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Yoni NazarathyGideon Weiss
University of Haifa
On the Asymptotic Variance Rate of the
Output Process of Finite Capacity Queues
Queueing Analysis, Control and GamesDecember 20,2007
Technion, Israel
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 2
•Poisson arrivals:
•Independent exponential service times:
•Finite buffer size:
•Jobs arriving to a full system are a lost.
•Number in system, , is represented by a finite state irreducible birth-death CTMC:
The M/M/1/K Queue
( )
( )
01e
K
* (1 )k
{ ( ), 0}Q t t
1
1 11
1 11
iK
i
K
K
kBuffer Server
M
0,...,i K
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 3
Traffic Processes
Counts of point processes:
• - The arrivals during
• - The entrances into the system during
• - The outputs from the system during
• - The lost jobs during
{ ( ), 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-Renewal
Poisson
Poisson Poisson Poisson
Non-Renewal
Renewal
( / /1)M M
[0, ]t
[0, ]t
[0, ]t
K
( )D t
( )L t
( )E t( )A t
M/M/1/K
Renewal
( ) ( ) ( )( ) ( ) ( )A t L t E tE t Q t D t
Book: Traffic Processes in Queueing Networks, Disney, Kiessler 1987.
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 4
•Some Attributes: (Disney, Kiessler, Farrell, de Morias 70’s)
•Not a renewal process (but a Markov Renewal Process).
•Expressions for .
•Transition probability kernel of Markov Renewal Process.
•A Markovian Arrival Process (MAP) (Neuts 1980’s).
•What about ?
D(t) – The Output process:
1( , )n nCov D D
( )Var D t
( )Var D t
t
V
( ) ( )Var D t V t o t
Asymptotic Variance Rate: V
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 5
Asymptotic Variance Rate of Outputs:
What values do we expect for ?V
?( )V
V
( , )K fixed
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 6
?( )V
K / / 1( )M M
( , )K fixed
Asymptotic Variance Rate of Outputs:V
What values do we expect for ?V
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 7
?( )V 40K
* (1 ) ???KV Similar to Poisson:
( , )K fixed
Asymptotic Variance Rate of Outputs:V
What values do we expect for ?V
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 8
Asymptotic Variance Rate of Outputs:
What values do we expect for ?V
?( )V
V
( , )K fixed
40K
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 9
( )V
( )fixed40K
23
M
Balancing Reduces Asymptotic Variance of Outputs
Asymptotic Variance Rate of Outputs:V
What values do we expect for ?V
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 10
Some Results
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 11
Results for M/M/1/K:
2
2
1 2 1
1 3
2 13 6 3(1 )(1 (1 2 ) (1 ) ) 1
(1 )
K K K
K
K KK KV
K
2lim3K
V
Other M/M/1/K results:
•Asymptotic correlation between outputs and overflows.
•Formula for y-intercept of linear asymptote when . 1
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 12
Calculating Using MAPs
V
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 13
C D Represented as a MAP (Markovian Arrival Process) (Neuts, Lucantoni et. al.)( )D t
* * 2 * 2 3 2( ) 2( ) 2 2( ) 2 ( )r btVar D 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 00 0
00
K
K
* De *[ ( )]E D t t
0 0
1 1 1
1 1 1
0 ( )
0 ( )0
K K K
K
Generator Transitions without events Transitions with events
1( )e
r0b
Asymptotic Variance Rate
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 14
Attempting 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
1 0
10K
1 1 0 2 0 30 4 0
1
1 0
2 0
3 0
4 0
1 1 0 2 0 30 4 0
1
10
20
30
40
40K
1 50 100 150 2 01
1
5 0
10 0
15 0
20 1
1 50 100 150 2 01
1
50
10 0
15 0
20 1
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 jK K
ijr
But This doesn’t get us far…
V
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 15
Main Theorem
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 16
1*
0
K
ii
V v
2
2 ii i
i
Mv Md
*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 i
0 1 1... K i
* 1V
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
and
or
If:
Then:
Calculation of : iv
(Asymptotic Variance Rate of Output Process)
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 17
Proof Outline
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 18
Use the Transition Counting Process
( ) ( ) ( )M t E t D t
( ) ( )Var M t M t o t
Lemma: 4M V
Proof:( ) 2 ( ) ( )M t D t Q t
( ) 4 ( ) ( ) 4 ( ), ( )Var M t Var D t Var Q t Cov D t Q t
( ), ( )1
( ) ( )
Cov D t Q t
Var D t Var Q t
( ) (1)Var Q t O ( ) ( )Var D t O t
( ), ( )Cov D t Q t O t
Q.E.D
- Counts the number of transitions in the state space in [0,t]
Asymptotic Variance Rate of M(t): M
Births Deaths
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 19
Idea of Proof of part (i):1
*
0
K
ii
V v
Whitt: Book: Stochastic Process Limits, 2001.
Paper: 1992 –Asymptotic Formulas for Markov Processes…
1) Look at M(t) instead of D(t).
2) The MAP of M(t) has an associated MMPP with same variance.
2) Results of Ward Whitt allow to obtain explicit expression for the asymptotic variance rate of MMPP with birth-death structure.
0iv Proof of part (ii), is technical.
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 20
More BRAVOBalancing Reduces Asymptotic Variance of Outputs
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 21
0 1 KK-1
Trying to understand what is going on….
M/M/1/K:
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 22
Intuition for M/M/1/K doesn’t carry over to M/M/c/K…But BRAVO does…
Vc
1 c
M/M/40/40
M/M/K/K
K=30 K=20K=10
M/M/c/40 c=1c=20
c=30
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 23
BRAVO also occurs in GI/G/1/K…
V
1 MAP is used to evaluate Var Rate for
PH/PH/1/40 queue with Erlang and Hyper-Exp
1
1
2
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 24
The “2/3 property” seems to hold for GI/G/1/K!!!
V
K
V 1 and increase K for different CVs
2 2CV
2 3/ 2CV
2 6 / 5CV
2 1/ 2CV
13
45
1
43
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Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 25
ThankYou