FUNDAMENTAL LAWS OF COMPUTER SYSTEM

FUNDAMENTAL

LAWS OF COMPUTER

J.P.Buzen for Research in Computing Technology Harvard University Cambridge, Mass., 02138 and BGS Systems, Inc. Box 128 Lincoln, Mass., 01773

Center

ABSTRACT A number

device

between

utilization,

ious other

sis.

related

These

the operational

throughput,

space-time

factors

performance.

modeling

response

products

to computer

laws are obtained

method

The operational

nificantly

Table

of laws are derived which establish

relationships

of computer method,

approach,

ved properties

of real computer errors,

consists

Assume

of i second;

of 3 seconds.

Except laws pre-

response

than the average

response response

sented in this paper apply with complete

preci-

sion to all collections

data,

Appendix

and they are similar

to fundamental

in other areas of engineering

K e y Words:

Computer

throughput,

response

operational

method.

laws found

and applied

performance

Most analyses

models.

of computer

benchmarks

Benchmarks,

programs,

simulations,

cisely results nature

specified

of the workload

to be processing, load definition ly different

TIME

RR(Q=2)

B

C

8 2/3

7 1/3

C

B

A

6

6 2/3

B

A

C

6 2/3

6 2/3

Table

In the second same except

sensitive

benchmark to the

and slight

may sometimes

changes produce

average

than average

second

set of benchmark

mark contains

in the work-

response

higher

strong preference

that the system is assumed

line of Table

response

for FCFS,

presents

conclusions.

200

is reversed.

time for RR is 11%

time for FCFS.

results

is the

The

thus indicate

a

even though this bench-

the same set of jobs as the first.

As a point of interest,

significant-

i, everything

that the order of arrival

In this case,

under a pre-

However,

i

of real

or trace driven

system behavior

can be surprisingly

RESPONSE

FCFS

system performance

programs

workload.

re-

preference

A

or probabilistic

are most useful when it is neces-

sary to determine

AVERAGE

FIRST SECOND THIRD

systems,

which may consist

synthetic

the benchmark

a definite

WORKLOAD

evaluation~

time, queueing

Thus

(see

for RR.

science.

i. INTRODUCTION

rely on either

A for details).

suits in line one indicate

time for

at approximate-

time for RR with a quantum of two seconds

of observational

and

The first

Note that the average

for FCFS is 18% higher

of

the order of arri-

val is "ABC" with all jobs arriving ly the same time.

for

that the workload

the three jobs in the case where

to obser-

algorithms

Job A, with a duration

line of Table i gives the average

stochastic

that an ana-

(RR) and "first

Job B, with a duration

Job C, with a duration

sig-

Suppose

scheduling

of three jobs:

7 seconds;

system analy-

the operational

example.

"round robin"

a central processor.

system

systems.

simplified

the crux of the problem with

come first served"(FCFS)

and var-

by using

and directly

i illustrates

lyst is comparing

is based on a set of concepts

naturally

for measurement

a highly

time,

which differs

from the conventional

that correspond

SYSTEM PERFOP~IANCE

yet another

the third line of Table

arrival

sequence

in which

i the

two scheduling average

response

Although plified, real.

algorithms

produce

exactly

and observed

time•

the example

in Table

1 is highly

the dangers which it illustrates

In particular,

specification detail:

benchmark

as a result,

the analyst

where his knowledge

is usually

to confusing

ly equivalent

require

imprecise. where

studies produce

only a single

results

in areas

reason

enable

directly with situations ledge of the workload

the analyst

to deal

Suppose,

case of Table l, that the workload

is known

represented

is uncertain.

with a probabilistic

lyst may be able to formulate tion i n t e r m s suppose

that it is reasonable

the Law of

Theorem~and

intervals)

imply

model,

all cases.

for constructing

This is the major

simulation

programs

structing

that run for long periods

benchmarks

The assertion

the ana-

to assume

that a particular

confidence

level--

enough"--

may be satisfactory

tain performance

solu-

in ~raetiee

assertions

if the experiment

evaluation

are clearly

oriented

to meet all

computer

the

"CBA" will occur 10% of the time,

and the

with sets of data that have been collected

sequence

"BAC" will occur

sequence

as follows:

FCFS response

.60 x 8~- + .I0 x 6

intervals

can be regarded

times can be computed Expected

rect measurement

In

of time.

interested

response

Such individuals

of actual

2 + .30 x 6~

quantities could,

the internal

C. Formulas

the analyst

can compute

-- FCFS response

that RR response

time.

The analyst

benchmark

tests

fail to confirm his prediction;

indeed,

he fully

expects response

to find RR response

higher

of existing

sets

the effect

that certain

to the system or the workload quantities

such as

throughput and response time. Note that empirically mance analysts

than FCFS

time for 10% of the arrival

determined.

data.

would have on measured

is not troubl-

ed by the fact that individual

but which

that can be used to verify

that predict

modifications

that -- "on the

time will be 10% higher

data to other

empirically

consistency

of measurement

Thus,

are basically

that were not measured

in principleabe

B. Relationships

= 7.07

by difinite

expressions which re-

time

RR response time i 2 2 = .60 x 7~ + .i0 x 6~ + .30 x 6~

during

These analysts

per-

usually deal

in:

late existing measurement

Expected

average"

systems

A. Precise mathematical

= 7.80

cer-

but such

"ABC" will occur 60% of the time,

30% of the time.

analysts.

re-

is run "long

for resolving

issues,

insufficient

the needs of empirically

that the

theoretical

at some desired

sequence

and the expected

of

OBJECTIVES

sequence

the arrival

that

time and con-

formance

this case,

pre-

of simulated

sult can he validated

For example,

as a random variable,

the

that

average values will be reason-

3. OPERATIONAL

can be

a meaningful

of random variables.

In such cases,

(e.g.,

run for long periods

to con-

If the uncer-

the order of arrival

In prac-

in the

sist of Jobs A, B, and C, but the order in which the jobs will arrive

sequence.

of

real time.

where only partial know-

exists.

by the im-

to include many three

theory

the Ergodic

in

consisted

in a single benchmark.

dicted and observed

different

MODELS

models

arrival expect

from probability

ably close in almost

2. PROBABILISTIC

tainty concerning

three-job

theory of confidence

final conclusions.

Probabilistic

that the benchmark

Large Numbers,

This in seeming-

times discussed

paragraph were magnified

assumption

job sequences

is often compel-

between predicted

response

tice one would normally

in complete

decisions

situations

benchmark

plicit

discrepancies

average

the preceding

sim-

are very

evaluations

of the system workload

led to make subtle but critical

turn leads

The potential

the same

lationships

sequences.

values

computer interested

between random variables

of random variabies;

terested

201

oriented

are not primarily

in relationships

perforin re-

or expected

r a t h e ~ they are in-

between measured

quan-

tities and quantities measured.

which can~in principle,

The remainder

be

Throughput

of this paper will deve-

lop a theory of such relationships. This theory is based on the concept tional variables. mal objects

Operational

defined

of opera-

variables

so that they correspond

systems.

that will be derived

Some of the results

using operational

variables

have direct

parts which can be expressed variables

and probabilistic

is important

to notice

are fundamentally

that operational

different

results:

that is, operational precision

vational

data,

whereas

apply precisely

results

probabilistic

(with probability

one)

case where

the length of the observation infinity.

al

results

artificial

independent

service

and identically

and inter-arrival

equilibrium,

specific

that are commonly babilistic

operational

to broad classes results

are similar mental

as universal

laws of computer in certain

statements etc.)

tion,

in this

provide

They

and electrical

to the funda-

of these similarities

engineering.

of other

a precise

applicable

Before proceeding is useful

to consider

wSth the main analysis, two specific

questions

are typical of a large class of problems arise in the study of computer The two questions, the throughput problem,

problem

it

possible

to each question.

in the problem

conform

to Furare

to the

statements,

regard-

characteristics

For future reference,

to the throughput

to

problem

these

note that is 51 jobs

to the response

time

is 5 seconds.

which

5. THE OPERATIONAL The solutions

system behavior.

problem and the response

appear

or

under considera-

it is entirely

answer

level

METHOD

that

which will be referred

are defined

factors which would

per hour and the solution

TWO BASIC PROBLEMS

size,

system overhead,

to all systems which

the solution 4.

of memory

operating

In addi-

the answers which can be obtained

systems may have.

of the presentation.

or the utilization

in the system.

less of any other particular

will be noted in the course

is insufficient

posed in each case.

to the basic problems

specifications

Some

that the information

is made in the problem

I/O devices

Nevertheless,

thermore,

laws found in such fields as basic mechanics,

thermodynamics,

number

What is the ave-

questions

there is no discussion

tion.

and funda-

system performance. respects

a

time for the

of most modern day com-

of CPU utilization

be relevant

the

of the system

service

statements

no mention

of multiprogramming,

the

sys-

Suppose

to the system disk per

it might appear

the specific

of any other

they apply without

that will be derived

the complexity

For example,

statistical

of systems,

a time sharing

terminals.

think time is 15 seconds,

of accesses

in the problem

to answer

assumptions

forms,

of

time of the system?

systems,

provided

in the case of pro-

Because

paper can be regarded mental

puter

distributed

distributional

required

results.

restriction

times,

Consider

25 active

system disk is 40 milliseconds.

is that operation-

mathematical

per pro-

What is the throughput

is 7, the utilization

rage response

can be derived without making any of

the seemingly (e.g.,

interaction

Considering

difference

a page

and the average number

disk is .35, and the average

in the

approaches important

that the average

only

limiting

that the

of back-

to process

(i.e.~ reads and writes)

Time Problem:

average number

of obserresults

the average amount

time required

tem which contains

results

interval

Another

Reponse

it

apply with

to all collections

store

the system?

from probabilistic

complete

is .85,

gram is equal to 6000.

counter-

However,

(i.e.,

of page transfers

in terms of random models.

Suppose

of the backing

fault is 10 milliseconds,

of real

a batch processing

the utilization

ing store service

natur-

properties

Consider

system with a paged virtual memory.

paging device)

are for-

ally and d i r e c t l y to observed

Problem:

presented

in Section

tained by using the operational

to as

system analysis.

time

definitions

as follows:

tended

and working

to reflect

gaged in empirical performance.

202

This method

method of computer

is based on a set of

assumptions

the viewpoint studies

4 were ob-

that are in-

of individuals

of computer

As stated in Section

system

3, such

en-

individuals

usually deal with collections

measurement

data that are obtained

system behavior

during

In operational

terms,

specific

ed an observation

interval,

that are measured vation interval variables.

tional method

involve

The operational

variable

as follows: (i)

( actually

providing

ing the observation

interval.

ed that the processors

the opera-

system are numbered

the following

i

service)

dur-

It is assum-

and devices in the

in some fashion

the term "processor-or-device

three steps:

so that

i" refers

to a unique entity.

A. Define a set of operational that correspond manner

completions

of time that device-or-processor

is busy

to as operational

which utilize

generally

B(i) ~ Amount

during an obser-

are referred

of job

X = J/T

is call-

and the quantities

or computed

Analyses

X is expressed

of time.

of time during

data is collected

(i.e.~number

per unit time).

by monitoring

intervals

an interval

which such measurement

X = Throughput

of

in a direct

to the intuitive

interest

(e.g.,

lization,

U(i) = Utilization

variables and natural

concepts

throughput,

of

C(i) = Total number

variables

characterize

single observation

during a

i during

completed the obser-

interval.

S(i) = Average

service

time for processor-or-de-

vice i. S(i) is expressed

as follows:

S(i) = B(i)/C(i)

relationships

such as those described

sub-headings

vation

requests

interval.

C. Apply these mathematical to problems

among

which

system behavior

U(i)

(2)

of service

by processor-or-device relationships

i.

as follows:

U(i) = B(i)/T

device uti-

etc.).

B. Derive mathematical operational

of processor-or-device

is expressed

A, B and C of Section

D(i) = Average

in

number

or-device

3.

(3)

of requests

i per job. D(i)

for processoris expressed

as follows: The next section of this paper defines of operational formulate

variables

a set

that can be used

a number of performance

D(i) = C(i)/J

to

evaluation 7. THE THROUGHPUT

problems. 6. OPERATIONAL

set of symbols

will be used throughout discussion.

stood in the context described

of this

Throughput

should be under-

of the operational

the time interval

the system was observed ments were taken).

method

service

The

and measure-

(i) through

U(i) S(i)'D(i)

all other time

interval.

during

the obDespite

A job is a basic

unit of work and may refer to a program,

Throughput

a job step, a job, or an interaction,

result.

depending

device per

(5)

is immediate

from

(4) since B(i) T

=

J/T

=

X

C(i) B(i)

Law represents

a useful

to the throughput

in this discussion.

J C(i)

of the proof,

Note that it can be used

ed earlier

203

Law

the trivial nature

tain the answer

on the system being studied.

of average

for any i.

of the Throughput

quantities.

of jobs completed

servation

proof

equatio~

that

is equal to device util-

Symbolically,

U(i) S(i), D(i)

X

the time units in which T is expressed

J = Number

Throughput

by the product

time and the average number of requests

interval during which

It is assumed

are also used to express

to

basic result.

job for that device.

T = Length of the observation

dependent

Law:

ization divided

above.

(i.e.,

the following

it is possible

and definitions

the remainder

All definitions

LAW

Given the above definitions,

VARIABLES

derive The following

(4)

the

and powerful to directly

ob-

problem present-

As specified

in

the problem u(i)

statement,

8. THE UTILIZATION

.85

=

Note that the Throughput

S(i) = i0 milliseconds D(i) = Thus,

any device

6000

or processor

the law were applied

X = .85/(6000 X i0) jobs per millisecond

X =

Since throughput

that it is independent

of a large number

tors which must normally lyses.

These

programming;

factors

be specified

include:

the service

device and processor

tion

of fac-

time distribution

gardless

Equation

For this reason,

the Throughput

ed as a universal mance.

A probabilistic

version

put Law, which is valid central Buzen

server model,

interval.

re-

system perfor-

Law bears certain

(a) for an interval

it travels

1 d =~.a

t

to all sysperformance.

Probabilistie

of the Utilization

In order

at a constant

of time

(t), the

previously

regarded

in the above relationship. to consider

ween distance,

a basic model

such systems operate.

in which

for describing

that will be employed

[8], and Muntz

Note that it

blocked)

probabi-

process

and time is signi-

than the argument Law.

bet-

However,

of this paper will demonstrate laws of greater mathematical

that

interest

also exist.

204

with it.

in think state

enters

active).

Response that

system state and the

instant that it next leaves

system state. leaves

a proeessing

has

The in-

(i.e.,

the instant

that an interactive

is said to occur.

the next

and each terminal

(i.e.,

process

[7],

of this model

There are a fixed num-

time between

process

in this discussion

associated

is either

or system state

(i.e., completes

required

1.

the

The model and

[i0], Kleinrock

The essence

terminals,

process

time is the elapsed an interactive

the relationship

in Figure

interactive

teractive

[9].

to the

it is first neces-

sary to provide

one

them as expected values

acceleration,

the Throughput

few sections

in a

[4].

techniques

systems,

manner

ber of interactive

true t h a t the mathematical

to derive

ficantly more complex

operational

Equality

[2] and,

TIME LAWS

to apply operational

is illustrated

as operational

since no underlying

It is, of course, required

by Buzen

are based on the work of Scherr

listic model is involved.

to derive

versions

of interactive

terminology

(d) is given by:

between this result and ip . Law is that dlstance~ acceleration

of random variables

argument

Equality

Law,

is applicable

analysis

if

2

and time are most natually

is not natural

some observation

law of computer

similarities

One point of similarity

variables

during

can

of a set

tems and is a u n i v e r s a l

Moore the Throughput

This equation

consistency

very general c o n t e x ~ by Chang and Lavenburg

of the

For example,

a body starts at rest and accelerates

distance

data collected

have been derived

of the Through-

to in this discus-

was derived earlier by

laws of motion.

acceleration

(7)

Equality.

the internal

9 RESPONSE

The Throughput

(6) that

As in the case of the Throughput

the Utilization

factors.

in the context

(7) will be referred

of empirical

[2].

to elementary

used in equa-

(5) and

U(j) S(j),D(j)

=

be used to verify

Law can be regard-

law of computer

(i) has a

of the device

from equations

sion as the Utilization

equilibrium;

Law is valid

of the status of these other

in equation

for each

of CPU and I/0 processing

the Throughput

(5), it follows

U(i) S(i)-D(i)

the fact that

within a single program is or is not permitted. In other words,

as defined

in other ana-

the system is or is not in statistical the fact that overlap

j with j # i, one

(6)

unique value regardless

Law is

the degree of multi-

in the system;

to device

to

If, for example,

U(j) S(j) 'D(j)

= 51 jobs per hour

aspect of the Throughput

Law can be'applied

in a system.

would obtain

= 3600000 X .85/(6000 X i0) jobs per hour

An important

EQUALITY

Each time

system state

request),

an interaction

amount

of think time

per transition

from think state to system Think State

Interactive Terminals

plying

the same reasoning

state).

Ap-

used to define

R, N

Z System State

1 Z z(k) J" k=l

=

System By combining N

these definitions

sented previously,

several

ponse time in interactive General

Response

Model

context,

it is possible

group of operational

to

Proof:

assumed

J = Number

terminals.

that N is constant

out the observation

observation

interval.

is simply a specific the definition

response

system is expressed

The derivation

k-th terminal)

during

as

ji j

Z

(i0)

of equation

(i0) utilizes

(the process spends in

associated

with the

think state,

plus uhe

total time that the k-th interactive process

the

in

This definition instance

S(i).D(i) U(i)

active process

through-

of the observation

of

interval.

That is

of J given previousz(k) + r(k) = T for all k

r(k) = Total

time that the k-th interactive

process

Thus, N

response

time

(i.e.,

interaction).

Since R is the average

Dividing

of system state time necessary

to generate computed

one interaction,

(by this

and then dividing

quantity by the total number during

and

both sides by J and using equations

(9) to simplify JIz + R = J

the

total time spent in system state

completed

(12)

(8)

the left hand side,

R can

by first obtaining

all processes)

actions

N

Z z(k) + E r(k) = N-T k=l k=l

average

of time in system state per

amount

(ii)

spends in system state.

R = Average amount

spends

system state, must be equal to the length

ly.

be

can be derived.

the fact that the total time that the k-th inter-

It is

interval.

of interactions

res-

vari-

ables. N = Number of interactive

systems

concerning

follows:

R = N define an additional

results

Time Law: The average

time of an interactive i. Basic Interactive

Given this general

with the set pre-

J

Response Time

Fig.

(9)

By equation J T

of inter-

Combining

the observa-

N T (13)

J

(i) and the Throughput

Law,

U(i) S(i),D(i) equations

(14) (13) and

(14),

tion interval. ~ Z Z + R = N S(i)'D(i) Thus,

i R = ~

z(k) = Total

J

R is given by

The proof

N Z r(k) k=l

(8)

A useful

jobs leave

state during

the observation

Z = .Average think time

(i.e.,

simplication

large with respect

of times that interactive

think state and enter

by subtracting

of the General

Time Law arises if it is assumed

spends in think state.

jr = Total number

is completed

jf 7 z

from both sides.

time that the k-th interactive

process

(15)

u(i)

system

interval.

205

that J is very

This will occur if the

observation

interval

is substantially

the average

response

time of the system.

cases,

average

to N.

JI/J~l

Response

since ~JI-J I ~ N.

longer

than

In such

present It is then approximately

true that

at the start of the interval

that arrived

at time zero).

the following

R =

N S(i)'D(i) U(i)

Z

(16)

(i.e.,

jobs

For each job, define

function:

f(k,t)

= Amount

of memory allocated

to

the k-th job at time t. Equation

(16) will be referred

t_ic Response

to as the Asympto-

Time Law, or simply,

the Response

Let

Time

y(k) = Space-time

Law. If it is assumed N, the Asymptotic

this paper. seconds,

that J is large with respect

Response

solve the response Recall

D(i)

to

defined

Time Law can be used to

time problem

stated earlier

that N = 25 terminals,

= 7 requests,

S(i) = .040 seconds.

Z = 15

U(i) = .35, and

Y = Average

Thus,

time

space-time

product

tic Response

versions

Time Law have been derived

investigators

for particular

models

systems

[7,8,10].

Response

Time Law for more general

models

Derivations

are presented

Muntz and Wong mary objects

pressed

processes,

Space-time

program performance to programs

in virtual

ly, a program's its execution amount

memory

M = Average

space-time

time multiplied

of memory allocated

cution.

Since space-time

and throughput

(i.e.,

allo-

variable

m(t)

is

f (k, t)

(19)

of memory

M is defined

By equations

(19) and

as follows: T ~ m(t) 0

is equal

T

M=~

at

(20)

(20),

charges

Essential-

i n use during

interval.

i

to

A

:

Z

O

k=l

f (k,t) dt

(21)

by the average Reversing

to it during its exeproducts,

the order of integration

and applying

response

equations

(17) and

and summation,

(18),

are all used as indicators

of system performance, mine the manner

amount

M = ~i

accounting

product

Z k=l

the observation

of random

systems.

in use

the pri-

are often used to evaluate

andassign

(18)

as follows:

re(t) =

[i] and by

PRODUCT LAWS

products

job), Y is

A

variables. i0. SPACE-TIME

of memory

interactive

values

(i.e.~ space-

A Z k= I y(k)

defined

and results were ex-

in terms of expected

per completed

The operational

of the Asymptotic

In all these cases,

product

cated to some job ) at time t.

by several

of analysis were random variables

and stochastic

= Amount

of interactive

by Boyse and Warn

[9].

m(t)

of the Asympto-

(17)

as follows:

i Y = ~

= 5 seconds

probabilistic

y(k) is

as follows:

R = 25 x .040 x 7/.25 - 15

Specific

for the k-th job.

variable

T y(k) -- f f(k,t)dt 0

in

defined

time,

product

The operational

it is interesting

in which

M = Y.J/T

to exa-

these quantities

are

Applying

equation

(22)

(i)

related. Begin by considering which may be operating or a batch processing variable arrive terval.

an arbitrary in either

mode.

Assume

X = M/Y Equation

Let the operational

A be defined as the number

at the systemlduring

system

of jobs which

the observation

that A includes

(23)

an interactive

in-

(23)~

Product

throughput

is equal to average

use divided

those jobs

206

which will be referred

Space-Time

Throughput

by average

Law,

amount

space-time

to as the

states

that the

of memory

product.

in

A similar

relationship

time product asymptotic tions

between average

and average

response

case can be obtained

(23),

(16), and

(5).

space-

average

time in t h e

by combining

response

Define equa-

time

(R) times one unit of space.

the operational

variable

L as followsf

L = Average number of jobs present

This yields

the system during

in

the observation

interval. R = N'Y/M - Z

Equation

(24)

(24) will be referred

Time Product

Response

variables

correspond

tospace-time

class~

are expressed quantities

service

correspond

to event and some

integrals.

interval,

Within

are expressed

completion" EVENT COUNTS

each

are expressed

into equation

space units,

and simplifying,

certain

(25) is the operational

is usually

referred

of course,

that equation

' TIi.iE ~ DURATIONS

to as Little's

rather

SPACE-TIME INTEGRALS

throughput

than arrival

assumed

theory

formula.

(25) is expressed

variables

(25) involves rather

counterpart

from queueing

lues of random variables.

basis.

the

(25)

of operational

on a "per

(23), cancelling

L = X.R

of the well known result

on a "per job" basis,

are expressed

Substituting

Equation

for the

certain quantities

on a "per unit time" basis,

and certain quantities

Note

to time durations,

certain quantities

entire observation

M = L x one unit of space

of the

in this paper.

variables

some correspond

paragraph,

Y = R x one unit of space.

the basic structure defined

that some operational counts,

to as the S_pace-

Time Law.

Table 2 illustrates operational

From the previous

that Note,

in terms

than expected

va-

Note also that equation (i.e.,

rate

departure

(i.e.~ % ).

that the system being

rate)

If it is

studied

is in equi-

librium in the sense that the same number of proJ A J' C(i)

ENTIRE INTERVAL

~

T B(i) r(k)

i

z (k)

T

grams are present

y(k)= ~f (k, t) dt

be equal T

I

PER UNIT TIME

J

X=~i

the observation

!U(i)=B$i)

special

N

i

A

i X R=7 ~ r(k)

D(i)=Cj (-i)

Y = 7

k=l PER SERVICE COMPLETION

can be deduced

k=l

of Little's

ments[5],[6]. objective

formula

However,

previous

_B(i) s(1)-c(i)

and included

•

Table 2

Operational

additional

12. RELATIONSHIP

Variables

system.

that each job has unit size

M will be equal

the average number of jobs present

to

TO PROBABILISTIC

(response

times one unit of space.

time)

it is useful

all

assumptions

RESULTS

results presented

in this

properties

for the

systems to

207

time.

which can be expressed models

to consider

analyses

of specific

analyses•

are concerned

behavior

between Essential-

with the

sequences which

follow during well defined Most results

in

and random variables,

the relationship

and operational

ly, operational

is

Thus Y is equal

terms of probabilistic

probabilistic

in the

in this case that y(k)

equal to the time-in-system k-th job

space),

(i.e.,

number of occupied memory units or, equi-

It also follows

and consequently

additional

FOrmULA

one unit of memory

the average

results

arguments.

paper have counterparts If it is assumed

valently,

argu-

was to derive

required

Since most operational ii. LITTLE'S

occupies

analysis

to operational

in all cases the ultimate

of the analysis

analyses

of deri-

have included

in terms of random variables,

k=l

i

counter-

as a

out that a number

steps that are very similar

N

rate will

case.

vations y(k)

and the end of

the departure

rate and a direct

formula

It should be pointed

O

PER JOB

to the arrival

part to Little's

M = l ~ m ( t ) dt

at the beginning

interval,

in this paper

intervals involve

of

opera-

tional variables

that represent

over such time intervals, these results

averages

behavior

difference

and conventional

between

(e.g.,

periodic

se-

care must be taken to avoid

behavior

that is incompatible

tical equilibrium).

a given set of average values

(lhis is the principal tional analyses

results

and as a consequence

apply to all possible

quences which produce

computed

opera-

parts exist, generality

deterministic

Thus,

operational

and simpler

strictly

with statis-

even when direct

counter-

laws benefit

from greater

interpretation

and applica-

tion.

analyses). Probabilistic (i.e.,

results

also apply to an ensemble

a set) of possible

a probabilistic of statistical

result

behavior

then the Ergodic

rem and the Law of Large Numbers

averages

defined

sequences,

riables

values

(i.e.,

babilistic

and which involve

laws that apply

to almost

ensemble

members

of infinite-time

pro-

the implications

possible

device

behavior

standpoint,

of these infinite-time

behavior

quest

se-

A. The number sing element

of arrivals

sing element

at each proces-

is unbounded.

ber of requests

pending

sense,

and can be assumed

to be

queues develop

the observation

interval.

processing arrival

element,

Thus,

rates must be equal;

the number

of departures

each processing

element

which involve

properties

that do

A and B above,

only constants

and

to probabilistic

volve constants

and expected values

results which in-

made in the case of the corresponding

is at least moderately

long,

and can be safely

apply

on a "per job" basis the operational

to all variables

(e.g.,

X,Y, and R).

definitions

presented

are collected

during

that

the observation

interval

and that "per job" and "per service averages

interval

by simple division.

are computed

This assumption

to conform well to existing performance

com-

at the end of the

measurement

practice

in

and evaluation

field.

of random

assumptions

if the ob-

pletion"

Note additional

of S(i).

interval

the computer

variables. In many cases,

at

Fortuantely,

appears

and average values,

will correspond

complete

interval will contribute

in this paper are based on the assumption

to

it will be true that operational

not contradict

in a a ser-

This will artificially

measurements

results which are based on assumptions

Likewise,

raise the value

In effect,

at

can be assumed

re-

to B(i) but not to C(i).

defined

equivalently,

be equal. In general,

reduced.

The same considerations

and

and arrivals

If a service

disregarded.

for each

the departure

completed

at the start of the ob-

these effects will be negligible

during

variables.

that average

the value of S(i) will,

be artificially

servation

zero. B. No unbounded

interval.

the end of the observation

for each proces-

to avoid

to total device busy

vice request which is only partially

Thus the num-

at the start of the interval

is insignificant

interval,

states.

operational

of requests

complete

and

understood

(3) states

time is equal

the observation

servation

are significant:

consistent

of operational

equation

is partially

laws themselves

of certain

misinterpretation

service

during

only two

of end

inter-

and terminal

should be carefully

For example,

of the

initial

time divided by the number

From the operational

quences

of the observation

of operational

for all possible

definitions

and

opera-

the problem

A c t u a l l ~ end effects do not in any way im-

However,

sequences.

properties

points

since these laws are internally valid

to operational

all

concerns

the state of the system at the ini-

pair the validity

to

Thus,

only constants

values will correspond

associated

val).

random va-

averages).

(i.e.,

tial and terminal

se-

correspond

of the associated

the ensemble

effects

for

laws which are based on equilibrium

assumptions expected

One question which arises when applying

the operational

and will

CONSIDERATIONS

tional laws in practice

Theo-

imply that,

for each of the individual

quences will be identical the expected

13. PRACTICAL

If

is based on the assumption

equilibrium,

almost all behavior

sequences.

that the above

to the problem

must be

probabilistic

208

meters

comments

of estimating

actually

intrinsic

on the basis of observations

pertain

system para-

taken during

finite intervals

of time.

both operational

and probabilistic

it is independent actually

This problem

arises

analyses,

of the analysis method

in

constant

and

For example,

that is

techniques

employed.

A somewhat

problem

of disk and drum subsystems

arises

which untilize

of equation service

rota-

space-time

(3) to such devices

times.

latency will not appear in S(i). this phenomenon

does not affect

the operational

laws derived

it does illustrate

Once again, the validity

in this paper,

of but

verify

the internal

definitions.

stant,

then the Space-Time

measurements

collected

numerical

values

algebraic

equations

solution

by

wise,

the Space-Time

plies

that average

generated

through

models

methods;

(e.g., device

in terms of other variables

In a somewhat

service

since,

to which response interactive system.

Like-

Time Law imby

value of

terminals

increases

in the Response

time,

time will increase are added

to

Once such systems reach cer-

see Scherr

in the number

(N) leave all other depen-

of this approach

and Muntz and Wong

in

which are easier

different

context,

can be helpful

Time Laws unchanged. to estimating

[i0], Kleinrock[7],

resMoore[8]

[9].

15. ACKNOWLEDGEMENTS

to

Many of t h e c o n c e p t s

by equation

(7),

the Utiliza-

in locating

presented

to calculate

for each device

values

The

16. REFERENCES

i.0, when

Equality,

also makes

i. Boyse,

in the system.

J. W., and Warn,

forward model

These

tion~

can then be substitu-

2. Buzen,

Comp.

Surveys

J. P.

in this paper can also be used

such as throughput these estimates, which variables

ACM, New York, 3. Buzen,

that certain modifications

and response

it is necessary

time.

indicators

J.P.

of the IEEE 4. Chang,

To make

209

(June 1975),

predicpp.73-93.

of system bottlenecks

network model= on Sys. Perf.

Proc.

Eval.

ACM

(April 1971)

pp. 82-103.

63,6

architecture~

(June 1975),

A., and Lavenburg,

dent servers.

A straight-

performance

I/O subsystem

closed queueing

to understand

will change and which will remain

7,2

Analysis

SIGOPS Workshop

time.

D.R.

for computer

using a queueing

and lower bounds on response

to a system will have on performance

by

the work of R. R. Muntz and L. Kleinrock.

throughput

the effect

P. J. Denning,

The author was also influenced

ted into other laws to obtain upper bounds on

The laws derived

conversations

the author and D. B. Rubin,

and E. Gelenbe.

the maximum utilization

and processor

maximum utilization

in this paper

during

must

have the largest value of U(i).

with the Utilization

between

bottle-

the device or pro-

fact that no value of U(i) can exceed

to estimate

Response

time will decrease

dent variables

time)

cessor with the largest value of S(i).R(i)

it possible

Law

by 25%.

Time Laws can also be used to esti-

were refined and sharpened

tion Equality

combined

Product

response

terminals

ponse

the explicit

measure.

necessarily

Throughput

con-

The laws can also be used to express

an unknown variable

necks

Product

will increase

of interactive

For examples

programs;

of systems

(i.e., M) approximately

tain levels of saturation,

of: performance

by simulation

derived

of probabilistic

equilibrium.

a time-sharing

they can be used to

empirical

in use

that throughput

The Response

in this paper are direct-

consistency

the average

load on the system to keep the

when additional

to all systems,

that

N'Y/Z.

14. APPLICATIONS Since the laws derived

installa-

to assume

an amount equal to 20% of the original

the need to fully understand

of all operational

the working

(i.e., Y) by 20%, and if there

of memory

implies

out.

restructuring

to reduce

at a particular

amount

mate the extent

ly applicable

that program

will also reduce

product

is a sufficient

Thus,

are carried

If it is reasonable

this modification

delay due to seek and rotational

the implications

suppose

make it possible

tion by 20%.

times which are approximate-

ly equal to average data transfer the additional

in the case

sensing and block multiplexing[3].

The application yields average

the modifications

set size of all programs

different

tional position

after

networks Oper.

Res.

S.S.

pp.871-879. Work rates in

with general 22

Proc.

(1974),

indepenpp.838-847.

5. Jewell, 15

W.S.

(1967),

6. Maxwell, L=%W.

A simple proof of: L=%W~

W.L.

On the generality

Oper.

7. Kleinrock,

Oper. Res.

APPENDIX

Res.

L.

18 (1970),

Proc,

of the equation

results

IFIP Congress

Operational

for time

C.

Univ.

1968,

C(i) =

Network models

systems.

TR-71-1,

R.R.,

Annual Princeton Sciences I0. Scherr, Systems.

J.

M.I.T.

Press,

Proc.

Eighth

A

of jobs

M = Average

ANALYSIS

times of individual time

(completion

and A-3 present

It is assumed

presented

that the quantum

robin scheduling

algorithm

= Amount

(or interactions)

number

r(k) = Total

in Table i.

of memory

terminals.

time.

time in system state for k-th inter-

S(i) = Average

is 2 seconds.

(k-th terminal).

service

AVERAGE

FCFS

RR

U(i) = Utilization

7 8 ii

ii 3 8

X = Throughput.

8 2/3

Table A-I Workload

y = Average

= ABC

time for device

Z = Average z(k) = Total

FCFS

RR

3 4 Ii

6 3 ii

AVERAGE

6

B A C

AVERAGE

Workload

= CBA

RR

1 8 ii

i" ii 8

Table A-3 = BAC

6 2/3

210

(k-th terminal).

Operational

Asymptotic

Response

Response

Workload

think time.

Principal

General

6 2/3

FCFS

6 2/3

Table A-2

per job.

for k-th job.

time in think state for k-th interactive

process

C B A

i.

product

product

i.

interval.

of device

space-time

y(k) = Space-time

7 1/3

the

in use at time t.

response

active process

size in the round

in use during

interval.

of interactive

Average

for

the

interval.

amount

T = Length of observation

A B C

completed

of jobs in system during

of memory

N = Number

response

of the entire benchmark

each of the three workloads

m(t)

the completion

jobs and the average

time)

to the k-th job

system state. L = Average

observation Tables A-l, A-2,

per job for

at time t. J = Number

Computer

(1967).

BENCHMARK

by

i.

observation APPENDIX

completed

during the observation interval. j'= Number of transitions from think state to

pp.348-3521

of Time Shared

of requests

= Amount of memory allocated

properties

on Information

(March 1974),

device f(k,t)

i is busy.

requests

i.

D(i) = Average number

Eng.,

(April 1971).

Asymptotic

Conference

An Analysis

scale time

Industrial

network models,

and Systems A.

Dept.

Ann Arbor

and Wong,

of closed queueing

for large

at system.

of time device

Number of service device

of Michigan,

9. Muntz,

Variables

B(i) = Amount

pp. 838-845.

sharing

GLOSSARY

A = Number of arrivals

pp.172-174.

Certain analytic

shared processors~

8. Moore,

B

pp.l109-1116.

Response

Laws

Time Law - Eq.

Time Law - Eq.

Time Law - Eq.

(16)

(I0)

(16)

Space-Time

Product

Response

Space~Time

Product

Throughput

Time Law - Eq.

Throughput

Law - Eq.

Utilization

Equality

(5) - Eq.

(7)

Law - Eq.

(24)

(23)