Classic DEVS An Introduction Using PythonPDEVS Yentl Van Tendeloo, - PowerPoint PPT Presentation

Autonomous (with output) red 𝑒𝑓𝑚𝑏𝑧 𝑠𝑓𝑒 𝑁 = , 𝑇, 𝑟 𝑗𝑜𝑗𝑢 , 𝜀 𝑗𝑜𝑢 , , 𝑢𝑏 𝑍 𝜇 𝑇 = {red, yellow, green} 𝜀 𝑗𝑜𝑢 = { red → green, !show_red green → yellow, yellow → red} 𝑟 𝑗𝑜𝑗𝑢 = (green, 0) 𝑢𝑏 = {red → 𝑒𝑓𝑚𝑏𝑧 𝑠𝑓𝑒 , yellow !show_green green → 𝑒𝑓𝑚𝑏𝑧 𝑕𝑠𝑓𝑓𝑜 , 𝑒𝑓𝑚𝑏𝑧 𝑧𝑓𝑚𝑚𝑝𝑥 yellow → 𝑒𝑓𝑚𝑏𝑧 𝑧𝑓𝑚𝑚𝑝𝑥 } 𝑍 : set of output events 𝑍 = {“ show_red ”, “ show_green ”, “ show_yellow ”} !show_yellow 𝜇 : 𝑇 → 𝑍 ∪ {𝜚} 𝜇 = { green → “ show_yellow ”, green 𝑓 = 0𝑡 yellow → “ show_red ”, 𝑒𝑓𝑚𝑏𝑧 𝑕𝑠𝑓𝑓𝑜 red → “ show_green ”}

Abstract Syntax Concrete Syntax atomic_out.py 𝑇 = {red, yellow, green} 𝑟 𝑗𝑜𝑗𝑢 = (green, 0) from pypdevs.DEVS import * 𝜀 𝑗𝑜𝑢 = { red → green, green → yellow, class TrafficLightWithOutput(AtomicDEVS): yellow → red} def __init__( self, …): 𝑢𝑏 = {red → 𝑒𝑓𝑚𝑏𝑧 𝑠𝑓𝑒 , AtomicDEVS.__init __(self, “light”) green → 𝑒𝑓𝑚𝑏𝑧 𝑕𝑠𝑓𝑓𝑜 , self.observe = self.addOutPort (“observer”) yellow → 𝑒𝑓𝑚𝑏𝑧 𝑧𝑓𝑚𝑚𝑝𝑥 } … 𝑍 = {“ show_red ”, “ show_green ”, … “ show_yellow ”} 𝜇 = {green → “ show_yellow ”, def outputFnc(self): yellow → “ show_red ”, state = self.state red → “ show_green ”} if state == “red”: return {self.observe: “ show_green ”} Operational Semantics elif state == “yellow”: time = 0 return {self.observe: “ show_red ”} current_state = initial_state elif state == “green”: last_time = -initial_elapsed return {self.observe: “ show_yellow ”} while not termination_condition(): time = last_time + ta(current_state) output( 𝜇 (current_state)) current_state = 𝜀 𝑗𝑜𝑢 (current_state) last_time = time

__ Current Time: 0.00 __________________________________________ INITIAL CONDITIONS in model <light> Initial State: green Next scheduled internal transition at time 57.00 __ Current Time: 57.00 __________________________________________ INTERNAL TRANSITION in model <light> New State: yellow Output Port Configuration: port <observer>: show_yellow Next scheduled internal transition at time 60.00 __ Current Time: 60.00 __________________________________________ INTERNAL TRANSITION in model <light> New State: red Output Port Configuration: port <observer>: show_red Next scheduled internal transition at time 120.00

X toManual red 𝑒𝑓𝑚𝑏𝑧 𝑠𝑓𝑒 ?toManual toAuto t !show_red ?toAuto S manual !show_green red yellow manual yellow 𝑒𝑓𝑚𝑏𝑧 𝑧𝑓𝑚𝑚𝑝𝑥 ∞ green ?toManual t !show_yellow Y show_green green ?toManual 𝑒𝑓𝑚𝑏𝑧 𝑕𝑠𝑓𝑓𝑜 𝑓 = 0𝑡 t

Reactive 𝑌 𝜀 𝑓𝑦𝑢 𝑁 = , 𝑍, 𝑇, 𝑟 𝑗𝑜𝑗𝑢 , 𝜀 𝑗𝑜𝑢 , , 𝜇, 𝑢𝑏 𝑍 = {“ show_red ”, “ show_green ”, “ show_yellow ”} red 𝑇 = {red, yellow, green, manual} 𝑒𝑓𝑚𝑏𝑧 𝑠𝑓𝑒 ?toManual 𝑟 𝑗𝑜𝑗𝑢 = (green, 0) 𝜀 𝑗𝑜𝑢 = {red → green, green → yellow, !show_red yellow → red} ?toAuto 𝜇 = {green → “ show_yellow ”, yellow → “ show_red ”, !show_green red → “ show_green ”} yellow manual 𝑢𝑏 = {red → 𝑒𝑓𝑚𝑏𝑧 𝑠𝑓𝑒 , 𝑒𝑓𝑚𝑏𝑧 𝑧𝑓𝑚𝑚𝑝𝑥 ∞ green → 𝑒𝑓𝑚𝑏𝑧 𝑕𝑠𝑓𝑓𝑜 , ?toManual yellow → 𝑒𝑓𝑚𝑏𝑧 𝑧𝑓𝑚𝑚𝑝𝑥 , manual → +∞ } !show_yellow 𝑌 : set of input events 𝑌 = {“ toAuto ”, “ toManual ”} green ?toManual 𝜀 𝑓𝑦𝑢 : Q × 𝑌 → 𝑇 𝑒𝑓𝑚𝑏𝑧 𝑕𝑠𝑓𝑓𝑜 𝑅 = 𝑡, 𝑓 𝑡 ∈ 𝑇, 0 ≤ 𝑓 ≤ 𝑢𝑏(𝑡) 𝜀 𝑓𝑦𝑢 = {( (*, *), “ toManual ”) → “manual”, ( (“manual”, *), “ toAuto ”) → “red”} 𝑓 = 0𝑡

Abstract Syntax 𝑍 = {“ show_red ”, “ show_green ”, “ show_yellow ”} Operational Semantics 𝑇 = {red, yellow, green, manual} time = 0 𝑟 𝑗𝑜𝑗𝑢 = (green, 0) current_state = initial_state 𝜀 𝑗𝑜𝑢 = {red → green, last_time = -initial_elapsed green → yellow, while not termination_condition(): yellow → red} next_time = last_time + ta(current_state) 𝜇 = {green → “ show_yellow ”, if time_next_ev <= next_time: yellow → “ show_red ”, e = time_next_ev – last_time red → “ show_green ”} time = time_next_ev 𝑢𝑏 = {red → 𝑒𝑓𝑚𝑏𝑧 𝑠𝑓𝑒 , current_state = 𝜀 𝑓𝑦𝑢 ((current_state, e), next_ev) green → 𝑒𝑓𝑚𝑏𝑧 𝑕𝑠𝑓𝑓𝑜 , else: yellow → 𝑒𝑓𝑚𝑏𝑧 𝑧𝑓𝑚𝑚𝑝𝑥 , time = next_time manual → ∞ } output( 𝜇 (current_state)) 𝑌 = {“ toAuto ”, “ toManual ”} current_state = 𝜀 _ 𝑗𝑜𝑢 (current_state) 𝜀 𝑓𝑦𝑢 = {( (*, *), “ toManual ”) → manual, last_time = time ( (manual, *), “ toAuto ”) → red}

Abstract Syntax Concrete Syntax atomic_ext.py 𝑍 = {“ show_red ”, “ show_green ”, “ show_yellow ”} from pypdevs.DEVS import * 𝑇 = {red, yellow, green, manual} 𝑟 𝑗𝑜𝑗𝑢 = (green, 0) class TrafficLight(AtomicDEVS): 𝜀 𝑗𝑜𝑢 = {red → green, def __init__( self, …): green → yellow, AtomicDEVS.__init __(self, “light”) yellow → red} self.interrupt = self.addInPort (“interrupt”) 𝜇 = {green → “ show_yellow ”, … yellow → “ show_red ”, … red → “ show_green ”} 𝑢𝑏 = {red → 𝑒𝑓𝑚𝑏𝑧 𝑠𝑓𝑒 , def extTransition(self, inputs): green → 𝑒𝑓𝑚𝑏𝑧 𝑕𝑠𝑓𝑓𝑜 , inp = inputs[self.interrupt] yellow → 𝑒𝑓𝑚𝑏𝑧 𝑧𝑓𝑚𝑚𝑝𝑥 , if inp == “ toManual ”: manual → ∞ } return “manual” 𝑌 = {“ toAuto ”, “ toManual ”} elif inp == “ toAuto ”: 𝜀 𝑓𝑦𝑢 = {( (*, *), “ toManual ”) → manual, if self.state == “manual”: ( (manual, *), “ toAuto ”) → red} return “red”

__ Current Time: 0.00 __________________________________________ INITIAL CONDITIONS in model <light> Initial State: green Next scheduled internal transition at time 57.00 __ Current Time: 57.00 __________________________________________ INTERNAL TRANSITION in model <light> New State: yellow Output Port Configuration: port <observer>: show_yellow Next scheduled internal transition at time 60.00 __ Current Time: 60.00 __________________________________________ INTERNAL TRANSITION in model <light> New State: red Output Port Configuration: port <observer>: show_red Next scheduled internal transition at time 120.00

𝑅 = 𝑡, 𝑓 𝑡 ∈ 𝑇, 0 ≤ 𝑓 ≤ 𝑢𝑏(𝑡) 𝑓 = 0 0 < 𝑓 < 𝑢𝑏(𝑡) 𝑓 = 𝑢𝑏 𝑡 ?

red 𝑒𝑓𝑚𝑏𝑧 𝑠𝑓𝑒 ?toManual/ !turn_off ?toAuto/ !show_red !show_red !show_green yellow manual 𝑒𝑓𝑚𝑏𝑧 𝑧𝑓𝑚𝑚𝑝𝑥 ∞ ?toManual/ !turn_off !show_yellow green ?toManual/ 𝑒𝑓𝑚𝑏𝑧 𝑕𝑠𝑓𝑓𝑜 !turn_off 𝑓 = 0𝑡

X toManual red 𝑒𝑓𝑚𝑏𝑧 𝑠𝑓𝑒 ?toManual/ toAuto !turn_off t ?toAuto/ !show_red S !show_red manual !show_green red yellow manual yellow 𝑒𝑓𝑚𝑏𝑧 𝑧𝑓𝑚𝑚𝑝𝑥 ∞ green ?toManual/ !turn_off t !show_yellow Y show_green green ?toManual/ show_red 𝑒𝑓𝑚𝑏𝑧 𝑕𝑠𝑓𝑓𝑜 !turn_off turn_off 𝑓 = 0𝑡 t

X toManual red 𝑒𝑓𝑚𝑏𝑧 𝑠𝑓𝑒 ?toManual/ toAuto !turn_off time = 0 current_state = initial_state t ?toAuto/ last_time = -initial_elapsed !show_red S !show_red while not termination_condition(): manual next_time = last_time + ta(current_state) !show_green if time_next_ev <= next_time: red yellow manual e = time_next_ev – last_time yellow 𝑒𝑓𝑚𝑏𝑧 𝑧𝑓𝑚𝑚𝑝𝑥 time = time_next_ev ∞ green current_state = 𝜀 𝑓𝑦𝑢 ((current_state, e), next_ev) ?toManual/ else: !turn_off t time = next_time !show_yellow output( 𝜇 (current_state)) Y current_state = 𝜀 _ 𝑗𝑜𝑢 (current_state) last_time = time show_green green ?toManual/ show_red 𝑒𝑓𝑚𝑏𝑧 𝑕𝑠𝑓𝑓𝑜 !turn_off turn_off 𝑓 = 0𝑡 t

X !show_red red toManual going_auto 0𝑡 𝑒𝑓𝑚𝑏𝑧 𝑠𝑓𝑒 toAuto !show_green t ?toAuto ?toManual !show_red S manual going_auto yellow manual going_manual 𝑒𝑓𝑚𝑏𝑧 𝑧𝑓𝑚𝑚𝑝𝑥 ∞ red green t !show_yellow !turn_off ?toManual Y show_green green going_manual show_red 𝑒𝑓𝑚𝑏𝑧 𝑕𝑠𝑓𝑓𝑜 0𝑡 turn_off 𝑓 = 0𝑡 t ?toManual

Full Atomic DEVS Specification 𝑁 = 𝑌, 𝑍, 𝑇, 𝑟 𝑗𝑜𝑗𝑢 , 𝜀 𝑗𝑜𝑢 , 𝜀 𝑓𝑦𝑢 , 𝜇, 𝑢𝑏 𝑌 : set of input events 𝑍 : set of output events 𝑇 : set of sequential states 𝑟 𝑗𝑜𝑗𝑢 : 𝑅 𝑅 = 𝑡, 𝑓 𝑡 ∈ 𝑇, 0 ≤ 𝑓 ≤ 𝑢𝑏(𝑡) 𝜀 𝑗𝑜𝑢 : 𝑇 → 𝑇 𝜀 𝑓𝑦𝑢 : 𝑅 × 𝑌 → 𝑇 𝜇 : 𝑇 → 𝑍 ∪ 𝜚 𝑢𝑏 : 𝑇 → ℝ 0,+∞ +

S (𝜀 𝑓𝑦𝑢 𝑡 𝑗 , 𝑓 , 𝑦 , 0) 𝜀 𝑓𝑦𝑢 𝑓 (𝑡 𝑗 , 0) output 𝜇(𝑡 𝑗 ) 𝑢𝑏 𝑡 𝑗 𝜀 𝑗𝑜𝑢 (𝜀 𝑗𝑜𝑢 (𝑡 𝑗 ), 0) t 𝑢 𝑗 + 𝑓 𝑢 𝑗 𝑢 𝑗 + 𝑢𝑏 𝑡 𝑗

Coupled Models

!show_red red going_auto 0𝑡 𝑒𝑓𝑚𝑏𝑧 𝑠𝑓𝑒 !show_green ?toAuto ?toManual !show_red yellow manual 𝑒𝑓𝑚𝑏𝑧 𝑧𝑓𝑚𝑚𝑝𝑥 ∞ !show_yellow !turn_off ?toManual green going_manual 𝑒𝑓𝑚𝑏𝑧 𝑕𝑠𝑓𝑓𝑜 0𝑡 𝑓 = 0𝑡 ?toManual

work 3600s !go_to_work !take_break idle 300s 𝑓 = 280𝑡

𝒕𝒇𝒎𝒈 𝐷 = 𝐸, 𝑁𝑇 𝑵 𝒎𝒋𝒉𝒊𝒖𝟐 𝑵 𝒒𝒑𝒎𝟐 𝑁𝑇 = 𝑁 𝑗 𝑗 ∈ 𝐸 𝑁 𝑗 = 𝑌 𝑗 , 𝑍 𝑗 , 𝑇 𝑗 , 𝑟 𝑗𝑜𝑗𝑢,𝑗 , 𝜀 𝑗𝑜𝑢,𝑗 , 𝜀 𝑓𝑦𝑢,𝑗 , 𝜇 𝑗 , 𝑢𝑏 𝑗 , ∀ 𝑗 ∈ 𝐸

𝐸 = {𝑚𝑗𝑕ℎ𝑢 1 , 𝑞𝑝𝑚 1 } 𝒕𝒇𝒎𝒈 𝐷 = 𝐸, 𝑁𝑇 𝑵 𝒎𝒋𝒉𝒊𝒖𝟐 𝑵 𝒒𝒑𝒎𝟐 𝑁𝑇 = 𝑁 𝑗 𝑗 ∈ 𝐸 𝑁 𝑗 = 𝑌 𝑗 , 𝑍 𝑗 , 𝑇 𝑗 , 𝑟 𝑗𝑜𝑗𝑢,𝑗 , 𝜀 𝑗𝑜𝑢,𝑗 , 𝜀 𝑓𝑦𝑢,𝑗 , 𝜇 𝑗 , 𝑢𝑏 𝑗 , ∀ 𝑗 ∈ 𝐸

𝒕𝒇𝒎𝒈 𝐷 = 𝑌 𝑡𝑓𝑚𝑔 , 𝑍 𝑡𝑓𝑚𝑔 , 𝐸, 𝑁𝑇 𝑵 𝒎𝒋𝒉𝒊𝒖𝟐 𝑵 𝒒𝒑𝒎𝟐 𝑁𝑇 = 𝑁 𝑗 𝑗 ∈ 𝐸 𝑁 𝑗 = 𝑌 𝑗 , 𝑍 𝑗 , 𝑇 𝑗 , 𝑟 𝑗𝑜𝑗𝑢,𝑗 , 𝜀 𝑗𝑜𝑢,𝑗 , 𝜀 𝑓𝑦𝑢,𝑗 , 𝜇 𝑗 , 𝑢𝑏 𝑗 , ∀ 𝑗 ∈ 𝐸

𝒕𝒇𝒎𝒈 𝐷 = 𝑌 𝑡𝑓𝑚𝑔 , 𝑍 𝑡𝑓𝑚𝑔 , 𝐸, 𝑁𝑇, 𝐽𝑇 𝑵 𝒎𝒋𝒉𝒊𝒖𝟐 𝑵 𝒒𝒑𝒎𝟐 𝑁𝑇 = 𝑁 𝑗 𝑗 ∈ 𝐸 𝑁 𝑗 = 𝑌 𝑗 , 𝑍 𝑗 , 𝑇 𝑗 , 𝑟 𝑗𝑜𝑗𝑢,𝑗 , 𝜀 𝑗𝑜𝑢,𝑗 , 𝜀 𝑓𝑦𝑢,𝑗 , 𝜇 𝑗 , 𝑢𝑏 𝑗 , ∀ 𝑗 ∈ 𝐸 𝐽𝑇 = 𝐽 𝑗 𝑗 ∈ 𝐸 ∪ 𝑡𝑓𝑚𝑔 ∀ 𝑗 ∈ 𝐸 ∪ 𝑡𝑓𝑚𝑔 ∶ 𝐽 𝑗 ⊆ 𝐸 ∪ 𝑡𝑓𝑚𝑔 ∀ 𝑗 ∈ 𝐸 ∪ 𝑡𝑓𝑚𝑔 ∶ 𝑗 ∉ 𝐽 𝑗 𝐽 𝑗 : Influencees of 𝑗

𝒕𝒇𝒎𝒈 𝐷 = 𝑌 𝑡𝑓𝑚𝑔 , 𝑍 𝑡𝑓𝑚𝑔 , 𝐸, 𝑁𝑇, 𝐽𝑇 𝑵 𝒎𝒋𝒉𝒊𝒖𝟐 𝑵 𝒒𝒑𝒎𝟐 𝑁𝑇 = 𝑁 𝑗 𝑗 ∈ 𝐸 𝑁 𝑗 = 𝑌 𝑗 , 𝑍 𝑗 , 𝑇 𝑗 , 𝑟 𝑗𝑜𝑗𝑢,𝑗 , 𝜀 𝑗𝑜𝑢,𝑗 , 𝜀 𝑓𝑦𝑢,𝑗 , 𝜇 𝑗 , 𝑢𝑏 𝑗 , ∀ 𝑗 ∈ 𝐸 𝐽𝑇 = 𝐽 𝑗 𝑗 ∈ 𝐸 ∪ 𝑡𝑓𝑚𝑔 ∀ 𝑗 ∈ 𝐸 ∪ 𝑡𝑓𝑚𝑔 ∶ 𝐽 𝑗 ⊆ 𝐸 ∪ 𝑡𝑓𝑚𝑔 ∀ 𝑗 ∈ 𝐸 ∪ 𝑡𝑓𝑚𝑔 ∶ 𝑗 ∉ 𝐽 𝑗 𝐽 𝑗 : Influencees of 𝑗 𝐽 𝑡𝑓𝑚𝑔 = {𝑚𝑗𝑕ℎ𝑢1} 𝐽 𝑞𝑝𝑚1 = {𝑚𝑗𝑕ℎ𝑢1} 𝐽 𝑚𝑗𝑕ℎ𝑢1 = {𝑡𝑓𝑚𝑔}

𝑍 𝑞𝑝𝑚1 = {𝑕𝑝_𝑢𝑝_𝑥𝑝𝑠𝑙, 𝑢𝑏𝑙𝑓_𝑐𝑠𝑓𝑏𝑙} 𝒕𝒇𝒎𝒈 𝐷 = 𝑌 𝑡𝑓𝑚𝑔 , 𝑍 𝑡𝑓𝑚𝑔 , 𝐸, 𝑁𝑇, 𝐽𝑇, 𝑎𝑇 𝑵 𝒎𝒋𝒉𝒊𝒖𝟐 𝑵 𝒒𝒑𝒎𝟐 𝑁𝑇 = 𝑁 𝑗 𝑗 ∈ 𝐸 𝑁 𝑗 = 𝑌 𝑗 , 𝑍 𝑗 , 𝑇 𝑗 , 𝑟 𝑗𝑜𝑗𝑢,𝑗 , 𝜀 𝑗𝑜𝑢,𝑗 , 𝜀 𝑓𝑦𝑢,𝑗 , 𝜇 𝑗 , 𝑢𝑏 𝑗 , ∀ 𝑗 ∈ 𝐸 𝐽𝑇 = 𝐽 𝑗 𝑗 ∈ 𝐸 ∪ 𝑡𝑓𝑚𝑔 ∀ 𝑗 ∈ 𝐸 ∪ 𝑡𝑓𝑚𝑔 ∶ 𝐽 𝑗 ⊆ 𝐸 ∪ 𝑡𝑓𝑚𝑔 ∀ 𝑗 ∈ 𝐸 ∪ 𝑡𝑓𝑚𝑔 ∶ 𝑗 ∉ 𝐽 𝑗 𝑎𝑇 = 𝑎 𝑗,𝑘 𝑗 ∈ 𝐸 ∪ 𝑡𝑓𝑚𝑔 , 𝑘 ∈ 𝐽 𝑗 𝑎 𝑡𝑓𝑚𝑔,𝑘 ∶ 𝑌 𝑡𝑓𝑚𝑔 → 𝑌 𝑘 , ∀ 𝑘 ∈ 𝐸 𝑎 𝑗,𝑡𝑓𝑚𝑔 ∶ 𝑍 𝑗 → 𝑍 𝑡𝑓𝑚𝑔 , ∀ 𝑗 ∈ 𝐸 𝑎 𝑗,𝑘 ∶ 𝑍 𝑗 → 𝑌 𝑘 , ∀ 𝑗, 𝑘 ∈ 𝐸 𝑌 𝑚𝑗𝑕ℎ𝑢1 = {𝑢𝑝𝑁𝑏𝑜𝑣𝑏𝑚, 𝑢𝑝𝐵𝑣𝑢𝑝}

𝒕𝒇𝒎𝒈 𝐷 = 𝑌 𝑡𝑓𝑚𝑔 , 𝑍 𝑡𝑓𝑚𝑔 , 𝐸, 𝑁𝑇, 𝐽𝑇, 𝑎𝑇 𝑵 𝒎𝒋𝒉𝒊𝒖𝟐 𝑵 𝒒𝒑𝒎𝟐 𝑁𝑇 = 𝑁 𝑗 𝑗 ∈ 𝐸 𝑁 𝑗 = 𝑌 𝑗 , 𝑍 𝑗 , 𝑇 𝑗 , 𝑟 𝑗𝑜𝑗𝑢,𝑗 , 𝜀 𝑗𝑜𝑢,𝑗 , 𝜀 𝑓𝑦𝑢,𝑗 , 𝜇 𝑗 , 𝑢𝑏 𝑗 , ∀ 𝑗 ∈ 𝐸 𝐽𝑇 = 𝐽 𝑗 𝑗 ∈ 𝐸 ∪ 𝑡𝑓𝑚𝑔 ∀ 𝑗 ∈ 𝐸 ∪ 𝑡𝑓𝑚𝑔 ∶ 𝐽 𝑗 ⊆ 𝐸 ∪ 𝑡𝑓𝑚𝑔 ∀ 𝑗 ∈ 𝐸 ∪ 𝑡𝑓𝑚𝑔 ∶ 𝑗 ∉ 𝐽 𝑗 𝑎𝑇 = 𝑎 𝑗,𝑘 𝑗 ∈ 𝐸 ∪ 𝑡𝑓𝑚𝑔 , 𝑘 ∈ 𝐽 𝑗 𝑎 𝑡𝑓𝑚𝑔,𝑘 ∶ 𝑌 𝑡𝑓𝑚𝑔 → 𝑌 𝑘 , ∀ 𝑘 ∈ 𝐸 𝑎 𝑗,𝑡𝑓𝑚𝑔 ∶ 𝑍 𝑗 → 𝑍 𝑡𝑓𝑚𝑔 , ∀ 𝑗 ∈ 𝐸 𝑎 𝑗,𝑘 ∶ 𝑍 𝑗 → 𝑌 𝑘 , ∀ 𝑗, 𝑘 ∈ 𝐸 take_break  toAuto go_to_work  toManual

𝒕𝒇𝒎𝒈 𝐷 = 𝑌 𝑡𝑓𝑚𝑔 , 𝑍 𝑡𝑓𝑚𝑔 , 𝐸, 𝑁𝑇, 𝐽𝑇, 𝑎𝑇, 𝑡𝑓𝑚𝑓𝑑𝑢 𝑵 𝒎𝒋𝒉𝒊𝒖𝟐 𝑵 𝒒𝒑𝒎𝟐 𝑁𝑇 = 𝑁 𝑗 𝑗 ∈ 𝐸 𝑁 𝑗 = 𝑌 𝑗 , 𝑍 𝑗 , 𝑇 𝑗 , 𝑟 𝑗𝑜𝑗𝑢,𝑗 , 𝜀 𝑗𝑜𝑢,𝑗 , 𝜀 𝑓𝑦𝑢,𝑗 , 𝜇 𝑗 , 𝑢𝑏 𝑗 , ∀ 𝑗 ∈ 𝐸 𝐽𝑇 = 𝐽 𝑗 𝑗 ∈ 𝐸 ∪ 𝑡𝑓𝑚𝑔 ∀ 𝑗 ∈ 𝐸 ∪ 𝑡𝑓𝑚𝑔 ∶ 𝐽 𝑗 ⊆ 𝐸 ∪ 𝑡𝑓𝑚𝑔 ∀ 𝑗 ∈ 𝐸 ∪ 𝑡𝑓𝑚𝑔 ∶ 𝑗 ∉ 𝐽 𝑗 𝑎𝑇 = 𝑎 𝑗,𝑘 𝑗 ∈ 𝐸 ∪ 𝑡𝑓𝑚𝑔 , 𝑘 ∈ 𝐽 𝑗 𝑎 𝑡𝑓𝑚𝑔,𝑘 ∶ 𝑌 𝑡𝑓𝑚𝑔 → 𝑌 𝑘 , ∀ 𝑘 ∈ 𝐸 𝑎 𝑗,𝑡𝑓𝑚𝑔 ∶ 𝑍 𝑗 → 𝑍 𝑡𝑓𝑚𝑔 , ∀ 𝑗 ∈ 𝐸 𝑎 𝑗,𝑘 ∶ 𝑍 𝑗 → 𝑌 𝑘 , ∀ 𝑗, 𝑘 ∈ 𝐸 𝑡𝑓𝑚𝑓𝑑𝑢 ∶ 2 𝐸 → 𝐸 ∀ 𝐹 ⊆ 𝐸, 𝐹 ≠ ∅: 𝑡𝑓𝑚𝑓𝑑𝑢 𝐹 ∈ 𝐹 take_break  toAuto go_to_work  toManual

Concrete Syntax trafficlight_system.py from pypdevs.DEVS import * from trafficlight import TrafficLight from policeman import Policeman def translate(in_evt): mapping = {“ take_break ”: “ toAuto ”, “ go_to_work ”: “ toManual ”} return mapping[in_evt] class TrafficLightSystem(CoupledDEVS): def __init__(self): CoupledDEVS.__init __(self, “system”) self.light = self.addSubModel(TrafficLight()) self.police = self.addSubModel(Policeman()) self.connectPorts(self.police.out, self.light.interrupt, translate) def select(self, immlist): if self.police in immlist: return self.police else: return self.light

__ Current Time: 0.00 __________________________________________ INITIAL CONDITIONS in model <system.light> Initial State: green Next scheduled internal transition at time 57.00 INITIAL CONDITIONS in model <system.policeman> Initial State: idle Next scheduled internal transition at time 20.00

__ Current Time: 20.00 __________________________________________ EXTERNAL TRANSITION in model <system.light> Input Port Configuration: port <interrupt>: toManual New State: going_manual Next scheduled internal transition at time 20.0 INTERNAL TRANSITION in model <system.policeman> New State: working Output Port Configuration: port <output>: go_to_work Next scheduled internal transition at time 3620.00

__ Current Time: 20.00 __________________________________________ INTERNAL TRANSITION in model <system.light> Output Port Configuration: port <observer>: turn_off New State: manual Next scheduled internal transition at time inf

__ Current Time: 3620.00 __________________________________________ EXTERNAL TRANSITION in model <system.light> Input Port Configuration: port <interrupt>: toAuto New State: going_auto Next scheduled internal transition at time 3620.00 INTERNAL TRANSITION in model <system.policeman> New State: idle Output Port Configuration: port <output>: take_break Next scheduled internal transition at time 3920.00

__ Current Time: 3620.00 __________________________________________ INTERNAL TRANSITION in model <system.light> Output Port Configuration: port <observer>: show_red New State: red Next scheduled internal transition at time 3680.00

__ Current Time: 3620.00 __________________________________________ EXTERNAL TRANSITION in model <system.light> Input Port Configuration: port <interrupt>: toAuto New State: going_auto Next scheduled internal transition at time 3620.00 INTERNAL TRANSITION in model <system.policeman> New State: idle Output Port Configuration: port <output>: take_break Next scheduled internal transition at time 3920.00

__ Current Time: 3920.00 __________________________________________ CONFLICT between models: <system.light> * <system.policeman> EXTERNAL TRANSITION in model <system.light> Input Port Configuration: port <interrupt>: toManual New State: going_manual Next scheduled internal transition at time 3920.00 INTERNAL TRANSITION in model <system.policeman> New State: work Output Port Configuration: port <output>: go_to_work Next scheduled internal transition at time 7520.00

Closure under Coupling

𝒕𝒇𝒎𝒈 𝑵 𝒎𝒋𝒉𝒊𝒖𝟐 𝑵 𝒒𝒑𝒎𝟐 take_break  toAuto go_to_work  toManual ?

? flatten 𝐷𝑁 = 𝑌 𝑡𝑓𝑚𝑔 , 𝑍 𝑡𝑓𝑚𝑔 , 𝐸, 𝑁𝑇, 𝐽𝑇, 𝑎𝑇 𝑔𝑚𝑏𝑢𝑢𝑓𝑜 𝐷𝑁 = 𝑌, 𝑍, 𝑇, 𝑟 𝑗𝑜𝑗𝑢 , 𝜀 𝑗𝑜𝑢 , 𝜀 𝑓𝑦𝑢 , 𝜇, 𝑢𝑏

CM 1 𝑌 𝐷𝑁 2 𝑇 1 𝑔𝑚𝑏𝑢𝑢𝑓𝑜 𝐷𝑁 = 𝑌, 𝑍, 𝑇, 𝑟 𝑗𝑜𝑗𝑢 , 𝜀 𝑗𝑜𝑢 , 𝜀 𝑓𝑦𝑢 , 𝜇, 𝑢𝑏 𝑇 2 𝑍 𝐷𝑁

CM 1 𝑌 𝐷𝑁 2 𝑇 1 𝑔𝑚𝑏𝑢𝑢𝑓𝑜 𝐷𝑁 = 𝑌, 𝑍, 𝑇, 𝑟 𝑗𝑜𝑗𝑢 , 𝜀 𝑗𝑜𝑢 , 𝜀 𝑓𝑦𝑢 , 𝜇, 𝑢𝑏 𝑇 2 𝑌 = 𝑌 𝐷𝑁 𝑍 = 𝑍 𝐷𝑁 𝑍 𝐷𝑁

CM 1 𝑌 𝐷𝑁 2 𝑓 1 𝑇 1 𝑡 1 , 𝑓 1 𝑔𝑚𝑏𝑢𝑢𝑓𝑜 𝐷𝑁 = 𝑌, 𝑍, 𝑇, 𝑟 𝑗𝑜𝑗𝑢 , 𝜀 𝑗𝑜𝑢 , 𝜀 𝑓𝑦𝑢 , 𝜇, 𝑢𝑏 𝑇 2 𝑓 2 𝑡 2 , 𝑓 2 𝑍 𝐷𝑁

CM 1 𝑌 𝐷𝑁 2 𝑓 1 𝑇 1 𝑡 1 , 𝑓 1 𝑔𝑚𝑏𝑢𝑢𝑓𝑜 𝐷𝑁 = 𝑌, 𝑍, 𝑇, 𝑟 𝑗𝑜𝑗𝑢 , 𝜀 𝑗𝑜𝑢 , 𝜀 𝑓𝑦𝑢 , 𝜇, 𝑢𝑏 𝑇 2 𝑓 2 𝑇 = × 𝑗∈𝐸 𝑅 𝑗 𝑡 2 , 𝑓 2 𝑍 𝐷𝑁

CM 1 𝑌 𝐷𝑁 2 𝑓 1 𝑡 1 , 𝑓 1 = 𝑟 𝑗𝑜𝑗𝑢,1 𝑇 1 𝑢𝑏 1 𝑡 1 𝑔𝑚𝑏𝑢𝑢𝑓𝑜 𝐷𝑁 = 𝑌, 𝑍, 𝑇, 𝑟 𝑗𝑜𝑗𝑢 , 𝜀 𝑗𝑜𝑢 , 𝜀 𝑓𝑦𝑢 , 𝜇, 𝑢𝑏 𝑇 2 𝑓 2 𝑡 2 , 𝑓 2 = 𝑟 𝑗𝑜𝑗𝑢,2 𝑢𝑏 2 𝑡 2 𝑍 𝐷𝑁

CM 1 𝑌 𝐷𝑁 2 𝑓 1 𝑡 1 , 𝑓 1 = 𝑟 𝑗𝑜𝑗𝑢,1 𝑇 1 𝑢𝑏 1 𝑡 1 𝑔𝑚𝑏𝑢𝑢𝑓𝑜 𝐷𝑁 = 𝑌, 𝑍, 𝑇, 𝑟 𝑗𝑜𝑗𝑢 , 𝜀 𝑗𝑜𝑢 , 𝜀 𝑓𝑦𝑢 , 𝜇, 𝑢𝑏 𝑇 2 𝑓 2 𝑟 𝑗𝑜𝑗𝑢 = 𝑡 𝑗𝑜𝑗𝑢 , 𝑓 𝑗𝑜𝑗𝑢 𝑡 2 , 𝑓 2 = 𝑟 𝑗𝑜𝑗𝑢,2 𝑢𝑏 2 𝑡 2 𝑍 𝐷𝑁

CM 1 𝑌 𝐷𝑁 2 𝑓 1 𝑡 1 , 𝑓 1 = 𝑟 𝑗𝑜𝑗𝑢,1 𝑇 1 𝑢𝑏 1 𝑡 1 𝑔𝑚𝑏𝑢𝑢𝑓𝑜 𝐷𝑁 = 𝑌, 𝑍, 𝑇, 𝑟 𝑗𝑜𝑗𝑢 , 𝜀 𝑗𝑜𝑢 , 𝜀 𝑓𝑦𝑢 , 𝜇, 𝑢𝑏 𝑇 2 𝑓 2 𝑟 𝑗𝑜𝑗𝑢 = 𝑡 𝑗𝑜𝑗𝑢 , 𝑓 𝑗𝑜𝑗𝑢 𝑡 2 , 𝑓 2 = 𝑟 𝑗𝑜𝑗𝑢,2 𝑡 𝑗𝑜𝑗𝑢 = … , 𝑡 𝑗𝑜𝑗𝑢,𝑗 , 𝑓 𝑗𝑜𝑗𝑢,𝑗 − 𝑓 𝑗𝑜𝑗𝑢 , … 𝑢𝑏 2 𝑡 2 𝑍 𝐷𝑁

CM 1 𝑌 𝐷𝑁 2 𝑓 1 𝑡 1 , 𝑓 1 = 𝑟 𝑗𝑜𝑗𝑢,1 𝑇 1 𝑢𝑏 1 𝑡 1 𝑔𝑚𝑏𝑢𝑢𝑓𝑜 𝐷𝑁 = 𝑌, 𝑍, 𝑇, 𝑟 𝑗𝑜𝑗𝑢 , 𝜀 𝑗𝑜𝑢 , 𝜀 𝑓𝑦𝑢 , 𝜇, 𝑢𝑏 𝑇 2 𝑓 2 𝑟 𝑗𝑜𝑗𝑢 = 𝑡 𝑗𝑜𝑗𝑢 , 𝑓 𝑗𝑜𝑗𝑢 𝑡 2 , 𝑓 2 = 𝑟 𝑗𝑜𝑗𝑢,2 𝑡 𝑗𝑜𝑗𝑢 = … , 𝑡 𝑗𝑜𝑗𝑢,𝑗 , 𝑓 𝑗𝑜𝑗𝑢,𝑗 − 𝑓 𝑗𝑜𝑗𝑢 , … 𝑢𝑏 2 𝑡 2 𝑓 𝑗𝑜𝑗𝑢 = min 𝑗∈𝐸 𝑓 𝑗𝑜𝑗𝑢,𝑗 𝑍 𝐷𝑁

CM 1 𝑌 𝐷𝑁 2 𝑓 1 𝑡 1 , 𝑓 1 = 𝑟 𝑗𝑜𝑗𝑢,1 𝑇 1 𝑢𝑏 1 𝑡 1 𝑔𝑚𝑏𝑢𝑢𝑓𝑜 𝐷𝑁 = 𝑌, 𝑍, 𝑇, 𝑟 𝑗𝑜𝑗𝑢 , 𝜀 𝑗𝑜𝑢 , 𝜀 𝑓𝑦𝑢 , 𝜇, 𝑢𝑏 𝑇 2 𝑓 2 𝑟 𝑗𝑜𝑗𝑢 = 𝑡 𝑗𝑜𝑗𝑢 , 𝑓 𝑗𝑜𝑗𝑢 𝑡 2 , 𝑓 2 = 𝑟 𝑗𝑜𝑗𝑢,2 𝑡 𝑗𝑜𝑗𝑢 = … , 𝑡 𝑗𝑜𝑗𝑢,𝑗 , 𝑓 𝑗𝑜𝑗𝑢,𝑗 − 𝑓 𝑗𝑜𝑗𝑢 , … 𝑢𝑏 2 𝑡 2 𝑓 𝑗𝑜𝑗𝑢 = min 𝑗∈𝐸 𝑓 𝑗𝑜𝑗𝑢,𝑗 𝑡 𝑗𝑜𝑗𝑢,𝑗 , 𝑓 𝑗𝑜𝑗𝑢,𝑗 = 𝑟 𝑗𝑜𝑗𝑢,𝑗 𝑍 𝐷𝑁

CM 1 𝑌 𝐷𝑁 2 𝑇 1 𝑢𝑏 1 𝑓 1 𝑔𝑚𝑏𝑢𝑢𝑓𝑜 𝐷𝑁 = 𝑌, 𝑍, 𝑇, 𝑟 𝑗𝑜𝑗𝑢 , 𝜀 𝑗𝑜𝑢 , 𝜀 𝑓𝑦𝑢 , 𝜇, 𝑢𝑏 𝑇 2 𝑢𝑏 2 𝑍 𝐷𝑁 𝑢

CM 1 𝑌 𝐷𝑁 2 𝑇 1 𝑢𝑏 1 𝑓 1 𝜏 1 𝑔𝑚𝑏𝑢𝑢𝑓𝑜 𝐷𝑁 = 𝑌, 𝑍, 𝑇, 𝑟 𝑗𝑜𝑗𝑢 , 𝜀 𝑗𝑜𝑢 , 𝜀 𝑓𝑦𝑢 , 𝜇, 𝑢𝑏 𝑇 2 + 𝑢𝑏 ∶ 𝑇 → ℝ 0,+∞ 𝑢𝑏 2 𝑢𝑏 𝑡 = min 𝑗∈𝐸 𝜏 𝑗 = 𝑢𝑏 𝑗 𝑡 𝑗 − 𝑓 𝑗 𝜏 2 𝑍 𝐷𝑁 𝑢

CM 1 𝑌 𝐷𝑁 2 𝑇 1 𝑢𝑏 1 𝑓 1 𝜏 1 𝑔𝑚𝑏𝑢𝑢𝑓𝑜 𝐷𝑁 = 𝑌, 𝑍, 𝑇, 𝑟 𝑗𝑜𝑗𝑢 , 𝜀 𝑗𝑜𝑢 , 𝜀 𝑓𝑦𝑢 , 𝜇, 𝑢𝑏 𝑇 2 + 𝑢𝑏 ∶ 𝑇 → ℝ 0,+∞ 𝑢𝑏 2 𝑢𝑏 𝑡 = min 𝑗∈𝐸 𝜏 𝑗 = 𝑢𝑏 𝑗 𝑡 𝑗 − 𝑓 𝑗 𝜏 2 𝐽𝑁𝑁 𝑡 = 𝑗 ∈ 𝐸 𝜏 𝑗 = 𝑢𝑏(𝑡) 𝑡𝑓𝑚𝑓𝑑𝑢(𝐽𝑁𝑁 𝑡 ) = 𝑗 ∗ 𝑍 𝐷𝑁 𝑢

CM 1 𝑌 𝐷𝑁 2 𝑇 1 𝑔𝑚𝑏𝑢𝑢𝑓𝑜 𝐷𝑁 = 𝑌, 𝑍, 𝑇, 𝑟 𝑗𝑜𝑗𝑢 , 𝜀 𝑗𝑜𝑢 , 𝜀 𝑓𝑦𝑢 , 𝜇, 𝑢𝑏 𝑇 2 𝑍 𝐷𝑁 𝑢

CM 1 𝑌 𝐷𝑁 2 𝑇 1 𝑧 2 𝑔𝑚𝑏𝑢𝑢𝑓𝑜 𝐷𝑁 = 𝑌, 𝑍, 𝑇, 𝑟 𝑗𝑜𝑗𝑢 , 𝜀 𝑗𝑜𝑢 , 𝜀 𝑓𝑦𝑢 , 𝜇, 𝑢𝑏 𝑇 2 𝜇 𝑡 = ൝ 𝑎 𝑗 ∗ ,𝑡𝑓𝑚𝑔 𝜇 𝑗 ∗ 𝑡 𝑗 ∗ 𝑗𝑔 𝑡𝑓𝑚𝑔 ∈ 𝐽 𝑗 ∗ 𝑗𝑔 𝑡𝑓𝑚𝑔 ∉ 𝐽 𝑗 ∗ ∅ 𝑍 𝐷𝑁 𝑢

CM 1 𝑌 𝐷𝑁 𝑢𝑏 𝑡 2 𝑇 1 𝑡 1 , 𝑢𝑏 1 𝑡 1 𝑡 1 , 0 𝑔𝑚𝑏𝑢𝑢𝑓𝑜 𝐷𝑁 = 𝑌, 𝑍, 𝑇, 𝑟 𝑗𝑜𝑗𝑢 , 𝜀 𝑗𝑜𝑢 , 𝜀 𝑓𝑦𝑢 , 𝜇, 𝑢𝑏 𝑇 2 ′ , 0 𝑡 2 𝑡 2 , 𝑓 2 𝑡 2 , 𝑓 2 + 𝑢𝑏 𝑡 = 𝑢𝑏 2 (𝑡 2 ) 𝑍 𝐷𝑁 𝑢

CM 1 𝑌 𝐷𝑁 𝑢𝑏 𝑡 2 𝑇 1 𝑡 1 , 𝑢𝑏 1 𝑡 1 𝑡 1 , 0 𝑔𝑚𝑏𝑢𝑢𝑓𝑜 𝐷𝑁 = 𝑌, 𝑍, 𝑇, 𝑟 𝑗𝑜𝑗𝑢 , 𝜀 𝑗𝑜𝑢 , 𝜀 𝑓𝑦𝑢 , 𝜇, 𝑢𝑏 ′ , … 𝑇 2 ′ , 𝑓 𝜀 𝑗𝑜𝑢 𝑡 = … , 𝑡 ′ , 0 𝑘 𝑘 𝑡 2 𝑔𝑝𝑠 𝑘 = 𝑗 ∗ 𝜀 𝑗𝑜𝑢,𝑘 𝑡 𝑘 , 0 𝑡 2 , 𝑓 2 ′ = 𝑔𝑝𝑠 𝑘 ∈ 𝐽 𝑗 ∗ ∖ 𝑡𝑓𝑚𝑔 ? ′ , 𝑓 𝑡 2 , 𝑓 2 + 𝑢𝑏 𝑡 = 𝑢𝑏 2 (𝑡 2 ) 𝑡 𝑘 𝑘 𝑡 𝑘 , 𝑓 𝑘 + 𝑢𝑏 𝑡 𝑓𝑚𝑡𝑓 𝑍 𝐷𝑁 𝑢

𝑦 2 CM 1 𝑌 𝐷𝑁 2 𝑧 1 𝑇 1 ′ , 0 𝑡 1 𝜀 𝑗𝑜𝑢 𝑡 1 , 𝑓 1 𝑡 1 , 𝑓 1 + 𝑢𝑏 𝑡 = 𝑢𝑏 1 (𝑡 1 ) 𝑔𝑚𝑏𝑢𝑢𝑓𝑜 𝐷𝑁 = 𝑌, 𝑍, 𝑇, 𝑟 𝑗𝑜𝑗𝑢 , 𝜀 𝑗𝑜𝑢 , 𝜀 𝑓𝑦𝑢 , 𝜇, 𝑢𝑏 ′ , … 𝑇 2 ′ , 𝑓 𝜀 𝑗𝑜𝑢 𝑡 = … , 𝑡 ′ , 0 𝑡 2 𝑘 𝑘 𝑔𝑝𝑠 𝑘 = 𝑗 ∗ 𝜀 𝑗𝑜𝑢,𝑘 𝑡 𝑘 , 0 𝜀 𝑓𝑦𝑢 𝑡 2 , 𝑓 2 𝑡 2 , 𝑓 2 + 𝑢𝑏 𝑡 ≤ 𝑢𝑏 2 𝑡 2 ′ = , 𝑎 𝑗 ∗ ,𝑘 𝜇 𝑗 ∗ 𝑡 𝑗 ∗ 𝑔𝑝𝑠 𝑘 ∈ 𝐽 𝑗 ∗ ∖ 𝑡𝑓𝑚𝑔 𝜀 𝑓𝑦𝑢,𝑘 𝑡 𝑘 , 𝑓 𝑘 + 𝑢𝑏 𝑡 , 0 ′ , 𝑓 𝑡 𝑘 𝑘 𝑡 𝑘 , 𝑓 𝑘 + 𝑢𝑏 𝑡 𝑓𝑚𝑡𝑓 𝑍 𝐷𝑁 𝑢

CM 𝑓 1 𝑌 𝐷𝑁 2 𝑇 1 𝑓 1 𝑔𝑚𝑏𝑢𝑢𝑓𝑜 𝐷𝑁 = 𝑌, 𝑍, 𝑇, 𝑟 𝑗𝑜𝑗𝑢 , 𝜀 𝑗𝑜𝑢 , 𝜀 𝑓𝑦𝑢 , 𝜇, 𝑢𝑏 𝑇 2 𝑍 𝐷𝑁 𝑢

CM 𝑓 1 𝑌 𝐷𝑁 2 𝑇 1 𝑓 1 𝑧 𝑡𝑓𝑚𝑔 𝑔𝑚𝑏𝑢𝑢𝑓𝑜 𝐷𝑁 = 𝑌, 𝑍, 𝑇, 𝑟 𝑗𝑜𝑗𝑢 , 𝜀 𝑗𝑜𝑢 , 𝜀 𝑓𝑦𝑢 , 𝜇, 𝑢𝑏 ′ , … 𝑇 2 ′ , 𝑓 𝑗 𝜀 𝑓𝑦𝑢 𝑡, 𝑓 , 𝑦 = … , 𝑡 𝑗 ′ = ቐ 𝜀 𝑓𝑦𝑢,𝑗 𝑡 𝑗 , 𝑓 𝑗 + 𝑓 , 𝑎 𝑡𝑓𝑚𝑔,𝑗 𝑦 , 0 𝑔𝑝𝑠 𝑗 ∈ 𝐽 𝑡𝑓𝑚𝑔 ′ , 𝑓 𝑗 𝑡 𝑗 𝑡 𝑗 , 𝑓 𝑗 + 𝑓 𝑓𝑚𝑡𝑓 𝑍 𝐷𝑁 𝑢

𝑔𝑚𝑏𝑢𝑢𝑓𝑜 𝐷𝑁 = 𝑌, 𝑍, 𝑇, 𝑟 𝑗𝑜𝑗𝑢 , 𝜀 𝑗𝑜𝑢 , 𝜀 𝑓𝑦𝑢 , 𝜇, 𝑢𝑏 𝑌 = 𝑌 𝐷𝑁 𝑍 = 𝑍 𝐷𝑁 𝑇 = × 𝑗∈𝐸 𝑅 𝑗 𝑅 = 𝑡, 𝑓 𝑡 ∈ 𝑇, 0 ≤ 𝑓 ≤ 𝑢𝑏(𝑡) 𝑟 𝑗𝑜𝑗𝑢 = 𝑡 𝑗𝑜𝑗𝑢 , 𝑓 𝑗𝑜𝑗𝑢 ∈ 𝑅 𝑡 𝑗𝑜𝑗𝑢 = (… , 𝑡 𝑗𝑜𝑗𝑢,𝑗 , 𝑓 𝑗𝑜𝑗𝑢,𝑗 − 𝑓 𝑗𝑜𝑗𝑢 , … ) 𝑓 𝑗𝑜𝑗𝑢 = min 𝑗∈𝐸 𝑓 𝑗𝑜𝑗𝑢,𝑗 𝑡 𝑗𝑜𝑗𝑢,𝑗 , 𝑓 𝑗𝑜𝑗𝑢,𝑗 = 𝑟 𝑗𝑜𝑗𝑢,𝑗 ′ , … ′ , 𝑓 𝜀 𝑗𝑜𝑢 𝑡 = … , 𝑡 𝑘 𝑘 𝑔𝑝𝑠 𝑘 = 𝑗 ∗ 𝜀 𝑗𝑜𝑢,𝑘 𝑡 𝑘 , 0 ′ = , 𝑎 𝑗 ∗ ,𝑘 𝜇 𝑗 ∗ 𝑡 𝑗 ∗ ′ , 𝑓 𝜀 𝑓𝑦𝑢,𝑘 𝑡 𝑘 , 𝑓 𝑘 + 𝑢𝑏 𝑡 , 0 𝑔𝑝𝑠 𝑘 ∈ 𝐽 𝑗 ∗ 𝑡 𝑘 𝑘 𝑡 𝑘 , 𝑓 𝑘 + 𝑢𝑏 𝑡 𝑓𝑚𝑡𝑓 ′ , … ′ , 𝑓 𝑗 𝜀 𝑓𝑦𝑢 𝑡, 𝑓 , 𝑦 = … , 𝑡 𝑗 ′ = ቐ 𝜀 𝑓𝑦𝑢,𝑗 𝑡 𝑗 , 𝑓 𝑗 + 𝑓 , 𝑎 𝑡𝑓𝑚𝑔,𝑗 𝑦 , 0 𝑔𝑝𝑠 𝑗 ∈ 𝐽 𝑡𝑓𝑚𝑔 ′ , 𝑓 𝑗 𝑡 𝑗 𝑡 𝑗 , 𝑓 𝑗 + 𝑓 𝑓𝑚𝑡𝑓 𝜇 𝑡 = ൝𝑎 𝑗 ∗ ,𝑡𝑓𝑚𝑔 𝜇 𝑗 ∗ 𝑡 𝑗 ∗ 𝑗𝑔 𝑡𝑓𝑚𝑔 ∈ 𝐽 𝑗 ∗ 𝑗𝑔 𝑡𝑓𝑚𝑔 ∉ 𝐽 𝑗 ∗ 𝜚 𝑗 ∗ = 𝑡𝑓𝑚𝑓𝑑𝑢(𝐽𝑁𝑁 𝑡 ) 𝐽𝑁𝑁 𝑡 = 𝑗 ∈ 𝐸 𝜏 𝑗 = 𝑢𝑏(𝑡) 𝑢𝑏 𝑡 = min 𝑗∈𝐸 𝜏 𝑗 = 𝑢𝑏 𝑗 𝑡 𝑗 − 𝑓 𝑗

Hierarchical Simulator

Root coordinator (i, t) (*, t) (done, t) (done, t) Coupled DEVS Coordinator (i, t) (*, t) (x, t) (y, t) (done, t) (done, t) (done, t) (done, t) Simulator Simulator Atomic DEVS Atomic DEVS

DEVS Semantics Operational Denotational Semantics Semantics Abstract Atomic DEVS [1] Simulator Hierarchical Closure under Coupled DEVS Simulator Coupling [1] Ashvin Radiya and Robert G. Sargent. A logic-based foundation of discrete event modeling and simulation. ACM Transactions on Modeling and Computer Simulation, 1(1):3-51, 1994.

Conclusions (*, t) (done, t) (i,t) (y, t) (x, t)  Atomic DEVS 𝒕𝒇𝒎𝒈 𝑵 𝒎𝒋𝒉𝒊𝒖𝟐 𝑵 𝒒𝒑𝒎𝟐  Coupled DEVS  Closure under coupling  Abstract Simulator take_break  toAuto go_to_work  toManual

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

Limitations of Classic DEVS  Parallel implementation  Parallel DEVS [1]  Select function is artificial  Parallel DEVS [1]  Dynamic Structure systems  Dynamic Structure DEVS [2] [1] A.C.-H. Chow. Parallel DEVS: A parallel, hierarchical, modular modeling formalism and its distributed simulator. Transactions of the Society for Computer Simulation International, 13(2):55-68, 1996. [2] F . Barros. The dynamic structure discrete event system specification formalism. Transactions of the Society for Computer Simulation International, 13(1):35-46, 1996.