1 System Dynamics Delays � Definition — characteristics — types & examples � Material Delays — pipeline — first-order — third-order … n th -order � Information Delays — first-order — third-order … n th -order 2 System Dynamics: Delays Definition Input Delay Output � The output of a delay lags behind the input — each input element with individual delay time — in the meantime residing inside the delay (=stock) i.e.: each delay contains a stock(s) — aggregated over all inputs: average delay time � Example (types) — Material � hungry students - students in mensa - satiated students — Informational (gradual adjustment of perceptions or beliefs) � actual number of bowls per student - prospected number of bowls UniOS-FB9-IMU-BWL/MSWI-Rieger-SS06-WiKyb-05: System Dynamics Basics (Delays) (1 / 8)
3 System Dynamics: Delays Excursion: PULSE-function Define non-recurring event (0/1) at <start> (time) for time period <duration>: y = PULSE(start,duration) Model equations : Example: hungry students=PULSE (12,0.1) - all students (100%) get hungry at 12°clock hungry students 1 0.75 0.5 0.25 0 0 2 4 6 8 10 12 14 16 18 20 22 24 Time (Hour) hungry students : Current 4 System Dynamics: Delays Pipeline Delay (DELAY_FIXED) � same delay time for each input eating and: each input served FIFO hungry satiated (at once) pipeline delay average eating time 2 Model equations: 1.5 average eating time=1 eating= INTEG (hungry-satiated,0) 1 hungry=PULSE(12, 0.01)/0.01 satiated=DELAY FIXED(hungry, average eating time, 0) 0.5 TIME STEP=0.01 0 11 12 13 14 15 16 Time (Hour) hungry : Current % satiated : Current % eating : Current % UniOS-FB9-IMU-BWL/MSWI-Rieger-SS06-WiKyb-05: System Dynamics Basics (Delays) (2 / 8)
5 System Dynamics: Delays 1st-order material delay (DELAY1I) � average delay time for all inputs wait and: inputs served randomly ! and eat hungry satiated average eating 1st-order material delay time 2 Model equations: average eating time=1 1.5 hungry=PULSE(12, 0.01)/0.01 wait and eat= INTEG (hungry-satiated,0) 1 satiated=wait and eat/average eating time oder: satiated=DELAY1I(hungry, 0.5 average eating time, 0) TIME STEP=0.01 0 11 12 13 14 15 16 Time (Hour) hungry : Current % satiated : Current % wait and eat : Current % 6 System Dynamics: Delays higher-order material delays (2 nd -order) � eating simply cascading waiting hungry served satiated 1st-order delays (assuming cascaded queues) 2nd-order material delay average eating time 2 Model equations: 1.5 average eating time=1 hungry=PULSE(12, 0.01)/0.01 waiting= INTEG (hungry-served,0) 1 served=waiting/(average eating time/2) eating= INTEG (served-satiated,0) 0.5 satiated=eating/(average eating time/2) TIME STEP=0.01 0 11 12 13 14 15 16 Time (Hour) hungry : Current % satiated : Current % waiting : Current % eating : Current % UniOS-FB9-IMU-BWL/MSWI-Rieger-SS06-WiKyb-05: System Dynamics Basics (Delays) (3 / 8)
7 System Dynamics: Delays 3rd-order material delay (DELAY3I) paying eating waiting hungry served paid satiated 3rd-order material delay average eating time 2 1.5 Model equations: average eating time=1 1 hungry=PULSE(12, 0.01)/0.01 waiting= INTEG (hungry-served,0) served=waiting/(average eating time/3) 0.5 paying= INTEG (served-paid,0) paid=paying/(average eating time/3) 0 eating= INTEG (paid-satiated,0) 11 12 13 14 15 16 satiated=eating/(average eating time/3) Time (Hour) oder: satiated=DELAY3I(hungry, hungry : Current % average eating time, 0) satiated : Current % waiting : Current % TIME STEP=0.01 paying : Current % eating : Current % 8 System Dynamics: Delays 9th-order material delay paying eating • 3 cascaded waiting paid hungry served satiated 3rd-order delays average eating time 9th-order material delay 2 Model equations: average eating time=1 1.5 hungry=PULSE(12, 0.01)/0.01 waiting= INTEG (hungry-served,0) 1 served=DELAY3I(hungry, average eating time/3, 0) paying= INTEG (served-paid,0) 0.5 paid=DELAY3I(served, average eating time/3, 0) eating= INTEG (paid-satiated,0) 0 satiated=DELAY3I(paid, average eating time/3, 0) 11 12 13 14 15 16 Time (Hour) TIME STEP=0.01 hungry : Current % satiated : Current % n th order with n →∞ ⇒ pipeline waiting : Current % paying : Current % eating : Current % UniOS-FB9-IMU-BWL/MSWI-Rieger-SS06-WiKyb-05: System Dynamics Basics (Delays) (4 / 8)
9 System Dynamics: Delays Alternative Delay Distributions (orders of delays) n th order with n →∞ ⇒ pipeline Alternative Delay Distributions (orders of delay) 2 1.5 1 0.5 0 11 12 13 14 15 16 Time (Hour) hungry : Current % satiated0 : Current % satiated1 : Current % satiated3 : Current % satiated9 : Current % 10 System Dynamics: Delays VenSim-PLE-Delay Macros � VenSimPLE-Delay-Macros: — DELAY FIXED pipeline delay (manual initialization) — DELAY1I 1st-order material delay (manual initiatization) — DELAY3I 3rd-order material delay (manual initialization) — DELAY3 3rd-order material delay (with integrated equilibrium initialization) � Delay Equilibrium (Initialization) — Input = Output — Delay Stock = Input * Average Delay Time — Initializing Delay in Equilibrium: � output = DELAY3I(input, avg_delay_time, input*avg_delay_time) = DELAY3(input, avg_delay_time) UniOS-FB9-IMU-BWL/MSWI-Rieger-SS06-WiKyb-05: System Dynamics Basics (Delays) (5 / 8)
11 System Dynamics: Delays Example Delay Equilibrium/Initialization Model Equations: avg time of additional additional beginners=0 studies beginners avg time of studies=9 beginners=500+STEP(additional beginners, 10) graduates=DELAY3(beginners, avg time of studies) … (s.left below) … students graduates beginners Example Delay equilibrium/initialization 1,000 6,000 500 3,000 Correct Equilibrium Initialization: students= INTEG (+beginners-graduates, beginners*avg time of studies) 0 Wrong Equilibrium Initialization: 0 students= INTEG (+beginners-graduates, 0 10 20 30 40 50 60 70 80 90 100 0) Time (Month) beginners : add100 graduates : add100 students : add100 students : init0 12 System Dynamics: Delays Information Delay (Exponential Smoothing) � gradual adjustment of perceptions or beliefs � perception/belief = delay stock! - Adjustment Adjust Time ment Perception Exponential Smooth 200 - + 170 Actual Gap Value + 140 Model equations: 110 Actual Value=100+STEP(50, 5) Adjustment=Gap/Adjustment Time Adjustment Time=6 80 Gap=Actual Value-Perception Perception= INTEG (Adjustment,Actual Value) 0 5 10 15 20 25 30 35 40 45 50 oder: Time (Month) Perception=SMOOTH(Actual Value, Actual Value : Current Adjustment Time) Perception : Current UniOS-FB9-IMU-BWL/MSWI-Rieger-SS06-WiKyb-05: System Dynamics Basics (Delays) (6 / 8)
13 System Dynamics: Delays VenSim-PLE-SMOOTH Macros � VenSimPLE-SMOOTH-Macros: — SMOOTH 1st-order exponential smoothing (with integrated initialization) — SMOOTHI 1st-order exponential smoothing (manual initialization) — SMOOTH3I 3rd-order exponential smoothing (manual initialization) � Exponential Smoothing Analogy — Output t = Output t-1 + (Input t -Output t-1 )/AdjustmentTime = Output t-1 + (Input t -Output t-1 )*(1/AdjustmentTime) — mit SmoothFactor = 1/AdjustmentTime: — Output t = SmoothFactor * Input t + (1-SmoothFactor) * Output t-1 14 System Dynamics: Delays Delay Behaviour Delay Comparison 200 delay time delay time delay time =6 =6 =6 170 95% 140 86% b 63% 110 a 80 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Time (Month) Actual Value : Current smooth1(t>=5) = a + b t * ( 1 - exp(-(t-5)/delaytime) ) smooth1 : Current smoth3 : Current UniOS-FB9-IMU-BWL/MSWI-Rieger-SS06-WiKyb-05: System Dynamics Basics (Delays) (7 / 8)
15 System Dynamics: Delays Zufluß Material Delays rate Summary LV1 Delay3 RT1 LV Delay1 Delay 3 Pipe LV2 Abfluß Abfluß Abfluß line Delay3 rate rate1 rate3 Delay1 RT2 Delay 3 LV3 Vergleich Delay1 und Delay3 Delay3 200 Abfluß rate Verzögerungszeit Delay3 175 150 125 100 0 3 6 9 12 15 18 Time (Month) Zuflußrate - BASIS Abflußrate Delay1 - BASIS Abflußrate Delay3 - BASIS 16 System Dynamics: Delays Input Information Delays RT1 S3 Summary LV1S3 Delta Smooth RT2 S3 Output Output LV Smooth Smooth LV2S3 Smooth 3 SMOOTH3: Die Glättung 1. bis 3. Ordnung 200 RT3 S3 175 Glättungs 150 faktor Verzögerungszeit LV3S3 125 100 0 3 6 9 12 15 18 Time (Month) Input - BASIS LV1S3 - BASIS LV2S3 - BASIS LV3S3 - BASIS Output Smooth3 - BASIS UniOS-FB9-IMU-BWL/MSWI-Rieger-SS06-WiKyb-05: System Dynamics Basics (Delays) (8 / 8)
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