Steady-State Simulation Steady-State Simulation Overview Reading: Ch. 9 in Law & Ch. 15 in Handbook of Simulation The Regenerative Method Regenerative processes Regenerative Simulation Peter J. Haas Delays The Batch Means Method Time-average limits CS 590M: Simulation Jackknifed Batch Means Spring Semester 2020 1 / 34 2 / 34 Steady-State Simulation Steady-State Performance Measures � � The setup for GSMP X ( t ) : t ≥ 0 with state space S Why do it? � t ◮ Output process Y ( t ) = f � � X ( t ) ◮ Quick approximation for cumulative cost C ( t ) = 0 Y ( s ) ds where f is a real-valued function ◮ Y ( s ) is output process of the simulation, e.g., Y ( t ) = f � � X ( t ) ◮ Let µ = initial distribution of GSMP where X ( t ) is system state and f is a real-valued function � t ◮ If time-average limit α = lim t →∞ (1 / t ) 0 Y ( s ) ds exists, then C ( t ) ≈ t α for large t A reminder: General notion of convergence in distribution ◮ Avoids arbitrary choice of time horizon ◮ Discrete-state case: ◮ X n ⇒ X if lim n →∞ P ( X n = s ) = P ( X = s ) for all s ∈ S ◮ Avoids arbitrary choice of initial conditions ◮ X ( t ) ⇒ X if lim t →∞ P � � X ( t ) = s = P ( X = s ) for all s ∈ S ◮ Note: E [ f ( X )] = � s ∈ S f ( s ) π ( s ), where Appropriate if � � π ( s ) = lim t →∞ P X ( t ) = s = P ( X = s ) ◮ No “natural” termination time for simulation ◮ Continuous-state case: ◮ No “natural” initial conditions ◮ Z n ⇒ Z if lim n →∞ P ( Z n ≤ x ) = P ( Z ≤ x ) ◮ Rapid convergence to (quasi-)steady state for all x where F Z is continuous � ¯ √ n (e.g., telecom w. nanosecond timescale observed every 5 min.) ◮ Ex: � X n − µ X ⇒ N (0 , 1) by CLT σ 3 / 34 4 / 34
Steady-State Performance Measures, Continued Steady-State Simulation Challenges Time-Average Limit of Y ( t ) process Autocorrelation problem � t � � ◮ For time-average limit, α such that P µ lim t →∞ (1 / t ) 0 Y ( u ) du = α = 1 for any µ � t α = lim t →∞ ¯ Y ( t ) = lim t →∞ (1 / t ) 0 Y ( u ) du ◮ Natural estimator of α is ¯ Y ( t ) for some large t Steady-State Mean of Y ( t ) process (obtained from one long observation of system) α = E [ f ( X )], where, for any µ , X ( t ) ⇒ X and E [ f ( X )] exists ◮ But Y ( t ) and Y ( t + ∆ t ) highly correlated if ∆ t is small ◮ So estimator is average of autocorrelated observations Limiting Mean of Y ( t ) process ◮ Techniques based on i.i.d. observations don’t work � � �� α = lim t →∞ E f X ( t ) for any µ Initial-Transient Problem ◮ “for any µ ” = for any member of GSMP family indexed by µ ◮ Steady-state distribution unknown, (with other building blocks the same) so initial dist’n is not typical of steady-state behavior ◮ E [ f ( X )] exists if and only if E [ | f ( X ) | ] < ∞ ◮ Autocorrelation implies that initial bias will persist ◮ If f is bounded or S is finite, then X ( t ) ⇒ X implies ◮ Very hard to detect “end of initial-transient period” � � �� lim t →∞ E X ( t ) = E [ f ( X )] (s-s mean = limiting mean) f 5 / 34 6 / 34 Estimation Methods Many alternative estimation methods ◮ Regenerative method Steady-State Simulation ◮ Batch-means method Overview ◮ Autoregressive method The Regenerative Method Regenerative processes ◮ Standardized-time-series methods Regenerative Simulation ◮ Integrated-path method Delays The Batch Means Method ◮ . . . Time-average limits Jackknifed Batch Means We will focus on: ◮ Regenerative method: clean and elegant ◮ Batch means: simple, widely used and the basis for other methods 7 / 34 8 / 34
The Regenerative Method Regenerative Processes: Formal Definition Definition: Stopping time A random variable T is a stopping time with respect to References: � � X ( t ) : t ≥ 0 if occurrence or non-occurrence of event { T ≤ t } is ◮ Shedler [Ch. 2 & 3], Haas [Ch. 5 & 6] � � completely determined by X ( u ) : 0 ≤ u ≤ t ◮ Recent developments: ACM TOMACS 25(4), 2015 Definition: Regenerative process Regenerative Processes � � The process X ( t ) : t ≥ 0 is regenerative if there exists an infinite ◮ Intuitively: � � X ( t ) : t ≥ 0 is regenerative if process � � sequence of a.s. finite stopping times T ( k ) : k ≥ 0 s.t. for k ≥ 1 “probabilistically restarts” infinitely often � X ( t ) : t ≥ T ( k ) � � X ( t ) : t ≥ T (0) � 1. is distributed as ◮ Restart times T (0) , T (1) , . . . called regeneration times or � � � � regeneration points 2. X ( t ) : t ≥ T ( k ) is independent of X ( t ) : t < T ( k ) ◮ Regeneration points are random ◮ Must be almost surely (a.s.) finite ◮ If T (0) = 0, process is non-delayed (else delayed) ◮ Ex: Arrivals to empty GI/G/1 queue ◮ Can drop stopping-time requirement, (more complicated def.) ◮ � � � � � � X ( t ) : t ≥ 0 regen. ⇒ f X ( t ) : t ≥ 0 regen. ◮ Analogous definition for discrete-time processes 9 / 34 10 / 34 Regenerative Processes: Examples Regenerative GSMPs Ex 3: GSMP with a single state ◮ ¯ s ∈ S is a single state if E (¯ s ) = { ¯ e } for some ¯ e ∈ E Ex 1: Successive times that CTMC hits a fixed state x ◮ Regeneration points: successive times that ¯ e occurs in ¯ s ◮ Formally, T (0) = 0 and ◮ Observe that for each k ≥ 1, T ( k ) = min { t > T ( k − 1) : X ( t − ) � = x and X ( t ) = x } ◮ New state s ′ at T ( k ) distributed according to p ( · ; ¯ s , ¯ e ) ◮ Observe that X � � T ( k ) = x for all k ◮ No old clocks ◮ The two regenerative criteria follow from Markov property ◮ Clock for new event e ′ distributed as F ( · ; s ′ , e ′ , ¯ s , ¯ e ) ◮ Regenerative property follows from Markov property for Ex 2: Successive times that CTMC leaves a fixed state x � ( S n , C n ) : n ≥ 0 � ◮ X � � T ( k ) distributed according to P ( x , · ) for each k ◮ Second criterion follows from Markov property Ex 4: GI/G/1 queue ◮ X ( t ) = number of jobs in system at time t Q: Is a semi-Markov process regenerative? ◮ � � X ( t ) : t ≥ 0 is a GSMP ◮ T ( k ) = time of k th arrival to empty system (why?) 11 / 34 12 / 34
Regenerative GSMPs, Continued Ex 5: Cancellation ◮ Suppose there exist ¯ s ′ , ¯ s ∈ S and ¯ e ∈ E (¯ s ) with s ′ ; ¯ s ′ ; ¯ e ) = ∅ p (¯ s , ¯ e ) r (¯ s , ¯ e ) > 0 such that O (¯ s , ¯ Steady-State Simulation ◮ T ( k ) = k th time that ¯ e occurs in ¯ s and new state is ¯ s ′ Overview The Regenerative Method Ex 6: Exponential clocks Regenerative processes Regenerative Simulation ◮ Suppose that Delays ◮ There exists ˜ E ⊆ E such that each e ∈ ˜ E is a simple event The Batch Means Method with F ( x ; e ) = 1 − e − λ ( e ) x Time-average limits e } ⊆ ˜ ◮ There exists ¯ s ∈ S and ¯ e ∈ E (¯ s ) − { ¯ s ) s.t. E (¯ E Jackknifed Batch Means ◮ T ( k ) = k th time that ¯ e occurs in ¯ s (memoryless property) Other (fancier) regeneration point constructions are possible ◮ E.g., if clock-setting distn’s have heavier-than-exponential � � tails or bounded hazard rates h ( t ) = f ( t ) / 1 − F ( t ) 13 / 34 14 / 34 Regenerative Simulation: Cycles Regenerative Simulation: Time-Average Limits � t ◮ Recall: ¯ Y ( t ) = (1 / t ) 0 Y ( u ) du Theorem Suppose that E [ | Y 1 | ] < ∞ and E [ τ 1 ] < ∞ . Then lim t →∞ ¯ Y ( t ) = α a.s., where α = E [ Y ] / E [ τ ]. ◮ So estimating time-average limit reduces to a ratio-estimation Regeneration points decompose process into i.i.d. cycles problem (can use delta method, jackknife, bootstrap) ◮ k th cycle: � X ( t ) : T ( k − 1) ≤ t < T ( k ) � (Most of) Proof ◮ Length of k th cycle: τ k = T ( k ) − T ( k − 1) � T ( k ) ◮ Set Y k = � T ( j ) T ( k − 1) Y ( u ) du � n � T ( n ) � n T ( j − 1) Y ( u ) du j =1 Y j 1 j =1 ¯ � � � = Y T ( n ) = Y ( u ) du = ◮ The pairs ( Y 1 , τ 1 ) , ( Y 2 , τ 2 ) , . . . are i.i.d as ( Y , τ ) � n � n � T ( n ) T ( j ) − T ( j − 1) j =1 τ j 0 j =1 ¯ ⇒ lim � � Y T ( n ) = α a.s. by SLLN Initial transient is not a problem! n →∞ 15 / 34 16 / 34
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