A Risk Hypothesis and Risk Measures
for Throughput Capacity in Systems
Author: James Bradley
From:IEEE Transactoins on Systems, Man and Cybernetics
Part A : Systems and Humans, Vol.32 No. 5, September 2002
報告人 : Wei-Chih Yin
Date:2008/11/08
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Outline
Section 1. Introduction
Section 2. Overview of risk equation and support
equations
Section 3. Risk and resources
Section 4. Risk and throughput capacity
Section 5. Concluding remarks
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Section 1. Introduction
 Complex systems increasingly have to cope with risk
in their operating environment. Some systems are
exposed to risk whose source is intrinsic.
 Such systems are all human-agent directed systems.
 In this paper, we present a hypothesis, expressed as a
risk equation, that relates system throughput capacity
to system resources and risk.
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Section 1. Introduction
 Definitions of Risk
In engineering:
Risk = ( probability of an accident ) *
( losses per accident)
In more general terms:
Risk = (probability of event occuring ) *
( impact of event occuring )
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Section 2. Overview of
risk equation and support equations
 Part A. Systems and System Throughput
Capacity
 Part B. Basic Resource Equation and
Axiomatic Foundation
 Part C. Resource-Sharing Equation
 Part D. Basic Risk Equation
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Section 2. Part A.
Systems and System Throughput Capacity
1. A system is considered to be any entity that generates
outputs from inputs.
2. Inputs can be another system or human.
3. The definitions are as below:
R : Resource
E : Environment
U : A output set is generated from a input set
per unit time.
(is also the system throughput)
I : I is the throughput capacity of the system
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Section 2. Part B. Basic Resource
Equation and Axiomatic Foundation
The basic resource equation is
I=K*R
where K is a constant.
It states that throughput increases linearly with resource
of the same type.
It seems to be implicit in the nature of things, and so, we
take it as axiomatic.
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Section 2. Part C.
Resource-Sharing Equation
Under sharing of the system resources, throughput
capacity is given by
I = K * R * (1+S)
0, there is no sharing and only one thread is active.
S (Sharing level) : 
n, means that thre ads are active.
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Section 2. Part D.
Basic Risk Equation
In this paper, we concerned only positive risk equation.
I = RK + Rcr(E) – Rr(E)
= R[K+(c-1)r(E)]
= R[K+br(E)].
I : a mean or expected throughput capacity
r(E) : risk per unit R; risk of loss of throughput capacity
c : constant, risk efficiency coefficient
b : constant, risk sensitivity coefficient
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Section 3. Risk and resources

To grasp the concept of risk, one must grasp the distinction
between exposure to the certainty of future loss and exposure to
merely the possibility of future loss.

Exposure to future loss is involved in both cases, but only
exposure to the possibility of future loss is a source of risk.
For example, a system may be exposed to a future throughput
capacity loss of 30, but if a future loss of 10 is certain, there is
only risk of loss of 20, at worst.
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Section 3. Risk and resources (cont.)

Risk Measures :
Suppose further that the statistics are constant or stationary and
that if we compute over a period of time long enough to be fully
representative of these statistics, we get a mean or expected
throughput capacity Im. Suppose further that over such a
representative time period, we get the following n throughput
capacity values:
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Section 3. Risk and resources (cont.)
Here, the downward deviations L1 ,L2 ,… are losses down from ,
Im and deviations G1, G2,… are gains up from Im , with n = i + j
, so that
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Section 3. Risk and resources (cont.)
 Standard-Deviation (SD) Risk Measure:
Compute the standard deviation of the deviations (L1,
L2,…G1,G2,.. ) from Im, and take this as a standard
deviation measure of possible loss with respect to Im.
This is the standard-deviation or SD-risk measure. Use
of twice the standard deviation gives an even more
conservative risk measure, called the 2-standarddeviations or 2-SD-risk measure.
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Section 3. Risk and resources (cont.)

MEL(mean expected loss)-Risk Measure:
We use the mean of these loss deviations from the hazard-free
throughput capacity as a risk measure. We call this measure the
mean expected loss (MEL) with respect to , or down from, the
throughput capacity obtaining in any time period where the
hazard risked does not occur at all.
However, there are two extreme possibilities with regard to what
is to be considered as for a hazard-free time period.
One is Natural, or explicit, hazard-free case; the other is
Artificial or implicit hazard-free case.
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Section 4. Risk and throughput capacity
 In this section, we derive the basic risk equation. We show that,
for a risk r(E), the expected throughput capacity is given by
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Section 4. Risk and throughput capacity
(cont.)
 Efficient Environments and Linear Relationship Between
Throughput and Risk
Consider now two systems environments E1 and E2. Suppose E1
is risk-free, in which the system, with resources R, has a stable
throughput capacity I=KR per time unit. Environment E2 is the
same as E1, except that in E2, there is risk.
Suppose gross throughput capacity in E2 is KR+G per time unit,
in each of one or more time periods, in which we run the risk in
E2, but where, by good fortune, it just happens that the hazard
we risked does not occur. We can therefore look at KR+G as the
hazard-free throughput capacity in the presence of risk.
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Section 4. Risk and throughput capacity
(cont.)
We could borrow additional R resources, at the cost KR of the risk-free
throughput from the borrowed R, and apply them to E2. The throughput
capacity will now be (2KR+2(G-Lr)-KR) for the system, or KR+2G2Lr, on average. So the extra throughput capacity for agent’s resources
will be 2(G-Lr), on average, with the MEL-risk being 2Lr.This is
consistent with throughput capacity increasing linearly with risk. It
follows that, in general, expected throughput capacity I
is given by
Here, is the MEL-risk, and and are independent agent controlled
variables, where . The factor will vary with the synthetic environment
chosen for the system by the agent.
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Section 5. Concluding remarks
 Since the risk equation and resource sharing equation are
valid only for agent-directed, nongrowth, nonevolving
Systems.
 An obvious area for future research is development of similar
equations for agent-directed systems that allow for system
growth and evolution.
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Thank You !
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