Business Dynamics and System ModelingChapter 11: Delays

Pard TeekasapSouthern New Hampshire University

Outline

1. Definition of the Delays2. Material Delays3. Information Delays4. Variable Delay Time5. Estimate the Delays

Pre-class Question

• Consider the market for agricultural commodities such as pork. What is the average delay between a rise in the price of pork and the resulting increase in pork supply?

• If there is an unanticipated increase in the inflation rate, how long will it take for the forecasts of the experts to adjust to the new rate?

Define the Delays

• A delay is a process whose output lags behind its input in some fashion

• Since the input differs from output, there must be a stock inside the process to accumulate the differences

• Material delay = a delay process of the physical flow of material

• Information delay = the gradual adjustment of perception or beliefs

Delays have stocks

Material Delays

• In many cases, the outflows are constrained by limited resources. These resources must be modeled explicitly

• The important aspects of delays– What is the average length of the delay?– What is the distribution of the output around the

average delay time?

What is the average length of the delay?

• How long, on average, does it take items to flow through the delay?

• What is the average residence time for a unit in the delay?

What is the distribution of the output?

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Time (multiples of average delay time)

Inflow

B

C

D

Outflow A

A = Pipeline delay; B = First-order delay; C = Third-order delay; D = Twelfth-order delay

Pipeline Delay

• Also known as transportation lag• The delay time is constant• The order of exit from the delay is precisely

the same as the order of entry• E.g. Auto assembly line• Outflow(t) = Inflow(t-D)

Structure of Pipeline Delay

First-Order Material Delay

• There is mixing and variation in the individual processing times, causing some variance in the distribution of delivery

• Perfect mixing = the order of entry is ignored• Outflow = Material in Transit/D

Structure of First-order Material Delay

Behavior of First-order Material Delay

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Delay Inflow and Outflow

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Time (multiples of average delay time)

Inflow Outflow

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Stock of Material in Transit

Mate

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Time (multiples of average delay time)

Material in Transit

Higher-order Material Delay

• The delay consists of multiple stages of processing in which items flow sequentially from one stage to the next, but where each stage introduces some mixing

• The same as cascading first-order delays together

Structure of Higher-order Delay

Nth-order Material Delay

• A delay with n stages, each with 1/n of the total delay time

• The higher the order of the delay, the less mixing and the smaller the variance of the output

• An infinite-order delay preserves the order of entry and permits no mixing = pipeline delay

Pulse response of a 3rd-order delay

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Delay Inflow and Outflow

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Time (multiples of average delay time)

Inflow Outflow

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Stock of Material in TransitM

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Time (multiples of average delay time)

Pulse response of 3rd-order delay by stage

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Delay Inflow and Outflow

Time (multiples of average delay time)

Inflow

Outflow

Stage 1Exit Rate

Stage 2Exit Rate

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Stock of Material in Transit

Time (multiples of average delay time)

Stage 3

Stage 1

Stage 2

Total Stock in Transit

Little’s Law

• How big will the stock in transit be for any given delay and inflow, suppose the inflow has been constant long enough for the delay to reach equilibrium?

• The equilibrium stock in transit for a delay is always DI units, regardless of the probability distribution of the outflow

Information Delays

• Why do perceptions and forecasts involve delays?

• It takes time to gather the information needed to form judgments

• People don’t change their mind immediately on the receipt of new information

• Need time to adjust emotionally to a new situation

Adaptive expectation

• Also known as Exponential smoothing• Simplest information delay and one of the

most widely used• The belief gradually adjusts to the actual value

of the variable• It’s also known as a first-order information

delay or first-order exponential smoothing

Structure of adaptive expectation

Response of adaptive expectations to a step change

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Perc

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Time (multiples of average delay time)

Output:Perceived Value

Input: Actual Value

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Time (multiples of average delay time)

Change in Perceived Value

Adaptive expectation eliminate short-term noise

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Un

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Expected Order Rate

Order Rate

Higher-order information delays

• In a first-order information delay, the output responds immediately to a change in the input

• In many cases, beliefs begin to respond after some time has passed

• The delay between the actual state and the decisions involves multiple stages

• Similar to the higher-order material delay, we cascade first-order smoothing structures

Structure of the 3rd-order info delay

Response of higher-order delays to a step input

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Perc

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Time (multiples of average delay time)

Output:Perceived Value

Input:Actual Value

1stOrder

3rdOrder

12thOrder

Variable Delay Time

• In many cases, the delay time is not fixed. It can varies both exogenously and endogenously

• If you’re the only one at the ATM, you can get a cash by about a minute

• However, the delay time to get cash increases if there are people ahead of you

• The rate at which people joint the line also depends on how many people are in line

Ratchet Effects

• People moves with positive things faster than negative things

• Aggregate consumption expenditures rise faster than they fell as income fluctuated over the business cycle

• People perceive job insecurity quickly and have a long memory of that

Nonlinear time constant in job insecurity

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Layo

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LayoffRate

Memory ofLayoffs

Change inLayoff

Memory

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MemoryAdjustment

Time

DI DD

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LayoffRate

Memory ofLayoffs

Estimating delays from data

• Econometric techniques can be used for estimating lags from time series include the Koyck or geometric lag, polynomial distributed lags, rational distributed lags. And ARIMA model

• You need to trade off the flexibility of the formulation against the number of parameters to be estimated in choosing the method

Don’t put the regression equation in the model

• You should replace the estimated distributed lag with the material or information delay that best matches the estimated lag because– Econometric techniques are for discrete time, but

system dynamics are for continuous time– The delay time may be incorporated as an

endogenous variable– Regression equations do not distinguish between

material and information delays

Estimation the lagged response of natural gas supply (3rd-order)

Construction lag for capital plant (2nd-order)

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Mean Delay(16.7 months)

2nd Order Delay

Estimating delays when no available data

• You must estimate these parameters from direct inspection of the delay process, experience with analogous delays in related systems, or judgment

• Judgmental estimation is quite unreliable and usually underestimate the duration

• The longer the delay, the greater the degree of underestimation

Decompose the process

• Decompose the process into various stages, then estimate the length of time required of each stage

• The more stages in delay, the tighter the output distribution will be and the smaller the initial response

Walk the Line

• Even when numerical data are available, direct inspection is important

• You should be suspicious of data and take the time to investigate the process firsthand