stochastic petri nets for discrete-event simulation - IBM Research

STOCHASTIC PETRI NETS FOR DISCRETE-EVENT SIMULATION Peter J. Haas IBM Almaden Research Center San Jose, CA

Petri Nets 2007

Part I Introduction

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Peter J. Haas

Petri Nets 2007

My Background I

Mid 1980’s: PhD student studying discrete-event simulation I

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“Regenerative simulation of stochastic Petri nets”

Kept working (in between Info. Mgmt. research) . . . I

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“Performance analysis using stochastic Petri nets”

Wrote PNPM85 simulation paper with Gerry Shedler I

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Under Donald Iglehart (Stanford) & Gerald Shedler (IBM)

Saw Michael Molloy 1982 paper in IEEE Trans. Comput.:

Modelling power for simulation [HS88] Prototypes: SPSIM, “Labelled” SPN simulator [JS89, HS90] Delays [HS93a,b] Standardized time series [Haa97,99a,99b] Transience and recurrence [GH06, GH07]

Gave this seminar at 2004 Winter Simulation Conference

Peter J. Haas

Petri Nets 2007

Complex Systems

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Peter J. Haas

Petri Nets 2007

Simulation and SPNs I

Assessment of system performance is difficult I I

Even modelling the system is hard! Model is usually analytically and numerically intractable I

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But can’t simulate blindly

SPNs can help I I I

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Huge state space and/or non-Markovian

Simulation is often the only available numerical method

An attractive graphically-oriented modelling framework Well suited to sample-path generation on a computer Solid mathematical foundation

Peter J. Haas

Petri Nets 2007

This Tutorial

Simulation theory for SPNs

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SPNs as a modelling framework for discrete-event systems

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Sample path generation for SPNs

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Steady-state output analysis: theory and methods

Peter J. Haas

Petri Nets 2007

Sources for This Tutorial

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“Law of large numbers and functional central limit theorem for generalized semi-Markov processes.” P. W. Glynn and P. J. Haas. Comm. Statist. Stochastic Models, 22(2), 2006, 201–232. On Transience and Recurrence in Irreducible Finite-State Stochastic Systems. P. W. Glynn and P. J. Haas. IBM Technical Report, 2007.

Peter J. Haas

Petri Nets 2007

Outline I

Simulation basics I I

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Modelling with SPNs I I

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The marking process Efficiency issues, parallelism

Steady-state estimation for SPNs I I

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Building blocks Modeling power for simulation

Sample-path generation I

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Discrete-event systems The simulation process

Conditions for long-run stability (recurrence, limit theorems) Output-analysis methods and their validity

Peter J. Haas

Petri Nets 2007

Goals

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Illustrate the rich behavior of non-Markvian SPNs

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Introduce you to some basic simulation methodology

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Explore foundational issues in modelling and analysis

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Connect modeling practice and simulation theory

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Stimulate your interest in SPNs as a simulation framework

Peter J. Haas

Petri Nets 2007

Part II Simulation Basics

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Peter J. Haas

Petri Nets 2007

What We Simulate: Discrete-Event Stochastic Systems

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System changes state when events occur I

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Underlying stochastic process { X (t) : t ≥ 0 } I I I

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X (t) = state of system at time t (a random variable) Piecewise-constant sample paths Typically not a continuous-time Markov chain

Modelling challenge: defining appropriate system state I I I

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Stochastic changes at random times

Compact for efficiency reasons Enough info to compute performance measures Enough info to determine evolution

Peter J. Haas

Petri Nets 2007

Why We Simulate: Performance Evaluation I

Steady-state performance measures I

Time-average limits: α = lim

t→∞

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1 t

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f X (u) du

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Steady-state means: α = E [f (X )], where X (t) ⇒ X I

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Want point estimate α ˆ (t) I I

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Unbiased: Eµ [ˆ α(t)] = α Strongly consistent: Pµ { limt→∞ α ˆ (t) = α } = 1

Want asymptotic 100p% confidence interval I

I (t) = [α ˆ (t) − H(t), α ˆ (t) + H(t)]

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Pµ { I (t) 3 α } ≈ p for large t CI width indicates precision of point estimate

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I.e., Pµ { X (t) = s } → P { X = s } as t → ∞

Peter J. Haas

Petri Nets 2007

Challenges in Performance Evaluation I

Is steady-state quantity α well-defined? I

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Is steady-state quantity independent of startup condition µ? I

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Ex: reducible Markov chain

Statistical challenges I I

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Ex: steady-state number in M/M/1 queue with ρ > 1

Autocorrelation problem Initialization bias problem

How to handle Delays? n−1 1X f (Dj ) n→∞ n

lim

j=0

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Peter J. Haas

Petri Nets 2007

The Simulation Process

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Peter J. Haas

Petri Nets 2007

How Modelling Frameworks Can Help

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But challenges, also: I

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Immediate transitions and markings

Peter J. Haas

Petri Nets 2007

Part III Modelling with SPNs

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Peter J. Haas

Petri Nets 2007

The SPN Graph d1

d2

s = (2, 1, 1) d3

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D = finite set of places

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E = finite set of transitions (timed and immediate)

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marking = assignment of token counts to places

Peter J. Haas

Petri Nets 2007

Transition Firing

p(s 0 ; s, e ∗ )

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The marking changes when an enabled transition fires

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Removes 1 token per place from random subset of input places

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Deposits 1 token per place in random subset of output places

Peter J. Haas

Petri Nets 2007

Clocks (Event Scheduling)

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One clock per transition: records remaining time until firing Enabled transitions compete to trigger marking change I I I

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At a marking change: three kinds of transitions I I I

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The clock that runs down to 0 first is the “winner” Can have simultaneous transition firing: p(s 0 ; s, E ∗ ) Numerical priorities: specify simultaneous-firing behavior New transitions: Use clock-setting distribution function Old transitions: Clocks continue to run down Newly-disabled transitions: Clock readings are discarded

Peter J. Haas

Petri Nets 2007

Clocks, Continued

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Clock-setting distribution depends on: I

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Clocks run down at marking-dependent speeds r (s, e) I I

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Old marking, new marking, trigger set Processor sharing Zero speeds: preempt-resume behavior

Peter J. Haas

Petri Nets 2007

Timed and Immediate Markings

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Immediate marking: ≥ 1 immediate transition is enabled

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An immediate marking vanishes as soon as it is attained

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Otherwise, marking is timed

Peter J. Haas

Petri Nets 2007

Example: Cyclic Queues with Feedback

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Peter J. Haas

Petri Nets 2007

Bottom-Up and Top-Down Modeling

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Peter J. Haas

Petri Nets 2007

Other Modeling Features

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Concurrency:

Synchronization:

Precedence:

Priority:

Peter J. Haas

Petri Nets 2007

Why This SPN Model?

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Conciseness: small set of building blocks Generality: subsumes GSPNs, etc. I

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Theory carries over

Modelling power: captures many discrete-event systems

Peter J. Haas

Petri Nets 2007

Modeling Power of SPNs I

Compare to Generalized semi-Markov processes (GSMPs) I I I I

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Strong mimicry I I I I

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Define X (t) = state of system at time t Define (Sn , Cn ) = (state, clocks) after nth state transition { X (t) : t ≥ 0 } processes have same dist’n (under mapping) { (Sn , Cn ) : n ≥ 0 } have same dist’n (under mapping)

Theorem: SPNs and GSMPs have same modeling power I I I

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Arbitrary state definition (s) Set E (s) of active events is a building block No restrictions on p(s 0 ; s, E ∗ ) No “immediate events”

Establishes SPNs as framework for discrete-event simulation Allows application of GSMP theory to SPNs Methodology allows other comparisons (e.g., inhibitor arcs)

Peter J. Haas

Petri Nets 2007

Part IV Sample-Path Generation

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Peter J. Haas

Petri Nets 2007

The Marking Process

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Marking process: { X (t) : t ≥ 0 } I I

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Defined in terms of Markov chain { (Sn , Cn ) : n ≥ 0 } I I I I

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X (t) = the marking at time t A very complicated process System observed after the nth marking change Sn = (Sn,1 , . . . , Sn,L ) = the marking Cn = (Cn,1 , . . . , Cn,M ) = the clock-reading vector Chain defined via SPN building blocks

Peter J. Haas

Petri Nets 2007

Definition of the Marking Process

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Peter J. Haas

Petri Nets 2007

Generation of the GSSMC { (Sn , Cn ) : n ≥ 0 } 1. [Initialization] Set n = 0. Select marking S0 and clock readings C0,i for ei ∈ E (S0 ); set C0,i = −1 for ei 6∈ E (S0 ). 2. Determine holding time t ∗ (Sn , Cn ) and firing set En∗ . 3. Generate new marking Sn+1 according to p( · ; Sn , En∗ ). 4. Set clock-reading Cn+1,i for each new transition ei according to F ( · ; Sn+1 , ei , Sn , En∗ ). 5. Set clock-reading Cn+1,i for each old transition ei as Cn+1,i = Cn,i − t ∗ (Sn , Cn )r (Sn , ei ). 6. Set clock-reading Cn+1,i equal to −1 for each newly disabled transition ei . 7. Set n ← n + 1 and go to Step 2. Can compute GSMP { X (t) : t ≥ 0 } from GSSMC

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Peter J. Haas

Petri Nets 2007

Implementation Considerations for Large-Scale SPNs I

Use event lists (e.g., heaps) to determine E ∗ I I

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Updating the state is often simpler in an SPN Efficient techniques for event scheduling [Chiola91] I

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Encode transitions potentially affected by firing of ei

Parallel simulation of subnets I I I

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O(1) computation of E ∗
O log m update time, where m = # of enabled transitions

E.g., Adaptive Time Warp [Ferscha & Richter PNPM97] Guardedly optimistic Slows down local firings based on history of rollbacks Peter J. Haas

Petri Nets 2007

Part V Stability Theory for SPNs

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Peter J. Haas

Petri Nets 2007

Stability and Simulation I

Focus on time-average limits: 1 r (f ) = lim t→∞ t

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f X (u) du

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1 X˜ f (Sn , Cn ) ˜r (f˜) = lim n→∞ n i=0

Functions (e.g. ratios) of such limits Cumulative rewards (impulse/continuous/mixed), gradients Steady-state means

Key questions: I

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Ex: long-run cost, availability, utilization Extensions: I

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When do such limits exist? When do various estimation methods apply? Can get weird behavior: limn E [ζn − ζn−1 ] = ∞ but explodes!

Approach: analyze the chain { (Sn , Cn ) : n ≥ 0 } Peter J. Haas

Petri Nets 2007

Harris Recurrence: A Basic Form of Stability I

Definition for general chain { Zn : n ≥ 0 } with state space Γ Pz { Zn ∈ A i.o. } = 1, z ∈ Γ

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Chain admits invariant probability measure π Pπ { Z1 ∈ A } = π(A) Implies stationarity when initial dist’n is π

When is { (Sn , Cn ) : n ≥ 0 } (positive) Harris recurrent? I

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φ is a recurrence measure (often “Lebesgue-like”) Every “dense enough” set is hit infinitely often w.p. 1 No “wandering off to ∞”

Positive Harris recurrence: I

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whenever φ(A) > 0

Fundamental question for steady-state estimation

Peter J. Haas

Petri Nets 2007

Some Stability Conditions I

Density component g of a cdf F : F (t) ≥ s0

p(s 0 ; s, e)

Rt 0

g (u) du

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s→

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s; either s → s 0 or s → s (1) → · · · → s (n) → s 0 Assumption PD(q):

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iff

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Marking set G is finite SPN is irreducible: s ; s 0 for all s, s 0 ∈ G All speeds are positive There exists x¯ ∈ (0, ∞) s.t. all clock-setting dist’n functions I I

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> 0 for some e

s 0:

Have finite qth moment Have density component positive on [0, x¯]

Assumption PDE: replace finite qth moment requirement by Z ∞ e ux dF (x) < ∞ for u ∈ [0, aF ] 0

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Peter J. Haas

Petri Nets 2007

Harris Recurrence in SPNs I

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Embedded chain: { (Sn , Cn ) : n ≥ 0 } observed only at transitions to timed markings ¯ φ({s} × A) = Lebesgue measure of A ∩ [0, x¯]M

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Theorem: If Assumption PD(1) holds, then the embedded chain is positive Harris recurrent with recurrence measure φ¯

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Implies Pµ { Sn = s i.o. } = 1 for all s ∈ S Proof:

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Alternate approach to recurrence: geometric-trials arguments I I

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First assume no immediate transitions ¯ Show that embedded chain is “φ-irreducible” Establish Lyapunov drift condition and apply MC machinery Extend to case of immediate transitions using strong mimicry Can drop positive-density assumption Use detailed analysis of specific SPN structure

Peter J. Haas

Petri Nets 2007

A Surprising Recurrence Result [Glynn and Haas 2007]

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Sn = marking just after nth marking change Conjecture: P { Sn = s i.o. } = 1 for each s if I I I

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CONJECTURE IS FALSE! I

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Marking set S is finite SPN is irreducible ∃ x¯ > 0 s.t. each F ( · ; e) has positive density on (0, x¯) In the presence of heavy-tailed clock-setting dist’ns

Peter J. Haas

Petri Nets 2007

The Counterexample I

S = { (2, 1, 1), (1, 2, 1), (1, 1, 2) }

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p(s 0 ; s, e ∗ ) = 0 or 1 (see schematic diagram) Clock-setting distributions:

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with β > 1/2 and α + β < 1 SPN hits marking s = (1, 2, 1) only if: I I

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F (t; e1 ) = 1 − (1 + t)−α F (t; e2 ) = 1 − (1 + t)−β F ( · ; e3 ) is Uniform[0, a]

e1 occurs and then e2 occurs No intervening occurrence of e3

Theorem: P { Sn = (1, 2, 1) i.o. } = 0

Peter J. Haas

Petri Nets 2007

Another Type of Stability: Limit Theorems I

Theorem (SLLN): If Assumption PD(1) holds, then for any f Z
1 t lim f X (u) du = r (f ) a.s. t→∞ t 0

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Theorem (FCLT): If Assumption PD(2) holds, then for any f Uν (f ) ⇒ σ(f )W

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R νt  Uν (f )(t) = ν −1/2 0 f X (u) − r (f ) du ⇒ denotes weak convergence on C [0, ∞) W = standard Brownian motion on [0, ∞) “Functional” form of CLT (ordinary CLT is a special case)

Note: r (f ) and σ(f ) are independent of initial conditions Follows from general result in [Glynn and Haas 2006] I

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as ν → ∞

Uses results for Harris recurrent MCs Peter J. Haas

Petri Nets 2007

FCLT Example: Donsker’s Theorem

Sn =

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Pn

i=0 Xi

Peter J. Haas

Petri Nets 2007

Part VI Steady-State Simulation

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Peter J. Haas

Petri Nets 2007

Regenerative Simulation: Regenerative Processes

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A regenerative process can be decomposed into i.i.d. cycles System “probabilistically restarts” at each Ti I

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Analogous definition for discrete-time process { Xn : n ≥ 0 } Extension: one-dependent cycles I

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Ex: successive arrival times to an empty GI/G/1 queue

Harris recurrent chains are od-regenerative (basis for previous SLLN and FCLT)

Peter J. Haas

Petri Nets 2007

Regenerative Simulation: The Ratio Formula I

Let

Ti

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f X (u) du

Yi =

and τi = Ti − Ti−1

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(Y1 , τ1 ), (Y2 , τ2 ), . . . are i.i.d. pairs

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It follows that Pn Z Tn
Yi Y¯n 1 E [Y1 ] def = → =r f X (u) du = Pi=1 n Tn 0 τ¯n E [τ1 ] i=1 τi almost surely as n → ∞ (need E [τ1 ] < ∞)

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Can show that 1 t

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Z

t


f X (u) du → r a.s. as t → ∞

0

If τ1 is “aperiodic”, then X (t) ⇒ X and E [f (X )] = r Peter J. Haas

Petri Nets 2007

Regenerative Simulation: The Regenerative Method I

Point estimate (biased): ˆrn = Y¯n /¯ τn :

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Confidence interval

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ˆrn → r a.s. as n → ∞ (strong consistency) Set Zi = Yi − r τi Z1 , Z2 , . . . i.i.d. with E [Zi ] = 0 and Var[Z1 ] = σ 2 Apply Central Limit Theorem (CLT) for i.i.d. random variables:

√ √ n ˆrn − r n ˆrn − r ⇒ N(0, 1) and ⇒ N(0, 1) σ/E [τ1 ] sn /¯ τn as n → ∞, where sn estimates σ (we assume σ 2 < ∞) 100p% asymptotic confidence interval:   zp sn zp sn ˆrn − √ , ˆrn + √ , τ¯n n τ¯n n where P { −zp ≤ N(0, 1) ≤ zp } = p, i.e., (1 + p)/2 quantile

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Many extensions: bias reduction, fixed-time or fixed-precision, generalized Y and τ , estimate α = g (E [Y ] , E [τ ]), . . . Peter J. Haas

Petri Nets 2007

Regenerative Simulation of SPNs I

A marking ¯s is a single state if E (¯s ) = { e¯ }

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Define θ(k) = kth marking change at which e¯ fires in ¯s

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Set Tk = ζθ(k) and τk = Tk − Tk−1

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Theorem: Suppose Assumption PD(2) holds and SPN has a single state ¯s I

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Can therefore apply standard regenerative method Variant theorems are available I I I

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Random times { Tk : k ≥ 0 } form sequence of regeneration points for marking process Finite expected cycle length: Eµ [τ1 ] < ∞ Finite variance constant for any f :
 R T σ 2 (f ) = Varµ T01 f X (u) du − r τ1 < ∞

Variants of single state (e.g., memoryless property) Other recurrence conditions (geometric trials) Discrete-time results Peter J. Haas

Petri Nets 2007

The Method of Batch Means I

Simulate system to (large) time t = mv (where 10 ≤ m ≤ 20)

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Divide into m batches of length v and compute batch means: 1 Y¯i = v

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f X (u) du

(i−1)v

Treat Y¯1 , Y¯2 , . . . , Y¯m as i.i.d., N(µ, σ 2 ): I I

Pm Point estimate: ˆrt = (1/m) i=1 Y¯i 100p% confidence interval:   tp,m−1 sm tp,m−1 sm ˆrt − √ , ˆrt + √ , m m where tp,m−1 = (1 + p)/2 quantile of Student’s T dist’n

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Peter J. Haas

Petri Nets 2007

Batch Means, Continued

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Why might batch means work? Formally, want to show I I I

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Consistency of ˆrt and validity of CI as t → ∞ For m fixed (standard batch means) What if m = m(t)? Overlapping batches?

Special case of standardized-time-series methods

Peter J. Haas

Petri Nets 2007

Standardized Time Series I

Consider a mapping ξ : C [0, 1] 7→ < such that I I

ξ(ax) = aξ(x) and ξ(x − be) = ξ(x), where e(t) = t P { ξ(W ) > 0 } = 1 and P { W ∈ D(ξ) } = 0

R νt

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Set Y¯ν (t) = (1/ν)

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Theorem: If Assumption PD(2) holds, then r exists and √ σW (1) W (1) ˆrν − r ν(ˆrν − r ) = √ ¯ ⇒ = , ¯ σξ(W ) ξ(W ) ξ(Yν ) ξ( ν(Yν − re))

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f X (u) du and ˆrν = Y¯ν (1)

so that an asymptotic 100p% confidence interval for r is   ˆrν − ξ(Y¯ν )zp , ˆrν + ξ(Y¯ν )zp ,

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where P{ −zp ≤ W (1)/ξ(W ) ≤ zp } = p Different choices of ξ yield different estimation methods I I

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batch means (fixed # of batches) STS area method, STS maximum method Peter J. Haas

Petri Nets 2007

Consistent-Estimation Methods (Discrete Time) I

Set ˆrn = (1/n)

Pn−1 ˜ j=0 f (Sj , Cj ) and suppose that √

lim ˆrn = ˜r a.s. and

n→∞ I

If we can find a consistent estimator Vn ⇒ σ ˜ 2 , then
√ n ˆrn − ˜r ⇒ N(0, 1) 1/2 Vn

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Then an asymptotic 100p% confidence interval for ˜r is " # 1/2 1/2 zp Vn zp Vn , ˆrn + √ , ˆrn − √ n n

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n ˆrn − ˜r ⇒ N(0, 1) σ ˜

where zp = (1 + p)/2 quantile of N(0, 1) Narrower asymptotic confidence intervals than STS methods Peter J. Haas

Petri Nets 2007

Consistent-Estimation Methods for SPNs I

Look at polynomially dominated functions: f˜(s, c) = O(1 + max1≤i≤M ciq ) for some q ≥ 0

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Require aperiodicity: no partition of marking set G s.t. G1 → G2 → · · · → Gd → G1 → G2 → · · · Focus on “localized quadratic-form variance estimators”

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Quadratic-form: Vn =

n X n X

(n) f˜(Si , Ci )f˜(Sj , Cj )qi,j

i=0 j=0 I

Localized: (n) |qi,j |

( a1 /n ≤ a2 (n)/n

if |i − j| ≤ m(n); if |i − j| > m(n)

where a2 (n) → 0 and m(n)/n → 0 50

Peter J. Haas

Petri Nets 2007

Exploiting Results for Stationary Output

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Theorem: For an aperiodic SPN, suppose that I I I I

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Then Vn ⇒ σ ˜ 2 for any initial distribution Proof: I I

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Assumption PDE holds (∃ invariant distribution π) { f˜(Sn , Cn ) : n ≥ 0 } obeys a CLT with variance constant σ ˜2 2 Vn is a localized quadratic-form estimator of σ ˜ Vn ⇒ σ ˜ 2 when initial distribution = π

{ (Sn , Cn ) : n ≥ 0 } couples with stationary version Localization: difference between Vn versions becomes negligible

Consequence: can exploit existing consistency results for stationary output

Peter J. Haas

Petri Nets 2007

Coupling Harris-Ergodic Markov Chains

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Peter J. Haas

Petri Nets 2007

Application to Specific Variance Estimators I

Variable batch means estimator of σ ˜2: I I

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Spectral estimator of σ ˜2: I I I I I

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b(n) batches of m(n) observations each VBM estimator is consistent if Assumption PDE holds, f˜ is polynomially dominated, b(n) → ∞, and m(n) → ∞. Pm−1 (S) ˆh Form of estimator: Vn = h=−(m−1) λ(h/m)R ˆ h = sample lag-h autocorrelation of { f˜(Sn , Cn ) : n ≥ 0 } R λ( · ) = “regular” window function (Bartlett, Hanning, Parzen) m = m(n) = spectral window length Spectral estimator is consistent if Assumption PDE holds, f˜ is polynomially dominated, m(n) → ∞, and m(n)/n1/2 → 0

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Overlapping batch means: asymp. equivalent to spectral

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Can extend results to continuous time (and drop aperiodicity)

Peter J. Haas

Petri Nets 2007

Estimation of Delays in SPNs Want to estimate limn→∞ (1/n)

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Delays D0 , D1 , . . . “determined by marking changes of the net” Specified as Dj = Bj − Aj

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j=0

f (Dj )

 Starts: Aj = ζα(j) : j ≥ 0 nondecreasing Terminations: Bj = ζβ(j) : j ≥ 0 Determined by { (Sn , Cn ) : n ≥ 0 }

Measuring lengths of delay intervals is nontrivial I

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Pn−1

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Must link starts and terminations Multiple ongoing delays Overtaking: delays need not terminate in start order Pn−1 Can avoid for limiting average delay limn→∞ (1/n) j=0 Dj

Measurement methods: tagging and start vectors

Peter J. Haas

Petri Nets 2007

Tagging

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Peter J. Haas

Petri Nets 2007

Start Vectors

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Assume # of ongoing delays = ψ(s) when marking is s

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Vn records starts for all ongoing delays at ζn

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Positions of starts = position of entities in system (usually)

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Use -1 as placeholder At each marking change:

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Insert current time according to iα (s 0 ; s, E ∗ ) Delete components according to iβ (s 0 ; s, E ∗ ) Permute components according to iπ (s 0 ; s, E ∗ ) Subtract deleted components from current time to compute delays (ignore -1’s)

Peter J. Haas

Petri Nets 2007

Start Vector Example

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Peter J. Haas

Petri Nets 2007

Regenerative Methods: The Easy Case

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Assume SPN has single state and “well behaved” cycles

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Use standard regenerative method

Peter J. Haas

Petri Nets 2007

Regenerative Methods: The Hard Case

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Assume SPN has single state and “well behaved” cycles

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Decompose delays into one-dependent cycles

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Use extended regenerative method or multiple-runs method

Peter J. Haas

Petri Nets 2007

Limiting Average Delay I

Under appropriate regularity conditions n−1

Eµ [Z1 ] 1X Dj = a.s. n→∞ n Eµ [δ1 ] lim

j=0

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δ1 = # R of starts in
regenerative cycle Z1 = cycle ψ X (t) dt ψ(s) = # of ongoing delays when marking is s (Z1 , δ1 ), (Z2 , δ2 ), . . . are i.i.d.

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Can use standard regenerative method

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No need to measure individual delays

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One proof of this result uses Little’s Law

Peter J. Haas

Petri Nets 2007

STS Methods for Delays I

Focus on “regular” start-vector mechanism

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Use polynomially-dominated functions f :