Yoni Nazarathy Gideon Weiss University of Haifa
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Yoni NazarathyGideon Weiss
University of Haifa
Yoni NazarathyGideon Weiss
University of Haifa
The Asymptotic Variance of the
Output Process of Finite Capacity Queues
The Asymptotic Variance of the
Output Process of Finite Capacity Queues
Meeting of the euro working group on stochastic modeling.Koc university, Istanbul. June 23-25, 2008
Meeting of the euro working group on stochastic modeling.Koc university, Istanbul. June 23-25, 2008
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 2
Sojourn Times
Queue Sizes
Server Idleness
Lost Job Rates
Variability of Outputs
Typical Queueing Performance MeasuresTypical Queueing Performance Measures
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Variability of OutputsVariability of Outputs
Sometimes related to
down-stream queue sizes
Aim for little
variability over [0,T]
Var ( )D t V
Queueing Networks Setting
Manufacturing Setting
( )Vt o t Asymptotic
Variance Rate of Outputs
t
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 4
Previous work – Asymptotic Variance Rate of OutputsPrevious work – Asymptotic Variance Rate of Outputs
Baris Tan, Asymptotic variance rate of the output in production lines with finite buffers, Annals of Operations Research, 2000.
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 5
In this work…In this work…
Analyze for simple finite queueing systems
• A surprising phenomenon
• Simple formula for
• Extensions
V
V
Results:e.g. M/M/1/K
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 6
The M/M/1/K QueueThe M/M/1/K Queue
K
* (1 )K
1
11
1
11
1
iK
i
K
KFiniteBuffer
0,...,i K
NOTE: output process D(t) is non-renewal.
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What values do we expect for ?V
?
( )V
Keep and fixed.K
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 8
?
( )V
K / / 1( )M M
What values do we expect for ?VKeep and fixed.K
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 9
?
( )V 40K
?* (1 )KV
Similar to Poisson:
What values do we expect for ?VKeep and fixed.K
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?
( )V
40K
What values do we expect for ?VKeep and fixed.K
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( )V
40K
2
3
Balancing
Reduces
Asymptotic
Variance of
Outputs
What values do we expect for ?VKeep and fixed.K
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**
* *
VV
V V
Output from M/M/1/KOutput from M/M/1/K
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 13
Calculating Using MAPs
Calculating Using MAPs
V
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 14
C DMAP (Markovian Arrival Process)(Neuts, Lucantoni et al.)MAP (Markovian Arrival Process)(Neuts, Lucantoni et al.)
* * 2 * 2 3 2Var ( ) 2( ) 2 2( ) 2 ( )r btD t D De t De O t e
0 0
1 1 1 1
1 1 1 1
( )
( )K K K K
K K
1
1
0 0
0 0
0
0K
K
* De *E[ ( )]D t t
0 0
1 1 1
1 1 1
0 ( )
0 ( )
0K K K
K
Generator Transitions without events Transitions with events
1( )e
, 0r b
Asymptotic Variance Rate
Birth-Death Process
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 15
Attempting to evaluate directlyAttempting to evaluate directly* * 2 12( ) 2 ( )V D e De
1 2 3 4 5 6 7 8 9 10
1
2
3
4
5
6
7
8
9
10
1 2 3 4 5 6 7 8 9 10
1
2
3
4
5
6
7
8
9
10
10K
1 10 20 30 40
1
10
20
30
40
1 10 20 30 40
1
10
20
30
40
40K
1 50 100 150 201
1
50
100
150
201
1 50 100 150 201
1
50
100
150
201
200K
For , there is a nice structure to the inverse.
2 2 3
2 3
( 2 ) ( 2 ) ( 1) 7( 1),
2( 1) 2( 1)ij
i i K j K j K Kr i j
K K
ijr
But This doesn’t get us far…
V
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Main TheoremMain Theorem
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1*
0
K
ii
V v
2
2 ii i
i
Mv M
d
*1i i iM D P
1
i
i jj
P
0
i
i jj
D d
Main Theorem
i i id
Part (i)
Part (ii)
0iv
1 2 ... K
0 1 1... K
*1
V
0 0
1 1 1 1
1 1 1 1
( )
( )K K K K
K K
*1KD
Scope: Finite, irreducible, stationary,birth-death CTMC that represents a queue.
0 10
1
ii
i
0 1
0 0 1
1iK
j
i j i
and
If
Then
Calculation of iv
(Asymptotic Variance Rate of Output Process)
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 18
Explicit Formula for M/M/1/KExplicit Formula for M/M/1/K
2
2
1 2 1
1 3
21
3 6 3
(1 )(1 (1 2 ) (1 ) )1
(1 )
K K K
K
K K
K KV
K
2lim
3KV
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 19
Idea of Proof Idea of Proof 1
*
0
K
ii
V v
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 20
Counts of TransitionsCounts of Transitions 1*
0
K
ii
V v
Book: 2001 - Stochastic Process Limits,.
Paper: 1992 - Asymptotic Formulas for Markov Processes…
1) Use above Lemma: Look at M(t) instead of D(t).
2) Use Proposition: The “Fully Counting” MAP of M(t) has associated MMPP with same variance.
3) Use Results of Ward Whitt: An explicit expression of asymptotic variance rate of birth-death MMPP.
( ) ( ) ( )M t E t D t
Var ( ) ( )M t M t o t
Lemma: 4M V
Asymptotic Variance Rate of M(t): ,MBirths Deaths
Observe: MAP of M(t) is “Fully Counting” – all transitions result in counts of events.
Proof OutlineProof Outline
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 21
More On BRAVO More On
BRAVOBalancing
Reduces
Asymptotic
Variance of
Outputs
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 22
0 1 KK – 1
Some intuition for M/M/1/KSome intuition for M/M/1/K
…
Use Lemma: 4M V
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Intuition for M/M/1/K doesn’t carry over to M/M/c/KIntuition for M/M/1/K doesn’t carry over to M/M/c/KV
c
But BRAVO doesBut BRAVO does
c
M/M/40/40
M/M/10/10
M/M/1/40
1
K=20K=30
c=30
c=20
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 24
BRAVO also occurs in GI/G/1/KBRAVO also occurs in GI/G/1/KMAP used for PH/PH/1/40 with Erlang
and Hyper-Exp distributions
1
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The “2/3 property”The “2/3 property”
• GI/G/1/K
• SCV of arrival = SCV of service
1
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M/M/1+ Impatient Customers - SimulationM/M/1+ Impatient Customers - Simulation
( 1, 1)
( )D t
( )L t
( )E t( )A t
V* *,
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Thank YouThank You
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 28
ExtensionsExtensions
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Counts of point processes:
• - Arrivals during
• - Entrances
• - Outputs
• - Lost jobs
Traffic ProcessesTraffic Processes
{ ( ), 0}A t t
{ ( ), 0}E t t
{ ( ), 0}D t t
{ ( ), 0}L t t
[0, ]t
1 K
( )A t
( )L t
( )E t
Poisson
K 1K
0 Renewal Renewal
( )D t
( ) ( )D t L t
( )A t
Non-RenewalPoisson
Poisson Poisson Poisson
Non-Renewal Renewal
( / /1)M M
K
( )D t
( )L t
( )E t( )A t
M/M/1/KM/M/1/K
Renewal
( ) ( ) ( )
( ) ( ) ( )
A t L t E t
E t Q t D t
Book: Traffic Processes in Queueing Networks, Disney, Kiessler 1987.
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Require:
•
• Stable Queues
Push-Pull Queueing Network(Weiss, Kopzon 2002,2006)Push-Pull Queueing Network(Weiss, Kopzon 2002,2006)
1 1
22
Server 2Server 1
PUSH
PULL
PULL
PUSH
0,0 1,0 2,0 3,0 4,0 5,0
0,1
0,2
0,3
0,4
n1
n2
1 1 1 1 1 1
1 1 1 1 1 1
2
2
2
2
2 2
2
2
2
2
0,0
1,3
2,0 3,0 4,0 5,0
0,1
0,2
0,3
0,4
n1
n2
11 1 1 1 1
1 1
1 1 1 1
2
2
2
2
2
2,1
2,2
2,3
2
2
22
2
2
3,1
3,2
3,3
2
2
2
2
2
2
4,1
4,2
4,3
2
2
2
2
2
2
5,1
5,2
5,3
2
2
22
2
2
1 1
1,0
1,4
1
1 1
2,4
0,5
2
1,5
1
1 1
2,5
2
2
2
1
1 1
4,5
2
Positive Recurrent Policies Exist!!!
* 1 1 2 21
1 2 1 2
( )
* 2 2 1 12
1 2 1 2
( )
1 2 1
Asymptotic Variance Rate of the
output processes?
i i i i
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Other Phenomena at Other Phenomena at 1
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 33
Asymptotic Correlation Between Outputs and OverflowsAsymptotic Correlation Between Outputs and Overflows
,
2 3
,
1
11
lim Corr( ( ), ( )) 15 5 3
4 12 2 4
1
K
t
K
R
KD t L t
K K K
R
1 1
2,
1 2 1 2 2 1
(1 )(1 3 ) (1 )(3 )
(1 )(1 (2 1)(1 ) )((1 )(1 ) 4( 1)(1 ) )
K K K K
KK K K K K
KR
K
0.139772 1
1lim Corr( ( ), ( )) 1
41
12
tD t L t
For Large K
( )D t
( )L t
M/M/1/KM/M/1/K
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Proposition: For ,
The y-intercept of the Linear AsymptoteThe y-intercept of the Linear Asymptote
4 3 2
2
7 28 37 18
180 360 180D
K K K KB
K K
M/M/1/K1
* * 2 * 2 3 2Var ( ) 2( ) 2 2( ) 2 ( )
D
r bt
BV
D t D De t De O t e , 0r b
1
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The variance function in the short rangeThe variance function in the short range
/ /1/ 40M M
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Lemma: 4M V
Proof:
( ) 2 ( ) ( )M t D t Q t
Var ( ) 4Var ( ) Var ( ) 4Cov ( ), ( )M t D t Q t D t Q t
Cov ( ), ( )1
Var ( ) Var ( )
D t Q t
D t Q t
Var ( ) (1)Q t O
Var ( ) ( )D t O t Cov ( ), ( )D t Q t O t
Q.E.D
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t
3 2 1
2 4 2
1 1 2
a b c
a
b
c
Fully Counting MAP and associated MMPPFully Counting MAP and associated MMPP
MMPP (Markov Modulated Poisson Process)
Example:
0 ( )N t
tabc
( )Q t
rate 4Poisson Process
rate 2
rate 3
rate 4
rate 2
rate 4
rate 3
rate 2
rate 3
rate 4
rate 2
1 0
1 0
E[ ( )] E[ ( )]
Var( ( )) Var( ( ))
N t N t
N t N t
Proposition
3 0 0 0 2 1
0 4 0 2 0 2
0 0 2 1 1 0
6 2 1 3 0 0
2 8 2 0 4 0
1 1 4 0 0 2
Transitions without events Transitions with events
1( )N tFully Counting MAP
1( ),N t
( )Q t
0 ( )N t