asaf cohen (joint work with rami atar) department of mathematics university of michigan financial...

A MULTICLASS QUEUEING MODEL IN THE MODERATE-DEVIATION HEAVY-TRAFFIC

REGIME

Asaf Cohen (joint work with Rami Atar) Department of Mathematics

University of Michigan

Financial Mathematics SeminarUniversity of Michigan

March 11, 2015

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Contents

The Differential Game

The Model

Varadhan’s Lemma

The Approximation

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Contents

The Differential Game

The Model

Varadhan’s Lemma

The Approximation

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The Model

Basic definitions

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size of thei-th buffer

arrival processfinite 2nd moment

controlled serviceprocessfinite 2nd moment

controlled rejectionprocess

buffers

buffer 1 buffer 2 buffer 3 buffer 4

The Model

Basic definitions

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• Moderate-deviation rate parameters

• Arrival rates• Departure rates • The system is critically loaded

size of thei-th buffer

arrival processfinite 2nd moment

controlled serviceprocessfinite 2nd moment

controlled rejectionprocess

buffers

Proportion of time

that server i is busy

The Model

Scaling

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(1) and (2) and

Assumption 1

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The sequencesatisfies the moderate-deviation principle with rate parameters and rate function . That is, for every and every closed and open sets one has

The Model

Sufficient condition:

The Model

Cost and value function

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Alternative costs:

Scaling implies Technical difficulties

Non-stationary solutions

Value:

Cost:

Leads to a stationary robust control

discount factor

The Model

Why Moderate-Deviation?

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• While there are many diffusion-scaling papers, there are very few papers with the large/moderate-deviations scaling.

• In diffusion scaling the probability of overflow is of order 1, while in the large/moderate deviation the order is very low.

• In the large/moderate deviation we get a robust control.

• Unlike the large-deviation, which suffers from high complexity. Under the moderate-deviation, we solve the problem completely.

TO ADD COLORS LATER ON

AND TO SIMPLIFY THE POLICY LIKE IN THE 20MIN

The Model

State space collapse

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where is called the workload.

Cost

The Model

Varadhan’s Lemma (intuition for the differential game)

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Suppose that (with support ) satisfies the large deviation principle with the good rate function and let be a continuous function.

Assume further (to simplify the proof) that is compact.Then,

Intuition:

The Model

State space collapse

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where

Cost

Deterministic differential game

Cost:

Value:

AnticipatingElliot-Kalton

is related to

Theorem

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The Model

Assume that and that Assumption 1 holds. Then,

where

To his end, we analyze the differential game…(We will refer only to the one dimensional case here and work with . )

Contents

The Differential Game

The Model

Varadhan’s Lemma

The Approximation

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The Differential Game

Intuition

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Cost:

Value:

AnticipatingElliot-Kalton

Properties of the game

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• Explicit expression for the value function .

• Optimal strategy for the minimizer

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

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The Differential Game

Properties of the game

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• Simple controls for the maximizer that achieves : - Stopping immediately as the dynamics hits zero, .

- until the first time the dynamics drop below and then decreases until the dynamics hit zero (solves PDE, independent of the minimizer choice). Also, the terminal time is smaller than .

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0.5

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• Under this control, if then the minimizer would prefer to reject immediately!

The Differential Game

Contents

The Differential Game

The Model

Varadhan’s Lemma

The Approximation

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Varadhan’s Lemma

Varadhan’s Lemma

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Suppose that (with support ) satisfies the Large deviation principle with the good rate function and let be a continuous function.

Assume further (to simplify the proof) that is compact.Then,

Intuition:

Varadhan’s Lemma (proof)

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Lower bound:Fix . There is s.t.So,

and

Since is arbitrary

Varadhan’s Lemma

Varadhan’s Lemma (proof)

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Upper bound:For every there is s.t.

Since is compact there is a finite cover ( independent of )

Varadhan’s Lemma

Contents

The Differential Game

The Model

Varadhan’s Lemma

The Approximation

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The Approximation

Deterministic differential game

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Cost:

Value:

anticipating

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We prove it by showing that for every sequence one has

and that, there is a sequence of policies for which

The Approximation

We will refer only to the one dimensional case here, and work with …

Theorem

Assume that and that Assumption 1 holds. Then,

where

Lower bound

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

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0.5

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1.5

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Notice that depends on the control. Now we consider asequence of controls . So, we consider .But how?

We would like to center around .

The Approximation

less than

Lower bound

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Take

(3) Fix . There is an interval s.t.

(1) s.t. stopping afterwards is too expensive.

(2) Divide into small intervals:

(4) Divide, the interval into small time intervals. Recall that termination time of the game is smaller than . (5) Now, continue centering around the second part of . There is an interval s.t.

The Approximation

Lower bound

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We focus on ‘s for which for a specific .Recall that there are only finitely many such .

The Approximation

is linear

Lower bound

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So,

The Approximation

Service (from workload to buffers): W.o.l.g. assume that

Asymptotically optimal policy

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Rejections: If overflow occurs (happens with low prob.) Otherwise, reject from buffer .

Low priority: let the cheapest buffer, which is not almost full (up to ) to be filled.Serve the others with rate higher than

The Approximation

Asymptotically optimal policy

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Buffer 1 Buffer 2 Buffer 3 Buffer 4 Buffer 5 Buffer 6

Assume and the buffers have the same sizes

The Approximation

cheapest buffer

THANK YOU!

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The Model

State space collapse

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where

Cost

Deterministic differential game

Cost:

Value:

anticipating is related to