PERTURBATION

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PERTURBATION ANALYSIS
Christos G. Cassandras CODES Lab. - Boston
University
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PERTURBATION ANALYSIS (PA)

• Special case of CONCURRENT SIMULATION when
• - parametric changes of interest are small
• - derivatives (sensitivities) are needed

• Historically, PA precedes CONCURRENT SIMULATION

• Sensitivity Information Gradient-based
  Optimization
• A very natural, powerful design/control
  mechanism

Christos G. Cassandras CODES Lab. - Boston
University
3
PERTURBATION ANALYSIS OVERVIEW
Brute Force Sensitivity Estimation
Finite Perturbation Analysis (FPA)
Infinitesimal or Smoothed Perturbation Analysis
(IPA, SPA)
Christos G. Cassandras CODES Lab. - Boston
University
4
PERTURBATION GENERATION

• DES parameter q perturbed by Dq
• Suppose q is a parameter of some cdf F(xq)
• OBSERVED sample path X(q)
• What is DX(q) ?
• What is ?

Christos G. Cassandras CODES Lab. - Boston
University
5
PERTURBATION GENERATION
F(xq)
1
U
X
Christos G. Cassandras CODES Lab. - Boston
University
6
PERTURBATION PROPAGATION
SERVER
QUEUE
A1, A2,
D1, D2,
Z1(q), Z2(q),
Suppose q is perturbed by Dq
?
Iterative relationship that depends ONLY on
observed sample path data !
Christos G. Cassandras CODES Lab. - Boston
University
7
PERTURBATION PROPAGATION
Four cases to consider After some algebra
where
(idle period length if gt 0)
Ik observed DZk calculated from Zk as described
earlier (PERTURBATION GENERATION)
Christos G. Cassandras CODES Lab. - Boston
University
8
PERTURBATION PROPAGATION
EXAMPLE Perturbations in mean service time ?
DZ1(q), DZ2(q),
2
I4
1
Z1
A1
A2
A3
A4
A5
D1
D2
D3
D4
Christos G. Cassandras CODES Lab. - Boston
University
9
INFINITESIMAL PERTURBATION ANALYSIS (IPA)
If Dq is so small as to ensure that DDk-1
Ik then
Perturbation Propagation
Perturbation Generation
Christos G. Cassandras CODES Lab. - Boston
University
10
INFINITESIMAL PERTURBATION ANALYSIS (IPA)
Using
we can obtain similar algorithms for performance
measures Mean System Time, Throughput, etc.

• IMPLEMENTATION
• Extremely simple and non-intrusive
• Overhead relative to nominal simulation is
  negligible

See http//vita.bu.edu/cgc/IPA/
Christos G. Cassandras CODES Lab. - Boston
University
11
IPA OPTIMIZATION
Christos G. Cassandras CODES Lab. - Boston
University
12
INFINITESIMAL PERTURBATION ANALYSIS (IPA)

• QUESTION Are IPA sensitivity estimators good ?
• Unbiasedness
• Consistency

• ANSWER Yes, for a large class of DES that
  includes
• G/G/1 queueing systems
• Jackson-like queueing networks (e.g., no
  blocking allowed)

NOTE IPA applies to REAL-VALUED parameters of
some event process distribution (e.g., mean
interarrival and service times)
Christos G. Cassandras CODES Lab. - Boston
University
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EXTENSIONS OF IPA

• There are several generalizations, at the expense
  of more simulation overhead (still, very minimal)
• e.g.,
• routing probabilities
• systems with real-time constraints
• some scheduling policies

IMPLEMENTATION Still very straightforward and
non-intrusive
Christos G. Cassandras CODES Lab. - Boston
University