CS4 Modelling and Simulation
LN2
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2.1
Introduction
Operational laws
are simple equations which may be used as an abstract representation or
model of the average behaviour of almost any system. One of the advantages of the laws
is that they are very general and make almost no assumptions about the behaviour of
the random variables characterising the system
1
. Another advantage of the laws is their
simplicity: this means that they can be applied quickly and easily by almost anyone.
Based on a few simple observations of the system the performance analyst can, by
applying these simple laws, derive more information. Using this information as input to
further laws the analyst gradually builds up a more complete picture of the behaviour
of the system. Note that although we will talk in this section about operational laws in
the context of systems, the laws are equally applicable to the observations obtained from
models and we will have occasion to use the laws in this way later in the course.
The foundation of the operational laws are
observable variables
. These are values which
we could derive from watching a system over a Fnite period of time. We assume that the
system receives
requests
from its environment. Each request generates a
job
or
customer
within the system. When the job has been processed the system responds to the environ
ment with the completion of the corresponding request.
System

Arrivals

Completions
±igure 1: An Abstract System
If we observed such an abstract system we might measure the following quantities:
T
,
the length of
time
we observe the system;
A
,
the number of request
arrivals
we observe;
C
,
the number of request
completions
we observe;
B
,
the total amount of time during which the system is
busy
(
B
≤
T
);
N
,
the average number of jobs in the system.
±rom these observed values we can derive the following four important quantities:
λ
=
A/T
,
the
arrival rate
;
X
=
C/T
,
the
throughput
or
completion rate
,
U
=
B/T
,
the
utilisation
;
S
=
B/C
,
the
mean service time
per completed job.
1
In contrast, Markovian analysis relies on very strong assumptions about the distribution function of
the random variables which are used.
9
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LN2
We will assume that the system is
job fow balanced
. This means that the number of
arrivals is equal to the number of completions during an observation period, i.e.
A
=
C
.
Obviously this assumption is not true in all observation periods, but it is a
testable
assumption
because an analyst can always test whether the assumption holds. Note that
if the system is job ﬂow balanced the arrival rate will be the same as the completion rate,
that is,
λ
=
X
.
2.2
Little’s Law
The best known and most commonly used operational law is Little’s law. It is named
after the man who published the Frst formal proof of the law in 1961, although it had
been widely used before that time. Little’s law is usually phrased in terms of the
jobs
in
a system and relates the average number of jobs in the system
N
to the
residence time
W
, the average time they spend in the system. Let
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 Spring '10
 MohammadAbdolahiAzgomiPh.D
 Law, analyst, CPU time

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