Balancing Reduces Asymptotic Variance of Outputs

Yoni Nazarathy*

EURANDOM, Eindhoven University of Technology,The Netherlands.

Based on some joint works with

Ahmad Al Hanbali, Michel Mandjes,Gideon Weiss and Ward Whitt

QTNA 2010, Beijing,July 26, 2010.

*Supported by NWO-VIDI Grant 639.072.072 of Erjen Lefeber

Overview• GI/G/1/K Queue (with or )

• number of customers served during

• Asymptotic variance

• Surprising results when

K K

( )D t [0, ]t

1

Var ( )limt

D tV

t

Balancing Reduces Asymptotic Variance of Outputs

The GI/G/1/K Queue

2, ac ( )D t2, sc

K

overflows

2 22

variance,meana sc c

* Load:

* Squared coefficient of variation:

* Assume (0) 0Q

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 for (simple) 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

Intermediate SummaryV

2ac

2sc

GI/G/1 V

2ac

2sc

GI/G/1/K

M/M/1V

V

M/M/1/K

??

? ?

Balancing Reduces Asymptotic Variance of Outputs

Theorem (Al Hanbali, Mandjes, N. , Whitt 2010):For the GI/G/1 queue with , under some further technical conditions:

2 221 ( )a sV c c

Theorem (N. , Weiss 2008): For the M/M/1/K queue with :

2

2 3 23 3( 1)

KVK

Conjecture (N. , 2009):

For the GI/G/1/K queue with , under furthertechnical conditions :

2 21 ( ) (1)3 a s KV c c o

1

1

1

2 2

2 2

21

1 (1)3

a s

a s K

c c KV

c c o K

BRAVO Summary for GI/G/1/KFor GI/G/1/K with :1

Proven:• : M/M/1/K• :

* M/M/1 * Assuming finite forth moments: *M/G/1 *GI/NWU/1 (includes GI/M/1) *Any GI/G/1 with 1/2( ) ( )P B x L x x

K K

Numerically Conjectured: GI/G/1/K with light tails

**

* *

VV

V V

Numerical Illustration: M/M/1/K

Numerical Illustration: M/M/1 (finite T)

0 1 KK-1

Some (partial) intuition for M/M/1/K

4 MV V

Asymptotic variance of number of transitionsMV

Easy to see:

References• Yoni Nazarathy and Gideon Weiss, The

asymptotic variance rate of the output process of finite capacity birth-death queues. Queueing Systems, 59(2):135-156, 2008.

• Yoni Nazarathy, 2009, The variance of departure processes: Puzzling behavior and open problems. Preprint, EURANDOM Technical Report Series, 2009-045.

• Ahmad Al-Hanbali, Michel Mandjes, Yoni Nazarathy and Ward Whitt. Preprint. The asymptotic variance of departures in critically loaded queues. Preprint, EURANDOM Technical Report Series, 2010-001.