CHAPTER 9
Representing Delays

Delays are crucial in creating the dynamic characteristics of information- feedback systems. Delays are themselves composed of the standard level and rate equations of Chapter 7. Some forms of delays -- exponential and pipeline -- are discussed in the present chapter. Because of the prevalence of delays, a "shorthand" notation is introduced to stand for the actual level and rate equations that are to be used in representing a delay.

PRECEDING chapters have stressed the important contribution of delays to creating the characteristics of an information-feedback system. This chapter discusses methods of representing in mathematical models the kinds of delays encountered in our industrial and economic processes.

In principle, delays exist in all flow channels. However, to introduce a time delay in every flow would lead to a vast amount of model detail, much of which would contribute little to system behavior. Two kinds of simplifications will always be used to reduce the number of points at which delays must be introduced into a model formulation. First, many system delays will be judged to be so short that their effect is negligible compared with the other longer or more significantly located delays that are to be incorporated. Second, delays that arise from separate, actual processes which are cascaded one after the other can often be combined into a single delay representation. In addition, delays in parallel branches entering a common channel may often be combined by shifting them into the common channel.


9.1 Structure of Delays

A delay is essentially a conversion process that accepts a given inflow rate and delivers a resulting flow rate at the output. The outflow may differ instant by instant from the inflow rate under dynamic circumstances where the rates are changing in value. This necessarily implies that the delay contains a variable amount of the quantity in transit. The content of the delay increases whenever the inflow exceeds the outflow, and vice versa.

A delay is a special, simplified category of the general concept of inventories or levels. All levels exist to permit the inflow rates to differ, over limited intervals, from the outflow rates. In the general concept of a level, there is no restriction on the factors that might control the outflow. For example, the outflow from an inventory can be influenced both by the level of the inventory and by the level of unfilled orders. By contrast, a delay is here thought of as a special class of level wherein the outflow is determined only by the internal level stored in the delay (and by certain descriptive constants). For example, a transportation system can often be adequately represented merely as a delay. If so, the internal level of goods in transit is the only variable. A constant describes the average delay, and a specified computing process is applied to the internal level to generate the appropriate type of transient relationships between the internal level and the output rate.

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