Stability using fluid limits: Illustration through an example

"Push-Pull" queuing network

Yoni Nazarathy*EURANDOM

Contains Joint work with Gideon Weiss and Erjen Lefeber

Universiteit GentOctober 14, 2010

* Supported by NWO-VIDI Grant 639.072.072 of Erjen Lefeber

KumarSeidmanRybkoStoylar

1 2

34

Purpose of the talkPart 1: Outline research on Multi-Class

Queueing Networks (with Infinite Supplies)

- N., Weiss, 2009- Ongoing work with Lefeber

Part 2: An overview of “the fluid limit” method for stability of queueing networks

Key papers:- Rybko, Stolyar 1992- Dai 1995- Bramson/Mandelbaum/Dai/Meyn… 1990-2000

Recommended Book:- Bramson, Stability of Queueing Networks, 2009

PART 1: MULTI-CLASS QUEUEING NETWORKS (WITH INFINITE SUPPLIES)

1( )Q t

2 ( )Q t

1S

2S

•Continuous Time, Discrete Jobs

• 2 job streams, 4 steps

•Queues at pull operations

• Infinite job supply at 1 and 3

• 2 servers

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:

Processing Times

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

1 2

34

1 2 1 2, 1 or , >1

1 3[ ] 1, [ ] 1 (for simplicity)E E

i.i.d.k

2 2 4 2[ ] , [ ]E E

Policies

1i Policy: Pull priority (LBFS)

Policy: Linear thresholds

1i

1 2

34

TypicalBehavior:

1( )Q t

2 ( )Q t

2,4

1S 2S

3

4

2 1

1,3

TypicalBehavior:

5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0

5

1 0

1 2 2Q Q

2 1 1Q Q

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

1S

2S

Push

Pull

Pull

Push

1,3

1Q

2Q

1Q

2Q

8

is strong Markov with state space .

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

( )X t

1 2

34

Queue Residual

Stability ResultsTheorem (N., Weiss): Pull-priority, , is PHR 1i ( )X t

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

Theorem (in progress) (Lefeber, N.): , pull-priority, is PHR if More generally, when there is a matrix such that is PHR when

e.g:

Theorem (Lefeber, N.): , pull-priority, if , is PHR 1i 2M

2 1M k 1i 11 1 k

( )X t

( )X t

1( ,..., )k k MA

spectral radius 1A ( )X t

Current work: Generalizing to servers2M

1i

1 1 1 2 3( 1)( 1)( 1)A 3M

Heuristic Modes Graph for M=3 Pull-Priority 1i

Heuristic Stable Fluid Trajectory of M=3 Pull-Priority Case1i

PART 2: THE “FLUID LIMIT METHOD” FOR STABILITY

Main IdeaEstablish that an “associated” deterministic system is “stable”

The “framework” then impliesthat is “stable”

Nice, since stability of is sometimes easier to establish than directly working

( )X t

( )X t

( )X t

( )X t

Stochastic Model and Fluid Model

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 t

T t T t t T t T t t

t S T t

Q q

Q t Q D t D t

2 4 1 1

0 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= t

k= ( )kT t

2 1 1 1 1 2 2 3

0 0

1 2 4 2 1 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

1 2

34

1S 2S

Comments on the Fluid Model• T is Lipschitz and thus has derivative almost everywhere

•Any Y=(Q,T) that satisfies the fluid model is called a solution

• In general (for arbitrary networks) a solution can be non-unique

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”Stable

Fluid ModelPositive Harris

Recurrence

Weakly StableFluid Model

Technical Conditions on

Markov Process (Pettiness)

Rate Stability:

Association of Fluid Model

To Stochastic System

1 2( ) ( )lim 0 a.s.t

Q t Q t

t

Association of Fluid Model and Stochastic System

fluid scalings

( , )( , )

nn Y ntY t

n

r

( ) ( ) ( ) is

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

fluid limit Y t Q t T t

r Y

is with

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

associ

el solution

atedY Y

Lyapounov Proofs for Fluid Stability

• When , it stays at 0.

• When , at regular

points of t, .

( )f t

Need: for every solution of fluid model:

( ) 0f t

( ) 0f t

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

( )f t

1:i

1:i

19

20

QUESTIONS?