concurrent simulation: a tutorial christos g. cassandras dept. of manufacturing engineering boston...

CONCURRENT SIMULATION:A TUTORIAL

Christos G. Cassandras

Dept. of Manufacturing EngineeringBoston UniversityBoston, MA 02215

[email protected]

• Scope of Concurrent Simulation

• Modeling Foundations

• The Sample Path Constructability Problem

• The Time Warping Algorithm (TWA)

Christos G. Cassandras CODES Lab. - Boston University

OUTLINE:

SCOPE OFCONCURRENT SIMULATION

SIMULATOR

DesignParameters

DesignParameters

Operating Policies,Control Parameters


PerformanceMeasures

PerformanceMeasures


CONVENTIONAL SIMULATION ANALYSIS

• Repeatedly change parameters/operating policies

• Test different conditions

• Answer multiple WHAT IF questions



CONTINUED

SIMULATOR

DesignParameters

DesignParameters

PerformanceMeasures

PerformanceMeasures



CONCURRENCYENGINE

WHAT IF…• Parameter p1 = a were replaced by p1 = b• Parameter p2 = c were replaced by p2 = d

WHAT IF…• Parameter p1 = a were replaced by p1 = b• Parameter p2 = c were replaced by p2 = d

PerformanceMeasuresunder allWHAT IFQuestions

PerformanceMeasuresunder allWHAT IFQuestions

ANSWERS TO MULTIPLE “WHAT IF”QUESTIONS AUTOMATICALLY PROVIDED


TYPICAL OPTIMIZATION PROCESS FOR COMPLEX SYSTEMS: Repeated Trial and Error


CONTINUED

SEARCH SPACE

SIMULATE SYSTEM

ESTIMATEPERFORMANCE

CHOOSE NEW POINT

SIMULATE SYSTEM

CHOOSE NEW POINT

ESTIMATEPERFORMANCE

etc…etc…etc

SIMULATE SYSTEM



CONTINUED

ESTIMATEPERFORMANCE

Estimate performance over many points from simulation at a single point

CONCURRENTSIMULATION

CHOOSE NEW POINT

SEARCH SPACE

SIMULATE SYSTEM

ESTIMATEPERFORMANCE

CONCURRENTSIMULATION

CHOOSE NEW POINT

RAPID

LEARNI

NG

TERMINOLOGY:IMPORTANT DISTINCTIONS1. BRUTE-FORCE SIMULATION:

sequentially simulate a system N times-- each time under a different setting

2. PARALLEL/DISTRIBUTED SIMULATION:

distribute simulation code over N processorsor a network of N workstations

3. CONCURRENT SIMULATION:

use data from simulation of one system toconcurrently simulate the system underN different settings


1 processor N runs N input-output data

N processors 1 run 1 input-output data

1 processor 1 run N input-output data

TERMINOLOGY:IMPORTANT DISTINCTIONS


1. CONCURRENT SIMULATION in SEQUENTIAL COMPUTER PROCESSING

• Sequential processing, N simulation threads unfolding in parallel• Main benefit: Sharing of data and some data structures

CONTINUED

2. CONCURRENT SIMULATION in PARALLEL COMPUTER PROCESSING

• One processor for each simulation thread

• Additional benefit: All state updates parallelized

BACKGROUND: PERTURBATION ANALYSIS


“Brute Force” Sensitivity Estimation:

J DES

DES J ]

dJ

d

J J

est

( ) ( )

Finite Perturbation Analysis (FPA):

J

J ]DES

dJ

d

J J

est

( ) ( )

FPAFPA

Infinitesimal or Smoothed Perturbation Analysis(IPA, SPA):

J DES

IPA/SPAIPA/SPAdJ

d est

MODELING FOUNDATIONS


DISCRETE EVENT SIMULATION:

The process of generating a sample path of a Stochastic Timed Automaton

AUTOMATON: (E, X, , f, x0)

: Event Set

X: State Space

(x) : Set of feasible or enabled events at

state x

f : State Transition Function (undefined for events )

x0 : Initial State,

f X E X: e x

x X0

e e1 2, , x x1 2, , f x e x, '

TIMED AUTOMATON


Add a Clock Structure V to the Automaton: (E, X, , f, x0 , V)

where:

and vi is a Clock or Lifetime Sequence:

one for each event i

V v i i E :

vi i iv v 1 2, ,

v v1 2, , x x1 2, , f x e x, ' '

NEXT EVENT

Need an internal mechanism to determine

NEXT EVENT e’ and hence

NEXT STATE

Need an internal mechanism to determine

NEXT EVENT e’ and hence

NEXT STATE x f x e' , '

HOW THE TIMED AUTOMATON WORKS...


CURRENT STATExX with feasible event set: (x)

CURRENT EVENTe that caused transition into x

CURRENT EVENT TIMEt associated with e

Associate a

CLOCK VALUE/RESIDUAL LIFETIME yi

with each feasible event i x



NEXT/TRIGGERING EVENT e’ :

e y

i xi' arg min

NEXT EVENT TIME t’ :

t t y

y yi x

i

' *

min

where: *

CONTINUED

NEXT STATE x’ :

x f x e' , '



CONTINUED

Detemine new CLOCK VALUESfor every event i x

yi

y

y y i x i x i e

v i x x e

otherwise

v i

i

i

ij

ij

* , ,

where: new lifetime for event

0

TIMED AUTOMATON - AN EXAMPLE


QUEUE SERVER

CustomerArrivals

CustomerDepartures

E a d X , , , , , 012

x a d x

a

, , for all 0

0

f x ex e a

x e d x,

,

1

1 0

Given input:

v va a a d d dv v v v 1 2 1 2, , , , ,

TIMED AUTOMATON - AN EXAMPLE


CONTINUED

t0

x0 = 0

a

e1 = a

x0 = 1

t1

ad

e2 = a

x0 = 2

t2

ad

A Typical Sample Path...

e3 = a

x0 = 3

t3

ad

x0 = 2

e4 = d

t4

STOCHASTICTIMED AUTOMATON

• Same idea with the Clock Structure consisting of Stochastic Processes

• Associate with each event i a Lifetime Distribution based on which vi is generated

Generalized Semi-Markov Process(GSMP)

• In a simulator, vi is generated through a pseudorandom number generator


THEORETICALFOUNDATIONS


THE CONSTRUCTABILITY PROBLEM:

Consider a Discrete Event System (DES) operating under parameter settings u um1 , ,

e tu

k k

1 1

1

, observed sample path under

• INPUT: = all event lifetimes u1 = a parameter setting

• OUTPUT:

u um2 , ,• PROBLEM:

Construct sample paths under all based only on

e tk k

1 1,The input The observed sample path

Observed state history

CONSTRUCTABILITY PROBLEM


DESDESu1

e tk k1 1,

DESDESu2 e tk k

2 2,

DESDESum e tk

mkm,

OBSERVED

CONSTRUCTED



CONTINUED

• OBSERVABILITY:

A sample path is observable with respect to

if

e tkj

kj

,

e tk k, x xkj

k for all k = 0,1,...

INTERPRETATION: For every state xkj

visited in the constructed sample path, all feasible

events are also feasible in the corresponding

observed state xk

i xkj

Ref: Cassandras and Strickland, 1989Ref: Cassandras and Strickland, 1989



CONTINUED

AGE CLOCK

zi yi

t

Define the conditional cdf of an event clock giventhe event age:

H t z P y t zi i i,

• CONSTRUCTABILITY:

A sample path is constructable with respect to

if

e tkj

kj

,

e tk k,

AND

•

x xkj

k for all k = 0,1,...

H t z H t z i x kji kj j

i k kj

, , , , ,, , for all 0 1



CONTINUED

Two simple cases of constructability:

Theorem: If all event processes are Markovian

(generated by Poisson processes), then

OBSERVABILITY CONSTRUCTABILITY

NOTE: Observability is a simple-to-check structural condition

NOTE: Observability is a simple-to-check structural condition

Theorem:

CONSTRUCTABILITY

x x kkj

k for all 0 1, ,

CONSTRUCTABILITY:AN EXAMPLE


A simple design/optimization problem

ADMISSIONCONTROL

da

Reject if Queue Length > K

PROBLEM: • How does the system behave under different choices of K? • What is the optimal K?

CONCURRENT SIMULATION APPROACH:• Choose any K

• Simulate (or observe actual system) under K

• Apply Concurrent Simulation to LEARN effect of all other feasible K


NOMINALSYSTEM

K = 3

NOMINALSYSTEM

K = 3

a a a

d d d

a

PERTURBEDSYSTEM

K = 2

PERTURBEDSYSTEM

K = 2

aa a

d d

AUGMENTED SYSTEM: Check for OBSERVABILITY


CONTINUED

a

d

a

d

a

a

a

d

d a

d



CONTINUED

a

a

d

d

a

a

d

d a

a

d

0 1 a a d,

OBSERVABILITY satisfied!

However, if roles of NOMINAL and PERTURBED are reversed, then OBSERVABILITY is not satisfied...

SOLVING THECONSTRUCTABILITY PROBLEM


• Constructability is not easily satisfied, in general

• However, the CONSTRUCTABILITY PROBLEM can be solved at some cost

SOLUTION METHODOLOGIES:

• STANDARD CLOCK (SC) methodology for Markovian systems

• AUGMENTED SYSTEM ANALYSIS (ASA) for systems with at most one non-Markovian event

• The TIME WARPING ALGORITHM (TWA) for arbitrary systems

Ref: Vakili, 1991Ref: Vakili, 1991

Ref: Cassandras and Strickland, 1989Ref: Cassandras and Strickland, 1989

Ref: Cassandras and Panayiotou, 1996Ref: Cassandras and Panayiotou, 1996

CONCURRENT SIMULATION FOR ARBITRARY SYSTEMS MAIN IDEA: MAXIMAL SAMPLE PATH COUPLING

• Observe event in a sample path under setting 0

• Save the lifetime of this event

• Wait until ALL sample paths constructed under2,…, M have used the lifetime at the right time

EXAMPLE: Queueing System with capacity K

Observed sample path: K = 2

Constructed sample path: K = 3

1

2

3

1

2

3

A D


DA

CONCURRENT SIMULATION FOR ARBITRARY SYSTEMS


CONTINUED

• OBSERVED event sequence under 0 after n events:

( ) : , ,n e j nj 1

• CONSTRUCTED event sequence under any after k events:

0

( ) : , , k e j kj 1

Define the SCORE of event i after n total events:

s ini

n number of occurrences of event after total events



CONTINUED

• Lifetime sequence of event i on OBSERVED sample path:

Vi i i inn v v s 1 , ,

• Lifetime sequence of event i on CONSTRUCTED sample path:

, , ,Vi i i ikk v v s k n 1

• A subsequence of Vi(n):

~, , ,Vi i i

ki i

nn k v s v s 1

LIFETIMES ALREADY OBSERVED,BUT NOT YET USEDIN CONSTRUCTED PATH



CONTINUED

• AVAILABLE Event Set:

A n k i i E s sin

ik, : ,

EVENTS ALREADY OBSERVED

IN n) WITH LIFETIMES YET UNUSED

IN CONSTRUCTED SAMPLE PATH

• MISSING Event Set:

M n k x x ek k k, 1

EVENTS NEEDEDTO CONTINUE CONSTRUCTION

EVENTS AVAILABLEFROM LAST STATE TRANSITION

NOTE: Constructed sample path is ACTIVE as long as MISSING Event Set is contained in AVAILABLE Event Set

CONCURRENT SIMULATION: STEP 1


CONTINUED

When event en+1 is observed:

1. Store observed event lifetime:

~,

~,~

,V

V

Vi

i i in

n

i

n kn k v s i e

n k

1

1 1if

otherwise

2. Add event to AVAILABLE Event Set:

A n k A n k en 1 1, ,

3. No change to MISSING Event Set:

M n k M n k 1, ,

4. Check if constructed sample path is ACTIVE:

M n k A n k 1 1, ,

CONCURRENT SIMULATION: STEP 2


CONTINUED

If constructed sample path is ACTIVE:

1. Discard “used up” observed event lifetime:

~,

~, ,

~,

VV

Vi

i i ik

i

n kn k v s i M n k

n k

1 1

1 1

1

if

otherwise

2. Delete from AVAILABLE Event Set:

A n k A n k i i M n k s sik

in 1 1 1 1 1 1, , , , :

3. Update MISSING Event Set:

M n k x x ek k k 1 1 1 1,

4. Check if constructed sample path is still ACTIVE:

M n k A n k 1 1 1 1, ,

TIME WARPING ALGORITHM (TWA)

Observe event en+1 and check ACTIVE condition:

M n k A n k 1 1, ,

If ACTIVE condition holds, then “Time Warp”:

• Construct desired state trajectory for events j = k, k+1,… as long as:

M n j A n j 1 1 1 1, ,

• When ACTIVE condition no longer satisfied, wait for next observed event and repeat

Observed sample path

Constructed sample path

n

k


k+1

TWA PERFORMANCE:SPEEDUP FACTORS


Simulation time for OBSERVEDsample path (N events)

TN0

Simulation time for OBSERVEDsample path (N events)+ TWA for CONSTRUCTEDsample path (K events)

TN K0

Simulation time for CONSTRUCTEDsample path (N events)

TN

ST N

KN

K

/

/SPEEDUP FACTOR:

NOTE:

• In conventional simulation, every event processed is discarded from the Event List

• In Concurrent Simulation, the event is saved and discarded only at the right time

TWA PERFORMANCE:SPEEDUP FACTORS


CONTINUED

TYPICAL NUMERICAL RESULTS:• For simple event lifetime distributions: S ~ 2• For complex event lifetime distributions: S > 20

Ref: Cassandras and Panayiotou, 1996Ref: Cassandras and Panayiotou, 1996

The TWA is particularly efficient for the class of systems that satisfy the following REGULARITYcondition:

k k k

kk

x x

k

max , , ,

1

1 2 NUMBER OF NEW EVENTSTRIGGERED WHEN AN EVENTTAKES PLACE

REGULARITY: 1

NOTE: A surprisingly large class of systems satisfies the REGULARITY condition...

THE ENDTHE END

ACKNOWLEDGEMENTS:

NSF, AFOSR, ARPA, NRL, United Technologies,Rome Lab ( Al Sisti)

Copies of these slides can be obtained byrequest and will be made available on the web:

http://cad.bu.edu/cgc

concurrent simulation: a tutorial christos g. cassandras dept. of manufacturing engineering boston...

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