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
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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
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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?
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