On Control of Queueing Networks and The Asymptotic Variance Rate of Outputs
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Transcript of On Control of Queueing Networks and The Asymptotic Variance Rate of Outputs
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On Control of Queueing Networks and The Asymptotic Variance Rate
of Outputs
Ph.d Summary Talk
Yoni NazarathySupervised by Prof. Gideon Weiss
Haifa Statistics Seminar,November 19, 2008
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PLANTOUTPUT
The Problem Domain
Finite Horizon [0,T]
Desired:
1. Low Holding Costs
2. Low Resource Idleness
3. Low Output Variability
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Queues and NetworksA Brief Survey
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Mean File Size
1 1 1
Phenomena of Queues
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Key Phenomena• Stability / instability
• Congestion increases with utilization
• Variability of primitives causes larger queues
• Steady state
• Little’s law
• Flashlight principle
• State space collapse
…
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Queueing Networks
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Multi-Class
=2
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Infinite Inputs
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Miracles
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PLANTOUTPUT
The Problem Domain
Finite Horizon [0,T]
Desired:
1. Low Holding Costs
2. Low Resource Idleness
3. Low Output Variability
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Sta
cked
Que
ue L
evel
s
time T
Q1
Q2Q3
Trajectory of a single job
Finished Jobs
Server 1Server 2
1
23
3
10
( )T
kk
Q t dtAttempt to minimize:
Near Optimal Finite Horizon Control
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1 2 3
0
1 1 1 1
0
2 2 1 1 2 2
0 0
3 3 2 2 3 3
0 0
1 3
2
min ( ) ( ) ( )
( ) (0) ( )
( ) (0) ( ) ( )
( ) (0) ( ) ( )
( ) ( ) 1
( ) 1
( ), ( ) 0
T
t
t t
t t
q t q t q t dt
q t q u s ds
q t q u s ds u s ds
q t q u s ds u s ds
u t u t
u t
u t q t
s.t.
Separated Continuous Linear Program (SCLP)
Fluid RelaxationServer 1Server 2
1
23
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• SCLP – Bellman, Anderson, Pullan, Weiss• Piecewise linear solution• Simplex based algorithm, finite time (Weiss)• Optimal Solution:
0 10 20 30 40
0
5
10
15
203 3
2 2
1 1
1 3
2
(0) (0) 15
(0) (0) 1
(0) (0) 8
1.0
0.25
40
Q q
Q q
Q q
T
3( )q t
2 ( )q t
1( )q t
Fluid Solution
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3
1
2
3
1
2
3
1
2
3
1
2
0 10 20 30 40
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15
20
25
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31 1 10 0 1 0 14 4 4 4
{1,2,3} {1,2,3} {1,3} {1}nK
0 { | ( ) 0, }nk nk q t t
{ | ( ) 0, }nk nk q t t
Fluid Tracking1 2 3 4
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0 10 20 30 400
500
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0 10 20 30 400
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1500
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0 10 20 30 400
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0 10 20 30 400
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0 10 20 30 400
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0 10 20 30 400
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0 10 20 30 400
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0 10 20 30 400
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1N
10N
100N
seed 1 seed 2 seed 3 seed 4
Asymptotic Optimality
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PLANTOUTPUT
The Problem Domain
Finite Horizon [0,T]
Desired:
1. Low Holding Costs
2. Low Resource Idleness
3. Low Output Variability
17 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
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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
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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
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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
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Policies
1 2
4 3
Inherently stable
Inherently unstable
Policy: Pull priority (LBFS)
Policy: Linear thresholds
1 2
4 3
1 2
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TypicalBehavior:
2 ( )Q t
4 ( )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
2 2 4Q Q
4 1 2Q Q
Server: “don’t let opposite queue go below threshold”
1S
2S
Push
Pull
Pull
Push
1,3
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KSRS
1 2
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Push pull vs. KSRS
Push Pull
KSRS with“Good” policy
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Stability Result
( ) Q(t), U(t)X t
1 2
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Queue Residual
is strong Markov with state space
( )X t
Theorem: Under assumptions (A1) and (A2), X(t) is positive Harris recurrent.
Proof follows framework of Jim Dai (1995)
2 Things to Prove:
1. Stability of fluid limit model
2. Compact sets are petite
Positive Harris Recurrence:
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PLANTOUTPUT
The Problem Domain
Finite Horizon [0,T]
Desired:
1. Low Holding Costs
2. Low Resource Idleness
3. Low Output Variability
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Example 1: Stationary stable M/M/1, D(t) is PoissonProcess:( )
Example 2: Stationary M/M/1/1 with . D(t) is RenewalProcess(Erlang(2, )):
Variability of OutputsVariability of Outputs(1)Vt B o
Asymptotic Variance Rate
of Outputs
t
1( , )D t
3( , )D t
t1( , )X t
3( , )X t 2( , )X t
2( , )D t
Var( ( ))D t
V
21 1 1Var( ( ))
4 8 8tD t t e
Var( ( ))D t t
2
3V
m
For Renewal Processes:
25Taken from Baris Tan, ANOR, 2000.
Previous Work: NumericalPrevious Work: Numerical
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**
* *
VV
V V
BRAVO Effect
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0 .2 0 .4 0 .6 0 .8 1 .0 1 .2
0 .2
0 .4
0 .6
0 .8
BRAVO Effect: A Phenomena
Using a “renewal-reward” method for regenerative simulation for .V
Queues with Restricted Accessibility (Perry et. al.)
V
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Summary of ResultsQueueing System Without Losses Finite Capacity Birth Death Queue
Push Pull Queueing Network Infinite Supply Re-Entrant Line
1*
0
K
ii
V v
stable
BRAVO (?) critical
instable
arrivals
service
V
V
V
1 2
Explicit Expressions
for , V V1
1
2
3
kk C
kk C
V
m
V
Diffusion LimitsDiffusion Limits
Matrix Analytic MethodsSimple
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Infinite Supply Re-entrant Line
4
2
1C
1 3
56
78
10 9
( )D t
2C 3C
4C
2
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1
: For any stable policy (e.g. LBFS): .k
k C
mkk C
Thm V
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1Infinite QueuesSupply
1
1
2 21
1
1 {2,..., } ... ,
1 .
Means: ,...,
Variances: ,...,
1, i=2,...,Ii
I
k
k
kk C
i kk C
K C C
C
m m
m
m
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“Renewal Like”
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2
1C
1 3
56
78
10 9
2C 3C
4C1
1
2
3
kk C
kk C
V
m
1C
1
6
8
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Renewal Output1 1 1 1 2 2 2 2 3 3 3 31 6 8 10 1 6 8 10 1 6 8 10
Job 1 Job 2 Job 3
, , , , , , , , , , , ,....x x x x x x x x x x x x
1 1 1 1 2 2 2 3 3 3 31 6 8 10 6 8 1 1 6 8 10
201, , , , , , , , , , , , ,...x x x x x x x x x x xx
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A Future Direction
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Finite QRate
1Infinite Q
Rate2
α
α
1
Steady State Total Mean Queue
Sizes
An Implication of BRAVO?
?
IT DOESN’T “WORK
”
Finite QRate1/4
Rate1/4
Finite Q
Finite Q Infinite QRate
2
Rate1/2
Infinite Q
Poisson(α)
Overflow
Overflows Priority
Infinite QRate
1
α
Steady State Mean Queue
Sizes
11/4
When rate exceeds ¼
overflows of first queue cause the second server to
mostly give priority to the fast
stream.
Non Monotonic Networks
?
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Now Lets Do!לחיים