[NOTE: This simply written book is meant as a tool, is profound in places, and requires study. However, readers who invest the time will gain a unique understanding of the nature of risk, and how to eliminate it. Please understand that our intention here is merely to give potential readers an accurate idea of the character of the book, by displaying brief excerpts that highlight the core ideas in the book. These brief excerpts should also reveal the quality of the writing and explanations, as well as the level of difficulty of the equations. We hope you find them interesting and enlightening to read nevertheless. Note that material in square brackets is not in the book, but has been added below to compensate for the necessarily out-of-context nature of the short excerpts, in order to create a readable, understandable and informative web page.]

The following two papers provide the foundation for the book:

Bradley, J. A Risk Hypothesis and Risk Measures for Throughput Capacity in Systems, IEEE Trans on Systems, Man and Cybernetics, Vol. 32, No. 5, Sept., 2002, pp 549-559.

Bradley, J. Time Period and Risk Measures in the General Risk Equation, Journal of Risk Research, Vol 10, Issues 3-4, April-June 2007, pp 355-369.

To read the abstracts see the "Elimination of Risk in Systems" page at Tharsis Books web site.

* * * * * * * *

        When we look at any system in terms of what it does, we soon realize that a system is simply an entity that generates outputs from inputs. This is a simple but very useful idea. A system can be looked at partly as a kind of box, which consists of the resources of the system, that is, the things needed to put the system together so that it can function, and partly as the inputs and outputs to and from that box.
       ...

        The resources of the system are important, for clearly, without them you do not have a system. You may have documents to copy and blank paper, but without a copying machine to do the copying, namely the generation of outputs from inputs, output cannot take place. The importance of this simple and obvious truth cannot be overemphasized, yet is often overlooked. The outputs of a system are usually very desirable too, and often essential for human survival. But without the system resources, there can be no output, and thus no production or throughput.
        There is even more to a system than just resources, and output being generated by those resources from input, however. There is an old ditty that illustrates this additional something very well. It goes like this:

        Big fleas have little fleas,
        On their backs to bite 'em.
        And little fleas have lesser fleas,
        And so on, ad infinitum.

        So it is with systems. A system is often made up of subsystems, that is, systems which, when operating together, make up the entire system. The subsystems are coupled, often in a complicated (takes a lot of words to describe) manner.
        But always the coupling follows a basic principle. When subsystems are coupled, at least some of the outputs from one or more subsystems will serve as the inputs to another subsystem.
       ...

        It is useful to view a business corporation as a system, since it is composed of physical resources and human resources, and takes in input in the form of energy, raw materials, parts and components, to produce useful outputs.

          - from Chapter One: What is a System?

* * * * * * * *

The basic throughput capacity relationship

If we increase system resources R (in a valid manner), then throughput capacity I will increase linearly, or proportionally, with R. For example, if we construct a system that is an m-fold replica of the original system, so that it has resources m times R, then throughput capacity will be m times I.
        We begin therefore, with an axiomatic proposition: We propose that where there is no coordinated resource-sharing, then for valid changes to R, the simple equation:

                I = KR

where K is a constant, must hold true. This is the basic relationship between throughput capacity and system resources.
       We cannot prove this assertion from any more fundamental propositions. It just seems to be implicit in the nature of things, and so we take it as axiomatic.
       It is obviously generally true. For example, suppose you have a hen house, ten chickens and a throughput capacity of six eggs per day. If you wanted eighteen eggs per day, you could simply build two more hen houses, each with ten chickens of the same kind. So if we triple the resources R, we get triple the output I, provided we triple the necessary input, in terms of feed, and so on.
       As another example, interest income I on capital resources R obviously also follows this rule, with K being the interest rate.

          - from Chapter Two: System Throughput and System Resources

* * * * * * * *

Risk of Loss
        Suppose the system is exposed to unpredictable losses and gains in throughput capacity, and that the statistics of these fluctuations are constant (or stationary, in statistical terminology). Assume that over a period of time, long enough to be representative of these statistics, the mean throughput capacity is M. Assume also that in a large number of fully representative time periods the actual throughput capacity values are:

                M - L1, M - L2, ..., M + G1, M + G2, ...

where L1, L2, ... are deviations downward (losses) from the average value M for throughput capacity I, and G1, G2 ... are deviations upward (gains) from M. Gains must equal losses, so that:

                (L1 + L2 + L3 + ...) = (G1 + G2 + G3 + ...)

We therefore assume that the same throughput capacity deviations can be expected to occur in the future, in unpredictable order, all equally likely.
...

Objective Risk Measures

        ...

        For a meaningful measure of risk there are now two choices, the traditional standard deviation (SD) measure, and a new (MEL) measure that in many cases is more suitable for systems in general.

SD-risk measure: Take the standard deviation of the deviations (L1, L2, ..., G1, G2, ...) from the mean throughput capacity M, as a standard deviation measure of possible loss with respect to (or down from) the mean M, to give the Standard Deviation (SD) risk measure.
        If we use twice the standard deviation we have an even stronger risk measure, the 2-Standard Deviations (2-SD) risk measure.
        ...

To deal with the problems and possibilities in arbitrary systems, where the distribution of gains and losses is not anything like normal, an additional and complementary risk measure is very useful. This is the MEL-risk measure defined below. The author therefore proposes:

MEL-risk measure. Suppose that for a system exposed to risk, there is at least one hazard-free time period, in which the hazard risked does not occur. Suppose that the gain with respect to the mean throughput capacity M in this hazard free time period is G, and that a gain exceeding G is thus not possible (but a gain under G with respect to M is very possible, as is a loss with respect to M). Thus, in the best-case scenario, the total hazard-free throughput capacity is M + G. Then all other throughput capacities, such as M - L1, M - L2, ..., M + G1, M + G2, ..., each in a time period where the hazard does occur in varying degrees of intensity, may be considered as exhibiting losses, or loss deviations, G + L1, G + L2,...G - G1, G - G2, ... down from, or with respect to, the best-case value of throughput capacity, namely M + G, in the hazard-free time period.
        We may use the mean of these loss deviations (down) from the best case throughput capacity of M + G as a measure of the risk, that is, a measure of expected losses in the future with respect to the throughput capacity for a hazard-free time period. In other words, we use the Mean Expected Loss (MEL) with respect to, or down from, the throughput capacity in any hazard-free time period, that is, the MEL-risk.
        Note that in specifying a MEL-risk, we must both specify the mean deviation of the losses, and specify with respect to what level, namely the best-case level.

Interpretation. An MEL-risk of L means that the average loss, with respect to the (best) value for throughput capacity in a time period where the hazard does not occur, is exactly L. However, there are two extreme possibilities with regard to what is the throughput capacity for a hazard-free time period:

(a) Natural, or explicit, hazard-free case. In this case, we can find a naturally-occurring, best-case, hazard-free throughput capacity M + G, that cannot be exceeded for the value of the system resources. This hazard free throughput capacity will thus occur in a time period when all goes well and no hazard occurs. Such a time period is certain to occur sometime. ...

(b) Artificial or implicit hazard-free case. In this case, the values in each time period fluctuate about the mean M. The distribution of the deviations from the mean follows some reasonably random and bell-shaped distribution. Large but usually improbable gain deviations from mean M do sometimes occur, but no explicit hazard-free throughput capacity can be determined. In such a case, we may define an artificial hazard-free case for throughput capacity M + G, by defining an imaginary hazard-free time period where the gain G is 2 standard deviations up from the mean M.

        In both cases, MEL-risk can therefore be quite simply viewed as the hazard-free deviation, either natural or artificial, up from the mean M, but also equal to the average loss to be expected in the future with respect to, or down from, throughput capacity M + G for the hazard-free time period (real or artificial).

          - from Chapter Four: Objective and Subjective Risk Measures

* * * * * * * *

Subjective measures of risk

The standard deviation (SD) risk measure, as we have seen, is a correct risk measure where throughput capacity, or return on investment, varies randomly. But because it is an average of the square root of the deviations about the mean, it can be confusing to those not versed in statistics, and is certainly not immediately conducive to gaining insight into the nature of risk. On the other hand, the mean expected loss (MEL) risk measure is simple, and is merely the average or mean (that is, expected) loss of throughput (or return on investment) with respect to the hazard-free, or best-case, throughput or return, where no loss occurs. Let us therefore concentrate on MEL-risk, to see what insight we can gain about the subjective reaction of a systems operator to risk.
        We will take a numerical example to keep the issue as clear as possible. We will also use a simple financial system, where the throughput capacity is in dollars returned per month from an investment (system). Let us suppose that the system will give us $100 per month if no hazard occurs, in other words, the hazard free throughput capacity of the system. Now let us suppose two quite different cases.

Case 1. Over a ten month period, five of the months each give a return of 100, so that in each month there is no loss. Another three months each give a return of 80, or a loss of 20 with respect to the hazard-free case of a return of 100. One month also gives a loss of 10, and one a loss of 30.
        Clearly, the average, mean or expected return is (5 x 100 + 3 x 80 + 90 + 70)/10 or 90, and the average loss with respect to the best case of 100 is 10 each month, for a MEL-risk of 10 per month. Remember that we cannot predict in advance which months will have no loss and which months will have the loss of 30. All we have is the statistics from the past and a guarantee that the future will be like the past. Note also that the losses and gains are not quite random, and thus not distributed normally about the mean of 90.
        The SD-risk would come in as the square root of [8 x 10 x 10 + 0 + 20 x 20]/10, or [1200]/10, or the square root of 120, or close to 11 per month.
        Note that the MEL-risk of 10 tells us what our average loss with respect to the best case of 100 will be, and is thus quite practical. The SD-risk is larger than 10, since standard deviations are larger than average deviations. But because the returns are not quite random and thus not normally distributed, there is no simple interpretation of the SD-measure.

Case 2. Over a ten month period, eight of the months each give a return 100, the hazard-free result, with a loss of zero in each month; one month gives a return of 90, for a loss of 10 in that month, and the other month gives a return of 10, for a (big) loss of 90 in that month.
        The average loss is (10 + 90)/10 or 10 per month, so that the average return is clearly 90 per month. The average loss of 10 is with respect to the best case of 100 in return, and is the MEL-risk, and the same as in Case 1. Remember again that we cannot predict in advance which months will have no loss and which months will have a loss, particularly which month will have that big loss of 90 in return. As in Case 1, all we have is the statistics from the past and a guarantee that the future will be like the past. The SD-risk would come in as the square root of [8 x 10 x 10 + 0 + 80 x 80]/10, or the square root of 7200/10, or 26.83 per month.
        Note that the MEL-risk of 10 is still quite practical, since it tells us correctly what our average loss per month will be, with respect to the best case of 100 per month; the SD-risk does not do that, but it tell us that the risk in the second case is much higher than in the first. But how can that be, if in both cases the average loss is the same? If we have two investors each running the system, one under the first scenario and the other under the second, both will end up with exactly the same income over the years, on average.

The two scenarios are the same with respect to a truly objective measure of the risk, namely the MEL risk of 10. If two robots, with no emotions, were running the two versions of the system, there would be no difference for the robots and no difference in the end result, and therefore the same risk.
        However, human beings are not robots. They have emotions, and negative emotions are triggered when a human being experiences a loss, and negative emotions hurt. The bigger the loss, the bigger the hurt. So from a human agent point of view, Case 1 is a much more comfortable scenario than Case 2. In Case 1, in the months where the small losses of 20 occur, even the loss of 30, there will be some minor pain perhaps, but nothing devastating. However, in Case 2, in that month where the loss of 90 occurs, there will likely be an intense pain for the investor.
        And how do humans react to intense negative emotions, feelings that hurt? They usually do one of two things. They either try to change the environment that is producing the loss that is causing the pain, or they flee from that environment-the famous fight or flight response.
        There is nothing wrong with this response. It is simply how humans are constructed, and they are constructed that way to ensure their survival long enough to reproduce and raise their young. Nature knows what it is about. If you are a primitive man and discover that a man-eating tiger has invaded your territory, you feel frightened, and with good reason. Your negative emotions will get you to drop what you are doing, to take the precaution of either fleeing the territory at once, or tracking down the beast and killing it, or, if you have the know-how, you might build a wall to keep it out instead, or set up an alarm system, to warn of its approach. Nature does not intend you stay exposed to great risk.
        Modern man is rarely exposed to being eaten by wild animals, but is exposed to losses from his systems. So, as a result of the way the human emotional system works, the risk associated with Case 2 above seems subjectively greater than with Case 1. But the objective losses are the same in both.
        Now we could argue that the intelligent investor should simply ignore his or her emotions, and figure the risk rationally in terms of average loss over a long period, as with the MEL-risk, and carry on. And many have tried to do that, desperately attempting to summon up the iron will not to be affected by a loss like the severe loss that occurs every so often in Case 2. However, such bravado is very difficult to keep up, and very stressful, so that most humans simply accept that Case 2 is worse that Case 1, or that it is "riskier".
        So obviously, a measure of risk that takes into account this human aversion to large losses would be useful in telling us which situations to avoid because they are likely to be too painful to allow us to operate our system rationally. There is a way to do that. In figuring out the expected average loss per month we could weight large losses more heavily than small losses, in order to get an average "loss" for Case 2 that is bigger than for Case 1.
        For example, we could take the average of some power of the losses, since this will give us a much bigger average "loss" when there are a few big losses, as compared to when there are many small losses, even if the true expected average loss per month is the same in each case. Two losses of 2 each are the same as one loss of 4 and one loss of zero over two months, for an average loss of 2 per month in each case. But the average of the square of the two losses of 2 each is (4 + 4)/2 or 4, and the average of square of the losses of 4 and 0 is (16 + 0)/2 or 8. If we then take the square root of these two losses we get 2 for the first, but 2.83 for the second, which could serve to warn us that the second case will be more painful than the first, and therefore riskier. This risk measure using a power of 2 for losses would clearly be subjective and arbitrary.
        Instead of giving more weight to big losses, or to large deviations, by using the square of the losses, we could equally well use the cube of the losses, that is, we could use a power of 3, in order to get a more extreme subjective measure. We could also go for a less extreme subjective measure and use a power of 1.7 or 1.5.
        The point is that taking average expected losses based on some power of the losses, assuming that the power is greater than one, will give a risk measure that is not the true average loss that can be expected. It will give a figure greater than that, and much greater when a few big losses are involved than when many small losses are involved, even if the average loss is the same in both cases.
        Use of a power of the losses is probably the best way to construct a subjective measure of risk that puts the emphasis on a few large losses as opposed to many small losses. This also tells us something important about the SD-risk measure of risk. Although it is widely assumed to be an objective measure of risk, we can now see that the SD measure of risk is actually a subjective measure, where we average the square of the losses and gains (or deviations) with respect to the mean.
        We might then ask if a subjective measure of risk based on losses to the power of 2 is intrinsically better, in the sense of being more objective, than one based on losses to the power of 3, or 1.7, or 1.3. The answer is no, it is not. A measure based on a power of 2 is entirely arbitrary. One reason it is entrenched in finance is that it leads immediately to the average "loss" being expressed as a standard deviation of the deviations from the mean throughput capacity, and standard deviations are amenable to all kinds of mathematical analysis. This power of 2 expected loss measure is thus the most mathematically convenient for a subjective risk measure, and hence the use of the SD-risk measure.
        Another reason for the popularity of the SD measure in finance is that in systems where the deviations from the mean are distributed normally about the mean (and so form a normal distribution), the SD-risk measure is very close to half the corresponding MEL-risk measure, which is objectively a true risk measure untainted by subjective considerations.
        Thus the SD-risk measure is entirely arbitrary and made up, and although it leads to no insight into the nature of risk, it is a good mathematical but fundamentally subjective risk measure, which, when loss deviations follow a normal distribution, gives a result equivalent to the fundamentally correct and objective MEL-risk measure. It will also serve to tell us which of two risks, which otherwise have the same average loss attached to them, will be more painful to endure. In the financial world, this is important, since investors are known to be risk averse. They will pay less for the investment that returns 100 for 8 months of ten, and 90 and 10 for the other two months, than they will for an investment that returns 100 for five months of ten and 80 for five months of ten, even though the average return is 90, the same in both cases.

          - from Chapter Four: Objective and Subjective Risk Measures

* * * * * * * *

Linear relationship between throughput and risk

Consider now two environments E1 and E2, in which the system can operate. Suppose E1 is a risk-free environment, in which the system has a throughput capacity given by I = KR per time unit for a system with resources R. (Assume that no resource-sharing procedure is in operation, as described in Chapter 3.) The environment E2 is the same as E1, except that in E2 the system is operating in an environment where there is a risk of loss of throughput capacity.
        Suppose gross throughput capacity in E2 is KR + G per time unit, in each of one or more time periods in which the risk in E2 is run but where it just happens (by good luck) that the hazard does not occur. Thus KR + G can be viewed as the hazard-free, or best-case, throughput capacity in the presence of risk, where the hazard risked does not occur.
        However, when a risk is run repeatedly, throughput capacity losses must occur over time. If the average (expected) throughput capacity loss per time unit, due to the hazard occurring, is L, then the net throughput capacity from running the risk in reality will on average be KR + G - L per time unit.
        But the expression KR + G - L must also give us the expected or average throughput capacity I. That means we can take L as the MEL-risk, with respect to the best case hazard-free capacity of I = KR + G. In general there are now two possibilities for this risk: it can be risk that it can pay to run repeatedly (positive risk), where best case gain G exceeds average loss L, or it can be risk it cannot pay to run repeatedly (negative risk), where G is less than L.
        Assume now that G - L is positive for E2, so that E2 exposes the system to a risk it can pay to run repeatedly. Now recall that environment E1 was risk free, with throughput capacity given by I = KR for resources R applied.
        It is clear that there will be an increase in throughput capacity by shifting the system (and resources R) from an environment E1 with no risk and throughput capacity KR, to environment E2, in which the throughput capacity varies from one period to another, but with average throughput capacity KR + G - L. The environment E2 differs from E1 only in E2 having a risk it can pay to run repeatedly.
        A simple illustrative example of this would be a system consisting of a fleet of ferry boats for conveying freight across a river. The quantity R is now a measure of the number of boats in the fleet. As R increases, throughput capacity I increases according to I = KR.
        Suppose there are two places to operate the fleet, E1 and E2. At E1 the river is broad but very deep, with no risk of a boat being delayed by running aground.
        At E2, the river is narrow, so that many more crossings per day are possible per boat, and thus a higher throughput capacity, but with a risk of running aground on occasion. If the boats are lucky and never run aground, the extra throughput capacity is G. Unfortunately, at E2, unpredictable sand bars can occur, causing a boat to run aground sometimes and be delayed, causing, on average a throughput capacity loss of L. If G exceeds L it can pay to operate the fleet at E2, and put up with a boat going aground occasionally-that is, the risk is positive. Notice that G must increase linearly with R, a measure of the number of boats, as must L. Thus it will be sometimes be more explicit to write G as G(R), indicating that it is a function of R, and L as L(R), indicating that it too is a function of R.
       ...

Of the three formulations of the risk equation, namely:

                I = R(K + cr(E) - r(E))
                I = R(K + (c-1)r(E)) , and
                I = R(K + br(E))

the author prefers the first, since it is more explanatory. In the third, we have an expression stating that the throughput capacity I will (possibly mysteriously) increase with the risk of the environment, without any inherent explanation of why. The constant b, the risk sensitivity coefficient, merely measures how sensitive I is to risk. [r(E) is L/R, i.e. the MEL risk L per unit of resources R in an environment E, e.g. the risk per ferry boat at a specific river crossing E, where R is now the number of boats in the fleet.]
        In contrast, the first equation is saying that there is an extra throughput capacity Rcr(E) to be gained by running the risk in E in cases where (by good luck) the hazard does not occur, and that this extra gain is reduced on average by the quantity Rr(E), which is a measure of the average loss in throughput capacity due to the hazard occurring sometimes. It is thus similar to the basic relationship from which it is derived, namely:

                I = KR + G - L

        [or:     I = KR + (G/L)L - L = KR +cL - L = KR + (c - 1)L = KR + (c - 1)Rr(E)
        where c = G/L, the risk efficiency coefficient,
        which measures the extra throughput capacity gained,
        per unit of risk run.]

        As we shall see in the next chapter, this way of looking at the gains due to running risk is the key to ways to reduce or eliminate the risk without losing the gross gain G due to running that risk.
       ...

Financial risk equation

In the finance arena, SD-risk r(E) is used; the risk equation above is also used, in a close equivalent of the formulation:

                I = R(K + br(E))

However, the more informative risk equation formulation:

                I = R(K + (c -1)r(E)) = R(K + cr(E) - r(E))

is unknown. This is due to the financial risk equation being derived in a manner quite different from the derivation above, as we shall see.
        With financial systems, using the risk equation formulation I = R(K + br(E)), the system resources R become the principal sum invested, and K becomes the risk-free per-unit dollar interest rate obtainable from (risk-free) Treasury bills. The constant b, or risk sensitivity coefficient, times the risk r(E) is the extra annual return (measured in per unit dollars), on average, for taking a risk r(E).

[Note that the risk equations above, and their derivation, although first published in this book, appeared shortly after as a technical paper. The reference is:

Bradley, J. A risk hypothesis and risk measures for throughput capacity in systems. IEEE Trans. on Systems, Man and Cybernetics. 32(5), 2002, p. 549-59. ]

          - from Chapter Five : Systm Throughput and Risk

* * * * * * * *

          - from Chapter Five : Systm Throughput and Risk

* * * * * * * *

To be posted

          - from Chapter Six: Risk Elimination Using preventive Resources

* * * * * * * *

To be posted

          - from Chapter Seven: Risk Elimination Using Precautionary Procedures

* * * * * * * *

To be posted

          - from Chapter Eight: Risk Elimination Using Monitoring Procedures

* * * * * * * *

To be posted

          - from Chapter Nine: Margin of safety and Ruinous Risk

* * * * * * * *

To be posted

          - from Chapter Nine: Margin of safety and Ruinous Risk

* * * * * * * *

IN THIS chapter we will look at a profound relationship between throughput capacity I, systems resources R, and the level of coordinated resource sharing going on. Once aware of this relationship, readers will see it in operation everywhere, for resource sharing in systems is very common. Usually this resource sharing is well hidden, however, under the guise of one or more of technological sophistication, efficiency, or waste elimination. The true cost of sharing resources is usually well hidden as well, as are the intrinsic risks associated with such sharing. The concept of resource sharing is not that simple, however, and needs to be thought about, particularly since there exists two very different kinds of resource sharing.
       ...

       Two types of resource sharing operation are possible, the simple one and the sophisticated one.
       The simple one involves sharing of system resources among complete task threads. This case was exemplified near the end of the previous chapter. The chief characteristics in this simple case are that interleaving of subtasks within task threads is not involved, nor are complex resource-sharing procedures. [An example is sharing a printer between four computers, where each computer is printing a book. The printing of a complete book is the task thread, and one book is printed before printing the next one begins.]
       In contrast, the complicated kind of resource sharing does involve sharing of system resources among subtasks from different task threads, and we call it coordinated inclusive resource-allocation sharing. Thus interleaving of subtasks within task threads is involved, as are complex resource-sharing procedures. [You rarely do this kind of sharing with a printer, since you are interspersing the printing from multiple jobs. But suppose four authors, each with a computer, and each printing a book on their shared printer. Printing a complete book is the task thread, and a chapter printing would be a subtask within a task thread. If you interleave the subtasks from four task threads for the four books, the chapters from the four books would come out of the printer all mixed up, and a time-consuming resource sharing procedure will have to be followed to ensure correct recovery of each book. You obviously risk a serious screw-up doing this, but printer idle time between chapter printing will be much reduced and throughput capacity will be increased.]
       In both cases there is resource sharing, but in each case the nature of the sharing is very different, and the difference is vital for understanding the operation of complex systems, and the risks that such complex resource sharing gives rise to.

          - from Chapter Three: System Throughput with Resource Sharing

* * * * * * * *

Equation relating throughput capacity [I] and the level of sharing [s] with undershared resources [R]

       In the general case of resource sharing, we would expect the following. As we increase the level of sharing, throughput capacity will increase, but the total resource-sharing administration time, during which resources are devoted to sharing coordination activities, will increase at an increasing rate. Initially, there will still be room for additional sharing of the system resources during the sharing period, and the system resources are under shared.
       As we further increase the sharing level, throughput capacity I will continue to climb, eventually reaching its peak, or fully-shared, throughput capacity, at the point where the system resources are first fully occupied with both thread processing and resource-sharing administration. The system resources are now fully shared and there is no more room [idle time to exploit] in the sharing period.
       If the sharing level continues to increase, past the point where the resources are first fully occupied, then, because the resource sharing administration time continues to increase, the throughput capacity will fall, but in all likelihood at a shallow rate. The system resources are now over-shared.
       If we increase the sharing level even further, throughput capacity will fall back to a very low level, in the neighborhood of the throughput capacity given by I = KR, when there was no sharing.
       In other words, if you share the system resources, initially you get an improvement in throughput capacity, which can be improved further by increasing the sharing level further. But if you overdo it, and over share, you will get less than if you had shared the resources less, and you may even get very low throughput capacity, in which state the system is said to be thrashing.
       This is obviously an important relationship. We call it the resource-sharing relationship. However, since the system behaves in two separate ways, in one way when the resources are under shared, and in another way when the resources are over shared, it is impossible to formulate a single, accurate equation for throughput capacity versus sharing level that covers both behaviors.
       We thus need two equations, one for the under-shared case and one for the over-shared case. Since it is both simple and the more important of the two, we give only the equation for the under-sharing case in this less technical section. In the next, more technical section, we will include the equation for over sharing.
       The following is the equation governing the relationship between throughput capacity and sharing level, for the case of under sharing:

        I = KR[1 + s]         resource under-sharing equation

Here I simply increases linearly with the level of sharing. This simple resource-sharing equation is the most useful one in practice.
       In the above resource under-sharing equation, a sharing level of s means that the resources are processing s + 1 threads at a time. Thus when s = 0, it means that is there is no resource sharing, and the equation reduces to I = KR, which we would expect. [If four authors were printing four books with a single shared printer, interleaving the subtasks of chapter printing, the printer sharing level s would be three.]

          - from Chapter Three: System Throughput with Resource Sharing

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Resource sharing versus increasing system resources

        The complexity and time consumption of the sharing procedure will likely hint at when it might be better to pay the financial price of increasing system resources R, and forego using resource-sharing to increase throughput capacity. The simplest way to improve throughput capacity, although not necessarily the least expensive way, will always be to increase system resources R. [Each author could print his own book on his own printer.]
        The more complex way to increase throughput capacity, attractive to computer scientists, is to look for and introduce a coordinated, resource-sharing procedure. Merely increasing system resources may appear to some to be too limiting. Indeed, since resources are always scarce, as each new type of system appears, it can confidently be expected that research, particularly computer science research, to find a way to increase system throughput capacity, by resource-sharing, will not be far behind.
        And a last word. It might be thought that there is no way for an investor to share a principal among many borrowers and get interest from all of them, in accordance with I = KR(1 + s), where K is now the per unit interest rate and R is the principal. Actually there is a way, but you have to be a bank. Banks are empowered to lend and re-lend the same money, thus sharing the resource among many borrowers and collecting interest from all of them!

          - from Chapter Three: System Throughput with Resource Sharing

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