The Asymptotic Variance of the
Output Process of Finite
Capacity Queues
Yoni Nazarathy
Gideon Weiss
University of Haifa
ORSIS Conference, Israel
April 18-19, 2008
Queueing Output Process
A Single Server Queue:


Buffer

State:


2
1
0




3

Server
4


5


…
6


M/M/1 Queue:
•Poisson Arrivals: 
•Exponential Service times: 
•State Process is a birth-death CTMC
D(t )
The Classic Theorem on M/M/1 Outputs:
Burkes Theorem (50’s):
Output process of stationary version is Poisson (  ).
Output
Process:
t
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008
1
The M/M/1/K Queue
K


 K
• Buffer size: K
Finite
Buffer
*   (1   K )
Server
“Carried load”
• Poisson arrivals: 
• Independent exponential service times: 
• Jobs arriving to a full system are a lost.
• Number in system,{Q(t ), t  0}, is represented by a finite state
irreducible birth-death CTMC.
•Assume{Q(t ), t  0} is stationary.

 

 







 (   ) 







 (   )  


  

 0
 e 1
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008
 i 1 
  1   K 1
 i  
1

 K  1


 1
i  0,..., K
 1
2
Traffic Processes
Counts of point processes:
M/M/1/K
• {A(t ), t  0} - Arrivals during [0, t ]
K
• {E (t ), t  0} - Entrances
E (t )
A(t )
D(t )
• {D(t ), t  0} - Outputs
L(t )
• {L(t ), t  0} - Lost jobs
K 
( M / M /1)
1 K  
K 1
A(t )  L(t )  E (t )
E (t )  Q(t )  D(t )
Poisson
A(t )
L(t )
0
Renewal
Renewal
E (t )
 A(t )
Non-Renewal
Renewal
D(t )
Poisson
Non-Renewal
Renewal
D(t )  L(t )
Poisson
Poisson
Poisson
Book: Traffic Processes in Queueing
Networks, Disney, Kiessler 1987.
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008
3
The Output process
• Some Attributes:
(Disney, Kiessler, Farrell, de Morias 70’s)
• Not a renewal process (but a Markov Renewal Process).
• Expressions for Cov(Tn , Tn1 ) .
• Transition probability kernel of Markov Renewal Process.
• A Markovian Arrival Process (MAP)
(Neuts 80’s)
• What about Var  D(t )  ?
Var  D(t ) 
V
Asymptotic Variance Rate: V
Var  D(t )   V t  o(t )
t
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008
4
What values do we expect for V ?
Keep K and  fixed.
V ( )

?



Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008



5
What values do we expect for V ?
Keep K and  fixed.
K 
V ( )
( M / M / 1)

?



Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008



6
What values do we expect for V ?
Keep K and  fixed.
K  40
V ( )

?
Similar to Poisson:
?
V     (1   K )



Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008
*



7
What values do we expect for V ?
Keep K and  fixed.
K  40
V ( )

?



Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008



8
What values do we expect for V ?
Keep K and  fixed.
K  40
V ( )

Balancing
Reduces
Asymptotic
Variance of
Outputs
2

3



Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008



9
Calculating V
Using MAPs
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008
10
MAP (Markovian Arrival Process)
(Neuts, Lucantoni et al.)


C
0
(1  1 ) 1
 K 1
   De
*






(K 1   K 1 ) K 1 

K
  K 

 0

 0






0
(1  1 ) 1
E[ D(t )]   * t
0






(K 1   K 1 ) K 1 

0
  K 

D
Transitions with events
Transitions without events
Generator
 0

 1








0

 1






0
0
0
 K 1
K






0

0 
(  e  )1
Var  D(t )     *  2( * ) 2  2  D  De  t  2( * ) 2  2    De  O(t 3r  2 e  bt )
r , b 0
Asymptotic Variance Rate
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008
11
Attempting to evaluate V directly
V  *  2(* )2  2 D (  e  )1 De
For
  ,
there is a nice structure to the inverse.
1
1
10
20
30
40
1
1
50
100
150
201
1
10
2
3
4
5
6
7
8
9
10
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
1
10
1
1
rij
50
50
20
20
30
30
40
100
150
150
201
3
4
5
6
7
8
9
10
K  10
40
1
100
2
10
20
30
40
K  40
201
1
50
100
150
201
K  200
i 2  i  ( K  2  j )2  ( K  2  j ) ( K  1)3  7( K  1)
rij 

,
2
3
2( K  1)
2( K  1)
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008
i j
12
Main Theorem
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008
13
Main Theorem
(Asymptotic Variance Rate of Output Process)
 0

 1






Part (i)
K 1
V   *   vi
i 0
Part (ii)
0
(1  1 ) 1
 K 1
   i 1
i  0
0
1   i
0 
and
0  1  ...  K 1
di   i i
1
j

i  0 j  0 i 1
K
i 1
i
Di   d j
j 0
Then
vi  0






(K 1   K 1 ) K 1 

K
  K 
Calculation of vi
1  2  ...  K
If
Scope: Finite, irreducible, stationary,
birth-death CTMC that represents a queue.

V
*
1
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008
M i  Di 1  * Pi
i
Pi    j
j 1
*  DK 1

M i2 
vi  2  M i 

d
i 

14
Explicit Formula for M/M/1/K
 2K 2  K

2
3
K
 6K  3

V 
 (1   K 1 )(1  (1  2 K )  K (1   )   2 K 1 )

(1   K 1 )3

 1
 1

  
 2

lim V  

K 
 3
    
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008
15
Proof Outline
(of part i)
K 1
V     vi
*
i 0
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008
16
Define The Transition Counting Process
M (t )
E(t )  D(t )
Births
- Counts the number of transitions in [0,t]
Deaths
Asymptotic Variance Rate of M(t): M , Var  M (t )   M t  o(t )
MAP of M(t) is “Fully Counting” – all transitions result in counts of events.
Lemma: M  4 V
Proof:
M (t )  2D(t )  Q(t )
Var  M (t )   4Var  D(t )   Var Q(t )   4Cov  D(t), Q(t) 
Var Q(t )   O(1)
Var  D(t )   O(t )
Cov  D(t ), Q(t ) 
Var  D(t )  Var  Q(t ) 
1

Cov  D(t ), Q(t )   O
 t
Q.E.D
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008
17
Proof Outline
K 1
V     vi
*
i 0
1) Lemma: Look at M(t) instead of D(t).
2) Proposition: The “Fully Counting” MAP of M(t) has
associated MMPP with same variance.
3) Results of Ward Whitt: An explicit expression of
asymptotic variance rate of birth-death MMPP.
Whitt: Book: 2001 - Stochastic Process Limits,.
Paper: 1992 - Asymptotic Formulas for Markov Processes…
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008
18
Fully Counting MAP and associated MMPP
Example:
Q(t )
a
Transitions with events
Transitions without events
b
c
a  3 2 1 


  b  2 4 2 
c  1 1 2 

 3 0 0 


0

4
0


 0 0 2 



 6 2 1 


 2 8 2 
 1 1 4 



0

2
1


3

0
0

2
0
1
0
4
0
1

2
0 
0

0
2 
Fully Counting MAP
N1 (t )
MMPP
N0 (t )
(Markov Modulated Poisson Process)
N1 (t ), N0 (t )
Proposition
E[ N1 (t )]  E[ N0 (t )]
Var( N1 (t ))  Var( N0 (t ))
t
Q(t )
c
b
a
rate 2
rate 2
rate 4
Poisson
Process
rate 4
rate 3
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008
rate 2
rate 2
rate 4
rate 4
rate 3
rate 3
t
19
More On
BRAVO
Balancing
Reduces
Asymptotic
Variance of
Outputs
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008
20
Some intuition for M/M/1/K


0

1





…




K–1



Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

K



21
Intuition for M/M/1/K doesn’t
carry over to M/M/c/K
But BRAVO does
V
c
M/M/1/40
c=20
c=30
K=30
K=20
M/M/10/10
M/M/40/40
   1
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

c
22
BRAVO also occurs in GI/G/1/K
V
MAP used for PH/PH/1/40 with Erlang
and Hyper-Exp distributions
2
1
   1
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

23
The “2/3 property”
• GI/G/1/K
• SCV of arrival = SCV of service
•
 1
V
2
2 4

3
3
3 2
 1
2 3
6 2 4
 
5
5 3
1 2 1
 
3
2 3
K
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008
24
Thank You
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008
25