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1 D-4614-F Generic Structures: Exponential Material Delays Single Stock outflow for single stockinflow for single stock AVERAGE DELAY INPUT First-Order Material Delay Prepared for the MIT System Dynamics in Education Project Under the Supervision of Prof. Jay W. Forrester by Stephanie Albin Copyright ©1998 by the Massachusetts Institute of Technology Permission granted to distribute for non-commercial educational purposes6 D-4614-F 3 Table of Contents 1. ABSTRACT 5 2. INTRODUCTION 3. DELAYS 3.1 EXAMPLE 1: OIL DEGRADATION (FIRST-ORDER DELAY) 8 3.2 EXAMPLE 2: LOTTERY PAYOFF SPENDING (FIRST-ORDER DELAY) 9 3.3 EXAMPLE 3: TREE HARVESTING (THIRD-ORDER DELAY) 10 4. THE GENERIC STRUCTURE 11 4.1 MODEL DIAGRAM 11 4.2 MODEL EQUATIONS 12 5. BEHAVIORS PRODUCED BY THE GENERIC STRUCTURE 15 5.1 PULSE INPUT 15 5.2 STEP INPUT 18 6. USING MATERIAL DELAYS IN MODELING 20 7. EXERCISES 7.1 EXERCISE 1: MATERIAL VS. INFORMATION DELAYS 23 7.2 EXERCISE 2: MAIL DELIVERY SYSTEM 23 7.3 EXERCISE 3: GENERIC MATERIAL DELAY ANALYSIS 24 8. SOLUTIONS TO EXERCISES 25 8.1 SOLUTION TO EXERCISE 1: MATERIAL VS. INFORMATION DELAYS 25 8.2 SOLUTION TO EXERCISE 2: MAIL DELIVERY SYSTEM 25 8.3 SOLUTION TO EXERCISE 3: GENERIC MATERIAL DELAY ANALYSIS 28 9. APPENDIX 29 6 234 D-4614-F 9.1 EQUATIONS FOR EXAMPLE 1: OIL DEGRADATION MODEL 29 9.2 EQUATIONS FOR EXAMPLE 2: LOTTERY PAYOFF SPENDING MODEL 31 9.3 EQUATIONS FOR EXAMPLE 3: TREE HARVESTING MODEL 31 9.4 EQUATIONS FOR SECTION 6: FACTORY-DISTRIBUTOR MODEL 335 D-4614-F 1. ABSTRACT This paper introduces the generic structure of a material delay. A material delay is a delay in a physical flow. In contrast, an information delay is a delay in perception. All delays are characterized by two parameters: the order of the delay and the average length of the delay. This paper uses Oil Degradation and Lottery Payoff Spending models as examples of first-order delays. A Tree Harvesting model is used as an example of a third-order delay. After presenting the generic model diagram and equations, this paper analyzes the behaviors produced by the generic structure with various inputs. The outflows to a first-, second-, third-, fourth-, and infinite-order delay with pulse and step inputs are examined. After presenting a brief section containing hints to using material delays in modeling, this paper concludes with three exercises.6 D-4614-F 2. INTRODUCTION Generic structures are relatively simple structures that recur in many diverse situations. In this paper, for example, spilled oil degradation, inheritance spending, and tree harvesting systems are shown to share the same basic structure. Transferability of structure between systems makes the study of generic structures a fundamental part of learning system dynamics. Road Maps contains a series of papers which use generic structures to develop an understanding of the relationship between the structure and behavior of a system. Such an understanding should help one refine intuition about surrounding systems and improve one’s ability to model system behaviors. Knowledge about a generic structure in one system is transferable to understand the behavior of other systems with the same structure. Thus, generic structures assist one in understanding systems never studied before. This paper introduces the generic structure of material delays. Many examples of systems containing the material delay generic structure will be used to illustrate both the transferability and functionality of the structure. Soon the reader will be able to recognize material delays in many new and different models. 3. DELAYS In real life the flow of non-physical information and physical material from one place to another takes time. Delivery of a package takes days and full comprehension of a new idea can take years. Delays are an inherent part of all flows. Including a time delay in every flow when building a system dynamics model, however, is impractical. Often, a system delay is so short that its effect is negligible in comparison to longer or more significantly located delays. For example, the delay in loading a delivery van is insignificant to the delay in driving the van from its origin in New York to its destination in California. Through practice, a modeler will learn to recognize which delays are significant and which are not.7 D-4614-F An delay is a stock-and-flow structure that accepts a given inflow and delivers a resulting outflow. A delay can be characterized as an exponential delay because, in the simplest form, the outflow equals the stock divided by the average delay, which characteristically produces exponential decay.1 Note that the outflow from the delay structure is determined only by the stocks of the delay and by the average delay time. The inflow to the delay is independent of the delay structure. Essentially, a delay modifies the time relationship between the inflow and outflow delay structure. Exponential delays in system dynamics are defined by two parameters: the order of the delay and the average length of the delay. The order of the delay is the number of stocks (or integrations) between the inflow and outflow of the delay structure. The average length of the delay is the average time between the initial input and the final output. The average length of the total delay is divided evenly among each stock in a higher-order delay. Exponential delays can be divided into two categories: material and information. A material delay modifies a physical flow, while an information delay is a delay in perception. Under most conditions with identical inputs, the output of a material delay is identical to the output of an information delay. The stock-and-flow structures are, however, fundamentally different. This paper will cover only material delays. The reader should keep in mind that a material delay does not lose or create any units of the flow traveling through the delay structure. Each unit that flows into the delay must either be stored in a stock of the delay, or flow out of the delay. Many software programs, such as STELLA, allow the modeler to define a converter as a delay using special functions without explicitly modeling the stock-and-flow structure. While such short-cuts are useful, the modeler must understand that all delays contain stocks as part of the delay structure. If the STELLA diagram contains a special function converter defined as a delay, then the diagram


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