New Stability and Diffusion Results for Multi-Class Queueing Networks

Yongjiang Guo1, Erjen Lefeber2, Yoni Nazarathy3,

Gideon Weiss4, Hanqin Zhang5.

Swinburne ORGANICS Seminar,

March 1, 2011

1 Beijing University of Post and Telecommunications2 Eindhoven University of Technology3 Swinburne University of Technology4 The University of Haifa5 National University of Singapore

INTRODUCTION: STABILITY OF MULTI-CLASS QUEUEING NETWORKS

The KumarSeidmanRybkoStoylar Queueing Network (90’s)

Priority

Priority

In this talk• Outline past research on Multi-Class Queueing

Networks with Infinite Supplies- N., Weiss, 2008-2009

• Overview of New Results- Stability of certain examples- Diffusion Limits

• Outlook- Stability of Queueing Networks is “hard”- Need methods to construct general policies

MULTI-CLASS QUEUEING NETWORKS WITH INFINITE SUPPLIES

1( )Q t

2 ( )Q t

1S

2S

•2 job streams, 4 steps

•Queues at pull operations

• Infinite job supply at 1 and 3

• 2 servers

Example: The Push-Pull Network

1 2

34

∞

∞

1S 2S

1 2( ), ( )Q t Q t•Control choice based on

• No idling, FULL UTILIZATION

• Preemptive resume

Push

Push

Pull

Pull

Push

Push

Pull

Pull

1Q

2Q

“interesting” Configurations:

ProcessingTimes

{ , 1,2,...}, 1, 2,3, 4jk k j kξ ξ= = =

1 2

34

∞

∞

1 2 1 2, 1 or , >1 ρ ρ ρ ρ<

1 3[ ] 1, [ ] 1 (for simplicity)E Eξ ξ= =

i.i.d.kξ

2 2 4 2[ ] , [ ]E Eξ ρ ξ ρ= =

Policies

1iρ <Policy: Pull priority (LBFS)

Policy: Linear thresholds

1iρ >

1 2

34

∞

∞

TypicalBehavior:

1( )Q t

2 ( )Q t

2,4

1S 2S

3

4

2 1

1,3

TypicalBehavior:

50 100 1

5

10

1 2 2Q Qκ=

2 1 1Q Qκ=

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

1S

2SPush

Pull

Pull

Push

1,3

1Q

2Q

1Q

2Q

is strong Markov with state space .

A Markov Process

( )( ) Q(t) U(t)X t =

( )X t

1 2

34

∞

∞Queue Residual

StabilityTheorem (N., Weiss): Pull-priority, , is PHR 1iρ < ( )X t

Theorem (N., Weiss): Linear thresholds, , is PHR 1iρ > ( )X t

Performance AnalysisTheorem (Kopzon, N., Weiss): Closed form for stationary distribution in specific cases and with memory-less assumptions

Diffusion (CLT) Limits of Outputs

Push-Pull Results (earlier work)

Main Tool For Stability ResultsEstablish that an “associated” deterministic system is “stable”

The “framework” then impliesthat is “stable”

Nice, since stability of is sometimes easier to establish

This “fluid framework” was pioneered and exploited in 90’s by Dai, Meyn, Stoylar, Bramson, Weiss, Chen ….

( )X t

( )X t

( )X t

Stochastic and Fluid Equations

1

1 4 2 3

k

k

1

Dynamics

( ) sup{ : }

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

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

nj

k kj

k 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 tQ qQ t Q D t D t

ξ=

−

= ≤

=+ = + === ≥= + −

∑

2 4 1 10 0

Pull priority policy

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

Q s dT s Q s dT s= =∫ ∫

( )1 2 1 2 3 4

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

Fluid

Fluid

k= tµ

k= ( )kT tµ

1 2

34

∞

∞

1S 2S

E.g. Lyapounov Proofs for Fluid Stability

• When , it stays at 0.

• When , at regular

points of t, .( )f t ε•

≤ −

Need: for every solution of fluid model:

( ) 0f t =

( ) 0f t >

2 4( ) ( ) ( )f t Q t Q t= +

( )f t =

1:iρ <

1:iρ >

OUTLINE OF “NEW RESULTS”

For a ring of M servers with Pull-Priority(generalizing Push-Pull)

Stable Fluid Trajectory of M=3 Pull-Priority1iρ >

A re-entrant line with infinite supply

∞

Two re-entrant lines in a push-pull

∞

∞

push

push

pull

pull

Diffusion Limit Results

Summary

• Specific cases of networks with infinite supplies (and full utilization) can be analysed with “some effort”

• General policies for stabilizing general networks remains an “open problem”

Recommended book on the subject: Bramson, 2009, Stability of queueing networks.

Association of Fluid Model and Stochastic System

fluid scalings( , )( , )

nn Y ntY tnωω =

( )r

( ) ( ) ( ) is

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

fluid limit Y t Q t T t

r Yω ω

=

→∞ ⋅ → ⋅

is with if w.p.1 every fluid limit is a fluid mod

associel solution

atedY Y

Stability of Fluid ModelDefinition: A fluid model is stable, if when ever,

there exists T, such that for all solutions, 1 2 1q q+ =

1 2( ) ( ) 0 t TQ t Q t+ = ∀ ≥

Definition: A fluid model is weakly stable, if when ever 1 2 0q q+ =

1 2( ) ( ) 0 t 0Q t Q t+ = ∀ ≥

Main Results of “Fluid Limit Method”

StableFluid Model

Positive Harris Recurrence

Weakly StableFluid Model

Technical Conditions on

Markov Process (Pettiness)

Rate Stability:

Association of Fluid ModelTo Stochastic

System+

+

⇒ 1 2( ) ( )lim 0 a.s.t

Q t Q tt→∞

+=

⇒