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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 Conference Athens, 21 – 23 October, 2008. Full Utilization Without Congestion. 1. 2. 3. 4. The Push-Pull Network. - PowerPoint PPT Presentation
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1
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
2
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
34
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
34
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
34
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
5
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
7
KSRS
1 2
34
8
Push pull vs. KSRS
Push Pull
KSRS with“Good” policy
9
Results
10
Contribution
1 2
4 3
Inherently stable Inherently unstablePull priority policy Linear threshold policies
1 2
4 3
1 2
34
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
11
Fluid Stability
12
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
13
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
14
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
15
Positive Harris Recurrence
16
is strong Markov with state space .
A Markov Process ( ) Q(t) U(t)X t
( )X t
1 2
34
Assume (A2), I.I.D. Queue Residual
17
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).
18
DiffusionLimit
19
Diffusion Scaling
20
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
21
Consequences of Diffusion Limit1 (Negative correlation of outputs
2 (Diffusion limit does not depend on policy!!!
22
Open Questions• Instability when push rate = pull rate• State space collapse • General MCQNs with infinite inputs
23
THANK YOU
24
Extensions (not in talk)
25
• Inherently stable network
• Inherently unstable network
• Unbalanced network
• Completely balanced network
Configuration 1 2
34
1 2
4 3
1 2
4 3
1 2 1 2
4 3 4 3
or
1 2
4 3
26
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
34
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
27
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
34
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
28
29
30
31
32