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