Discrete-Event Systems and Generalized Semi-Markov Processes - PowerPoint PPT Presentation

Discrete-Event Systems and Generalized Semi-Markov Processes Reading: Section 1.4 in Shedler or Section 4.1 in Haas Peter J. Haas CS 590M: Simulation Spring Semester 2020 1 / 27

Discrete-Event Systems and Generalized Semi-Markov Processes Discrete-Event Stochastic Systems The GSMP Model Simulating GSMPs Generating Clock Readings: Inversion Method Markovian and Semi-Markovian GSMPs 2 / 27

Discrete-Event Stochastic Systems Stochastic state transitions occur at an increasing sequence of random times X(t) t � � How to model underlying process X ( t ) : t � 0 ? I Generalized semi-Markov processes (GSMPs) I Basic model of a discrete-event system 3 / 27

GSMP Overview I Events associated with a state “compete” to trigger next state transition I Each event has own distribution for determining the next state I New events I Associated with new state but not old state, or I Associated with new state and just triggered state transition I Clock is set with time until event occurs (runs down to 0) I Old events I Associated with old and new states, did not trigger transition I Clock continues to run down I Canceled events I Associated with old state, but not new state I Clock reading is discarded I Clocks can run down at state-dependent speeds 4 / 27

Clock-Reading Plot clock : a given for :* ' " i ← cancellation . eject occurs 5 / 27

GSMP Building Blocks I S : a (finite or countably infinite) set of states I E = { e 1 , e 2 , . . . , e M } : a finite set of events I E ( s ) ✓ E : the set of active events in state s 2 S I p ( s 0 ; s , E ⇤ ): probability that new state = s 0 when events in E ⇤ simultaneously occur in s I Write p ( s 0 ; s , e ⇤ ) if E ⇤ = { e ⇤ } (unique trigger event) I r ( s , e ): the nonnegative finite speed at which clock for e runs down in state s I Typically r ( s , e ) = 1 I Set r ( s , e ) = 0 to model “preempt resume” service discipline I F ( · ; s 0 , e 0 , s , E ⇤ ): cdf of new clock-reading for e 0 after state E ⇤ ! s 0 transition s � I µ : initial distribution for state and clock readings D D I Assume initial state s ⇠ ν and clock readings ⇠ F 0 ( · ; e , s ) - 6 / 27

New and Old Events 7 / 27

Example: GI/G/1 Queue - Assume that interarrival-time dist’n F a and service-time dist’n F s are continuous (no simult. event occurrences) - Assume that at time t = 0 a job arrives to an empty system X ( t ) = # of jobs in service or waiting in queue at time t Can define ( X ( t ) : t � 0) as a GSMP: - r - I " completion of service I S = { , I , b 0 " " arrival , ee { er , ed e , " I E = , en } if s > 0 e Sei - o ; EG sell it s , e) =o otherwise - I E ( s ) = pest ; s - l ; s , edel , - I , pls , e.) - pls ti ; s - ez I p : ' and fix ) ite - faux ) if E 's e , I F ( x ; s 0 , e 0 , s , e ⇤ ) : sea , s ) : Falk ) for all , e s I fix ) to I I r ( s , e ) = , Wako Hs # f. ex ; VC D= I I Initial dist’n: 8 / 27

A More Complex Example: Patrolling Repairman See handout for details I Provides an example of how to concisely express GSMP building blocks Specifying a GSMP can be complex and time-consuming, so why do it? I Direct guidance for coding (helps catch “corner cases”) I Communicates model at high level (vs poring through code) I Theory for GSMPs can help in establishing important properties of the simulation I Stability (i.e., convergence to steady state), so that steady-state estimation problems are well defined I Validity of specific simulation output-analysis methods, so that estimates are correct 9 / 27

GSMPs and GSSMCs - fei , - u , en ) events E � � GSMP formally defined in terms of GSSMC ( S n , C n ) : n � 0 I S n = state just after n th transition I C n = ( C n , 1 , C n , 2 , . . . , C n , M ) = clock readings just after n th transition I See Haas or Shedler books for definition of P � � ( s , c ) , A and µ 10 / 27

GSMP Definition reading clock for ei I Define I Holding time: t ⇤ ( s , c ) = min { i : e i 2 E ( s ) } c i / r ( s , e i ) I n th state-transition time: ζ n = P n � 1 k =0 t ⇤ ( s , c ) I # of state transitions in [0 , t ]: N ( t ) = max { n � 0 : ζ n  t } Let ∆ 62 S and set ( S N ( t ) if N ( t ) < 1 ; X ( t ) = if N ( t ) = 1 ∆ 11 / 27

GSMP Definition in a Picture N t ( ) = 2 S 2 N t ( ) = 3 S 3 N t ( ) = 0 S 0 N t ( ) = 1 S 1 t S *( , C ) t S *( , C ) t S *( , C ) t S *( , C ) 0 0 1 1 2 2 3 3 ζ 0 ζ 1 ζ 2 ζ 3 ζ 4 t = 0 X t ( ) = 3 S = Such 12 / 27

Discrete-Event Systems and Generalized Semi-Markov Processes Discrete-Event Stochastic Systems The GSMP Model Simulating GSMPs Generating Clock Readings: Inversion Method Markovian and Semi-Markovian GSMPs 13 / 27

Sample Path Generation GSMP Simulation Algorithm (Variable Time-Advance) 1. (Initialization) Select s D ⇠ ν . For each e i 2 E ( s ) generate a D clock reading c i ⇠ F 0 ( · ; e i , s ). Set c i = 0 for e i / 2 E ( s ). 2. Determine holding time t ⇤ ( s , c ) and set of trigger events E ⇤ = E ⇤ ( s , c ) = { e i : c i / r ( s , e i ) = t ⇤ ( s , c ) } . 3. Generate next state s 0 D ⇠ p ( · ; s , E ⇤ ). D 4. For each e i 2 N ( s 0 ; s , E ⇤ ), generate c 0 ⇠ F ( · ; s 0 , e i , s , E ⇤ ). i 0 5. For each e i 2 O ( s 0 ; s , E ⇤ ), set c i = c i � t ⇤ ( s , c ) r ( s , e i ). 6. For each e i 2 ( E ( s ) � E ⇤ ) � E ( s 0 ), set c 0 i = 0 (i.e., cancel event e i ). 7. Set s = s 0 and c = c 0 , and go to Step 2. (Here c = ( c 1 , c 2 , . . . , c M ) and similarly for c 0 .) 14 / 27

Sample Path Generation, Continued Algorithm generates sequence of states ( S n : n � 0) , clock-reading vectors ( C n : n � 0) , and holding times � � t ⇤ ( S n , C n ) : n � 0 Transition times ( ζ n : n � 0) and continuous-time process tt � � X ( n ) : n � 0 computed as described previously ⇥ � �⇤ Use usual techniques to estimate quantities like E X ( t ) f or even Z t  1 � � � α = E f X ( u ) du t 0 N ( t ) � 1 " !# 1 X f ( S n ) t ⇤ ( S n , C n ) + f ( S N ( t ) ) � � = E t − ζ N ( t ) t n =0 Flow charts and diagrams can be helpful (see Law, p. 30–32 for an example) 15 / 27

Discrete-Event Systems and Generalized Semi-Markov Processes Discrete-Event Stochastic Systems The GSMP Model Simulating GSMPs Generating Clock Readings: Inversion Method Markovian and Semi-Markovian GSMPs 16 / 27

HEE ' Generating Clock Readings: Example de Exponential distribution with rate (intensity) λ cdf pdf ( ( λ e � λ x if x � 0; 1 � e � λ x if x � 0; f ( x ; λ ) = and F ( x ; λ ) = 0 if x < 0 0 if x < 0 Mean = 1 / λ Claim: If U D , then V D ⇠ Uniform(0 , 1) and V = � ln U ⇠ exp( λ ) λ - Pan using ⇒ x ) Proof: " - ein ) : e- tx - ¥ Ptv > x ) =P ( - = Pcu 17 / 27

⇒ The Inversion Method: Special Case FT 'm Spose that cdf F ( x ) = P ( V  x ) is increasing and continuous Claim: If U D ⇠ Uniform(0 , 1) and V = F � 1 ( U ), then V D ⇠ F xD Proof: - ' latex )=PfrCf" CUD - Pff pcvsx ) - Plus Fox ) ) FIX ) 18 / 27

I - EdX=U Example: Exponential Distribution - U U' II - U e- H F ( x ) = 1 � e � λ x - I - - Helmuth ) - u ) - bull - braid F � 1 ( u ) = T x . t ' ) - lulu = f inversion method case of special 19 / 27

The Inversion Method: General Case Generalized inverse F � 1 ( u ) = min { x : F ( x ) � u } F(x) u X w co o x -1 F (u) Claim still holds: F � 1 ( u )  x , u  F ( x ) by definition Exercise: Show that inversion method = naive method for discrete RVs 20 / 27

Discrete-Event Systems and Generalized Semi-Markov Processes Discrete-Event Stochastic Systems The GSMP Model Simulating GSMPs Generating Clock Readings: Inversion Method Markovian and Semi-Markovian GSMPs 21 / 27

pdf " " " E " EEE r . ' ' Ii Markovian GSMPs , ↳ xx in Properties of the Exponential Distribution If X D ∼ exp( λ ) and Y D ∼ exp( µ ) then 1. min( X , Y ) D ∼ exp( λ + µ ) [indep. of whether min = X or Y ] λ 2. P ( X < Y ) = λ + µ 3. P ( X > a + b | X > a ) = e � λ b [memoryless property] Properties 1 and 2 generalize to multiple exponential RVs Simple GSMP event e 0 F ( · ; s 0 , e 0 , s , E ⇤ ) ≡ F ( · ; e 0 ) and F 0 ( · ; e 0 ; s ) ≡ F ( · ; e 0 ) 22 / 27

Markovian GSMPs, Continued Suppose that all events in a GSMP are simple with exponential clock-setting distn’s Key observation: By memoryless property, whenever GSMP jumps into a state s , clock readings for events in E ( s ) are mutually independent and exponentially distributed Simplified Simulation Algorithm (No clock readings needed) 1. (Initialization) Select s D ∼ ν 2. Generate holding time t ⇤ D ∼ exp( λ ), where λ = λ ( s ) = P e i 2 E ( s ) λ i 3. Select e i ∈ E ( s ) as trigger event with probability λ i / λ 4. Generate the next state s 0 D ∼ p ( · ; s , e i ) 5. Set s = s 0 and go to Step 2 23 / 27