Encounters with the BRAVO Effect in Queueing Systems
Yoni NazarathySwinburne University of Technology
Based on some joint papers with Ahmad Al-Hanbali, Daryl Daley, Yoav Kerner,
Michel Mandjes, Gideon Weiss and Ward Whitt
University of Queensland Statistics SeminarDecember 2, 2011
Outline
• Queues and Networks
• Variance of Outputs
• BRAVO Effect
• BRAVO Results (Theorems)
• Summary
Queues and Networks
The GI/G/1/K Queue
2, ac ( )D t2, sc
KOverflows
2 22
variance,meana sc c
Load:
Squared coefficients of variation:
or K K Buffer:
OutputsArrivals
Variance of Outputs
Variance of Outputs( )tVt o
t
Var ( )D t
Var ( )D T TV
* Stationary stable M/M/1, D(t) is PoissonProcess( ):
* Stationary M/M/1/1 with , D(t) is RenewalProcess(Erlang(2, )):
21 1 1( )
4 8 8tVar D t t e
( )Var D t t V
4V
2 1 23V m cm
* In general, for renewal process with :
* The output process of most queueing systems is NOT renewal
2,m
Asymptotic Variance
Var ( )limt
VD tt
Simple Examples:
Notes:
Asymptotic Variance when 1
( ) ( ) ( )
( ) ( ) ( ) ( ), ( ) 2
D t A t Q tVar D t Var A t Var Q t Cov A t Q t
t t t t
2aV c
2sV c
, 1K
After finite time, server busy forever…
is approximately the same as when or 1 K V
, 1K
K
1
Look now at the Asymptotic Variance of M/M/1/K (for any Value of )
?( )V 40K
?* (1 )KV
Similar to Poisson:
What values do we expect for ?V
M/M/1/K
?( )V
40K
M/M/1/K
What values do we expect for ?V
( )V
40K
23
Balancing Reduces Asymptotic Variance of Outputs
M/M/1/K
What values do we expect for ?V
0 1 KK – 1
Some Intuition…
…
4M V
Do we Have the Same BRAVO Effect in
M/M/1 ?
M/M/1 (Infinite Buffer)
Key BRAVO Results
Balancing Reduces Asymptotic Variance of Outputs
Theorem (N. , Weiss 2008): For the M/M/1/K queue with :
2
2 3 23 3( 1)
KVK
Conjecture (N. , 2011):For the GI/G/1/K queue with :
2 2
(1)3
a sK
c cV o
1
1
Theorem (Al Hanbali, Mandjes, N. , Whitt 2010):For the GI/G/1 queue with ,under further conditions:
2 2 2( ) 1a sV c c
1
A Bit More on the GI/G/1 Result
Reminder: Uniform Integrability (UI)
{| | }lim sup [| |1 ] 0tt Z MM t
E Z
A family of RVs, , is UI if:{ }tZ
A sufficient condition is:
1sup [| | ] for some ttE Z
If (in distribution) and is UI then: { }tZtZ Z
[ ] [ ]tE Z E Z
Theorem : Assume that is UI,
then , with
2
0( ) ,Q t t tt
Q
VarV D
Theorem : 2 2 2Var ( ) 1a sD c c
Theorem : Assume finite 4’th moments,then, is UI under the following cases:(i) Whenever and L(.) bounded slowly varying. (ii) M/G/1(iii) GI/NWU/1 (includes GI/M/1)(iv) D/G/1 with services bounded away from 0
The GI/G/1 Result:
1/2( ) ( )P B x L x x
2 21 20 1
inf ( ) (1 )a stD c B t c B t
2 2 2( ) 1a sV c c
Q
Summary• BRAVO:
Balancing Reduces Asymptotic Variance of Outputs
• Different “BRAVO Constants”: Finite Buffers:
Infinite Buffers:
• Further probabilistic challenges in establishing full UI conditions
• In future: Applications of BRAVO and related results in system identification (model selection)
21
13
BRAVO References
• Yoni Nazarathy and Gideon Weiss, The asymptotic variance rate of the output process of finite capacity birth-death queues. Queueing Systems, 59, pp135-156, 2008.
• Yoni Nazarathy, The variance of departure processes: Puzzling behavior and open problems. Queueing Systems, 68, pp 385-394, 2011.
• Ahmad Al-Hanbali, Michel Mandjes, Yoni Nazarathy and Ward Whitt. The asymptotic variance of departures in critically loaded queues. Advances in Applied Probability, 43, 243-263, 2011.
• Yonjiang Guo, Erjen Lefeber, Yoni Nazarathy, Gideon Weiss, Hanqin Zhang, Stability and performance for multi-class queueing networks with infinite virtual queues, submitted.
• Daryl Daley, Yoni Nazarathy, The BRAVO effect for M/M/c/K+M systems, in preperation.
• Yoni Nazarathy and Gideon Weiss, Diffusion Parameters of Flows in Stable Queueing Networks, in preparation.
• Yoav Kerner and Yoni Nazarathy, On The Linear Asymptote of the M/G/1 Output Variance Curve, in preparation.
Extra Slides
C DMAP (Markovian Arrival Process)
* * 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 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
, 0r b
Asymptotic Variance Rate
Birth-Death Process
Attempting to Evaluate Directly* * 2 12( ) 2 ( )V D e De
1 2 3 4 5 6 7 8 9 1 0
1
2
3
4
5
6
7
8
9
10
1 2 3 4 5 6 7 8 9 1 0
1
2
3
4
5
6
7
8
9
1 0
10K
1 1 0 2 0 3 0 4 0
1
10
20
30
40
1 1 0 2 0 3 0 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
V
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
Finite B-D Result
i i id
Part (i)
Part (ii)
0iv
1 2 ... K
0 1 1... K
* 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
If
Then
Calculation of iv
(Asymptotic Variance Rate of Output Process)
Vc
Other Systems
c
M/M/40/40
M/M/10/10
M/M/1/40
1
K=20K=30
c=30
c=20
Using a Brownian BridgeTheorem:
2 2 2Var ( ) 1a sD c c
Proof Outline:
2 22 2 1 2 1 1 2 20 1
1 1 2 2
1 1 2 2
inf ( ) min( , )| (1) , (1)
1 min( , )t
P b c c c B t x x b c b cP D x B b B b
x b c b c
1 1 2 22 2
1 2
( ) ( ) (1) b c b cB t B t t Bc c
2 21 20 1
inf ( ) (1 )a stD c B t c B t
Brownian Bridge: