## Introduction

This chapter has been reserved for a variety of important quantitative relationships so that they can be found in one location in this book. Both systems engineering and project management depend on the mastery of basic quantitative relationships, with an emphasis on the former. That is, it is really not possible to understand the details of systems engineering without mastering these relationships.

Several earlier chapters introduced key quantitative relationships, for example, the chapters containing discussions of requirements analysis, technical performance measurement, simulation, and modeling. Some of these relationships are reiterated here.

A central motivation for examining the relationships set forth in this chapter is to predict the performance of systems. In the early stages of system development and engineering, there is no real system that can be tested to determine its performance. Thus, we resort to ''pencil and paper'' studies that purport to tell us how the designed system will, we hope, perform in the real world. Today, and for the foreseeable future, the pencils and papers have turned into computers. In that sense, we must also master a variety of computer applications that contain the quantitative relationships appropriate to the task at hand.

This chapter is almost exclusively devoted to probability relationships, in distinction to other classes of mathematics (e.g., differential equations, transformation calculus, control systems theory, etc.). The basic reason for this is that many of the technical performance measures of systems are expressed in terms of probabilities. A few examples are:

• The detection and false-alarm probabilities for a radar system

• The response time probability for an on-line transaction-processing system (OLTP)

• The probability of a call going through in a telephone system

• The probability of having a system available to operate when called on to do so

• The probability of successful operation of a system over its lifetime

• The kill probability for a weapons system

• The probability of experiencing some type of disastrous failure that places peoples' lives in jeopardy (e.g., a nuclear plant incident)

• The circular error probability for a guidance system

• The distribution of trip times for a transportation system

As difficult as it might be to calculate these probabilities, we often have little choice because many of the analyses and trade-offs that we must carry out as systems engineers are based on these measures. This chapter contains but a small sampling of the theory available to us in this regard.