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Yoni NazarathyGideon Weiss
University of Haifa
Yoni NazarathyGideon Weiss
University of Haifa
The Asymptotic Variance Rate of the Departure Process of the M/M/1/K
Queue
The Asymptotic Variance Rate of the Departure Process of the M/M/1/K
Queue
The XXVI International Seminaron Stability Problems for Stochastic Models
October 24, 2007, Nahariya, Israel
The XXVI International Seminaron Stability Problems for Stochastic Models
October 24, 2007, Nahariya, Israel
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 CTMC:
The M/M/1/K QueueThe M/M/1/K Queue
( )
( )
0
1e
K
* (1 )k
{ ( ), 0}Q t t
1
1
1
1
1
iK
i
K
K
kBuffer Server
M
0,...,i K
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 3
Traffic ProcessesTraffic Processes
Counts of point processes:
• - The arrivals during
• - The entrances into the system during
• - The departures 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/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.
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 4
•A Markov Renewal Process (Cinlar 1975).
•A Markovian Arrival Process (MAP) (Neuts 1980’s).
•Some Attributes: (Disney, Kiessler, Farrell, de Morias 70’s):
•Not a renewal process.
•Expressions for .
•Transition probability kernel of the Markov Renewal Process.
•Departures processes of M/G/. Models.
•What about ?
D(t) – The Departure process of M/M/1/KD(t) – The Departure process of M/M/1/K
1( , )n nCov D D
( )Var D t
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 5
Asymptotic Variance RateAsymptotic Variance Rate
( ) ( )Var D t V t o t For a given system ( ), what is ?, ,K V
?
( )V
V
( )fixed
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 6
Asymptotic Variance RateAsymptotic Variance Rate
( ) ( )Var D t V t o t For a given system ( ), what is ?, ,K V
?
( )V
V
( )fixed
K / / 1( )M M
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 7
Asymptotic Variance RateAsymptotic Variance Rate
( ) ( )Var D t V t o t For a given system ( ), what is ?, ,K V
?
( )V
V
( )fixed40K
* (1 ) ???KV Similar to Poisson:
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 8
Asymptotic Variance RateAsymptotic Variance Rate
( ) ( )Var D t V t o t For a given system ( ), what is ?, ,K V
( )V
V
( )fixed40K
2
3
OUR MAINRESULT
M
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 9
An Explicit FormulaAn Explicit Formula
2 2 1 1 2 2 2 2 1
1 1 2
2
2
( 2 1) 2 2 ( 1) 2 ( 1)
( )( )
2
3 6 3
K K K K K K K K K K
K K K
K K K K
VK K
K K
Theorem:
Corollary:
2lim
3KV
Corollary:
*
V
Is minimal over all when .,
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 10
What is going on?What is going on?
( ) ( ) ( )M t E t D t
( )M t - The number of movements on the state space during [0, ]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
0 1 KK-1
The State Space
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 11
What is going on? (continued…)What is going on? (continued…)
0 1 KK-1
Observation: When , is minimal. 0 K
As a result the “modulation” of M(t) is minimal.
Rate of M(t), depends on
current state of Q(t)
And thus the “modulation” of D(t) is minimal.
M
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 12
Calculationsand Proof Outline
Calculationsand Proof Outline
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 13
Represented as a MAP (Markovian Arrival Process) (Neuts, Lucantoni et. al.) Represented as a MAP (Markovian Arrival Process) (Neuts, Lucantoni et. al.)
( )D t
* * 2 1( ) 2( ) 2 ( ) ( )
V
Var D t D e De t o t
( )
( )
C D 0 0
0 0
0
0
* (1 )KDe
*[ ( )] ( )E D t t o t
0 ( )
0 ( )
0
Note: We may similarly represent M(t), E(t), L(t) and we may also use similar methods (MMAP) to find cross-covariances.
Generator Transitions without “arrivals” Transitions with “arrivals”
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 14
Calculation of : Calculation of :
* * 2 12( ) 2 ( )V D e De
V
Option 1: Invert Numerically
Option 2: For only, we have the explicit structure of the inverse…
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
1 10 20 30 40
1
10
20
30
40
1 10 20 30 40
1
10
20
30
40
1 50 100 150 201
1
50
100
150
201
1 50 100 150 201
1
50
100
150
201
10K
40K
200K
Option 3: Find an associated Markov Modulated Poisson Process (MMPP) to the MAP of (Proof of the explicit formula for any ).
( )M t
2 2 2
2
3 2( 1) ( 1) 2( 1) 7
6( 1)ij
i i j j K j K Kr i j
K
,
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 15
Proof Outline:Proof Outline:
Option 3: Find an associated Markov Modulated Poisson Process (MMPP) to the MAP of (Proof of the explicit formula).
( )M t
1) M(t) is “fully counting”: It exactly counts the number of movements in the state-space during [0,t].
2) “Decoupling Theorem” (stated loosely): There exists a MMPP that has the same expectation and variance as a fully counting MAP.
3) Combined results of Ward Whitt (2001 book and 1992 paper) are used to find explicit formulas for the asymptotic variance rate of birth-death type MMPPs.
Note: This technique can be used to find similar explicit formulas for the asymptotic variance rate of departures from M/M/c/K and other structured finite birth and death queues.
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 16
Open Questions:Open Questions:
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 17
Open Questions:Open Questions:
1) The limiting value of 2/3 also appears in the asymptotic variance rate of the losses (e.g. Whitt 2001). What is the connection?
2) Non-Exponential Queueing systems. Is minimization of the characteristic attribute of the “dip” in the asymptotic variance rate?
3) Asymptotic variance rate of departures from the null-recurrent M/M/1?
4) Variance of departure processes from more complex queueing networks (our initial motivation).
0 K
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 18
ThankYou
ThankYou