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Classic DEVSPractical Modelling Techniques Using PythonPDEVS

Yentl Van Tendeloo, Hans Vangheluwe

IntroductionBrief recap of introductory DEVS tutorial

Bernard P. Zeigler.

Theory Of Modelling And Simulation.

1st ed. Wiley, 1976.

Bernard P. Zeigler, Herbert Praehofer,

and Tag Gon Kim.

Theory Of Modelling And Simulation.

2nd ed. Academic Press, 2000.

Bernard P. Zeigler.

Multifacetted Modelling and

Discrete Event Simulation.

1st ed. Academic Press, 1984.

Yentl Van Tendeloo and Hans Vangheluwe.

An Evaluation of DEVS Simulation Tools.

Simulation: Transactions of the Society

for Modeling and Simulation International.

2017, 93(2): 103-121

Yentl Van Tendeloo and Hans Vangheluwe.

An Overview of PythonPDEVS.

In Proceedings of JournΓ©es DEVS

Francophones (JDF), pages 59-66, 2016.

Our presentation uses initialized DEVS models, which

contain an initial total state. This was left implicit in

the original DEVS specification.

Yentl Van Tendeloo and Hans Vangheluwe. Extending the DEVS

Formalism with Initialization Information, 2018. ArXiv e-prints.

https://arxiv.org/abs/1802.04527

𝑀 = 𝑋, π‘Œ, 𝑆, π‘žπ‘–π‘›π‘–π‘‘, 𝛿𝑖𝑛𝑑, 𝛿𝑒π‘₯𝑑, πœ†, π‘‘π‘Ž

𝑋 : set of input events

π‘Œ : set of output events

𝑆 : set of sequential states

π‘žπ‘–π‘›π‘–π‘‘ : 𝑄𝑄 = 𝑠, 𝑒 𝑠 ∈ 𝑆, 0 ≀ 𝑒 ≀ π‘‘π‘Ž(𝑠)

𝛿𝑖𝑛𝑑 : 𝑆 β†’ 𝑆𝛿𝑒π‘₯𝑑: 𝑄 Γ— 𝑋 β†’ π‘†πœ† : 𝑆 β†’ π‘Œ βˆͺ πœ™π‘‘π‘Ž : 𝑆 β†’ ℝ0,+∞

+

Atomic DEVS Specification

𝑀 = 𝑋, π‘Œ, 𝑆, π‘žπ‘–π‘›π‘–π‘‘, 𝛿𝑖𝑛𝑑, 𝛿𝑒π‘₯𝑑, πœ†, π‘‘π‘Ž

𝑋 : set of input events

π‘Œ : set of output events

𝑆 : set of sequential states

π‘žπ‘–π‘›π‘–π‘‘ : 𝑄𝑄 = 𝑠, 𝑒 𝑠 ∈ 𝑆, 0 ≀ 𝑒 ≀ π‘‘π‘Ž(𝑠)

𝛿𝑖𝑛𝑑 : 𝑆 β†’ 𝑆𝛿𝑒π‘₯𝑑: 𝑄 Γ— 𝑋 β†’ π‘†πœ† : 𝑆 β†’ π‘Œ βˆͺ πœ™π‘‘π‘Ž : 𝑆 β†’ ℝ0,+∞

+

Atomic DEVS Specification

S

t

(𝑠𝑖 , 0)

π‘‘π‘Ž 𝑠𝑖

(𝛿𝑒π‘₯𝑑 𝑠𝑖 , 𝑒 , π‘₯ , 0)

(𝛿𝑖𝑛𝑑(𝑠𝑖), 0)

𝛿𝑒π‘₯𝑑

𝛿𝑖𝑛𝑑

output πœ†(𝑠𝑖)

𝑑𝑖 𝑑𝑖 + 𝑒 𝑑𝑖 + π‘‘π‘Ž 𝑠𝑖

𝑒

𝐢 = 𝑋𝑠𝑒𝑙𝑓, π‘Œπ‘ π‘’π‘™π‘“, 𝐷,𝑀𝑆, 𝐼𝑆, 𝑍𝑆, 𝑠𝑒𝑙𝑒𝑐𝑑

𝐼𝑆 = 𝐼𝑖 𝑖 ∈ 𝐷 βˆͺ π‘ π‘’π‘™π‘“βˆ€ 𝑖 ∈ 𝐷 βˆͺ 𝑠𝑒𝑙𝑓 ∢ 𝐼𝑖 βŠ† 𝐷 βˆͺ 𝑠𝑒𝑙𝑓

βˆ€ 𝑖 ∈ 𝐷 βˆͺ 𝑠𝑒𝑙𝑓 ∢ 𝑖 βˆ‰ 𝐼𝑖

𝑍𝑆 = 𝑍𝑖,𝑗 𝑖 ∈ 𝐷 βˆͺ 𝑠𝑒𝑙𝑓 , 𝑗 ∈ 𝐼𝑖𝑍𝑠𝑒𝑙𝑓,𝑗 ∢ 𝑋𝑠𝑒𝑙𝑓 β†’ 𝑋𝑗 , βˆ€ 𝑗 ∈ 𝐷

𝑍𝑖,𝑠𝑒𝑙𝑓 ∢ π‘Œπ‘– β†’ π‘Œπ‘ π‘’π‘™π‘“, βˆ€ 𝑖 ∈ 𝐷

𝑍𝑖,𝑗 ∢ π‘Œπ‘– β†’ 𝑋𝑗 , βˆ€ 𝑖, 𝑗 ∈ 𝐷

𝑠𝑒𝑙𝑒𝑐𝑑 ∢ 2𝐷 β†’ π·βˆ€ 𝐸 βŠ† 𝐷, 𝐸 β‰  βˆ…: 𝑠𝑒𝑙𝑒𝑐𝑑 𝐸 ∈ 𝐸

𝑀𝑆 = 𝑀𝑖 𝑖 ∈ 𝐷

𝑀𝑖 = 𝑋𝑖 , π‘Œπ‘– , 𝑆𝑖 , π‘žπ‘–π‘›π‘–π‘‘,𝑖 , 𝛿𝑖𝑛𝑑,𝑖 , 𝛿𝑒π‘₯𝑑,𝑖 , πœ†π‘– , π‘‘π‘Žπ‘– , βˆ€ 𝑖 ∈ 𝐷

Coupled DEVS Specification

(done, t)

(*, t)

(y, t)

(x, t)(*, t)

(done, t) (done, t)

Root coordinator

Coordinator

SimulatorSimulator

Coupled DEVS

Atomic DEVS Atomic DEVS

(i, t)

(done, t)

(done, t)

(i, t)

(done, t)

Closure Under CouplingDenotational semantics for Coupled DEVS models

π‘΄π’π’Šπ’ˆπ’‰π’•πŸπ‘΄π’‘π’π’πŸ

𝒔𝒆𝒍𝒇

take_break toAuto

go_to_work toManual

?

?𝐢𝑀 = 𝑋𝑠𝑒𝑙𝑓 , π‘Œπ‘ π‘’π‘™π‘“ , 𝐷,𝑀𝑆, 𝐼𝑆, 𝑍𝑆 π‘“π‘™π‘Žπ‘‘π‘‘π‘’π‘› 𝐢𝑀 = 𝑋, π‘Œ, 𝑆, π‘žπ‘–π‘›π‘–π‘‘, 𝛿𝑖𝑛𝑑, 𝛿𝑒π‘₯𝑑, πœ†, π‘‘π‘Ž

flatten

π‘“π‘™π‘Žπ‘‘π‘‘π‘’π‘› 𝐢𝑀 = 𝑋, π‘Œ, 𝑆, π‘žπ‘–π‘›π‘–π‘‘, 𝛿𝑖𝑛𝑑, 𝛿𝑒π‘₯𝑑, πœ†, π‘‘π‘Ž

CM

21𝑋𝐢𝑀

π‘ŒπΆπ‘€

𝑆1

𝑆2

𝑋𝐢𝑀

π‘ŒπΆπ‘€

𝑆1

𝑆2

π‘“π‘™π‘Žπ‘‘π‘‘π‘’π‘› 𝐢𝑀 = 𝑋, π‘Œ, 𝑆, π‘žπ‘–π‘›π‘–π‘‘, 𝛿𝑖𝑛𝑑, 𝛿𝑒π‘₯𝑑, πœ†, π‘‘π‘Ž

CM

21

𝑋 = π‘‹πΆπ‘€π‘Œ = π‘ŒπΆπ‘€

π‘“π‘™π‘Žπ‘‘π‘‘π‘’π‘› 𝐢𝑀 = 𝑋, π‘Œ, 𝑆, π‘žπ‘–π‘›π‘–π‘‘, 𝛿𝑖𝑛𝑑, 𝛿𝑒π‘₯𝑑, πœ†, π‘‘π‘Ž

CM

21𝑋𝐢𝑀

π‘ŒπΆπ‘€

𝑆1

𝑆2

𝑠1, 𝑒1

𝑠2, 𝑒2𝑒2

𝑒1

π‘“π‘™π‘Žπ‘‘π‘‘π‘’π‘› 𝐢𝑀 = 𝑋, π‘Œ, 𝑆, π‘žπ‘–π‘›π‘–π‘‘, 𝛿𝑖𝑛𝑑, 𝛿𝑒π‘₯𝑑, πœ†, π‘‘π‘Ž

𝑆 = Γ—π‘–βˆˆπ· 𝑄𝑖

CM

21𝑋𝐢𝑀

π‘ŒπΆπ‘€

𝑆1

𝑆2

𝑠1, 𝑒1

𝑠2, 𝑒2𝑒2

𝑒1

π‘“π‘™π‘Žπ‘‘π‘‘π‘’π‘› 𝐢𝑀 = 𝑋, π‘Œ, 𝑆, π‘žπ‘–π‘›π‘–π‘‘, 𝛿𝑖𝑛𝑑, 𝛿𝑒π‘₯𝑑, πœ†, π‘‘π‘Ž

CM

21𝑋𝐢𝑀

π‘ŒπΆπ‘€

𝑆1

𝑆2

𝑠1, 𝑒1 = π‘žπ‘–π‘›π‘–π‘‘,1

𝑠2, 𝑒2 = π‘žπ‘–π‘›π‘–π‘‘,2𝑒2

𝑒1

π‘‘π‘Ž1 𝑠1

π‘‘π‘Ž2 𝑠2

π‘“π‘™π‘Žπ‘‘π‘‘π‘’π‘› 𝐢𝑀 = 𝑋, π‘Œ, 𝑆, π‘žπ‘–π‘›π‘–π‘‘, 𝛿𝑖𝑛𝑑, 𝛿𝑒π‘₯𝑑, πœ†, π‘‘π‘Ž

CM

21

π‘žπ‘–π‘›π‘–π‘‘ = 𝑠𝑖𝑛𝑖𝑑, 𝑒𝑖𝑛𝑖𝑑

𝑋𝐢𝑀

π‘ŒπΆπ‘€

𝑆1

𝑆2

𝑠1, 𝑒1 = π‘žπ‘–π‘›π‘–π‘‘,1

𝑠2, 𝑒2 = π‘žπ‘–π‘›π‘–π‘‘,2𝑒2

𝑒1

π‘‘π‘Ž1 𝑠1

π‘‘π‘Ž2 𝑠2

π‘“π‘™π‘Žπ‘‘π‘‘π‘’π‘› 𝐢𝑀 = 𝑋, π‘Œ, 𝑆, π‘žπ‘–π‘›π‘–π‘‘, 𝛿𝑖𝑛𝑑, 𝛿𝑒π‘₯𝑑, πœ†, π‘‘π‘Ž

CM

21

π‘žπ‘–π‘›π‘–π‘‘ = 𝑠𝑖𝑛𝑖𝑑, 𝑒𝑖𝑛𝑖𝑑

𝑠𝑖𝑛𝑖𝑑 = … , 𝑠𝑖𝑛𝑖𝑑,𝑖 , 𝑒𝑖𝑛𝑖𝑑,𝑖 βˆ’ 𝑒𝑖𝑛𝑖𝑑 , …

𝑋𝐢𝑀

π‘ŒπΆπ‘€

𝑆1

𝑆2

𝑠1, 𝑒1 = π‘žπ‘–π‘›π‘–π‘‘,1

𝑠2, 𝑒2 = π‘žπ‘–π‘›π‘–π‘‘,2𝑒2

𝑒1

π‘‘π‘Ž1 𝑠1

π‘‘π‘Ž2 𝑠2

𝑒𝑖𝑛𝑖𝑑 = minπ‘–βˆˆπ·

𝑒𝑖𝑛𝑖𝑑,𝑖

π‘“π‘™π‘Žπ‘‘π‘‘π‘’π‘› 𝐢𝑀 = 𝑋, π‘Œ, 𝑆, π‘žπ‘–π‘›π‘–π‘‘, 𝛿𝑖𝑛𝑑, 𝛿𝑒π‘₯𝑑, πœ†, π‘‘π‘Ž

CM

21

π‘žπ‘–π‘›π‘–π‘‘ = 𝑠𝑖𝑛𝑖𝑑, 𝑒𝑖𝑛𝑖𝑑

𝑠𝑖𝑛𝑖𝑑 = … , 𝑠𝑖𝑛𝑖𝑑,𝑖 , 𝑒𝑖𝑛𝑖𝑑,𝑖 βˆ’ 𝑒𝑖𝑛𝑖𝑑 , …

𝑋𝐢𝑀

π‘ŒπΆπ‘€

𝑆1

𝑆2

𝑠1, 𝑒1 = π‘žπ‘–π‘›π‘–π‘‘,1

𝑠2, 𝑒2 = π‘žπ‘–π‘›π‘–π‘‘,2𝑒2

𝑒1

π‘‘π‘Ž1 𝑠1

π‘‘π‘Ž2 𝑠2

𝑒𝑖𝑛𝑖𝑑 = minπ‘–βˆˆπ·

𝑒𝑖𝑛𝑖𝑑,𝑖

π‘“π‘™π‘Žπ‘‘π‘‘π‘’π‘› 𝐢𝑀 = 𝑋, π‘Œ, 𝑆, π‘žπ‘–π‘›π‘–π‘‘, 𝛿𝑖𝑛𝑑, 𝛿𝑒π‘₯𝑑, πœ†, π‘‘π‘Ž

CM

21

π‘žπ‘–π‘›π‘–π‘‘ = 𝑠𝑖𝑛𝑖𝑑, 𝑒𝑖𝑛𝑖𝑑

𝑠𝑖𝑛𝑖𝑑,𝑖 , 𝑒𝑖𝑛𝑖𝑑,𝑖 = π‘žπ‘–π‘›π‘–π‘‘,𝑖

𝑠𝑖𝑛𝑖𝑑 = … , 𝑠𝑖𝑛𝑖𝑑,𝑖 , 𝑒𝑖𝑛𝑖𝑑,𝑖 βˆ’ 𝑒𝑖𝑛𝑖𝑑 , …

𝑋𝐢𝑀

π‘ŒπΆπ‘€

𝑆1

𝑆2

𝑠1, 𝑒1 = π‘žπ‘–π‘›π‘–π‘‘,1

𝑠2, 𝑒2 = π‘žπ‘–π‘›π‘–π‘‘,2𝑒2

𝑒1

π‘‘π‘Ž1 𝑠1

π‘‘π‘Ž2 𝑠2

𝑋𝐢𝑀

π‘ŒπΆπ‘€

𝑆1

𝑆2

π‘‘π‘Ž1

π‘‘π‘Ž2

𝑑

𝑒1 π‘“π‘™π‘Žπ‘‘π‘‘π‘’π‘› 𝐢𝑀 = 𝑋, π‘Œ, 𝑆, π‘žπ‘–π‘›π‘–π‘‘, 𝛿𝑖𝑛𝑑, 𝛿𝑒π‘₯𝑑, πœ†, π‘‘π‘Ž

CM

21

π‘‘π‘Ž ∢ 𝑆 β†’ ℝ0,+∞+

π‘‘π‘Ž 𝑠 = minπ‘–βˆˆπ·

πœŽπ‘– = π‘‘π‘Žπ‘– 𝑠𝑖 βˆ’ 𝑒𝑖

𝜎1

𝜎2

π‘“π‘™π‘Žπ‘‘π‘‘π‘’π‘› 𝐢𝑀 = 𝑋, π‘Œ, 𝑆, π‘žπ‘–π‘›π‘–π‘‘, 𝛿𝑖𝑛𝑑, 𝛿𝑒π‘₯𝑑, πœ†, π‘‘π‘Ž

CM

21𝑋𝐢𝑀

π‘ŒπΆπ‘€

𝑆1

𝑆2

π‘‘π‘Ž1

π‘‘π‘Ž2

𝑑

𝑒1

π‘‘π‘Ž ∢ 𝑆 β†’ ℝ0,+∞+

π‘‘π‘Ž 𝑠 = minπ‘–βˆˆπ·

πœŽπ‘– = π‘‘π‘Žπ‘– 𝑠𝑖 βˆ’ 𝑒𝑖

𝐼𝑀𝑀 𝑠 = 𝑖 ∈ 𝐷 πœŽπ‘– = π‘‘π‘Ž(𝑠)𝑠𝑒𝑙𝑒𝑐𝑑(𝐼𝑀𝑀 𝑠 ) = π‘–βˆ—

π‘“π‘™π‘Žπ‘‘π‘‘π‘’π‘› 𝐢𝑀 = 𝑋, π‘Œ, 𝑆, π‘žπ‘–π‘›π‘–π‘‘, 𝛿𝑖𝑛𝑑, 𝛿𝑒π‘₯𝑑, πœ†, π‘‘π‘Ž

CM

21

𝜎1

𝜎2

𝑋𝐢𝑀

π‘ŒπΆπ‘€

𝑆1

𝑆2

π‘‘π‘Ž1

π‘‘π‘Ž2

𝑑

𝑒1

π‘“π‘™π‘Žπ‘‘π‘‘π‘’π‘› 𝐢𝑀 = 𝑋, π‘Œ, 𝑆, π‘žπ‘–π‘›π‘–π‘‘, 𝛿𝑖𝑛𝑑, 𝛿𝑒π‘₯𝑑, πœ†, π‘‘π‘Ž

CM

21𝑋𝐢𝑀

π‘ŒπΆπ‘€

𝑆1

𝑆2

𝑑

𝑋𝐢𝑀

π‘ŒπΆπ‘€

𝑆1

𝑆2πœ† 𝑠 = ࡝

π‘π‘–βˆ—,𝑠𝑒𝑙𝑓 πœ†π‘–βˆ— π‘ π‘–βˆ—

βˆ…

𝑖𝑓 𝑠𝑒𝑙𝑓 ∈ πΌπ‘–βˆ—

𝑖𝑓 𝑠𝑒𝑙𝑓 βˆ‰ πΌπ‘–βˆ—

𝑑

π‘“π‘™π‘Žπ‘‘π‘‘π‘’π‘› 𝐢𝑀 = 𝑋, π‘Œ, 𝑆, π‘žπ‘–π‘›π‘–π‘‘, 𝛿𝑖𝑛𝑑, 𝛿𝑒π‘₯𝑑, πœ†, π‘‘π‘Ž

CM

21

𝑦2

π‘“π‘™π‘Žπ‘‘π‘‘π‘’π‘› 𝐢𝑀 = 𝑋, π‘Œ, 𝑆, π‘žπ‘–π‘›π‘–π‘‘, 𝛿𝑖𝑛𝑑, 𝛿𝑒π‘₯𝑑, πœ†, π‘‘π‘Ž

CM

21𝑋𝐢𝑀

π‘ŒπΆπ‘€

𝑆1

𝑆2

𝑑

𝑠1, 0

𝑠1, π‘‘π‘Ž1 𝑠1

𝑠2, 𝑒2𝑠2, 𝑒2 + π‘‘π‘Ž 𝑠 = π‘‘π‘Ž2(𝑠2)

𝑠2β€² , 0

π‘‘π‘Ž 𝑠

𝛿𝑖𝑛𝑑 𝑠 = … , 𝑠𝑗′, 𝑒𝑗

β€² , …

𝑠𝑗′, 𝑒𝑗

β€² =

𝛿𝑖𝑛𝑑,𝑗 𝑠𝑗 , 0 π‘“π‘œπ‘Ÿ 𝑗 = π‘–βˆ—

? π‘“π‘œπ‘Ÿ 𝑗 ∈ πΌπ‘–βˆ— βˆ– 𝑠𝑒𝑙𝑓

𝑠𝑗 , 𝑒𝑗 + π‘‘π‘Ž 𝑠 𝑒𝑙𝑠𝑒

π‘“π‘™π‘Žπ‘‘π‘‘π‘’π‘› 𝐢𝑀 = 𝑋, π‘Œ, 𝑆, π‘žπ‘–π‘›π‘–π‘‘, 𝛿𝑖𝑛𝑑, 𝛿𝑒π‘₯𝑑, πœ†, π‘‘π‘Ž

CM

21𝑋𝐢𝑀

π‘ŒπΆπ‘€

𝑆1

𝑆2

𝑑

𝑠1, 0

𝑠1, π‘‘π‘Ž1 𝑠1

𝑠2, 𝑒2𝑠2, 𝑒2 + π‘‘π‘Ž 𝑠 = π‘‘π‘Ž2(𝑠2)

𝑠2β€² , 0

π‘‘π‘Ž 𝑠

𝑑

𝑋𝐢𝑀

π‘ŒπΆπ‘€

𝑆1

𝑆2 𝛿𝑖𝑛𝑑 𝑠 = … , 𝑠𝑗′, 𝑒𝑗

β€² , …

𝑠𝑗′, 𝑒𝑗

β€² =

𝛿𝑖𝑛𝑑,𝑗 𝑠𝑗 , 0 π‘“π‘œπ‘Ÿ 𝑗 = π‘–βˆ—

𝛿𝑒π‘₯𝑑,𝑗 𝑠𝑗 , 𝑒𝑗 + π‘‘π‘Ž 𝑠 , π‘π‘–βˆ—,𝑗 πœ†π‘–βˆ— π‘ π‘–βˆ— , 0 π‘“π‘œπ‘Ÿ 𝑗 ∈ πΌπ‘–βˆ— βˆ– 𝑠𝑒𝑙𝑓

𝑠𝑗 , 𝑒𝑗 + π‘‘π‘Ž 𝑠 𝑒𝑙𝑠𝑒

𝑠1, 𝑒1𝑠1, 𝑒1 + π‘‘π‘Ž 𝑠 = π‘‘π‘Ž1(𝑠1)

𝑠2, 𝑒2𝑠2, 𝑒2 + π‘‘π‘Ž 𝑠 ≀ π‘‘π‘Ž2 𝑠2

𝛿𝑖𝑛𝑑

𝛿𝑒π‘₯𝑑

𝑠1β€² , 0

𝑠2β€² , 0

π‘“π‘™π‘Žπ‘‘π‘‘π‘’π‘› 𝐢𝑀 = 𝑋, π‘Œ, 𝑆, π‘žπ‘–π‘›π‘–π‘‘, 𝛿𝑖𝑛𝑑, 𝛿𝑒π‘₯𝑑, πœ†, π‘‘π‘Ž

CM

21𝑦1

π‘₯2

π‘“π‘™π‘Žπ‘‘π‘‘π‘’π‘› 𝐢𝑀 = 𝑋, π‘Œ, 𝑆, π‘žπ‘–π‘›π‘–π‘‘, 𝛿𝑖𝑛𝑑, 𝛿𝑒π‘₯𝑑, πœ†, π‘‘π‘Ž

CM

21𝑋𝐢𝑀

π‘ŒπΆπ‘€

𝑆1

𝑆2

𝑑

𝑒

𝑒1

𝛿𝑒π‘₯𝑑 𝑠, 𝑒 , π‘₯ = … , 𝑠𝑖′, 𝑒𝑖

β€² , …

𝑠𝑖′, 𝑒𝑖

β€² = ቐ𝛿𝑒π‘₯𝑑,𝑖 𝑠𝑖 , 𝑒𝑖 + 𝑒 , 𝑍𝑠𝑒𝑙𝑓,𝑖 π‘₯ , 0 π‘“π‘œπ‘Ÿ 𝑖 ∈ 𝐼𝑠𝑒𝑙𝑓

𝑠𝑖 , 𝑒𝑖 + 𝑒 𝑒𝑙𝑠𝑒

π‘“π‘™π‘Žπ‘‘π‘‘π‘’π‘› 𝐢𝑀 = 𝑋, π‘Œ, 𝑆, π‘žπ‘–π‘›π‘–π‘‘, 𝛿𝑖𝑛𝑑, 𝛿𝑒π‘₯𝑑, πœ†, π‘‘π‘Ž

CM

21

𝑦𝑠𝑒𝑙𝑓

𝑋𝐢𝑀

π‘ŒπΆπ‘€

𝑆1

𝑆2

𝑑

𝑒

𝑒1

𝑆 = Γ—π‘–βˆˆπ· 𝑄𝑖

π‘‘π‘Ž 𝑠 = minπ‘–βˆˆπ·

πœŽπ‘– = π‘‘π‘Žπ‘– 𝑠𝑖 βˆ’ 𝑒𝑖

𝐼𝑀𝑀 𝑠 = 𝑖 ∈ 𝐷 πœŽπ‘– = π‘‘π‘Ž(𝑠)

πœ† 𝑠 = ΰ΅π‘π‘–βˆ—,𝑠𝑒𝑙𝑓 πœ†π‘–βˆ— π‘ π‘–βˆ—

πœ™

𝑖𝑓 𝑠𝑒𝑙𝑓 ∈ πΌπ‘–βˆ—

𝑖𝑓 𝑠𝑒𝑙𝑓 βˆ‰ πΌπ‘–βˆ—

𝛿𝑖𝑛𝑑 𝑠 = … , 𝑠𝑗′, 𝑒𝑗

β€² , …

𝑠𝑖′, 𝑒𝑖

β€² = ቐ𝛿𝑒π‘₯𝑑,𝑖 𝑠𝑖 , 𝑒𝑖 + 𝑒 , 𝑍𝑠𝑒𝑙𝑓,𝑖 π‘₯ , 0 π‘“π‘œπ‘Ÿ 𝑖 ∈ 𝐼𝑠𝑒𝑙𝑓

𝑠𝑖 , 𝑒𝑖 + 𝑒 𝑒𝑙𝑠𝑒

π‘“π‘™π‘Žπ‘‘π‘‘π‘’π‘› 𝐢𝑀 = 𝑋, π‘Œ, 𝑆, π‘žπ‘–π‘›π‘–π‘‘, 𝛿𝑖𝑛𝑑 , 𝛿𝑒π‘₯𝑑, πœ†, π‘‘π‘Ž

𝑒𝑖𝑛𝑖𝑑 = minπ‘–βˆˆπ·

𝑒𝑖𝑛𝑖𝑑,𝑖

π‘žπ‘–π‘›π‘–π‘‘ = 𝑠𝑖𝑛𝑖𝑑 , 𝑒𝑖𝑛𝑖𝑑 ∈ 𝑄

𝑠𝑗′, 𝑒𝑗

β€² =

𝛿𝑖𝑛𝑑,𝑗 𝑠𝑗 , 0 π‘“π‘œπ‘Ÿ 𝑗 = π‘–βˆ—

𝛿𝑒π‘₯𝑑,𝑗 𝑠𝑗 , 𝑒𝑗 + π‘‘π‘Ž 𝑠 , π‘π‘–βˆ—,𝑗 πœ†π‘–βˆ— π‘ π‘–βˆ— , 0 π‘“π‘œπ‘Ÿ 𝑗 ∈ πΌπ‘–βˆ—

𝑠𝑗 , 𝑒𝑗 + π‘‘π‘Ž 𝑠 𝑒𝑙𝑠𝑒

𝑋 = π‘‹πΆπ‘€π‘Œ = π‘ŒπΆπ‘€

𝑠𝑖𝑛𝑖𝑑,𝑖 , 𝑒𝑖𝑛𝑖𝑑,𝑖 = π‘žπ‘–π‘›π‘–π‘‘,𝑖

𝑠𝑖𝑛𝑖𝑑 = (… , 𝑠𝑖𝑛𝑖𝑑,𝑖 , 𝑒𝑖𝑛𝑖𝑑,𝑖 βˆ’ 𝑒𝑖𝑛𝑖𝑑 , … )

𝑄 = 𝑠, 𝑒 𝑠 ∈ 𝑆, 0 ≀ 𝑒 ≀ π‘‘π‘Ž(𝑠)

π‘–βˆ— = 𝑠𝑒𝑙𝑒𝑐𝑑(𝐼𝑀𝑀 𝑠 )

𝛿𝑒π‘₯𝑑 𝑠, 𝑒 , π‘₯ = … , 𝑠𝑖′, 𝑒𝑖

β€² , …

Applications of DEVSDEVS in practice

http://irh.inf.unideb.hu/user/jsztrik/education/09/english/3f.html

http://irh.inf.unideb.hu/user/jsztrik/education/09/english/3f.html

π‘π‘œπ‘ π‘‘ = 10 Γ— #π‘™π‘–π‘”β„Žπ‘‘π‘  + π‘Žπ‘£π‘”(π‘‘π‘‘π‘Ÿπ‘Žπ‘£π‘’π‘™)

Generator StatisticsQueue

Notifier

Poller

…

Queue

Notifier

Poller

…

Custom Atomic DEVS ModelsExtending a DEVS repository

𝑀 = 𝑋, π‘Œ, 𝑆, π‘žπ‘–π‘›π‘–π‘‘, 𝛿𝑖𝑛𝑑, 𝛿𝑒π‘₯𝑑, πœ†, π‘‘π‘Ž

𝑋 : set of input events

π‘Œ : set of output events

𝑆 : set of sequential states

π‘žπ‘–π‘›π‘–π‘‘ : 𝑄𝑄 = 𝑠, 𝑒 𝑠 ∈ 𝑆, 0 ≀ 𝑒 ≀ π‘‘π‘Ž(𝑠)

𝛿𝑖𝑛𝑑 : 𝑆 β†’ 𝑆𝛿𝑒π‘₯𝑑: 𝑄 Γ— 𝑋 β†’ π‘†πœ† : 𝑆 β†’ π‘Œ βˆͺ πœ™π‘‘π‘Ž : 𝑆 β†’ ℝ0,+∞

+

Atomic DEVS Specification

𝑀 = 𝑋, π‘Œ, 𝑆, π‘žπ‘–π‘›π‘–π‘‘, 𝛿𝑖𝑛𝑑, 𝛿𝑒π‘₯𝑑, πœ†, π‘‘π‘Ž

𝑋 : set of input events

π‘Œ : set of output events

𝑆 : set of sequential states

π‘žπ‘–π‘›π‘–π‘‘ : 𝑄𝑄 = 𝑠, 𝑒 𝑠 ∈ 𝑆, 0 ≀ 𝑒 ≀ π‘‘π‘Ž(𝑠)

𝛿𝑖𝑛𝑑 : 𝑆 β†’ 𝑆𝛿𝑒π‘₯𝑑: 𝑄 Γ— 𝑋 β†’ π‘†πœ† : 𝑆 β†’ π‘Œ βˆͺ πœ™π‘‘π‘Ž : 𝑆 β†’ ℝ0,+∞

+

Atomic DEVS Specification

Pattern 1: Ignore an Event

X

Pattern 1: Ignore an Event

0.1 seconds

later

X Y

Pattern 1: Ignore an Event

π‘‘π‘Ž(𝑠𝑖)

Pattern 1: Ignore an Event

π‘‘π‘Ž(𝑠𝑖)

Pattern 1: Ignore an Event

π‘‘π‘Ž(𝑠𝑖′)𝑒

π‘‘π‘Ž 𝑠𝑖′ = π‘‘π‘Ž 𝑠𝑖 βˆ’ 𝑒

Pattern 1: Ignore an Event

class RailwaySegmentState():

def __init__(self):

self.timer = INFINITY

class RailwaySegment(AtomicDEVS):

def extTransition(self, inputs):

self.state.timer -= self.elapsed

…

def timeAdvance(self):

return self.state.timer

Pattern 2: Simulated Time

π‘₯ seconds

later

𝑑 = 𝑏 𝑑 = 𝑏 + π‘₯

Pattern 2: Simulated Time

π‘₯ seconds

later

𝑑 = 𝑏 𝑑 = 𝑏 + π‘₯

π‘‘π‘‘π‘Ÿπ‘Žπ‘›π‘ π‘–π‘‘ = π‘₯

Pattern 2: Simulated Time

π‘₯ seconds

later

𝑑 = 𝑏 𝑑 = 𝑏 + π‘₯

π‘‘π‘‘π‘Ÿπ‘Žπ‘›π‘ π‘–π‘‘ = π‘₯

𝑑 = ?

Pattern 2: Simulated Time

π‘‘π‘ π‘–π‘š = π‘₯

Pattern 2: Simulated Time

π‘‘π‘Ž(𝑠𝑖)

π‘‘π‘ π‘–π‘š = π‘₯

Pattern 2: Simulated Time

π‘‘π‘Ž(𝑠𝑖)

π‘‘π‘ π‘–π‘š = π‘₯

π‘‘π‘ π‘–π‘š = π‘₯ + π‘‘π‘Ž 𝑠𝑖

Pattern 2: Simulated Time

𝑒

π‘‘π‘ π‘–π‘š = π‘₯

Pattern 2: Simulated Time

𝑒

π‘‘π‘ π‘–π‘š = π‘₯

π‘‘π‘ π‘–π‘š = π‘₯ + 𝑒

Pattern 2: Simulated Time

class GeneratorState:

def __init__(self):

self.t_sim = 0.0

…

class Generator(AtomicDEVS):

def __init__(self):

self.state = GeneratorState()

…

def intTransition(self):

self.state.t_sim += self.timeAdvance()

…

def extTransition(self, inputs):

self.state.t_sim += self.elapsed

…

Pattern 3: Multiple Timers

Pattern 3: Multiple Timers

π‘‘π‘‘π‘Ÿπ‘Žπ‘–π‘›

π‘‘π‘žπ‘’π‘’π‘Ÿπ‘¦

?

π‘‘π‘Ž(𝑠𝑖)

Pattern 3: Multiple Timers

π‘‘π‘‘π‘Ÿπ‘Žπ‘–π‘›

π‘‘π‘žπ‘’π‘’π‘Ÿπ‘¦

?

π‘‘π‘Ž(𝑠𝑗)π‘‘π‘Ž(𝑠𝑖)

Pattern 3: Multiple Timers

π‘‘π‘‘π‘Ÿπ‘Žπ‘–π‘›

π‘‘π‘žπ‘’π‘’π‘Ÿπ‘¦

?

π‘‘π‘Ž(𝑠𝑗)

Pattern 3: Multiple Timers

class RailwaySegmentState:

def __init__(self):

self.t_query = INFINITY

self.t_train = INFINITY

class RailwaySegment(AtomicDEVS):

def timeAdvance(self):

return min(self.state.t_query, \

self.state.t_train)

def extTransition(self, inputs):

self.state.t_query -= self.elapsed

self.state.t_train -= self.elapsed

…

…

def intTransition(self):

self.state.t_query -= self.timeAdvance()

self.state.t_train -= self.timeAdvance()

if (self.state.t_query == 0):

… # process query

elif (self.state.t_train == 0):

… # process train

Pattern 4: Statistics Gathering

[1815.7; 1901.1; 1801.4; 1867.3; 1911.8; 1846.4; 1844.3; 1873.5; …]

𝑛 trains = 𝑛 doubles = 8𝑛 bytes

Pattern 4: Statistics Gathering

[1815.7; 1901.1; 1801.4; 1867.3; 1911.8; 1846.4; 1844.3; 1873.5; …]

π‘ π‘’π‘š =

𝑖

π‘‘π‘‘π‘Ÿπ‘Žπ‘£π‘’π‘™ = 36,902,140.45

π‘π‘œπ‘’π‘›π‘‘ = π‘Žπ‘Ÿπ‘Ÿπ‘–π‘£π‘Žπ‘™π‘  = 20,000

π‘Žπ‘£π‘” =π‘ π‘’π‘š

π‘π‘œπ‘’π‘›π‘‘

𝑛 trains = 𝑛 doubles = 8𝑛 bytes

𝑛 trains = 1 double + 1 long long int = 16 bytes

Pattern 4: Statistics Gathering

class CollectorState():

def __init__(self):

self.counter = 0

self.accum = 0.0

self.t_sim = 0.0

class Collector(AtomicDEVS):

def extTransition(self, inputs):

left = inputs[self.train].left

transit = self.state.t_sim – left

self.accum += transit

self.counter += 1

Pattern 5: Complex State/Event

𝑆 𝑋, π‘Œ

Pattern 5: Complex State/Event

𝑆 𝑋, π‘Œ

𝑆 = π‘‘π‘–π‘šπ‘’π‘Ÿπ‘‘π‘Ÿπ‘Žπ‘–π‘›, π‘‘π‘–π‘šπ‘’π‘Ÿπ‘žπ‘’π‘’π‘Ÿπ‘¦ , π‘£π‘‘π‘Ÿπ‘Žπ‘–π‘›, π‘‘π‘‘π‘Ÿπ‘Žπ‘–π‘›, π‘Žπ‘‘π‘Ÿπ‘Žπ‘–π‘› π‘‘π‘–π‘šπ‘’π‘Ÿπ‘‘π‘Ÿπ‘Žπ‘–π‘› ∈ ℝ+, …

𝑋, π‘Œ = 𝑑, 𝑣, π‘Ž 𝑑 ∈ ℝ+, … βˆͺ query βˆͺ π‘π‘œπ‘™π‘œπ‘’π‘Ÿ π‘π‘œπ‘™π‘œπ‘’π‘Ÿ ∈ π‘Ÿπ‘’π‘‘, π‘”π‘Ÿπ‘’π‘’π‘›

Pattern 5: Complex State/Event

class Train:

def __init__(self, t, a):

self.t = t

self.a = a

self.v = 0.0

class Query:

def __init__(self):

pass

class QueryAck:

def __init__(self, colour):

self.colour = colour

class RailwaySegmentState:

def __init__(self):

self.train = None

class RailwaySegment(AtomicDEVS):

def __init__(self):

self.state = RailwaySegmentState()

…

def extTransition(self, inputs):

…

self.state.train = inputs[self.train_in]

…

Conclusion

http://msdl.cs.mcgill.ca/projects/PythonPDEVS

http://msdl.cs.mcgill.ca/projects/PythonPDEVS

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