stability using fluid limits: illustration through an example "push-pull" queuing network...
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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
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KumarSeidmanRybkoStoylar
1 2
34
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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
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PART 1: MULTI-CLASS QUEUEING NETWORKS (WITH INFINITE SUPPLIES)
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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
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“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
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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
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8
is strong Markov with state space .
A Markov Process ( ) Q(t) U(t)X t
( )X t
1 2
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Queue Residual
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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
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Heuristic Modes Graph for M=3 Pull-Priority 1i
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Heuristic Stable Fluid Trajectory of M=3 Pull-Priority Case1i
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PART 2: THE “FLUID LIMIT METHOD” FOR STABILITY
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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
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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
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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
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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
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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
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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
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QUESTIONS?