on heavy-tailed rare event analysis · design of scheduling algorithms, simulation algorithms, etc...

1

On Heavy-Tailed Rare Event Analysis

Chang-Han Rhee

Centrum Wiskunde & Informatica, Amsterdam

2

Rare Events

2

Rare Events

Although rare, rare events matter.

Need for understanding ‘how often?’ & ‘why?’

3

Rare Events depend on “Tail Behaviors”

Light-Tailed Distributions

• Extreme Values are Very Rare

• Normal, Exponential, etc

Heavy-Tailed Distributions

• Extreme Values are Frequent

• Power Law, Weibull, etc

Structural difference in the way systemwide rare events arise.

3





Systemwide rare events

arise because

EVERYTHING goes wrong.

(Conspiracy Principle)





3






arise because

EVERYTHING goes wrong.

(Conspiracy Principle)





arise because of

A FEW Catastrophes.

(Catastrophe Principle)


4

Light-tailed rare-event analysis has a long & successful history.

• Large Deviations Theory answers ‘How rare?’,

• Design of Scheduling Algorithms, Simulation Algorithms, etc

Varadhan won Abel Prize in 2007

Heavy-tailed rare events are NOT understood well.

4


• Large Deviations Theory answers ‘How rare?’, ’Why?’




4


• Large Deviations Theory answers ‘How rare?’, ‘Most likely scenario?’




4


• Large Deviations Theory answers ‘How rare?’, ‘Most likely scenario?’




In fact, ‘The only plausable scenario!’

5

But, Heavy Tails are Everywhere:Heavy tails are everywhere!

Computer systems Finance

delays, files, … losses

1 𝑃 𝑠𝑖𝑧𝑒 > 𝑛 ≈ 1

𝑛𝛼

Social networks Energy Systems

popularity, contagion blackouts

B. Zwart (CWI) Heavy tails 4 / 43

6

For Example: Open Problem Posed by Whitt (2000)

Congestion of Multiple Server Queue:

d

2

1

How many big jobs are needed to create large queue lengths?

In case of Weibull tails, NOT EVEN a conjecture!

7

Goal: Systematic Tools for Heavy-Tailed Systems

In many applications, one can write the rare event of interest as

{Sn ∈ A}

where Sn is the whole trajectory of a random walk.

We want to understand P(Sn ∈ A) and P(Sn ∈ ·|Sn ∈ A)

for as general A as possible.

7



{Sn ∈ A}




How rare?

7



{Sn ∈ A}




How rare? Most likely scenario?

7



{Sn ∈ A}




sample path︷︸︸︷How rare? Most likely scenario?

8

Goal: Systematic Tool for Heavy-Tailed Systems

• Sk , X1 + · · ·+Xk.

• Xi: iid, EXi = 0,

P(Xi ≥ x) = x−(α+1), α > 0 or P(Xi ≥ x) = exp(−xα), α ∈ (0,1).

yolo

• P(Sn ∈ A)→ 0

Goal - How fast? P(Sn ∈ A)∼ ?

- What is the most likely scenario? P(Sn ∈ ·|Sn ∈ A)→?

- How to compute? P(Sn ∈ A) = ?

8


• Sk , X1 + · · ·+Xk.



By Law of Large Numbers, P(Sn/n > ε)→ 0.

• P(Sn ∈ A)→ 0




Heavy-Tailed

8


• Sk , X1 + · · ·+Xk.



• By Law of Large Numbers, P(Sn/n > ε)→ 0.

• P(Sn ∈ A)→ 0




Heavy-Tailed

8


• Sk , X1 + · · ·+Xk.



• Sn(t), Sbntc/n, t ∈ [0,1] (scaled process)

• P(Sn ∈ A)→ 0




Heavy-Tailed

8


• Sk , X1 + · · ·+Xk.




t

(t)00S5

0 5

• P(Sn ∈ A)→ 0




Heavy-Tailed

8


• Sk , X1 + · · ·+Xk.




t

00S5(t)

0 1

• P(Sn ∈ A)→ 0




Heavy-Tailed

8


• Sk , X1 + · · ·+Xk.




t

0S10(t)

0 1

• P(Sn ∈ A)→ 0




Heavy-Tailed

8


• Sk , X1 + · · ·+Xk.




t

0S30(t)

0 1

• P(Sn ∈ A)→ 0




Heavy-Tailed

8


• Sk , X1 + · · ·+Xk.




t

S100(t)

0 1

• P(Sn ∈ A)→ 0




Heavy-Tailed

8


• Sk , X1 + · · ·+Xk.




t

S500(t)

0 1

• P(Sn ∈ A)→ 0




Heavy-Tailed

8


• Sk , X1 + · · ·+Xk.




t

0 S∞(t)

0 1

• P(Sn ∈ A)→ 0




↙Typical Sn

Heavy-Tailed

8


• Sk , X1 + · · ·+Xk.




• P(Sn ∈ A)→ 0 if A does not contain 0




Heavy-Tailed

8


• Sk , X1 + · · ·+Xk.





Goal

- How fast? P(Sn ∈ A)∼ ?



Heavy-Tailed

8


• Sk , X1 + · · ·+Xk.








Heavy-Tailed

8


• Sk , X1 + · · ·+Xk.




• P(Sn ∈ A)→ 0 for “general” A⊆ D




Heavy-Tailed

9

Outline

Part 1. Large Deviations for Power Law Tails

R., Blanchet, Zwart (2016)

Under second round review at Annals of Probability

Part 2. Heavy-Tailed Rare Event Simulation

Chen, Blanchet, R., Zwart (2017)

Mathematics of Operations Research

Part 3. Large Deviations for Weibull Tails

Bazbha, Blanchet, R., Zwart (2017)

Submitted to Annals of Applied Probability

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Part 1. Large Deviations for Power Law Tails

i.e., P(Xi ≥ x) = x−(α+1), α > 0

11

What’s already known: principle of a single big jump

“In heavy-tailed systems, rare events arise due to one big anomaly.”

12


t

f (t)

0 1

a

• A = { f ∈ D : f crosses level a on [0,1]}

• P(Sn ∈ A)→ 0α−1 (Ruin Probability of Insurance Firm)

• P(Sn ∈ ·|Sn ∈ A)→ ?

Hult, Lindskog, Mikosch, Samorodnitsky (2005)

↙ Typical Sn

12


t

f (t)

0 1

a


• P(Sn ∈ A)→ 0→ 0α−1 (Ruin Probability of Insurance Firm)

• P(Sn ∈ ·|Sn ∈ A)→ ?


↙ Typical Sn

12


t

f (t)

0 1

a


• P(Sn ∈ A) ∼ ?α−1 (Ruin Probability of Insurance Firm)

• P(Sn ∈ ·|Sn ∈ A)→ ?


↙ Typical Sn

12


t

f (t)

0 1

a



• P(Sn ∈ ·|Sn ∈ A)→ ?


12


t

f (t)

0 1

a



• P(Sn ∈ ·|Sn ∈ A)→ ?


↖This is typical Sn|Sn ∈ A

12


t

f (t)

0 1

a


• P(Sn ∈ A) ∼ cn−αsupt∈[0,1] (Ruin Probability of Insurance Firm)

• P(Sn ∈ ·|Sn ∈ A)→ ?



12


t

f (t)

0 1

a



• P(Sn ∈ ·|Sn ∈ A)→ P(Z1[U≤t] ∈ ·) for some r.v.-s Z and U



12


t

f (t)

0 1

a

U

Z






12


t

f (t)

0 1

a

U

Z






Principle of a single big jump

13

Are all heavy-tailed rare events due to a single big jump?

No, by no means, absolutely not:

• Multiple server queues

• Queueing networks

• Re-insured insurance line

• Down-and-in barrier option

• Many more

13

Principle of a single big jump is just a tip of the iceberg!

14

Illustration: What if Large Claims are Reinsured?

t

f (t)

0 1

a

b?

• A = { f ∈ D : f crosses level a on [0,1] & jump sizes ≤ b}

• P(Sn ∈ A) ?∼ cn−α

• P(Sn ∈ ·|Sn ∈ A) ?→ P(Z1[U≤t] ∈ ·) for some r.v.-s Z and U

14


t

f (t)

0 1

a

b

> b


• P(Sn ∈ A) ?∼ cn−α


14


t

f (t)

0 1

a

b

> b


• P(Sn ∈ A) 6∼ cn−α


14


t

f (t)

0 1

a

b

> b


• P(Sn ∈ A) 6∼ cn−α

• P(Sn ∈ ·|Sn ∈ A) 6→ P(Z1[U≤t] ∈ ·) for some r.v.-s Z and U

14


t

f (t)

0 1

a

b

≤ b

≤ b

≤ b

≤ b


• P(Sn ∈ A) 6∼ cn−α


14


t

f (t)

0 1

a

bb

b

b


• P(Sn ∈ A) 6∼ cn−α


14


t

f (t)

0 1

a

bb

b

b


• P(Sn ∈ A) 6∼ cn−α ?⇒ (cn−α)3


14


t

f (t)

0 1

a

bb

b

b


• P(Sn ∈ A) 6∼ cn−α ?⇒ (cn−α)3

• P(Sn ∈ ·|Sn ∈ A) 6→ P(Z1[U≤t] ∈ ·)?⇒ P

(∑

3i=1 Zi1[Ui≤t] ∈ ·

)

14


t

f (t)

0 1

a

bb

b

b


• P(Sn ∈ A) 6∼ cn−α ?⇒ (cn−α)3

• P(Sn ∈ ·|Sn ∈ A) 6→ P(Z1[U≤t] ∈ ·)?⇒ P

(∑

3i=1 Zi1[Ui≤t] ∈ ·

)

Principle of multiple big jumps?

15

Exact Asymptotics for Heavy-tailed Random Walks

P(Xi ≥ x) = x−(α+1), α > 0

Theorem (R., Blanchet, Zwart, 2017)

For “general” A⊆ D

C(A◦)≤ liminfn→∞

P(Sn ∈ A)n−αJ (A)

≤ limsupn→∞

P(Sn ∈ A)n−αJ (A)

≤ C(A−).

• J (A): min #jumps for step functions to be inside A

• C(·): a measure

15


P(Xi ≥ x) = x−(α+1), α > 0



P(Sn ∈ A)∼ n−αJ (A)



15


P(Xi ≥ x) = x−(α+1), α > 0



P(Sn ∈ A)∼ n−αJ (A)



↙ LD power index

16

Back to Our Reinsurance Example: Conjecture Confirmed!

t

f (t)

0 1

a

bb

b

b


• P(Sn ∈ A)∼ n−da/beα

• P(Sn ∈ ·|Sn ∈ A)→ P(

∑da/bei=1 Zi1[Ui≤t] ∈ ·

)

J (A) = da/be= 3

17

Conspiracy vs Catastrophy

A = { f ∈ D : f crosses level a on [0,1] & jump sizes ≤ b}

t

f (t)

0 1

a

b

t

f (t)

0 1

a

bb

b

b

Light-Tailed Claim Size

Reinsurance makes no difference.

Heavy-Tailed Claim Size

Reinsurance helps!

18

Two-sided Random Walks

P(Xi ≥ x) = x−(α+1), α > 0 and P(Xi ≤−x) = x−(β+1), β > 0.


For “general” A⊆ D,

P(Sn ∈ A)∼ n−{αJ (A)+βK(A)}.

• J (A): # of upward jumps

• K(A): # of downward jumps

of step functions that minimize the cost of staying inside A

↙ Cost of Jumps︷︸︸︷

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Connection to Impulse Control Problem

The “rate of decay” is determined by a discrete optimization problem:

αJ (A)+βK(A) = minj,k

αj+βk

subject to ( j,k) ∈ Z2+

Dj,k∩A 6= /0

where

Dj,k = {step functions w/ j upward jumps and k downward jumps}

Different from variational calculus that arises in light-tailed case!

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Example: Barrier Option

t

f (t)

0 1

a

b

• A = { f ∈ D : f is below b at some point & end up above a}

• P(Sn ∈ A)∼ ?n−(α+β )

• P(Sn ∈ ·|Sn ∈ A)→ ?

P(Z11[U1≤t]−Z21[U2≤t] ∈ ·

)

K(A) = 1

J (A) = 1

20


t

f (t)

0 1

a

b


• P(Sn ∈ A)∼ n−(1α+1β )

• P(Sn ∈ ·|Sn ∈ A)→ ?

P(Z11[U1≤t]−Z21[U2≤t] ∈ ·

)

K(A) = 1

J (A) = 1

20


t

f (t)

0 1

a

b


• P(Sn ∈ A)∼ n−(1α+1β )

• P(Sn ∈ ·|Sn ∈ A)→ P(Z11[U1≤t]−Z21[U2≤t] ∈ ·

)

K(A) = 1

J (A) = 1

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Example: Sausage

t

f (t)

0 1

• A = { f ∈ D : f stays between the two curves}

• P(Sn ∈ A)∼ ?n−αn−β

• P(Sn ∈ ·|Sn ∈ A)→ ?

P(

∑3j=1 Z+

j 1[U+j ≤t]−∑

2k=1 Z−k 1[U−k ≤t] ∈ ·

)

J (A) = ?

K(A) = ?

21

Example: Sausage

t

f (t)

0 1



• P(Sn ∈ ·|Sn ∈ A)→ ?

P(

∑3j=1 Z+

j 1[U+j ≤t]−∑

2k=1 Z−k 1[U−k ≤t] ∈ ·

)

J (A)≥ 1

K(A) = ?

21

Example: Sausage

t

f (t)

0 1



• P(Sn ∈ ·|Sn ∈ A)→ ?

P(

∑3j=1 Z+

j 1[U+j ≤t]−∑

2k=1 Z−k 1[U−k ≤t] ∈ ·

)

J (A)≥ 2

K(A) = ?

21

Example: Sausage

t

f (t)

0 1



• P(Sn ∈ ·|Sn ∈ A)→ ?

P(

∑3j=1 Z+

j 1[U+j ≤t]−∑

2k=1 Z−k 1[U−k ≤t] ∈ ·

)

J (A)≥ 2

K(A)≥ 1

21

Example: Sausage

t

f (t)

0 1



• P(Sn ∈ ·|Sn ∈ A)→ ?

P(

∑3j=1 Z+

j 1[U+j ≤t]−∑

2k=1 Z−k 1[U−k ≤t] ∈ ·

)

J (A)≥ 2

K(A)≥ 2

21

Example: Sausage

t

f (t)

0 1



• P(Sn ∈ ·|Sn ∈ A)→ ?

P(

∑3j=1 Z+

j 1[U+j ≤t]−∑

2k=1 Z−k 1[U−k ≤t] ∈ ·

)

J (A)≥ 3

K(A)≥ 2

21

Example: Sausage

t

f (t)

0 1



• P(Sn ∈ ·|Sn ∈ A)→ ?

P(

∑3j=1 Z+

j 1[U+j ≤t]−∑

2k=1 Z−k 1[U−k ≤t] ∈ ·

)

J (A) = 3

K(A) = 2

21

Example: Sausage

t

f (t)

0 1


• P(Sn ∈ A)∼ n−(3α+2β )

• P(Sn ∈ ·|Sn ∈ A)→ ?

P(

∑3j=1 Z+

j 1[U+j ≤t]−∑

2k=1 Z−k 1[U−k ≤t] ∈ ·

)

J (A) = 3

K(A) = 2

21

Example: Sausage

t

f (t)

0 1


• P(Sn ∈ A)∼ n−(3α+2β )

• P(Sn ∈ ·|Sn ∈ A)→ P(

∑3j=1 Z+

j 1[U+j ≤t]−∑

2k=1 Z−k 1[U−k ≤t] ∈ ·

)

J (A) = 3

K(A) = 2

21

Example: Sausage

t

f (t)

0 1


• P(Sn ∈ A)∼ exp(−nI∗)

I∗: sol’n of optimization problem over absolutely continuous functions.

in case Xi’s are light tailed

21

Example: Sausage

Conspiracy

Connection to variational problems

(continuous optimization)

Catastrophy

Connection with impulse control

(discrete optimization)

22

Levy Processes and Multidimensional Processes

• Same results for Levy processes.

• Similar results for vector valued processes with independentcomponents as well (for both Levy processes and random walks).

Analysis of Many Server Queues and even Queueing Networks!

23

Example - Stochastic Fluid Network

Queue 1

Queue 2

Queue 3µ1 = 1

µ2 = 1

µ3

80%

80%

10%

10%

ρ1 = 0.8

ρ2 = 0.8

ρ3 = 1

• Job size distribution: Pareto with α1 = 1, α2 = 1, α3 = 3

• Queue 3 experiences congestion because of what?

• Optimization problem reduces to a knapsack type problem

23


0.8

0.1

Queue 1

Queue 2

Queue 3µ1 = 1

µ2 = 1

µ3

80%

80%

10%

10%

ρ1 = 0.8

ρ2 = 0.8

ρ3 = 1




23


0.8

0.720.1

Queue 1

Queue 2

Queue 3µ1 = 1

µ2 = 1

µ3

80%

80%

10%

10%

ρ1 = 0.8

ρ2 = 0.8

ρ3 = 1




23


0.8

0.72Congestion!

(if µ3 < 1+0.8+0.72 = 2.52)

Cost = α1 = 1

0.1

Queue 1

Queue 2

Queue 3µ1 = 1

µ2 = 1

µ3

80%

80%

10%

10%

ρ1 = 0.8

ρ2 = 0.8

ρ3 = 1




23


Queue 1

Queue 2

Queue 3µ1 = 1

µ2 = 1

µ3

80%

80%

10%

10%

ρ1 = 0.8

ρ2 = 0.8

ρ3 = 1




23


0.8

0.8

Queue 1

Queue 2

Queue 3µ1 = 1

µ2 = 1

µ3

80%

80%

10%

10%

ρ1 = 0.8

ρ2 = 0.8

ρ3 = 1




23


0.8

0.8Congestion!

(if µ3 < 1+0.8+0.8 = 2.6)

Cost = α1 +α2 = 2

Queue 1

Queue 2

Queue 3µ1 = 1

µ2 = 1

µ3

80%

80%

10%

10%

ρ1 = 0.8

ρ2 = 0.8

ρ3 = 1




23


Queue 1

Queue 2

Queue 3µ1 = 1

µ2 = 1

µ3

80%

80%

10%

10%

ρ1 = 0.8

ρ2 = 0.8

ρ3 = 1




23


Congestion!

Cost = α3 = 3

Queue 1

Queue 2

Queue 3µ1 = 1

µ2 = 1

µ3

80%

80%

10%

10%

ρ1 = 0.8

ρ2 = 0.8

ρ3 = 1




24

Part 2. Heavy-Tailed Rare Event Simulation

Computing P(Sn ∈ A) for finite n

25

Challenge of Rare Event Simulation

Monte Carlo simulations as repetitive random experiments:

e.g. Coin flip: want to estimate P(Head)

Suppose≈ 10−6

• Flip the coin 100 times

• Count the number of head

• Divide by 100 and report the number

Should be reasonably close to 1/2

25



e.g. Coin flip: want to estimate P(Edge)

Suppose≈ 10−6


• Count the number of Edge


Should be roughly 1/2

25




Suppose≈ 10−6


• Count the number of Edge : most likely to be 0



25




Suppose≈ 10−6



• Divide by 100 and report the number : and hence, likely to be 0


25




Suppose≈ 10−6




Is 0 a useful answer?

No.

e.g., Nuclear Meltdown, Large-Scale Blackout, Large Financial Loss

25




Suppose≈ 10−6




Is 0 a useful answer? No.

e.g., Nuclear Meltdown, Large-Scale Blackout, Large Financial Loss

25



e.g. Coin flip: want to estimate P(Edge)Suppose≈ 10−6





25




• Flip the coin 100 a few million times


• Divide by the total number of flips and report the number


25




• Flip the coin 100 a few million times


• Divide by the total number of flips and report the number

Much harder than P(Head)

26

Importance Sampling

for P(Sn ∈ A)

• Construct an alternative universe

- i.e., Consider an ”importance distribution” Q

• Perform experiments there

- i.e., Sample I(1)Edge, . . . ,I(m)Edge from Q

i.e., as well as the likelihood ratio(

dPdQ

)(1), . . . ,

(dPdQ

)(m)

• Recover the true probability using the relationship between the twoparallel universes

- i.e., Report1m

m

∑i=1

I(i)Edge(

dPdQ

)(i)

Finding a good alternative universe Q is crucial.

26

Importance Sampling

for P(Sn ∈ A)






dPdQ

)(1), . . . ,

(dPdQ

)(m)


- i.e., Report1m

m

∑i=1

I(i)Edge(

dPdQ

)(i)as an estimate of P(Edge)


26

Importance Sampling for P(Sn ∈ A)






dPdQ

)(1), . . . ,

(dPdQ

)(m)


- i.e., Report1m

m

∑i=1

I(i)Edge(

dPdQ

)(i)as an estimate of P(Edge)


26



- i.e., Consider an ”importance distribution” Qn


- i.e., Sample I(1){Sn∈A}, . . . ,I(m)

{Sn∈A} from Qn


dPdQn

)(1), . . . ,

(dP

dQn

)(m)


- i.e., Report1m

m

∑i=1

I(i){Sn∈A}

(dP

dQn

)(i)as an estimate of P(Sn ∈ A)

Finding a good alternative universe Qn is crucial.

26



- i.e., Consider an ”importance distribution” Qn


- i.e., Sample I(1){Sn∈A}, . . . ,I(m)

{Sn∈A} from Qn


dPdQn

)(1), . . . ,

(dP

dQn

)(m)


- i.e., Report1m

m

∑i=1

I(i){Sn∈A}

(dP

dQn

)(i)as an estimate of P(Sn ∈ A)

Finding a good alternative universe Qn is crucial.

︸︷︷︸I{Sn∈A}

dPdQn

: IS estimator

27

Goal: Strongly Efficient IS Estimator

I{Sn∈A}dP

dQnis a strongly efficient estimator for P(Sn ∈ A), if

EQn

(I{Sn∈A}

dPdQn

)2

∼ P(Sn ∈ A)2

⇒ Number of simulation runs required remains bounded.

Notoriously Hard for Heavy-Tailed Processes.

27


I{Sn∈A}dP


EQn

(I{Sn∈A}

dPdQn

)2

∼ P(Sn ∈ A)2



Error 2

27


I{Sn∈A}dP


EQn

(I{Sn∈A}

dPdQn

)2

∼ P(Sn ∈ A)2



Error 2Target Quantity2

28

What is a good alternate universe Qn for P(Sn ∈ A)?

General principle for making I{Sn∈A}dP

dQnan efficient estimator:

- Choose Qn(·) as close to P(·|Sn ∈ A) as possible.

- Make sure that dPdQn

does not blow up.

29

First All-Purpose Simulation Scheme for Heavy-Tails

• Bγ , {paths w/ at least J (A) jumps of size > γ}

⇒ J (A) = J (Bγ)

• Fix w ∈ (0,1) and define

Qn( ·), wP( ·)+(1−w)P( · | Sn ∈ Bγ)

• If we choose γ so that J (A\Bγ)≥ 2J (A),

EQn

(1{Sn∈A}

dPdQn

)2

≤ 1w

P(Sn ∈ A\Bγ)+P(Sn ∈ A) ·P(Sn ∈ Bγ)

∼(n−αJ (A))2 ∼

(P(Sn ∈ A)

)2

Zn , 1{Sn∈A}dP

dQnis strongly efficient for P(Sn ∈ A)!

Chen, Blanchet, R., and Zwart (2017)

29



⇒ J (A) = J (Bγ)


Qn( ·), wP( ·)+(1−w)P( · | Sn ∈ Bγ)


EQn

(1{Sn∈A}

dPdQn

)2

≤ 1w


∼(n−αJ (A))2 ∼

(P(Sn ∈ A)

)2

Zn , 1{Sn∈A}dP



dPdQn

not too big

↑

29



⇒ J (A) = J (Bγ)


Qn( ·), wP( ·)+(1−w)P( · | Sn ∈ Bγ)


EQn

(1{Sn∈A}

dPdQn

)2

≤ 1w


∼(n−αJ (A))2 ∼

(P(Sn ∈ A)

)2

Zn , 1{Sn∈A}dP



dPdQn

not too big

↑ Qn(·) close to P(·|Sn ∈ A)↗

29


• Bγ , {paths w/ at least J (A) jumps of size > γ} ⇒ J (A) = J (Bγ)


Qn( ·), wP( ·)+(1−w)P( · | Sn ∈ Bγ)


EQn

(1{Sn∈A}

dPdQn

)2

≤ 1w


∼(n−αJ (A))2 ∼

(P(Sn ∈ A)

)2

Zn , 1{Sn∈A}dP



dPdQn

not too big

↑ Qn(·) close to P(·|Sn ∈ A)↗

29



⇒ J (A) = J (Bγ)


Qn( ·), wP( ·)+(1−w)P( · | Sn ∈ Bγ)


EQn

(1{Sn∈A}

dPdQn

)2

≤ 1w


∼(n−αJ (A))2 ∼

(P(Sn ∈ A)

)2

Zn , 1{Sn∈A}dP



n−αJ (A\Bγ )

29



⇒ J (A) = J (Bγ)


Qn( ·), wP( ·)+(1−w)P( · | Sn ∈ Bγ)


EQn

(1{Sn∈A}

dPdQn

)2

≤ 1w


∼(n−αJ (A))2 ∼

(P(Sn ∈ A)

)2

Zn , 1{Sn∈A}dP



n−αJ (A\Bγ ) n−αJ (A)

29



⇒ J (A) = J (Bγ)


Qn( ·), wP( ·)+(1−w)P( · | Sn ∈ Bγ)


EQn

(1{Sn∈A}

dPdQn

)2

≤ 1w


∼(n−αJ (A))2 ∼

(P(Sn ∈ A)

)2

Zn , 1{Sn∈A}dP



n−αJ (A\Bγ ) n−αJ (A) n−αJ (Bγ )

29




Qn( ·), wP( ·)+(1−w)P( · | Sn ∈ Bγ)


EQn

(1{Sn∈A}

dPdQn

)2

≤ 1w


∼(n−αJ (A))2 ∼

(P(Sn ∈ A)

)2

Zn , 1{Sn∈A}dP




29




Qn( ·), wP( ·)+(1−w)P( · | Sn ∈ Bγ)


EQn

(1{Sn∈A}

dPdQn

)2

≤ 1w


∼(n−αJ (A))2

∼(P(Sn ∈ A)

)2

Zn , 1{Sn∈A}dP




29




Qn( ·), wP( ·)+(1−w)P( · | Sn ∈ Bγ)


EQn

(1{Sn∈A}

dPdQn

)2

≤ 1w


∼(n−αJ (A))2 ∼

(P(Sn ∈ A)

)2

Zn , 1{Sn∈A}dP




30

How to ensure J (A\Bγ)≥ 2J (A): Reinsurance Example

t

f (t)

0 1

a

b

γ≤ b

≤ b

b≥

• A = {paths that cross level a on [0,1] & jump sizes ≤ b}

• Bγ = {paths with at least 3 jumps of size > γ}

• A\Bγ = {paths that cross level a on [0,1]& all jump sizes ≤ b & at most 2 jumps of size > γ}

30


t

f (t)

0 1

a

b

γ≤ b

≤ b

b≥




∈ A

30


t

f (t)

0 1

a

b

γ≤ b

≤ b

b≥




∈ Bγ

30


t

f (t)

0 1

a

b

γ≤ b

≤ b

b≥




/∈ A\Bγ

30


t

f (t)

0 1

a

b

γ≤ b

≤ b

b≥≤ γ

≤ γ

≤ γ




/∈ A\Bγ

30


t

f (t)

0 1

a

b

γ≤ b

≤ b

≤ γ

≤ γ

≤ γ

≤ γ




∈ A\Bγ

30


t

f (t)

0 1

a

b

γ≤ b

≤ b

≤ γ

≤ γ

≤ γ

≤ γ




7 = J (A\Bγ)> 2J (A) = 6

31

Numerical Experiments for Reinsurance Example

2000 4000 6000 8000 10000

0.08

0.10

0.12

0.14

0.16

0.18

Numerical results − reinsurance policy

n

leve

l of p

reci

sion

β=1.2

β=1.4

β=1.6

β=2.0

32

Numerical Results for Queueing Network

2000 4000 6000 8000 10000 12000

0.10

0.15

0.20

0.25

Numerical results − fluid queue

n

leve

l of p

reci

sion

β=(1.2,1.2,2.3)

β=(1.6,1.4,3.1)

β=(1.8,2.0,3.0)

β=(2.3,2.0,3.5)

33

Part 3. Large Deviations for Weibull Tails

P(Xi ≥ x) = exp(−xα), α ∈ (0,1)

34

Rate function depends on the size of the jump

Theorem (Bazhba, Blanchet, R., Zwart 2017)

Suppose that P(Xi ≥ x) = exp(−xα), α ∈ (0,1). Then

P(Sn ∈ A)∼ exp(−nα inff∈A

I( f ))

where

I( f ) =

{∑t(

f (t)− f (t−))α

, if f is a nondecreasing step function

∞, otherwise.

35

Back to our old example

t

f (t)

0 1

a

bb

b

a−2b


• P(Sn ∈ A

)∼ exp

(−nα I∗

)where I∗ = bα +bα +(a−2b)α .

35

Back to our old example

t

f (t)

0 1

a

bb

b

a−2b


• P(Sn ∈ A

)∼ exp

(−nα I∗

)where I∗ = bα +bα +(a−2b)α .

Jump Sizes Matter

36

Queue Length Asymptotics for the Weibull GI/GI/d Queue

In case service time distribution is Weibull, i.e., P(S > x) = exp(−xα),

d

2

1

Tail Asymptotics? Most likely scenario?

So far, NOT EVEN a reasonable conjecture!

36



P(Queue length exeeds n)∼ exp(−c∗nα)

where

c∗ = mind

∑i=1

xαi s.t.

λ s−d

∑i=1

(s− xi)+ ≥ 1 for some s ∈ [0,γ],

x1, ...,xd ≥ 0 .

• Special case of Lα -norm minimization problem.

• We have explicit solution.

Solution to Open Problem Posed by Whitt (2000)

36



P(Queue length exeeds n)∼ exp(−c∗nα)

where

c∗ = mind

∑i=1

xαi s.t.

λ s−d

∑i=1

(s− xi)+ ≥ 1 for some s ∈ [0,γ],

x1, ...,xd ≥ 0 .

• Special case of Lα -norm minimization problem.

• We have explicit solution.

Solution to Open Problem Posed by Whitt (2000)

Nontrivial Tradeoff between

Size and Number of Jumps

37

Summary

• Systematic tools for rare-event analysis of heavy-tailed systems

• The principle of multiple (least expensive) big jumps

• Solved longstanding open problems in simulation and queueing theory

• New connections btwn rare event analysis and (discrete) optimization

e.g., impulse control, knapsack problem, Lα -norm minimization α < 1

on heavy-tailed rare event analysis · design of scheduling algorithms, simulation algorithms, etc...

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