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Positive Harris Recurrence and Diffusion Scale Analysis of

a Push-Pull Queueing Network

Yoni Nazarathy and Gideon Weiss University of Haifa

ValueTools ConferenceAthens, 21 – 23 October, 2008

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Full UtilizationWithout

Congestion

3 2 ( )Q t

4 ( )Q t

1S

2S

• 2 job streams, 4 steps

• Queues at 2 and 4

• Infinite job supply at 1 and 3

• 2 servers

The Push-Pull Network

1 2

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1S 2S

2 4( ), ( )Q t Q t• Control choice based on

• No idling, FULL UTILIZATION

• Preemptive resume

Push

Push

Pull

Pull

Push

Push

Pull

Pull

2Q

4Q

4

Configurations• Inherently stable network

• Inherently unstable network

Assumptions(A1) SLLN

(A2) I.I.D. + Technical assumptions

(A3) Second moment

Processing Times

Previous Work (Kopzon et. al.):

{ , 1,2,...}, 1, 2,3,4jk k j k

1 2

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1 1lim , a.s. 1, 2,3,4

nj

kj

nk

kn

2 1 2Var( ) , 1,2,3,4k k kc k

1 ~ exp( ), 1, 2,3, 4k k k

1 2

4 3

1 2

4 3

5

Policies

1 2

4 3

Inherently stable

Inherently unstable

Policy: Pull priority (LBFS)

Policy: Linear thresholds

1 2

4 3

1 2

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TypicalBehavior:

2 ( )Q t

4 ( )Q t

2,4

1S 2S

3

4

2 1

1,3

TypicalBehavior:

50 1 00 1 50 2 00 2 50 3 00

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1 0

2 2 4Q Q

4 1 2Q Q

Server: “don’t let opposite queue go below threshold”

1S

2SPush

Pull

Pull

Push

1,3

6

Similar to KSRSBut different

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KSRS

1 2

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Push pull vs. KSRS

Push Pull

KSRS with“Good” policy

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Results

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Contribution

1 2

4 3

Inherently stable Inherently unstablePull priority policy Linear threshold policies

1 2

4 3

1 2

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Results:Assumptions:

(A3 )Second moments

Thm 1: Fluid limit model stability

Thm 2: Positive Harris recurrence

Thm 3: Diffusion limit

(A1 )SLLN

(A2 )I.I.D. + technical

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Fluid Stability

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Stochastic Model and Fluid Limit Model

1

1 4 2 3

k

k

1

Dynamics

( ) sup{ : }

(0) 0, ( )( ) ( ) , ( ) ( )

D ( ) ( ( ))(0) 0, Q (t) 0( ) (0) ( ) ( )

nj

k kj

k k

k k

k

k k k k

S t n t

T T tT t T t t T t T t t

t S T tQQ t Q D t D t

4 1 2 10 0

Pull priority policy

( ) ( ) 0 ( ) ( ) 0t t

Q s dT s Q s dT s 4 1 2 1 2 2 4 30 0

2 4 4 4 2 21 20 0

Linear thresholds policy

{0 ( ) ( )} ( ) 0 {0 ( ) ( )} ( ) 0

1 1{ ( ) ( )} ( ) 0 { ( ) ( )} ( ) 0

1 1

1 1

t t

t t

Q s Q s dT s Q s Q s dT s

Q s Q s dT s Q s Q s dT s

2 4 1 2 3 4

Network process( ) ( ), ( ), ( ), ( ), ( ), ( )Y t Q t Q t T t T t T t T t

or

Assume (A1), SLLN

fluid scalings

( , )( , )nn Y ntY tn

r

( ) ( ) ( ) is

if exists and : Y ( , ) ( ), u.o.c.

fluid limit Y t Q t T t

r Y

Fluid limits exists and w.p. 1, satisfy the fluid limit model

Fluid

Fluid

k= t

k= ( )kT t

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Fluid Stability

Thm 1: Under assumption (A1), the fluid limit model is stable.

Definition: A fluid limit model is stable if there exists such that for every fluid solution, whenever then for any .

>0| (0) | 1Q ( ) 0Q t t

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Lyapounov Proof1 2

4 3

Inherently stable

Pull priority policy

Inherently unstableLinear threshold policies

1 2

4 3

2 4( ) ( ) ( )f t Q t Q t

( )f t 2 4( ), ( )Q t Q t

2 ( )Q t

4 ( )Q t

• When , it stays at 0.

• When , at regular

points of t, .

( )f t

For every solution of fluid model:

( ) 0f t

( ) 0f t

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Positive Harris Recurrence

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is strong Markov with state space .

A Markov Process ( ) Q(t) U(t)X t

( )X t

1 2

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Assume (A2), I.I.D. Queue Residual

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Positive Harris Recurrence

Thm 2: Under assumptions (A1) and (A2), the state process is positive Harris recurrent.

Proof follows framework of Jim Dai (1995).

2 Things to Prove:

1. Stability of fluid limit model (Thm 1).

2. Compact sets are petite (minorization).

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DiffusionLimit

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Diffusion Scaling

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Diffusion Limit

Thm 3: Under assumptions (A1), (A2), (A3),

With .

( ) ( ) ( ) B(0, )

n wn nD t T t Q t

10 dimensional Brownian motion

Expressions of are simple, yield asymptotic variance rate of outputs.

( ) 0

wnQ t

Proof Outline: Use positive Harris recurrence to show, , simple calculations along with functional CLT for renewal processes yields the result.

( ) 0

wnQ t

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Consequences of Diffusion Limit1 (Negative correlation of outputs

2 (Diffusion limit does not depend on policy!!!

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Open Questions• Instability when push rate = pull rate• State space collapse • General MCQNs with infinite inputs

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THANK YOU

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Extensions (not in talk)

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• Inherently stable network

• Inherently unstable network

• Unbalanced network

• Completely balanced network

Configuration 1 2

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1 2

4 3

1 2

4 3

1 2 1 2

4 3 4 3

or

1 2

4 3

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Calculation of Rates

1

2

2 3 41 2 1

1 3 2 4

4 1 23 4 3

1 3 2 4

( )

( )

1 4 3 21 , 1

1 2

4 3

1 1 2 2

4 4 3 3

1 2

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Corollary: Under assumption (A1), w.p. 1,

every fluid limit satisfies: .

k - Time proportion server works on k

k -Rate of inflow, outflow through k

Full utilization:

Stability:

( ) , ( )k kk kT t t D t t

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Memoryless Processing(Kopzon et. al.)

1 2

4 3

Inherently stable

Inherently unstable

Policy: Pull priority

Policy: Generalized thresholds

1 2

4 3

1 2

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1S 2S

Alternating M/M/1 Busy Periods

Results:Explicit steady state:

Stability (Foster – Lyapounov)

- Diagonal thresholds

2 ( )Q t

4 ( )Q t

- Fixed thresholds

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