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On Control of Queueing Networks and The Asymptotic Variance Rate of Outputs. Ph.d Summary Talk Yoni Nazarathy Supervised by Prof. Gideon Weiss. Haifa Statistics Seminar, November 19, 2008. The Problem Domain. PLANT. OUTPUT. Desired: Low Holding Costs Low Resource Idleness - PowerPoint PPT Presentation
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1
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
2
PLANTOUTPUT
The Problem Domain
Finite Horizon [0,T]
Desired:
1. Low Holding Costs
2. Low Resource Idleness
3. Low Output Variability
3
Queues and NetworksA Brief Survey
4
Mean File Size
1 1 1
Phenomena of Queues
5
Key Phenomena• Stability / instability
• Congestion increases with utilization
• Variability of primitives causes larger queues
• Steady state
• Little’s law
• Flashlight principle
• State space collapse
…
6
Queueing Networks
7
Multi-Class
=2
8
Infinite Inputs
9
Miracles
10
PLANTOUTPUT
The Problem Domain
Finite Horizon [0,T]
Desired:
1. Low Holding Costs
2. Low Resource Idleness
3. Low Output Variability
11
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
12
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
13
• 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
14
3
1
2
3
1
2
3
1
2
3
1
2
0 10 20 30 40
5
10
15
20
25
30
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
15
0 10 20 30 400
500
1000
1500
2000
0 10 20 30 400
500
1000
1500
2000
0 10 20 30 400
500
1000
1500
2000
0 10 20 30 400
500
1000
1500
2000
0 10 20 30 400
50
100
150
200
0 10 20 30 400
50
100
150
200
0 10 20 30 400
50
100
150
200
0 10 20 30 400
50
100
150
200
0 10 20 30 400
5
10
15
20
0 10 20 30 400
5
10
15
20
0 10 20 30 400
5
10
15
20
0 10 20 30 400
5
10
15
20
1N
10N
100N
seed 1 seed 2 seed 3 seed 4
Asymptotic Optimality
16
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
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
18
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
19
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:
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
20
KSRS
1 2
34
21
Push pull vs. KSRS
Push Pull
KSRS with“Good” policy
22
Stability Result
( ) Q(t), U(t)X t
1 2
34
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:
23
PLANTOUTPUT
The Problem Domain
Finite Horizon [0,T]
Desired:
1. Low Holding Costs
2. Low Resource Idleness
3. Low Output Variability
24
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
26
**
* *
VV
V V
BRAVO Effect
27
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
28
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
29
Infinite Supply Re-entrant Line
4
2
1C
1 3
56
78
10 9
( )D t
2C 3C
4C
2
13
1
: For any stable policy (e.g. LBFS): .k
k C
mkk C
Thm V
1
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
30
“Renewal Like”
4
2
1C
1 3
56
78
10 9
2C 3C
4C1
1
2
3
kk C
kk C
V
m
1C
1
6
8
10
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
31
A Future Direction
32
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
?
34
Now Lets Do!לחיים