CONCURRENT SIMULATION:
A TUTORIAL
Christos G. Cassandras
Dept. of Manufacturing Engineering
Boston University
Boston, MA 02215
[email protected]
OUTLINE:
• Scope of Concurrent Simulation
• Modeling Foundations
• The Sample Path Constructability Problem
• The Time Warping Algorithm (TWA)
Christos G. Cassandras
CODES Lab. - Boston University
SCOPE OF
CONCURRENT SIMULATION
Design
Parameters
SIMULATOR
Performance
Measures
Operating
Policies,
Control
Parameters
CONVENTIONAL SIMULATION ANALYSIS
• Repeatedly change parameters/operating policies
• Test different conditions
• Answer multiple WHAT IF questions
Christos G. Cassandras
CODES Lab. - Boston University
SCOPE OF
CONCURRENT SIMULATION
CONTINUED
Design
Parameters
SIMULATOR
Operating
Policies,
Control
Parameters
Performance
Measures
CONCURRENCY
ENGINE
WHAT IF…
• Parameter p1 = a were replaced by p1 = b
• Parameter p2 = c were replaced by p2 = d
Performance
Measures
under all
WHAT IF
Questions
ANSWERS TO MULTIPLE “WHAT IF”
QUESTIONS AUTOMATICALLY PROVIDED
Christos G. Cassandras
CODES Lab. - Boston University
SCOPE OF
CONCURRENT SIMULATION
CONTINUED
TYPICAL OPTIMIZATION PROCESS FOR
COMPLEX SYSTEMS: Repeated Trial and Error
SIMULATE SYSTEM
ESTIMATE
PERFORMANCE
CHOOSE NEW POINT
SIMULATE SYSTEM
ESTIMATE
PERFORMANCE
CHOOSE NEW POINT
etc…etc…etc
Christos G. Cassandras
CODES Lab. - Boston University
SCOPE OF
CONCURRENT SIMULATION
CONTINUED
Estimate performance over many points from
simulation at a single point
SIMULATE SYSTEM
CONCURRENT
SIMULATION
ESTIMATE
PERFORMANCE
CHOOSE NEW POINT
SIMULATE SYSTEM
CONCURRENT
SIMULATION
ESTIMATE
PERFORMANCE
CHOOSE NEW POINT
Christos G. Cassandras
CODES Lab. - Boston University
TERMINOLOGY:
IMPORTANT DISTINCTIONS
1. BRUTE-FORCE SIMULATION:
sequentially simulate a system N times
-- each time under a different setting
1 processor
N runs
N input-output data
2. PARALLEL/DISTRIBUTED SIMULATION:
distribute simulation code over N processors
or a network of N workstations
N processors
1 run
1 input-output data
3. CONCURRENT SIMULATION:
use data from simulation of one system to
concurrently simulate the system under
N different settings
1 processor
Christos G. Cassandras
1 run
N input-output data
CODES Lab. - Boston University
TERMINOLOGY:
IMPORTANT DISTINCTIONS
CONTINUED
1. CONCURRENT SIMULATION
in SEQUENTIAL COMPUTER PROCESSING
• Sequential processing, N simulation threads
unfolding in parallel
• Main benefit: Sharing of data and some data
structures
2. CONCURRENT SIMULATION
in PARALLEL COMPUTER PROCESSING
• One processor for each simulation thread
• Additional benefit: All state updates parallelized
Christos G. Cassandras
CODES Lab. - Boston University
BACKGROUND:
PERTURBATION ANALYSIS
“Brute Force” Sensitivity Estimation:
q
DES
q+Dq
DES
J q 
J q + Dq 
]
J( q + Dq )  J( q )
 dJ 
  
Dq
 dq  est
Finite Perturbation Analysis (FPA):
q
q+Dq
DES
J q 
J q + Dq 
FPA
]
J( q + Dq )  J( q )
 dJ 
  
Dq
 dq  est
Infinitesimal or Smoothed Perturbation Analysis
(IPA, SPA):
q
DES
IPA/SPA
Christos G. Cassandras
J q 
 dJ 
 dq 
est
CODES Lab. - Boston University
MODELING FOUNDATIONS
DISCRETE EVENT SIMULATION:
The process of generating a sample path of a
Stochastic Timed Automaton
AUTOMATON: (E, X, G, f, x0)
E : Event Set
X : State Space
G(x) : Set of feasible or enabled events at state x
f:X E  X
f : State Transition Function
(undefined for events e  G  x  )
x0 : Initial State, x0  X
e1 ,e2 ,
Christos G. Cassandras
f  x ,e  x'
x1 , x2 ,
CODES Lab. - Boston University
TIMED AUTOMATON
Add a Clock Structure V to the Automaton:
(E, X, G, f, x0 , V)
where:
V  vi : i  E 
and vi is a Clock or Lifetime Sequence:
vi  vi1 , vi 2 ,
one for each event i
v1 ,v2 ,
f  x , e'   x'
x1 , x2 ,
NEXT EVENT
Need an internal mechanism to determine
NEXT EVENT e’ and hence
NEXT STATE x'  f  x ,e' 
Christos G. Cassandras
CODES Lab. - Boston University
HOW THE TIMED
AUTOMATON WORKS...
CURRENT STATE
xX with feasible event set: G(x)
CURRENT EVENT
e that caused transition into x
CURRENT EVENT TIME
t associated with e
Associate a
CLOCK VALUE/RESIDUAL LIFETIME yi
with each feasible event i  G  x 
Christos G. Cassandras
CODES Lab. - Boston University
HOW THE TIMED
AUTOMATON WORKS...
CONTINUED
NEXT/TRIGGERING EVENT e’ :
e'  arg min  yi 
i G  x 
NEXT EVENT TIME t’ :
t'  t + y *
where: y*  min  yi 
i G  x 
NEXT STATE x’ :
x'  f  x ,e' 
Christos G. Cassandras
CODES Lab. - Boston University
HOW THE TIMED
AUTOMATON WORKS...
CONTINUED
Detemine new CLOCK VALUES yi
for every event i  G  x 
 y  y * i G  x  , i G  x  , i  e
 i
yi  vij
i  G  x   G  x   e
0
otherwise

where: vij  new lifetime for event i
Christos G. Cassandras
CODES Lab. - Boston University
TIMED AUTOMATON
- AN EXAMPLE
QUEUE
SERVER
Customer
Arrivals
Customer
Departures
E  a ,d ,
X  0,1,2 ,
G  x   a ,d ,
G  0   a
for all x  0
 x + 1 e  a
f  x ,e  
 x  1 e  d , x  0
Given input:
va  va1 ,va 2 ,, vd  vd 1 ,vd 2 ,
Christos G. Cassandras
CODES Lab. - Boston University
TIMED AUTOMATON
- AN EXAMPLE
CONTINUED
A Typical Sample Path...
e1 = a
x0 = 0
e2 = a
x0 = 1
t0
t1
e3 = a
x0 = 2
e4 = d
x0 = 3
t2
t3
x0 = 2
t4
a
a
d
a
d
a
d
Christos G. Cassandras
CODES Lab. - Boston University
STOCHASTIC
TIMED 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
Christos G. Cassandras
CODES Lab. - Boston University
THEORETICAL
FOUNDATIONS
THE CONSTRUCTABILITY PROBLEM:
Consider a Discrete Event System (DES) operating
under parameter settings u1 , , um 
• INPUT:
w = all event lifetimes
u1 = a parameter setting
• OUTPUT:
observed sample path
 e ,t   under
u
1
1
k
k
1
• PROBLEM:
Construct sample paths under all
based only on
The input w

1 1
The observed sample path e k ,t k
u2 ,,um

Observed state history
Christos G. Cassandras
CODES Lab. - Boston University
CONSTRUCTABILITY
PROBLEM
w
u1
u2
um
Christos G. Cassandras


DES


DES


e1k ,t k1
DES
ek2 ,t k2
ekm ,t km
CODES Lab. - Boston University
CONSTRUCTABILITY
PROBLEM
CONTINUED
• OBSERVABILITY:

j
j

A sample path ek ,t k is observable with respect to
ek ,tk 
if
G
   G  x  for all k = 0,1,...
j
xk
k
j
INTERPRETATION: For every state x k
visited in the constructed sample path, all feasible
events i G
  are also feasible in the corresponding
j
xk
observed state xk
Ref: Cassandras and Strickland, 1989
Christos G. Cassandras
CODES Lab. - Boston University
CONSTRUCTABILITY
PROBLEM
CONTINUED
Define the conditional cdf of an event clock given
the event age:
H t , zi  P yi  t zi

 

AGE
CLOCK
zi
yi
t
• CONSTRUCTABILITY:

j
j

A sample path ek ,t k is constructable with respect to
ek ,tk 
 
if
G xk  G  xk  for all k = 0,1,...
j
AND

j


• H j t , zi ,k  H j t , zi ,k
Christos G. Cassandras

 
for all i G xk , k  0,1,
j
CODES Lab. - Boston University
CONSTRUCTABILITY
PROBLEM
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
 
Theorem: G xkj  G  xk  for all k  0,1,
 CONSTRUCTABILITY
Christos G. Cassandras
CODES Lab. - Boston University
CONSTRUCTABILITY:
AN EXAMPLE
A simple design/optimization problem
a
ADMISSION
CONTROL
d
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
Christos G. Cassandras
CODES Lab. - Boston University
CONSTRUCTABILITY:
AN EXAMPLE
a
a
a
1
0
d
d
a
a
0
d
PERTURBED
SYSTEM
K=2
a
2
1
d
NOMINAL
SYSTEM
K=3
a
3
2
CONTINUED
d
AUGMENTED SYSTEM: Check for OBSERVABILITY
d
00
10
a
a
d
d
21
11
a
a
d
Christos G. Cassandras
22
d
a
a
32
CODES Lab. - Boston University
CONSTRUCTABILITY:
AN EXAMPLE
CONTINUED
G  0  a  G 1  a ,d 
d
00
10
a
a
d
d
21
11
a
a
d
22
d
a
a
32
OBSERVABILITY satisfied!
However, if roles of NOMINAL and PERTURBED
are reversed, then OBSERVABILITY is not satisfied...
Christos G. Cassandras
CODES Lab. - Boston University
SOLVING THE
CONSTRUCTABILITY
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
Ref: Vakili, 1991
• AUGMENTED SYSTEM ANALYSIS (ASA)
for systems with at most one non-Markovian event
Ref: Cassandras and Strickland, 1989
• The TIME WARPING ALGORITHM (TWA)
for arbitrary systems
Ref: Cassandras and Panayiotou, 1996
Christos G. Cassandras
CODES Lab. - Boston University
CONCURRENT SIMULATION
FOR ARBITRARY SYSTEMS
MAIN IDEA: MAXIMAL SAMPLE PATH COUPLING
• Observe event in a sample path under setting q0
• Save the lifetime of this event
• Wait until ALL sample paths constructed under
q2,…, qM have used the lifetime at the right time
EXAMPLE: Queueing System with capacity K
Observed sample path:
K=2
3
2
A
D
1
3
Constructed sample path:
K=3
2
1
D
Christos G. Cassandras
A
CODES Lab. - Boston University
CONCURRENT SIMULATION
FOR ARBITRARY SYSTEMS
CONTINUED
• OBSERVED event sequence under q0 after n events:


 ( n )  e j : j  1, , n
• CONSTRUCTED event sequence under any q  q0
after k events:

( k )  e j : j  1, , k

Define the SCORE of event i after n total events:
sin  number of occurrences of event i
after n total events
Christos G. Cassandras
CODES Lab. - Boston University
CONCURRENT SIMULATION
FOR ARBITRARY SYSTEMS
CONTINUED
• Lifetime sequence of event i on
OBSERVED sample path:

 
Vi  n  vi 1 , , vi sin
• Lifetime sequence of event i on
CONSTRUCTED sample path:

 ,
  k   v 1 , , v s k
V
i
i
i i
k n
• A subsequence of Vi(n):
 
 
~
Vi n , k   vi sik + 1, , vi sin
LIFETIMES ALREADY OBSERVED,
BUT NOT YET USED
IN CONSTRUCTED PATH
Christos G. Cassandras
CODES Lab. - Boston University
CONCURRENT SIMULATION
FOR ARBITRARY SYSTEMS
CONTINUED
• AVAILABLE Event Set:

An, k   i : i  E , sin  sik

EVENTS ALREADY OBSERVED
IN n) WITH LIFETIMES YET UNUSED
IN CONSTRUCTED SAMPLE PATH
• MISSING Event Set:


M n, k   G  xk   G  xk 1  ek 
EVENTS NEEDED
TO CONTINUE CONSTRUCTION
EVENTS AVAILABLE
FROM LAST STATE TRANSITION
NOTE: Constructed sample path is ACTIVE as long
as MISSING Event Set is contained in AVAILABLE
Event Set
Christos G. Cassandras
CODES Lab. - Boston University
CONCURRENT SIMULATION:
STEP 1
CONTINUED
When event en+1 is observed:
1. Store observed event lifetime:


~

Vi n , k  + vi sin + 1
~
Vi n + 1, k   
~

V
i n , k 

if i  en+1
otherwise
2. Add event to AVAILABLE Event Set:
An + 1, k   An , k   en+1
3. No change to MISSING Event Set:
M n + 1, k   M n, k 
4. Check if constructed sample path is ACTIVE:
M n + 1, k   An + 1, k 
Christos G. Cassandras
CODES Lab. - Boston University
CONCURRENT SIMULATION:
STEP 2
CONTINUED
If constructed sample path is ACTIVE:
1. Discard “used up” observed event lifetime:
 
~

Vi n + 1, k   vi sik
~
Vi n + 1, k + 1  
~

V
i n + 1, k 

if i  M n + 1, k 
otherwise
2. Delete from AVAILABLE Event Set:


An + 1, k + 1  An + 1, k   i : i  M n + 1, k , sik +1  sin+1
3. Update MISSING Event Set:


M n + 1, k + 1  G  xk +1  G  xk   ek +1
4. Check if constructed sample path is still ACTIVE:
M n + 1, k + 1  An + 1, k + 1
Christos G. Cassandras
CODES Lab. - Boston University
TIME WARPING ALGORITHM
(TWA)
Observe event en+1 and check ACTIVE condition:
M n + 1, k   An + 1, k 
If ACTIVE condition holds, then “Time Warp”:
• Construct desired state trajectory for
events j = k, k+1,… as long as:
M n + 1, j + 1  An + 1, j + 1
• When ACTIVE condition no longer satisfied,
wait for next observed event and repeat
n
Observed
sample path
k
Constructed
sample path
Christos G. Cassandras
k+1
...
CODES Lab. - Boston University
TWA PERFORMANCE:
SPEEDUP FACTORS
TN0  Simulation time for OBSERVED
sample path (N events)
TN0 +  K  Simulation time for OBSERVED
sample path (N events)
+ TWA for CONSTRUCTED
sample path (K events)
TN  Simulation time for CONSTRUCTED
sample path (N events)
SPEEDUP FACTOR:
TN / N
S
K / K
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
Christos G. Cassandras
CODES Lab. - Boston University
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, 1996
The TWA is particularly efficient for the class of
systems that satisfy the following REGULARITY
condition:
 k  G xk  G xk 1
NUMBER OF NEW EVENTS
TRIGGERED WHEN AN EVENT
TAKES PLACE
   
  max k , k  1,2 ,
k
REGULARITY:
 1
NOTE: A surprisingly large class of systems
satisfies the REGULARITY condition...
Christos G. Cassandras
CODES Lab. - Boston University
ACKNOWLEDGEMENTS:
NSF, AFOSR, ARPA, NRL, United Technologies,
Rome Lab ( Al Sisti)
Copies of these slides can be obtained by
request and will be made available on the w
http://cad.bu.edu/cgc
THE END