is not within the scope of this guide. However, it

is important to note that the forecasting process is an

integral part of the life-cycle management of an ADP system.

The life-cycle of an ADP system is generally defined as a collection of several phases, progressing from the

requirements study to system operation. In general, an ADP

life-cycle encompasses four main phases as shown in Figure

1.

Without an on-going workload forecast, the computing

resources (system capacity) needed to perform the workload

cannot effectively be estimated and planned. Figure 2

depicts the workload growth at a hypothetical organization.

Assume, the organization of an ADP system took place in year

1 (point A) . In year 4 (point B) the system - in its

present configuration - reaches the saturation point. At

this point, if possible, the system needs to be tuned/

upgraded in order to handle the workload growth. Let's

assume, that the system reaches its maximum practical

capacity in year 8 (point D) , that is, no additional tuning

will provide better service. From year 6 (point C) , in the

absence of a capacity planning (including forecasting)

activity, this organization would not be able to provide the

needed resources to fulfill the organization's workload

requirements. This could result in it not being able to

support the organization's mission.

C. Overview

This section provides a brief outline of this guide.

SECTION II. Description of Workload Forecasting Techniques

gives a brief description of those forecasting techniques

which are most widely used. Also, several criteria for

selecting a given technique are discussed.

SECTION III. Workload Forecasting Steps describes a step-by-step approach to the workload forecasting process.

The following four procedural steps are identified:

STEP 1. Analyze Historical Data and Collect Future

Requirements. This step discusses the importance of

the participation of different levels of users in the

historical data collection process, and in the future

3
t

PLANNING:

STUDIES AND APPROVALS

REQUIREMENTS ANALYSIS

i

DESIGN:

TECHNICAL SPECIFICATIONS

REQUEST FOR PROPOSAL

i

DEVELOPMENT:

VENDOR EVALUATION

AND SELECTION

SYSTEM INSTALLATION

ACCEPTANCE

\

OPERATION:

SYSTEM OPERATION

TUNING, UPGRADES

I

FIGURE 1. ADP SYSTEM LIFE CYCLE

4
1980 1981 1982 1983 1984 1985 1986 1987

Time Periods

Workload Requirements

System Capacity

FIGURE 2. WORKLOAD REQUIREMENTS VERSUS SYSTEM CAPACITY
requirements identification procedure.

STEP 2. Select Forecasting Technique. This step

identifies the criteria to be met in order to select a given technique,

STEP 3. Perform Workload Forecast. This step gives a practical example of how to apply each of the

forecasting techniques discussed in the Overview of

Workload Forecasting Techniques section of this guide.

, STEP 4. Analyze Forecast Results. This step discusses

. the importance of analyzing the forecast results before

: any actions are taken.

6
II. OVERVIEW OF WORKLOAD FORECASTING TECHNIQUES

A. Forecasting in General

The distinction between planning and forecasting is

often confused. For the purpose of this guide, forecasting

is defined as a planning tool to predict what will

"actually" happen or to identify possible outcomes.

Planning is defined as a process that formulates objectives,

and identifies the actions to be taken to achieve these

objectives in light of the agency's mission. With every

forecasting situation, in varying degrees, uncertainties are

associated. Forecasting results should not be treated as

absolutes. Forecasts need to be carefully evaluated, before

decisions can be made based on their results. The terms

forecasting and predicting are used interchangeably in this

guide.

In general, forecasting techniques can be divided into

two groups:

1. quantitative methods, and

2. qualitative methods.

The quantitative forecasting methods rely on the

existence of historical data and on those variables that can

be quantified. With this method of forecasting, those

variables that are non-quantifiable are often ignored or

given very little consideration.

For the qualitative methods of forecasting, the

forecasts are based on judgements and expert opinions and

usually with no direct reliance on historical data. The

qualitative methods are especially useful for long-range

forecasting, when no historical data is available or a trend

is expected to change significantly. With these methods, it

is usually assumed that forecasts can be made by analyzing

the probable outcomes of cause-effect relations in a given

situation under certain conditions. The Delphi Method

[LI75] is one of the most widely used qualitative

forecasting techniques. This method of forecasting takes

the best judgement estimates from individuals determined to

be expert in a given area. The refined consensus of these

estimates becomes the forecast.

In this guide, only those quantitative techniques will

be discussed which are most widely used in the workload

forecasting process.

7
1. Forecast Spans

The forecast span is defined in this guide as the time

horizon for which forecasting is to be performed. In this

guide three forecast spans are identified:

1. Short-range forecast span

.

The short-range forecast

span in the workload forecasting process is defined to be in

the range of from one week to two years.

2. Medium- range forecast span

.

The medium-range forecast

span in this guide is defined as a time horizon between two

to five years.

3 . Long-range forecast span

.

in the workload forecasting

horizon of five or more years.

2. Data Patterns

The long-range forecast span

process is defined as a time

With the exception of the Box-Jenkins method described

below, the workload forecasting techniques discussed in this

guide assume that a certain underlying pattern can be

identified in the historical data. In general, four basic

patterns can be observed in historical data:

1. Trend Data Pattern?

2. Cyclical Data Pattern;

3. Seasonal Data Pattern;

4. Stationary Data Pattern.

It should be noted, that these patterns are not

mutually exclusive; often they can be observed together.

For predicting the workload it is sometimes desirable to

separate the effects of each of the pattern components on

the forecast value. This can be done by decomposition

techniques, such as the Classical Decomposition Method (see

SECTION III . ) .

2.1. Trend Data Pattern

A trend data pattern refers to a change where the

historical workload increases or decreases over a longer

period of time. In practice, the workload has the tendency

to increase due to such factors as increase in the number of

transactions and increase in program development. Figure 3

8
FIGURE 3. TREND DATA PATTERN

Time Period

FIGURE 4. CYCLICAL DATA PATTERN
shows a trend data pattern.

2.2. Cyclical Data Pattern

A cyclical data pattern refers to the long range

fluctuations in the workload requirements. The length of

the pattern is usually longer than one year, and the

fluctuation (pattern) is not repeated at identical time

intervals. Cyclical data patterns are usually caused by

factors, such as introduction of new application systems and

elimination of others. Figure 4 depicts a cyclical data

pattern.

2.3. Seasonal Data Pattern

A seasonal data pattern is similar to a cyclical data

pattern except that the fluctuation is caused by some

seasonal factor and the fluctuation is repeated at almost

identical time intervals. That is, the seasonal pattern

repeats itself usually within a year's interval. Seasonal

data patterns can be found at many organizations where

certain tasks need to be performed daily, weekly, monthly,

etc., and can cause seasonal patterns in the workload data.

Figure 5 shows a seasonal data pattern.

2.4. Stationary Data Pattern

Stationary data pattern refers to a series of

observations that do not exhibit any systematic increase or

decrease. The time series behaves as random fluctuations

about a mean value and no trend can be identified. Figure 6

is an example of a stationary data pattern.

B. Criteria for Selecting Forecasting Techniques

In general, the amount of effort associated with using

a forecasting technique should be approximately proportional

to the benef it/criticality of the forecast to the

organization as a whole. There are four main criteria in

choosing forecasting techniques:

1 . Data requi rements and pattern of the

data. Most of the quantitative forecasting

techniques require the availability of

reliable historical data. The success of a forecast process is significantly affected by

the selection of the technique that can most

10
1980 1981 1982 1983

Time Period

FIGURE 5. SEASONAL DATA PATTERN

Mean

Tine Period

FIGURE 6. HORIZONTAL (STATIONARY) DATA PATTERN

11
effectively handle a given pattern type.

2. Accuracy versus cost of us ing the

technique . In general, the more

sophisticated the technique and the more

factors considered, the more accurate and the

more costly are the forecasts compared to

those derived from simpler, less costly

methods. However, when the reliability of

the data is questionable or subjective, the

usage of sophisticated techniques will not

add to the accuracy of the forecast results.

3. Forecast span . Some forecasting

techniques, such as simple moving averages

are appropriate only for short-range

forecasting. Others, like regression

analysis are suitable for short- to

long-range forecasting.

4. Management ' s conf idence in the technique . Since management has the final authority on

the course of actions to be taken, it cannot

be emphasized enough that the communicabi 1 ity

of the forecast results and management's

confidence in the technique have a major

impact on whether the results will be used.

C. Forecasting Techniques

This section discusses six quantitative techniques that

are most commonly used for forecasting future workload

requirements. Following a brief general description, the

techniques are viewed from the perspective of the first

three selection criteria presented in Section B.

"Management's confidence in the technique" as one of

the criteria for the selection of a forecasting technique

will not be further discussed, since this criterion is

dependent on the background and preferences of the manager

involved in the decision making process. Some managers

might feel confident with a given technique, while others

might not.

The techniques to be examined are:

1. Simple moving averages;

2. Exponential smoothing;

3. Classical decomposition method;

12
4. Simple linear regression analysis;

5. Multiple linear regression analysis; and

6. Box-Jenkins method.

1. Simple Moving Averages

In this method of forecasting, historical data is used

to compute the forecast value. The average of the known

observations (values) becomes the forecast for the next time

period. When this forecast value then becomes a known value

(an actual observation) the average is then recalculated for

the next period. Hence, this method is called moving

averages. Moving average methods are also referred to as

"smoothing techniques," where extreme values of historical

data are smoothed by averaging. The smoothing effect of

this technique becomes greater with the increase in the

number of observations included in the averaging.

Data requirements and pattern of the data. Historical data

Is required over several time periods (^0-30), depending on

the smoothing effect desired. The number of time periods

selected should cover any periods with random fluctuations,

if the effect of the extreme values is to be evaluated. The

basic pattern of the data should be nearly stationary and

this same pattern is assumed for the future.

Accuracy versus cost of using the technique . The advantages

of using this method are that it is simple and inexpensive.

However, simple moving averages are limited to one time

period at a time into the future. If the pattern of the

historical data shows only little variation and the change

is nearly stationary, the confidence level using this

technique is usually high.

Forecast span . The forecast span is determined by the

interval of the time periods of the historical data. For

example, if the historical data is available for every three

months, then the forecast can be made for three months into

the future. This technique is mostly used for short-range

forecasting.

2. Exponential Smoothing

The underlying theory of exponential smoothing is the

same as for moving averages: historical data provides the

average which becomes the forecast value and which in turn

becomes a new known value. The difference is that more

recent observations are given exponentially increasing

13
smoothing weights. This is based on the assumption that

more recent observations provide a better indication of what

will happen in the future. The smoothing weight is

determined to be between zero and one.

Data requirements and pattern of the data. In order to

calculate the initial forecast value, historical data is

needed over several time periods (20-30) and the smoothing

weight must be determined. For subsequent forecasts, no

historical data are needed. The only data required are the

most recent observed value, the most recent forecast and the

value for the smoothing weight. The pattern of historical

data should be stationary and with no major changes expected

in the future.

Accuracy versus cost of using the techn ique . The major

advantages of using this technique are the low costs

associated with the development of the forecast, and its

ease of use. It is more accurate than the simple moving

averages. However, it also requires a stationary pattern in

the data.

Forecast span . As with simple moving averages, the forecast

span IS determined by the intervals (e.g., weekly , monthly

)

of the historical data.

3. Classical Decomposition Method

In the workload forecasting process, it is sometimes

desirable to distinquish the individual pattern components

from the underlying pattern in the historical data, i.e., to

identify what part of the increase in the requirements is

caused by seasonal, cyclical, or random fluctations (about

the trend) , and what portion represents an overall increase

(trend). The identification of the effect of the pattern

components can provide valuable information to the operation

personnel in the current scheduling process, in addition to

forecasting. The major advantages of this method are that

it is easy to use and interpret.

Data requirements and pattern of the data. For determining

the different pattern components for a given set of time

series data, historical data is needed over several

seasonal/cyclical periods (6-7). Time series data is

defined as historical data presented in uniform terms and

covering a specified period of time. As noted above, this

method can handle time series data with trend, seasonal, and

cyclical components.

Accuracy versus cost of using this technique . A relatively

accurate forecast can be achieved if sufficient historical

data is available (e.g., covering six seasons/cycles). The

development of this technique is expensive, but once the

14
model is developed, the application of the method is

inexpensive

Forecast span . This method of forecasting can be used from

short- to long-range forecasting, although, it is most

applicable for short-range.

4. Simple Linear Regression Analysis

Simple regression analysis is used to determine the

linear relationship between two variables. That is, a mathematical relationship is determined between the

dependent variable and the independent variable. This

relationship is expressed in the regression coefficient of

the regression equation. The statistical method of least

squares is used to determine the equation of the

relationship using historical data. The future values of

the dependent variable can then be extrapolated or

calculated using the equation determined by the least

squares method.

Data requi rements and pattern of the data . A large number

of accurate historical data points (30-40) for both

dependent and independent variables is required to determine

the linearity of the relationship. In contrast with

forecasting techniques discussed in the previous sections,

this technique not only assumes that an underlying pattern

exists in the historical data, but also that the basic

pattern is linear. That is, if the historical data is

plotted, the data points would fall along a straight line.

Accuracy versus cost of using the technique . This technique

allows for testing of the statistical significance of the

regression equation and of the individual coefficients, and

obtaining bounds for the predicted values. High accuracy

can be expected when no changes are anticipated to occur

which may modify the linear relationship of the variables.

The cost of developing the model for this technique can be

expensive if the data is not readily available.

Forecast span . Simple regression methods can be used for

any forecast span as long as a linear relationship exists

between the variables and the same relationship can be

expected for the future.

15
5. Multiple Linear Regression Analysis

This forecast method is similar to simple regression

analysis. However, with this method, the linear

relationship is defined among several independent variables

to the dependent variable. This relationship is expressed

in the regression coefficients of the regression equation.

Using observations of historical data for the variables,

correlation coefficients can be obtained by least squares

method for the independent variables. The future values of

the dependent variable can be computed (if future estimates

for independent variables are available) or extrapolated

using the equation determined by the least squares method.

Data requi rements and pattern of the data. To successfully

apply this technique, a large number of historical data

points (30-40) is required to assure the linear relationship

of the independent variables to the dependent variable.

This technique can handle any kind of pattern as long as a linear relationship exists between the dependent and

independent variables.

Accuracy versus cost of using the technique . This technique

allows for testing of the statistical significance of the

regression equation and of the individual coefficients, and

obtaining bounds for the predicted values. These measures

enable the forecaster to achieve a relatively high degree of

accuracy. Collecting the required large amount of

historical data to develop the initial regression equation

can be an expensive undertaking, but forecasting the future

values of the dependent variable using the regression

equation is inexpensive.

Forecast span . There is no limit on the forecast span.

6. Box-Jenkins Method

In cases where the pattern of the historical data is

readily apparent, methods discussed in the previous sections

are generally less expensive to use and more easily

understood than the more sophisticated Box-Jenkins method.

This method is a relatively recent quantitative

forecasting technique to handle those complex time series

data where the pattern is not easily identifiable. This

forecasting method is based on pattern identification. It

identifies the pattern in the historical data and uses this

pattern as the basis for forecasting future requirements.

Pattern recognition is accomplished by computing the

autocorrelations among values of the same variable. The

autocorrelation is a measure of the dependence among values

of the same variable at different time periods. High

16
autocorrelation indicates seasonal and/or cyclical patterns.

Like forecasting techniques discussed in the previous

sections, the Box-Jenkins method forecasts a single

variable. This method is a powerful forecasting tool where

the variable to be forecasted is a complex function of a variety of factors.

The recognition of the pattern in the data is

accomplished by multiple iterations of the data, with each

iteration providing information that allows the next

iteration to more closely approximate the actual pattern of

the data.

Data requirements and pattern of the data . Since this

technique is developed to handle complex time series data

and/or where the basic pattern is not apparent, the only

requirement is historical data over several time periods

(covering 6-7 seasonal/cyclical periods).

Accuracy versus cost of using the technique . This technique

IS considered by many experts in the field as one of the

most powerful and accurate forecasting techniques available

today. However, it is complex and expensive to use in

comparison to other techniques discussed.

Forecast span . This technique is mostly used for short- to

medium-range forecasting.

17
III. WORKLOAD FORECASTING STEPS

STEP 1. Analyze Historical Data and Collect Future

Requirements

As indicated in the previous sections, workload

forcasting plays a central role in the life-cycle management

of an ADP system and should be an on-going process at an

organization. One of the basic criteria for successful

forecasting is the availability of reliable historical data.

The historical data requirements are mostly determined by

the degree of criticality associated with the effect of the

decisions being made, based on the forecast results.

1.1 Identify Forecasting Objectives

The forecasting objectives must be identified by the

decision makers, if the forecasting process is to succeed.

The forecasting objectives might include the determination

of items such as: the criticality of the decisions to be

made based on the forecast results, level of effort to be

expended, desired forecast span, etc.

1.2 Organize Forecasting Team

A forecasting team will vary in size from agency to

agency and from one forecasting activity to another. There

are several factors that determine the size and composition

of the forecasting team. These factors include the variety

of existing and new functions/application systems, and

whether historical data is available or needs to be

collected. The forecasting team should include those

personnel who are knowledgeable about every aspect of the

workload description at a given workload level. The

decision makers and personnel responsible for the planning

at the organization should also be included in the

forecasting team.

1.3 Collect Data by Workload Levels

The workload forecasting process requires input from

several sources. At this point it seems appropriate to

describe an agency's workload on different levels. Figure 7

depicts the different workload levels at a given

organization.

18
Q<i:

•a:

I

19
Each of these workload levels can be associated with

different levels of management and/or users. As described

in STEP 3 ("Perform Workload Forecast"), this description of

the workload by levels provides the necessary input for

translating mission requirements into ADP resource needs.

On the first four levels the workload is described in

user oriented terms. In fact, it is almost imperative that

these measures are used when forecasting workload

requirements. These user oriented or functional measures

can then be translated into hardware/software requirements

whether the workload be performed on micros, mainframes,

etc. To collect data and to perform forecasts for new

functions/applications with no history, initial estimates

can be based on similar operational applications.

The functional users (Level 3) usually can identify to

the forecaster performing the survey, those personnel to be

contacted for obtaining workload data on different workload

levels. After identifying the personnel responsible for

providing the workload data for the different workload

levels, the forecaster should organize teams by functional

areas. Also, the personnel performing the planning function

should be included in the team. It is important to note,

that before the actual data collection takes place on the

different workload levels the variables/forecast units must

be identified, which is not always trivial. This is the

main reason why the participation of the personnel

knowledgeable in every aspect of the workload description on

the given workload level is essential to the success of the

workload forecasting process. Several criteria need to be

considered for selecting the variables/forecast units, for

example, objective of the forecast, data availability and

accuracy, etc.

1.3,1 Level 1. Agency Mission

The agency's mission is dictated by national interests

and needs. The identification of the agency's mission

assumes that an agency has a definite structure based on

goals and objectives. The agency's mission takes form in

the agency functions.

1.3.2 Level 2. Agency Functions

On this level, the workload is described by the

functions necessary to support the agency's mission. These

might include, for example, payroll, engineering, inventory,

budget, transportation, etc. On this level, the workload

can be associated with the top management. They are usually

the decision makers, and the sources of information which

20
enable the users to describe (and forecast) their workload

requirements on the next level.

The data that can be obtained on this level includes

such important information as:

type of functions to be performed,

budget ceilings,

frequency of reports generated,

changes in number of employees.

This data and other agency-dependent information and

decisions should be obtained to the extent that they impact

future computing resource needs.

While collecting information on past, present, and

future workloads, the agency functions should be evaluated

in relation to the agency mission. This is done to see if

any new functions are needed to support the agency's

mission, and whether the present functions need to be

continued into the future.

1.3.3 Level 3. Quantifiable Events

On this level, the workload can be described by those

quantifiable events that can be associated with a given

agency function. The quantifiable events are also referred

to as "Natural Forecast Units" (NFU's). For example, the

number of paychecks are the quantifiable events associated

with the payroll function; the number of student loans are

associated with the loan function. On this level, the

workload description can be associated with functional

users, such as personnel preparing the payroll. Information

from these functional users is then provided to the direct

users for describing their workload on the next level (Level

4). The functional users can provide the relationship

between quantifiable events and agency functions. These are

the users that can usually provide information such as:

number of paychecks produced,

number and type of transactions processed,

frequency and schedule of production runs,

number of loans processed.

21
' 1.3.4 Level 4. ADP Operations Performed

The workload on this level can be described in terms of

ADP operations which must be performed in order to produce

the given quantifiable events which support an agency

function and ultimately a given agency mission. These users

usually are those personnel who provide the programming and

other support to the functional user. These users can

provide the relationship between ADP operations and

quantifiable events. The data that can be obtained on this

level includes information such as:

number of records sorted,

number of records updated,

number of database queries,

for each quantifiable event.

1.3.5 Level 5. Resources Consumed

On this level, the workload can be described by the

computing resources it consumes. For example, a job/jobstep, or a transaction processed, can be described by

a set of resource demands. This information is usually

provided by the operations personnel through the use of

accounting logs. The resource usage data can usually be

obtained from accounting logs. If the data is not readily

available, the collection of the historical resource usage

data by applications/application systems can be a lengthy

procedure. Those organizations where the workload

forecasting is an on-going process only need to update the

resource usage information. The resource usage data is

usually collected with the help of operations personnel.

1.4 Analyze Historical Data

Having collected the workload from the different

levels, the forecaster needs to analyze the data. The

historical data should be analyzed in two stages:

1. by application systems; and

2. by the total workload.

22
1.4.1 Analyze Historical Data by Application Systems

The analysis of the historical data by application

systems can reveal important information necessary to build

a model which will be used to forecast a given application

system. The historical data collected from the different

levels of users can be plotted using simple histograms and

scatter diagrams. These simple techniques can provide the

forecaster valuable insight into relationships among

different levels of workload, such as:

the effect of budget change on the number of

quantifiable events,

the effect of the increase in the number of

quantifiable events on the volume of an

associated ADP operation performed,

change in resource requirements due to

variations in the number of quantifiable

events

,

how the rates of growth compare on the

different workload levels,

the pattern of the historical data for the

application systems by workload levels in

terms of some identifiable unit of measure.

1.4.2 Analyze Historical Data for the Total Workload

In order to determine the total workload requirements

at a given organization, the aggregate workload requirements

for all the application systems need to be obtained. The

most frequently used unit for determining the total workload

requirements is in terms of system accounting units (SAU's).

The SAU can be found on most third-generation computers;

examples of such units are CRU, SUP, SRU. In some cases,

where the workload is homogeneous, the total workload

requirements might be expressed in terms of functional

measures such as the number of transactions processed.

Simple analysis tools, such as graphs, can be used to

plot the total historical workload requirements against time

periods (weekly, monthly, etc.) to observe the behavior of

the total workload requirements at an organization.

The analysis of the graphs provides valuable

information to the forecaster for identifying:

the pattern of workload requirements in terms

of resource usage, and

23
the underlying trend of the historical data.

Also, the graph for the total workload requirements can

be used in conjunction with the graphs obtained for the

application systems by workload levels to obtain information

such as:

which of the applications/application systems

are causing a certain pattern in the resource

consumption,

what are the effects of the workload

composition changes in the resource

consumption.

This information can be useful for technical personnel

involved in benchmark construction and capacity planning on

the present system.

STEP 2. Select Forecasting Technique

As described in the Overview of Workload Forecasting

Techniques section, there are several criteria to be

considered in selecting a forecasting technique for a given

situation including:

1. data availability and the pattern of the

historical data;

2. accuracy versus cost of using the

technique;

3. desired forecast span.

The analysis of the historical data and the future

requirements (performed in STEP 1) provides a critical input

for selecting a given forecasting technique. This analysis

reveals the identifiable pattern of the historical data.

Also, the availability of the future requirements gives a good indication whether causal models are appropriate or

time-series analysis methods should be used.

As mentioned earlier, management's confidence in the

forecasting technique to be selected plays an essential role

in the success of the forecasting process. Management

should be aware of the accuracy that can possibly be

achieved, and also, the limitations of a technique intended

to be employed.

It is important to note that the more sophisticated

techniques do not necessarily lead to more accurate results.

In fact, if the validity of the historical data is

24
questionable or the uncertainties associated with future

requirements are too great, the use of complex techniques

would not benefit the organization as a whole.

In addition to the criteria identified in Table 1,

there are several other factors to be considered when

selecting a forecasting technique for a given situation.

Organizations where no formal forecasting is in

existence should always start with the simplest technique

appropriate and progress to the more complex as necessary.

The criticality of the forecast results should also be

taken into consideration when selecting forecasting

techniques, that is, what the effect of the forecast results

will be on the decision making.

STEP 3. Perform Workload Forecasting

At this point, it is assumed, the data are collected,

organized, and analyzed. Also, the decisions have been made

concerning the criticality of the forecast results, which

determines the required accuracy and the amount of resources

to be expended on the forecasting effort. The forecast

unit(s) and the required forecast span should also have been

determined.

Before applying any given model for forecasting

workload requirements, the model derived through a forecasting technique needs to be validated. A way to

perform this validation is by using only part of the

historical data (if sufficient) to build the model, and

exercise the model on the remaining data to see how the

forecast values compare to the actual values.

In most cases, short- and medium-range forecasting

quantitative techniques can be used. However, for

long-range forecasting, in addition to quantitative

techniques, qualitative evaluation of the forecast results

should be performed through expert opinions and judgements.

3.1 Organizational Approach to Forecasting

The application of the forecasting techniques does not

depend on the kind of data (e.g.; resource usage or ADP

operations) used. In some cases the workload forecast is

performed by using data from only one workload level, in

others by using data from two different levels. The usage

of the information from more than one workload level

provides a tool by which an organization's missions/plans

can be translated into processing demands expressed in terms

25
26
of ADP operations/resource demands. These resource demands

can then be used as input to the agency's ADP planning

process to determine the actions (e.g., tuning, upgrade,

replacement, etc.) to be taken to fulfill the agency's

mission requirements. Figure 7 depicts this translation

procedure (organizational approach) in the workload

forecasting process.

The ultimate goal of an agency's ADP facility is to

provide the computing resources needed to fulfill the

agency's mission requirements in the most cost-effective

manner. Therefore, the workload forecasting process should

also be performed in view of the benefit to the organization

as a whole. The result of the workload forecasting process

should provide the input for determining these computing

resource needs.

Usually, Level 2 is the highest level to be considered

by the workload forecasting team. However, the information

that is available from Level 1 is useful in the analysis of

the forecast results, especially for long-range forecasting.

A method of translating an agency's mission requirements

into computing resource requirements is described in the

following sections.

Usually, several functions (Level 2) need to be

performed by an agency in order to fulfill its mission. For

illustration purposes, the payroll function is used to show

how the translation can take place (see Figure 8).

The methods used for workload forecasting through the

different levels are largely determined by the data

availability and the relationship that can be established

among the workload levels. In general, the workload

forecasting can be performed by either using these

relationships (if they are known or can be determined)

through causal models or forecasting the workload

independently on a given workload level. With causal models

the effects (dependent variable) can be forecasted on the

basis of their causes (independent variable(s) ) . In

general, the causes can be forecasted in a relatively more

reliable manner than the effects. However, in either case,

the workloads need to be forecasted on the individual

levels, either by using relationships that can be identified

on a given level among the variables or forecasting a single

variable as the function of time, using historical data and

assuming that the past trend will continue into the future.

The workload forecast can usually be performed on three

different levels of the workload:

Level 3. quantifiable events;

Level 4. ADP operations performed;

Level 5. resources consumed.

27
LEVEL 2.

LEVEL 3.

LEVEL 4.

LEVEL 5.

FIGURE 8.

PAYROLL FUNCTION

(Number of Employees)

1

NO. OF PAYCHECKS

1

e.g, NO. OF UPDATES

I

SYSTEM ACCOUNTING

UNITS

PAYROLL DESCRIPTION FOR THE

FORECASTING PROCESS

28
3.1.1 Forecast Level 3 Workload Requirements

As noted earlier, the workload on Level 3 can be

described by the quantifiable events. As an example, the

number of paychecks are the quantifiable events for the

payroll function. They can be forecasted by using the

relationships between Levels 2 and 3. For example, a simple

linear regression model can be developed using historical

data to determine these relationships. In this example the

number of paychecks are the dependent variables and, for

example, the personnel budget ceiling is the independent

variable. If more than one independent variable can be

identified on Level 2, and relationships can be established

between these variables (independent) and the quantifiable

events as the dependent variables, multiple linear

regression as a causal model can be developed and used to

forecast the quantifiable events. If no relationships can

be identified, trend analysis techniques might be used to

forecast the quantifiable events as a function of time.

3.1.2 Forecast Level 4 Workload Requirements

The workload on Level 4 can be described by the ADP

operations necessary to produce a given quantifiable event.

For example, the generation of a single paycheck might

require one or more sorts, file updates, etc. On this level

the workload requirements can be forecasted as on Level 3,

either by causal models or by forecasting a given ADP

operation as the function of time using trend analysis

techniques

.

3.1.3 Forecast Level 5 Workload Requirements

A computer system can be considered as a collection of

resources upon which the workload places demands. The

demands on these resources can be used to describe the

workload on this level. This is the level at which the most

quantifiable data exists. The workload requirements on this

level can be forecasted in three (3) different ways;

1. by causal models, using data from Level 3 (as

independent variables) and Level 5 data as the

dependent variable (forecast unit);

2. by causal models, using data from Level 4 (as

independent varibles) and the Level 5 data as the

dependent variable (forecast unit);

3. by trend analysis techniques, with the forecasting

being performed on a single variable (on Level 5) as

29
the function of time using only historical data.

3.2 Apply Forecasting Techniques

In this section a practical example will be given on

how to use each of the forecasting techniques discussed in

the "Overview of Workload Forecasting Techniques" section.

Table 1 can be used as a guide to determine the

applicability of a given forecasting technique for the

desired situation.

There are several commercially available forecasting

software packages [DA83] to perform the necessary

statistical computations.

3.2.1 Forecasting with Simple Moving Averages

The simple moving averages are most applicable for

forecasting workload requirements during the operational

phase of the ADP life-cycle. For illustration purposes,

assume that an agency has twenty observations available for

CPU utilization on a weekly basis. Table 2 is a list of the

historical data for CPU utilization over the 20 weekly

periods. For short term scheduling purposes, the operations

personnel might be interested in what the CPU utilization

would be in the 21st week.

As noted earlier, this method of forecasting can handle

only stationary data. The stationary nature of the data can

be determined by plotting the CPU utilization against the

time periods. Figure 9 depicts that the CPU utilization for

this example is nearly stationary.

After the stationary nature of the data has been

determined, the decision has to be made concerning the

number of observations (periods) to be included in the

computation of the moving averages. The moving averages can

be computed using formula:

F

n n

Where

forecast value for time t,

number of observations included in the moving

average

,

actual (observation) value at time t.

30
60

50

40

30

20

10

I I I I I I I

1 2 3 4 5 6 7 9 10 11 12 13 14 15 16 17 18 19 20

Time Periods

FIGURE 9. WEEKLY CPU UTILIZATION

31
PERIOD OBSERVATION FORECAST FORECAST ERROR % ERROR

1 71

2 76

3 79

4 75 75.33 -0.33 - 0.33

5 72 76.67 -4.67 - 6.48

6 72 75.33 -3.33 - 4.62

7 73 73.00 0.00 0.00

8 69 72.33 -3.23 - 4.68

9 73 71.33 + 0.67 + 2.28

10 77 71.67 + 5.33 + 6.92

11 74 73.00 + 1.00 + 1.35

12 73 74.67 -1.67 - 2.29

13 75 74,67 + 0.33 + 0.44

14 75 74.00 + 1.00 + 1.33

15 71 74.33 -3.33 - 4.69

16 74 73.67 + 0.33 + 0.45

17 70 73.33 -3.33 - 4.76

18 70 71.67 -1.67 - 2.39

19 65 71.33 -6.33 - 9.74

20 60 68.33 -8.33 -13.88

TABLE 2. THREE MONTHS MOVING AVERAGE RESULTS

FOR CPU UTILIZATION

32
The number of periods to be included in the computation of

the moving averages can be determined by varying the number

of periods and choosing the one which gives the smallest

mean squared error on the forecast. The mean squared error

(MSE) can be computed using formula:

MSE = N

Where

= actual (observations) value at time t, = forecast value at time t,

N = number of forecast values.

In this example, three observations (a test was made with

three, five, and seven) in the moving average gives the

smallest mean square error on the forecast. Table 2 shows

the results of using three observations in the moving

average to compute the forecast values.

In order to estimate the error on forecast values

mean absolute percentage error (MAPE) can be computed:

the

MAPE =

N

I

(Yx - FO

t=1

N

(roo).

In this example the mean absolute percentage error on the

forecast is 3.93%.

Also, the forecaster might want to know if any

consistent under- or over-estimation might occur. That is

if a bias can be expected in the forecast results. This can

be estimated by the mean percentage error (MPE) using

formula

MPE =

N

t =1

(

N

.( »oo)

The mean percentage error in this example is: -2.42%. That

is a -2.42% underestimation can be expected in the forecast

results

.

It shold be noted that MSE, MPE, and MAPE can also be

used with other forecasting techniques.

33
The next step is to perform the forecast for period 21,

using formula:

P

. Yx Vt-n _

'^t + 1- +Fj , n n

~ 6 5.00 .

Where

n = number of observations included in the moving

average

,

= forecast for time t,

Y^. = actual (observed) value at time t.

The forecast value of CPU utilization for time period 21 is

65%.

The major limitation of this technique is that a new

forecast cannot be made until the forecast value for time

period t+1 becomes a known value (actual observation).

3.2.2 Forecasting with Exponential Smoothing

One of the major problems with this method of

forecasting is the selection of the appropriate weight. The

smoothing weight is determined to be between zero and one.

There is no best method for choosing the proper smoothing

weight. In most cases, this is done experimentally by using

different weights to decide which one is the most

appropriate

.

The exponential smoothing technique, like simple moving

averages, is most appropriate for forecasting workload

requirements during the operational phase of the ADP

life-cycle. The method of performing workload forecasting

with exponential smoothing is similar to the simple moving

averages method. However, with this method of forecasting

the more recent observations are given exponentially

increasing weight [KE76].

As with moving averages, the time-series data

(observations) need to be plotted in order to determine the

stationary nature of the series. The next step is to

determine the smoothing weight which gives the smallest mean

squared error (MSE) on the forecast (See section 3.2.1).

This can be accomplished by trial and error, giving

34
different values to the smoothing weights and choosing the

one which gives the smallest mean squared error on the

forecast. The determination of the most appropriate

smoothing weight can be performed on the historical data,

where the forecast values can be compared to the actual

(historical) values. Table 3 shows the results of using

0.2, 0.5, and 0.7 as the smoothing weights in the

exponential smoothing to compute the forecast values. Once

the smoothing weight is determined, this method of

forecasting requires only the latest observed value and the

forecast for this same time period. The following equation

can be used to perform the forecast: [MA78].

= forecast for time t,

= actual (observed) value at time t,

a = weighting (smoothing) value.

It is found that, using 0.7 as the smoothing weight

gives the smallest mean squared error among the three

smoothing weights tried. The mean absolute percentage error

on the forecast is 3.91%, using 0.7 as the smoothing weight.

Now, the forecast can be made for time period 21. The

forecast value of the CPU utilization for time period 21 is

3.2.3 Forecasting with Classical Decomposition Method

For the purpose of forecasting, it is assumed that the

historical data can be described as:

S = T * C * I

S = forecast value,

T = trend factor,

C = cyclical factor,

I = seasonal factor,

* = denotes multiplication.

In the real world, randomness always exists in time

series data. However, randomness cannot be predicted.

Therefore, it is not included in the above equation. In

order to use this equation in forecasting workload

requirements the components (T, C, and I) must be

Where

60.90%.

Where

35
FORECAST VALUE

PERIOD OBSERVATION a= 0.2 a=0.5 a=0.7

1 71

2 76 71.0 71.0 71.0

3 79 72.0 73.5 73.4

4 75 73.4 74.3 77.3

5 72 73.7 73.1 75.7

6 72 73.4 76.6 73.1

7 73 73.1 72.8 72.3

8 69 73.1 72.9 72.8

9 73 72.3 71.0 70.1

10 77 72.4 72.0 72.1

11 74 73.3 74.5 75.5

12 73 73.4 74.2 74.4

13 75 73.3 73.6 73.5

14 75 73.6 74.3 74.5

15 71 73.9 74.6 74.8

16 74 73.3 72.8 72.1

17 70 73.4 73.4 73.4

18 70 72.7 71.7 71.0

19 65 72.2 70.8 70.3

20 60 70.7 67.9 63.1

TABLE 3. FORECAST RESULTS BY EXPONENTIAL SMOOTHING

36
determined, A detailed description for computing these

components is given in the following paragraphs.

Table 4 shows monthly historical data over a five-year

period for the number of transactions processed. The

analysis of the historical data in Table 4 suggests that in

this time series a seasonal pattern exists. This time

series, if plotted, does not show a definite trend. The

cyclical factor is not as apparent; its existence can

usually be determined only after the decomposition of the

time series is performed.

In the classical decomposition, first the seasonal

factor is determined. This can be accomplished by:

1. Computing the 12-month moving average (MA), shown in

Table 5. Each of these values are computed as the average

of the preceeding six (6) historical data and the six (6)

historical data which are following it.

2. As it can be seen in Table 5, the moving averages fall

between each pair of months. It is necessary to adjust

these moving averages so they will be in accord with the

historical data. This can be accomplished through

centering, by computing the 2-month moving average of the

12-month moving average. This is the 12-month centered

moving average.

3. The next step is to compute the seasonal factor (Table

5). The seasonal factor is computed by expressing each

historical data (Table 4) as a percentage of the

corresponding 12-month centered moving average (Table 5).

Then the seasonal index is computed for each of the 12

months by taking the average of the seasonal factors. The

seasonal index is shown in Table 6.

The next step is to determine the trend factor in the

historical data (Table 4). This can be accomplished by

using a simple linear regression model (see 3.2.4.1 of

Section III.). The trend equation is:

= 592.74 + 0.61X^.

The trend values can then be computed for each month using

this equation. For the simplicity of computation, in this

example 593 is used as the trend value.

After the trend values have been computed, the cyclical

factor can be computed next. The cyclical factor is

calculated by dividing the centered 12-month moving average

values by the trend values and the results are multiplied by

100. Next, the forecasting can be done for the desired

point into the future using the identified pattern

components. Assume the forecast is desired for January

1982. The seasonal index for January is 132.72. The trend

37
1976 1977 1978 1979 1980

January 582 1022 822 724 790

February 478 780 624 610 612

March 468 716 542 536 538

April 490 766 586 578 610

May 482 694 520 542 546

June 434 688 514 520 550

July 536 716 566 618 662

August 488 636 488 474 492

September 482 620 436 490 602

October 528 634 566 472 628

November 702 652 532 552 750

December 812 740 650 696 1100

TABLE 4. MONTHLY DATA FOR THE NUMBER OF TRANSACTIONS

38
CENTERED

12 -MONTH 12 -MONTH

MOVING MOVING SEASONAL CYCLICAL

AVERAGE AVERAGE FACTOR FACTOR

ly / fa January

February

March

Apr i 1

May

June

b4U . 1 / July ceo 558 . 50 95

.

97 94 . 18

576 .83

August 589. 42 82. 79 99. 39

faUz . UU

September fa 14 . 3 3 lo . 4b 1 A 103 . 1 1 C 7

October fa3fa . 1 / O I . Q Q

y y

1 A 107 . 7 Q lo

C A t; <X 7

November CCA fa54 . C A 50 1 A T 10 7. lb 1 1 A 110. 7 1 37

C CI fafai . 33

December fa / 3 . y z

1 T A IIU . A Q 4 y110 113 . C A fa4

ly / / C Q /I . bU

January Q T

fay 2 . A A 00 1/17 14 / . fay 1 1 £r Ufa . fay

Q Q

fayy . dU

February 7 A C /Ob

.

C 7

fa /

1 1 A 1 lU . b3 1 1 Q iiy

.

A A

U U

"7 11 /II . O 3

Marcn 7 11/ 11 . bo 1 A A lUU . faz 1 1 Q 1 ly

.

Q Q y y

7 0 "3 Apr 1

1

TOT III. 1 b

1 A C 10 b. 7 £r Z fa 17 7 111 , 7 7/ I

1 7

May 7 O A /30 . A 1 0 1 y b

.

A 7

0 / 17 7 1 Z 3 .

1 A 1 U

. (JO

June T o c

1 lo

,

A A 00 Q A y4

.

Q A y u 17 7 111 . 7 C I fa TOO 111 . 00

July fay 7 . A O 4 1

1 A C

1 (J b

.

3o 11/1 114. C 7 b /

fa Jfo . o3

August 664. 58 98. 67 112 . 07

692 .33

September 685. 08 90. 50 115. 53

677 .83

October 670. 33 94. 58 113. 04

662 .83

November 655. 58 99. 45 110 . 55

648 .33

December 666. 08 111. 10 112. 32

TABLE 5. MOVING AVERAGES AND CYCLICAL FACTORS FOR THE

NUMBER OF TRANSACTIONS PROCESSED

39
1978

12 -MONTH

MOVING

AVERAGE

683.83

CENTERED

12 -MONTH

MOVING

AVERAGE

SEASONAL

FACTOR

CYCLICAL

FACTOR

January 652. 58 125. 96 110.04

621. 33

February 615. 17 101. 44 103.74

609.00

March 603. 34 89. 83 101.74

597.67

Apr i 1 592. 84 98. 85 99.97

588.00

May 583, 00 89. 19 98.31

578.00

June ^ 574. 25 89. 51 96.84

570 . 50

July 566. 42 99. 93 95.52

562 .33

August 561. 75 86. 87 94.73

561.17

September 560

.

92 77. 73 94. 59

560.67

October 560

.

33 101. 01 94.49

560.00

November 562

.

92 94. 56 94.93

561 .83

December 562

.

08 115. 64 94 .79

1979

562.33

January 564. 50 128. 26 95.19

566.67

February 566. 09 107. 26 95.46

565.50

March 567. 75 94. 41 95.74

570.00

Apr i 1 566. 09 102. 10 95.46

562.17

May 563. 00 96. 27 94.94

563.83

TABLE 5. MOVING AVERAGES AND CYCLICAL FACTORS FOR THE

NUMBER OF TRANSACTIONS PROCESSED

40
1979

June

July

August

September

October

November

December

1980

January

February

March

April

May

June

12 -MONTH

MOVING

AVERAGE

567.67

573.17

573.17

573.50

576.17

576.50

CENTERED

12 -MONTH

MOVING

AVERAGE

SEASONAL

FACTOR

CYCLICAL

FACTOR

579.00

582.67

584.17

553.50

606.50

623.00

656.67

565. 75 91. 75 95. 40

57042 108

34 9619

573. 17 82, 70 96. 66

573. 34 85. 46 96. 68

574. 84 82. 11 96. 94

576. 34 95. 78 97. 19

577. 75 120. 47 97. 43

580. 84 136. 01 97. 95

583. 42 104. 90 98. 38

568. 84 94. 58 95. 92

580. 00 105. 17 97. 81

614. 75 88. 82 103. 67

639. 84 85. 96 107. 90

TABLE 5. MOVING AVERAGES AND CYCLICAL FACTORS FOR THE

NUMBER OF TRANSACTIONS PROCESSED

41
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42
value is approximately 593. The cyclical factor needs to be

estimated by the forecaster. The forecaster should examine

expected changes in the workload such as: introduction of

new application systems, elimination of application systems

and any other factors that might effect the cyclical

component in the forecasting process. The cyclical factor

in this example is fairly smooth, does not exibit any

radical change over a short period of time. In this example

a somewhat subjectively higher cyclical factor is used

(1.09) than the last computed cyclical factor, 107.90 (for

June 1980).

The model to be used in forecasting is:

S = 593 * 1.32 * 1.09.

The forecast value for January 1982 is: 853.21.

3.2.4 Forecasting with Simple Linear Regression

The simple regression method can be used for

forecasting workload requirements throughout the life-cycle

of an ADP system. The simple regression analysis technique

can be used either as a trend analysis tool or as a causal

model. When it is used as a trend analysis technique, the

values of the independent variable are the time periods, and

the dependent variable is the unit of measure for which the

trend is to be determined. In the case of a causal model,

the independent variable in the regression equation is a variable which causes a linear rate of change in the

dependent variable (forecast unit of measure).

3.2.4.1 Simple Linear Regression as a Causal Model

As noted earlier, simple regression analysis is a useful tool for quantifying the relationship between two

variables. Correlation analysis can be used to determine

the linearity of this relationship. This relationship can

be used for forecasting the dependent variable. When using

simple linear regression as a causal model, a linear

relationship is assumed between the dependent (Y) variable

and the independent (X) variable). Table 7 contains 30

observations for the number of transactions processed (as

dependent variable), and the number of orders received (as

the independent variable) . The most commonly used regression method to find the

best fit line to the points is the least squares fit

technique. The simple linear regression equation between

43
PERIOD No. OF TRANSACTIONS NO. OF ORDERS

1 46 20

2 48 14

3 66 14

128 38

5 130 36

6 154 32

7 164 48

8 166 50

9 172 56

10 176 52

11 198 42

12 210 48

13 302 80

14 324 78

15 754 186

16 390 100

17 398 96

18 412 116

19 420 104

20 438 110

21 442 120

22 448 120

23 508 140

24 512 146

25 562 148

26 576 136

27 694 192

28 738 188

29 754 186

30 784 192

TABLE 7. HISTORICAL DATA OVER 30 PERIODS

44
two variables Y (dependent variable) and X (independent

variable) is denoted:

Y = a + bX.

Where "a" is the Y intercept (the value of Y when X = 0),

"b" is the regression coefficient (or the slope of the line)

which indicates the change in Y (dependent variable) when X

(independent variable) changes by one unit.

This procedure is simple and can be dealt with without

computers. Several advanced hand calculators have the

capability to perform these statistical computations.

The parameters "a" and "b" can be determined by the

least squares fit method. With the least squares fit method

the distance between the observations and the corresponding

points on the straight line is minimized. The parameters

can be obtained by the following equations:

b =

N N

-X

1=1 N N

-X Ix

i =1 1=1

a = Y-bX

Using these equations and the historical data from Table 7,

the following values are obtained for the parameters:

a = -7.81,

b = 3.93.

Where

N = number of observations,

"X = mean value of Xs,

7 = mean value of Ys.

The causal model (regression equation) derived by the least

squares fit method is:

Ft = -7.81 + 3.93X.

Where

= the computed (fitted) values from the regression

equation.

45
Before the regression equation can be used for

forecasting the number of transactions processed,

correlation analysis needs to be performed to determine the

degree of association between the two variables (dependent

and independent). The linear association between two

variables can also be visually detected by using a scatter

diagram, where each pair of observations (dependent and

independent variable) is plotted as a point. A straight

line could be manually fitted to these points, but this line

would not necessarily be the best fitted one.

This association can be measured by the correlation

coefficient of the regression line. The correlation

coefficient ranges between -1 and +1. The closer the

correlation coefficient is to +1 (for our purpose), the

better the association is between the two variables. It is

important to note that this measure is not meaningful unless

sufficient historical data is available.

The equation that can be used for computing the

correlation coefficient is:

N

Where:

= the observed values (dependent variables),

= the computed (fitted) values from the regression

equation,

7 = mean value of Ys,

N = number of observations.

The correlation coefficient is: 0.99. Since a large number

of observations (30) was used to determine the correlation

coefficient, it is reasonable to assume that the high

correlation (0.99) is significant in this example.

Also, a statistical significance test can be performed

on the regression coefficient. The significance of the

regression coefficient can be determined by its standard

error. It can be computed using [MA78]:

46
The standard error of the regression coefficient (b) is

0.09.

In order to obtain confidence in the regression

coefficient and for the predicted values, statistical

significance tests need to be performed. After the standard

error of the regression coefficient is computed, a t-test

should be performed to determine whether the regression

coefficient differs significantly from zero. The t-test for

"b" can be computed as follows:

b 3.9 3 ^, t-test. = ^= ^ =^3 . b SEb 0.09

By comparing the computed value of t to the appropriate

value in a t-table, the significance level can be determined

for the regression coefficient. Since the t-table value

[SP75] is greater than two and the computed t-test for "b"

is greater than the critical t-table value, it can be

concluded (with almost 100% certainty) that the regression

coefficient is significantly different from zero.

In addition, the "goodness of fit" should also be

determined. This can be measured by the coefficient of

determination (square of the correlation coefficient). The

coefficient of determination measures the proportion of

total variation about the mean Y that is accounted for in

the regression equation (explained variation). This can be

computed as:

N

i (Yrvf

1=1

that is, the sum of the squares of the deviations of the

computed (estimated) values from their mean.

In this example, the coefficient of determination is

0.98. That is 98% of the variation from the mean value of Y

is explained by the model. The significance (at 95%

confidence level) of this explained variation can be

obtained by the F-test:

47
N

I {Y,-Frl(N-K)

Where

N = number of observations,

K = number of variables.

This value can be compared to the appropriate entry

{intersection of 1 and 28 in the F-table) [SP75] which is

4.20. Since the computed F-test is greater than five (as a general rule) [MA78], and it is greater than the critical

value obtained from the F-table, the forecaster can be 95%

certain that the regression equation explains 98% of the

total variation from the mean value of Y.

The unexplained variation is the sum of the squares of

the deviations of the Y values from their computed

(estimated) values:

1=1

These unexplained variations are called the residuals. An

examination of the residuals (difference between observed

and predicted values) also needs to be performed. At least

three criteria need to be satisfied:

1. the mean value of residuals must be zero;

2. the residuals must be normally distributed; and

3. the variance of residuals X { f^—i ) must

be constant. i=i

In most cases, the forecaster and the decision makers

are interested in the confidence interval (bounds) on the

forecast. This can be obtained by computing the standard

error on the forecast. The standard error on the forecast

can be computed by using formula:

SEf =

SEf =

N-2

24 628

28

1 (Xi-X)

M N 2

1=1

1+ —L JSO- 96.27)

30 97671

= 4.16

48
when 50 orders are received, the number of transactions

processed (at 95% confidence level) would be:

Yf = -7.81 + 3.93(50) ± (4.16)

bound on the forecast.

That is, the number of transactions processed for 50 orders

received would be between 180 and 197 with 95% certainty.

3.2.4.2 Trend Analysis with Simple Linear Regression

The method of using simple regression for trend

analysis is similar to the use of simple regression as a causal model. The only exception is that, with this method,

the values of the independent variable are the time periods.

3.2.5 Forecasting with Multiple Linear Regression

The multiple regression method, like simple regression,

can be used for forecasting workload requirements throughout

the life-cycle of an ADP system. The forecasting with

multiple regression in basic concept is similar to the

method of forecasting with simple regression. However, in

the case of multiple regression, the dependent variable

(forecast unit of measure) is defined as a function of

several independent variables. The independent variables

are assumed to have linear relationships with the dependent

variable.

In multiple linear regression, the theoretical

relationship between the dependent and independent variables

is assumed to be of the form:

Y = a +b^x^+b2X2+ ... +b^x^^ oc . Where, Y is the dependent variable, XI, X2 , Xp are the

independent variables, and a is a normally distributed

random error (residual) with zero mean and constant

variance. For the relationship to be modeled in workload

forecasting, in most cases it is reasonable to assume that

the relationships may be of this form. The assumption of

the linear relationships allows the building of a model for

making predictions. Using historical data (observations)

for the variables by the least squares fit method, the

coefficients for the independent variables can be estimated.

The residual can be computed as the difference between the

estimated value of Y and the observed value of Y.

49
The choice of the independent variables should be based

on two criteria:

1. expected linear relationship with the dependent

variable; and

2. availability of estimates for the future.

Although non-linear relationships can be transformed

into linear, the model might not be as successful in

forecasting as it would be if linear relationships existed.

Simple techniques, such as scatter-diagrams , can be used to

initially determine the linearity.

For the relationships to be modeled in the area of

workload forecasting it is reasonable to assume, in most

cases, that the relationships may be in this form. For

example, the processing requirements to support a payroll

system may reasonably be expected to be a linear function of

the number of people the system supports.

It should be noted that the forecaster planning to

employ this technique should obtain a statistical package to

perform the necessary computations.

In order to develop a multiple regression model for the

purpose of forecasting workload requirements, the forecast

unit of measure and the candidate independent variables need

to be identified.

The selection of the variables can be performed by

stepwise regression [DR66]. This procedure enters the

independent variables into the regression equation one at a time. At each step those variables are entered which are

making the largest contribution among those not yet entered.

With the introduction of the next variable, the contribution

of some of the previously introduced variables might become

negligible. That is, they can be eliminated from the

equation. Hence, after each step, the contribution of the

independent variables are reevaluted.

Usually, in practice, the procedure begins by computing

the simple correlation of each independent variable with the

dependent variable and then first introducing the

independent variable with the highest correlation into the

regression equation. Those variables which have the largest

partial F-statistic (F-test) are then added one at a time.

The procedure terminates when the F-statistic of the

variables not yet introduced are statistically not

signi f icant

.

The example given below uses the results of a previous

study [P078]. The dependent variable to be modeled is the

total number of I/O's, and the candidate independent

50
variables are shown in Table 8.

In modeling the total number of I/O's, stepwise

regression is performed with all the independent variables

shown in Table 8. In Table 8 it can be seen that Step 1

selects Airmen as the best single independent variable. The

coefficient of determination for this regression is 0.53;

the standard error is 4.1x10®; the mean value of total

I/O's is 22.7x10®. Adding the Account Control in the second

step improves the "goodness of fit" considerably, the

coefficient of determination increases to 0.76 and decreases

the standard error to 2.93x10?. The procedure terminates at

this point if the inclusion of the variables is restricted

to those variables whose coefficients are significant at the

0.01 level, as measured by the partial F-statistic.

Allowing a slightly less significant variable, the procedure

next enters Civil Engineering, which increases the

coefficient of determination to 0.78 and lowers the standard

error to 2.82x10®. The coefficient of the added variable

(Civil Engineering) has a partial F-statistic of 6.2,

significant at 0.02 level. The fourth step adds Medical

Material, bringing the determination of coefficient to just

under 0.80 and reducing the standard error of the previous

equation by two percent. The least significant coefficient,

of the Medical Material variable just entered, has a partial

F-statistic of 4.1, significant at the 0.05 level.

The procedure would stop at this point if the level for

entry were set at the 0.05 level. If less significant terms

are allowed to enter, steps 5, 6, and 7 add Data Control,

Fighter Pilots, and Travel. Each brings a slight increase

in the coefficient of determination and decrease in the

standard error. Steps 8 and 9 then remove the Medical

Material and Data Control variables, leaving five variables

in the equation, achieving 0.82 for the coefficient of

determination and standard error of 2.65x10®. Each variable

of this equation is significant at the 0.01 level.

As can be seen, the stepwise regression procedure is a useful tool for selecting the most appropriate model. The

equation in step 4 provides the best model. The partial

F-statistic of the coefficients of the variables are all

significant at the 0.05 level, all but one being significant

at the 0.01 level.

In practice, the selection of the most appropriate

model is not always straightforward. The examination of all

possible equations does not provide an incisive answer. It

should be remembered that the independent variables included

in the equation to forecast the values for the dependent

variable are themselves the result of a separate forecasting

or experts' opinions. That is, when selecting the most

appropriate model, the degree of uncertainties associated

with the independent variables need to be taken into account

in addition to the statistical significance tests performed.

51
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3.2.6 Forecasting with the Box-Jenkins Method

The procedure for the Box-Jenkins method in general can

best be described by a schematic diagram, as depicted in

Figure 10.

As noted earlier, and depicted by Figure 10, this method of

forecasting can handle any data pattern. The identification

of the pattern in the data is accomplished by the method

itself. The pattern identification leads to the model to be

employed. The primary aid in the identification of the

model to be utilized is the autocorrelation. High

autocorrelation indicates seasonal or cyclical pattern, and

where the autocorrelation is low or does not exist, it

indicates random data. The Box-Jenkins method uses this

information to arrive at the optimal model to be used in

forecasting. Further information on the use of

autocorrelation can be found in [B076]. The autocorrelation

can be computed by the following formula [KE76]:

k

^ j-T

Where

n = the number of observations,

Xt = the value of the variable at time t,

k = the length of time,

X =the mean value of all of the observations.

Three general classes of models can be utilized in the

Box-Jenkins method of forecasting:

1. Autoregressive (AR)

,

2. Moving Average (MA), and

3. Autoregressive-Moving Average (ARMA)

.

The following is a brief description of these general

classes of models.

The Autoregressive Model (AR(p) )

.

The autoregressive model can be characterized by the

autocorrelation function which decays exponentially to zero

or exhibits an exponentially damped sine wave. The

53
IDENTIFY

TENTATIVE MODEL

ESTIMATE

PARAMETERS IN

TENTATIVE MODEL

USE MODEL FOR

FORECAST I NG

FIGURE 10. BOX-JENKINS METHOD OF FORECASTING

54
autocorrelation coefficient versus time period can be

plotted on a correlogram to observe this behavior. A graph

of the autocorrelation function is referred to as the

correlogram. An autoregressive model of order p describes

the current observation as a weighted sum of "p" previous

observations plus a random perturbation or residual that

cannot be explained by the model. The p order

autoregressive model is in the form of:

Where

Xt = observations

$ = the weighting parameter,

= the residual.

The procedure for computing the parameters can be found in

[NE73].

The Mov i ng Average Model ( MA ( q )

)

A moving average model of order q can be characterized by

the autocorrelation function which if plotted on a correlogram exhibits an abrupt cut off after the time period

q. In the moving average model, the current value

(dependent variable) is computed as the sum of the current

residual plus the weighted sum of previous residuals.

Mathematically, this can be expressed as:

Where:

Xt = the value of the variable at time t,

ot = the weighting parameter,

= the residual,

q = the number of previous residual in the weighted

sum.

The Autoregressive-Moving Average Model ( ARMA ( p , q )

)

Sometimes it might be desirable to combine the

autoregressive AR(p) and the moving average MA(q) models to

describe a series of observations. Such a model is referred

55
to as autoregressive-moving average model. This model when

used in forecasting bases the forecast value both on past

observations and on the residuals. This model can be

mathematically expressed as:

p q

j=1 kr1

Where:

§ = the weighting parameter.

Since this model is a mix of the autoregressive and moving

average models, it can be characterized by an

autocorrelation function which decays exponentially or

exhibits an exponentially damped sine wave and/or shows an

abrupt cut off after p-q periods.

This method of forecasting is applicable for

forecasting workload requirements in the operational phase

of an ADP system life-cycle. The Box-Jenkins method of

forecasting does not require prior assumptions about the

underlying pattern in the series.

The first step in the Box-Jenkins method of forecasting

is the identification of the tentative model. This can be

accomplished by the analysis of the autocorrelations.

Table 9 is a list of autocorrelations for illustration

purposes. The examination of the autocorrelations shows

peaks near time periods (lags) seven and fourteen. This

indicates the possibility of a seven-day cycle (seasonal

pattern), which suggests that seasonal differencing is

necessary. By taking the seasonal difference of period

seven it can be seen (Table 10) that all of the sample

autocorrelations are near zero, except for period seven.

This is an indication that the model is in the form

Xt = Xt-7 + - 5Zt-7,

which is a seasonal model. It should be noted that the

identification of the model is not as simple as it might

seem from the above example. Users attempting to apply this

method would find the following references [B076 ,KU78 ,NE73

]

useful. Also, there are several software packages available

to perform the required statistical computations.

The next step is to determine the $ parameter. This

can be accomplished by minimizing the sum of squares on the

residuals (Z^) . It is found to be 0.6144.

56
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57
The model to be used in forecasting is thus:

^t " ^t'"^ + Zt - 0.6144Zt-7 . Before this model can be used in forecasting,

diagnostic checks need to be performed to determine the

adequacy of the model. This can be accomplished by the

analysis of autocorrelations of the estimated residuals. If

the residuals do not show any pattern and they all are

nearly zero the model seems adequate.

Since the model is seasonal, one more diagnostic check

needs to be performed on the residuals using a cumulative

periodogram [B076], Figure 11 [KU78]. By a careful

inspection of the cumulative periodogram plot of the

residuals it can be determined whether any periodic

components exists in the series which are not explained by

the model. Kolmogorov-Smirnof f goodness of fit [B076] can

be performed to determine whether there are any

periodicities in the residuals. The cumulative periodogram

(Figure 11) does not show any periodicity in the residuals.

Thus it can be assumed that the model is adequate.

STEP 4. Analyze Forecast Results

The results of the forecasting process need to be

carefully analyzed and evaluated before any decisions can be

made based on the forecast results. The results of the

forecasting process present the decision makers with

information on whether adequate computing resources are

available to perform the users workload. The actions to be

taken might involve one or more of the following activities:

tune the current system,

upgrade system components,

obtain outside resources,

replace the current system.

In the analysis and evaluation step of the forecasting

process several important activities should take place.

The purpose of the workload forecasting process is to

provide information on future processing requirements in

order to provide adequate computing resources to perform the

user's workload and ultimately to fulfill the agency's

mission. Therefore, it must be determined whether the

applications/application systems included in the forecast

process are the ones which are necessary to' support the

agency's mission. This can be accomplished by submitting

58
0.10 0.20 0.30 0.40 0.50

Frequency

FIGURE 11. CUMULATIVE PERIODOGRAM OF RESIDUALS

59
the forecast results to personnel who have been surveyed in

the data collection process.

Concerning the forecasting techniques used, several

factors need to be analyzed. One of the most important

considerations is the examination of the model used in the

forecasting. That is, whether the conditions under which

the model was derived are still the same. If not, the model

needs to be updated. Also, if causal models (e.g., multiple

regression) were used in the forecasting process, it is

desirable to perform several forecasts by varying the values

(sensitivity analysis) of the independent variables

(predictors). This is usually desirable when the users have

doubt about the certainty of their future requirements,

and/or if the applications/application systems involved play

a highly important role in fulfilling the agency's mission.

The forecast results should be presented to the

decision makers by identifying the outcomes of the different

techniques applied, the results of the alternative

techniques, and sensitivity analysis, if one is performed.

Also, the likely solutions (e.g., upgrade) to overcome any

possible shortcomings in the computing resources, should be

submitted to the decision makers.

SUMMARY

This guide attempted to provide practical guidance on

workload forecasting in a step-by-step fashion. Some

agencies may find that the sequence of these steps is not

suited to their particular needs, in this case the Table of

Contents can be used as a reference to the specific sections

of interest. However these steps should at least be

considered, especially by those agencies where no formal

forecasting currently exists.

It is hoped that this guide will help agencies to

obtain up-to-date information on their future workload

requirements upon which informed management decisions can be

made regarding computer resource needs to meet these future

workload requirements.

60
APPENDIX A

SUGGESTED READINGS AND REFERENCES

[B076] Box, George E.P.; Jenkins, Gwilym M. "Time Series

Analysis, Forecasting and Control." Revised Edition.

Holden-Day, Inc. 1976.

[CH74] Chambers, John C; Mullick, Satinder K.; Smith,

Donald D. "An Executive's Guide to Forecasting." John Wiley

and Sons. 1974.

[DA83] Datapro Research Corporation, "Management Sciences,"

Delran, NJ. 08075.

[DR66] Draper, N. R. "Applied Regression Analysis." John

Wiley and Sons. 1966.

[KE76] Kendall, Maurice. "Time Series." Charles Griffin and

Company LTD. 1976.

[KU78] Kulp, Richard W. ; Melendez, Kenneth. "An

Application of Time Series Analysis in Computer Performance

Evaluation." Computer Performance Evaluation Users Group:

CPEUG 14th Meeting; 1978.

[LI75] Linstone, Harold A.; Turoff, Murray, eds. "The

Delphi Method: Techniques and Applications."

Addisonp-Wesley , Reading, Ma. 1975.

[MA78] Makridakis, Spyros; Wheelright, Steven C.

"Interactive Forecasting Univariate and Multivariate

Methods." Holden-Day, Inc. 1978.

[NB80] National Bureau of Standards. Federal Information

Processing Standards (FIPS) Publication. "Guideline on

Constructing Benchmarks for ADP System Acquisitions." NBS.

1980 September 18; FIPS PUB 75.

[NE73] Nelson, Charles R. "Applied Time Series Analysis for

Managerial Forecasting." Holden-Day, Inc. 1973.

[P078] Poggio, Eugene C; Ainslie, Naomi. "Forecasting

Processing Requirements of Standard USAF Base Level Computer

Systems." 1978.

[SP75] Spiegel, Murray R. "Theory and Problems of

Probability and Statistics." McGraw-Hill Book Company, 1975.

[SA79] Sarna, David E.Y. "Forecasting Resource Utilization

Using Key Volume Indicators." Proceedings National Computer

Conference; 1979.

61
[SH67] Shiskin, Julius; Young, Allen H.; Musgrave, John C.

"The X-11 Variant of the Census Method II Seasonal

Adjustment Program." U.S. Department of Commerce, Bureau of

Census. Technical Paper No. 15, 1967.

6U.S. GOVERNMENT PRINTING OFFICE: 1985-461-105 •• 20025

62
NBS-n4A iREv. 2-ec) U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO. NBS/SP-500/123

2. Performing Organ. Report No. 3. Publ ication Date

March 19R5

4. TITLE AND SUBTITLE

Computer Science and Technology:

Guide on Workload Forecasting

5. AUTHOR(S)

Helen Letmanyi

6. PERFORMING ORGANIZATION (If joint or other than NBS. see in struct/on sj

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

GAITHERSBURG, MD 20899

7. Contract/Grant No.

8. Type of Report & Period Covered

Fi nal

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State. ZIP)

Same as in item 6 above,

10. SUPPLEMENTARY NOTES

Library of Congress Catalog Card Number: 85-600504

[^J Document describes a computer program; SF-185, FlPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual sumanary of most si gnifi cant information. If document includes a si gnifleant bibliography or literature survey, mention it here)

The purpose of this guide is to provide ADP managers and

technical personnel with useful quantitative techniques for

forecasting future workload requirements. It additionally

provides a step-by-step approach to the forecasting process.

Readers can then, in a timely manner, provide the computing

resources needed to perform the user's workload at required

service levels throughout the life-cycle of an ADP system.

These techniques are described so that readers with little

or no training in statistics should find them useful.

However, this guide does not intend to give an exhaustive

treatment of the techniques discussed. Readers requiring

more information are referred to Appendix A ("Suggested

Readings and References").

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

causal models; forecast spans; time-series models; workload forecasting techniques;

workload levels; workload transition.

13. AVAILABILITY

QX] Unl imited

I I For Official Distribution. Do Not Release to NTIS

rX] Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402.

Order From National Technical Information Service (NTIS), Springfield, VA. 2216 I

14. NO. OF

PRINTED PAGES

68

15. Price

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