levels and rates

Dennis T. Beng Hui, De La Salle University-Manila

Levels and Rates

The Bath tub Example


Stock Flow Diagram (Flow Diagrams)

Stock and flow diagrams are ways of representing the structure of a system with more information than a simple causal loop diagram.Stocks (levels) are fundamental to generating behavior in a system.


Stock Flow Diagram (Flow Diagrams)

Flows (rates) causes stocks to change.Stock and flow diagram is a common step toward building a simulation model because they help define the types of variables that are important in causing behavior.


Stock Flow DiagramStocks or levels

Flows or Rates

Auxiliary

Table Function

Constant

Exogenous Variable

Variable not defined in diagram

Information Link

Material Link

Source or Sink of material


Population Stock Flow Diagram

PopulationBirth Death

% Birth WomenGiving Birth

% of Populationdying


Aids Stock Flow Model

AIDSHIVIncubation RateInfection Rate Death Rate


Stock Flow of Ordering system

Amount ofInvtyDelivery Orders

Demand RateAmount toreplenish


Stock Flow of Ordering system (Alternative)

Amount ofInventory

Net of Orders andDelivery

Order

Deliver

Demand


Coffee Temperature Stock Flow Diagram

Coffee TemperatureChange in Temp

Heat Loss

Room Temp


Stock Flow of Household Expenditure

AvailableMoney Allowance

Utilities

Income

Amount ofOvertime


Problem and Addiction Stock Flow diagram

AddicitionLevelChange in level

Amount ofProblemNew Problems Solved Problems


Classes of EquationsLevel equations Rate equations Auxiliary equations Supplementary equations Initial-value equations


Level equationsLevel equations have varying contents of reservoirs of the system. They would exist even if the system is in rest and no flows existed.Examples are stocks, inventories and others.New values of levels are calculated at each of the closely spaced solution intervals.


Level equationsLevels are assumed to change at a constant rate between solution times, but no values are calculated between those times.Levels determine rates Example:

L INVTY.K=INVTY.J+DT(MAKES.JK- SALES.JK)


Rate equationsRate equations are decision functions.Defines the rates of flow between the levels of the system.A rate equation is evaluated from presently existing values of levels in the system, very often, including the level from which the rate comes and the one into which it goes.


Rate equationsThe rate in turn cause the changes in the levels.Rates determine levels.Example:

R BIRTH.KL = POPN.K*0.20


Auxiliary equationsAuxiliary Equations are components of a rate equation. These are equations that assist but are incidental. Helps in keeping the model in close correspondence to the actual system.


Auxiliary equationsThese equations can be substituted forward into one another and hence into rate equations. Unlike rate equations, auxiliary equations must be evaluated in proper order.


Auxiliary equationsExample:

A DRUGS.K = POPN.K * 0.1R USERS.KL = DRUGS.K * 0.2L AIDS.K = AIDS.J + DT(BIRTH.JK + USERS.JK)


Supplementary equationsSupplementary Equations are used to define variables which are not actually part of the model structure but arise in printing and plotting values of interest about the model. These equations are denoted y “S”.


Initial-value equationsInitial-Value Equations are used to define initial values of all levels and some rates that must be given before the first cycle of model equation computation can begin. These also be values of some constants from other constants.Example:

N INVTY = 100


Computational Interval (Solution Interval)

DT represents “Delta Time”It is the model time elapsing between computations in the simulation model.


Computational Interval (Solution Interval)

The solution interval must be short enough so that its value does not seriously affect the computed results. It should also be long enough as permissible to avoid unnecessary digital-computer timeDT should be between one-half to one-tenth of the smallest time constant in the model.Common values are 0.50, 0.25, and 0.125)


Coffee Cooling Model using Dynamo

*Coffee Cooling Temperature

NOTE COFTEMP.K = Coffee Temperature in CelsiusL COFTEMP.K=COFTEMP.J+DT*(COOL.JK) N COFTEMP=100

NOTE COOL.KL = Cooling Rate of CoffeeR COOL.KL=K(ROOM-COFTEMP.K)C ROOM=25NOTE K is a constantC K=.01

SPEC DT=.25/SAVPER=.25/LENGTH=5NOTE Time is in minutesSAVE COFTEMP,COOL,ROOM

levels and rates

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