of Queues and Networks Yoni Nazarathy Swinburne University of Technology Melbourne Based on collaborations with Ahmad Al Hanbali Daryl Daley Michel Mandjes Gideon Weiss and Ward Whitt ID: 797185
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Slide1
Departure Process Variabilityof Queues and Networks
Yoni NazarathySwinburne University of Technology, Melbourne.Based on collaborations with Ahmad Al Hanbali, Daryl Daley, Michel Mandjes,Gideon Weiss and Ward Whitt
IFORS 2011, Melbourne,
July
15, 2011.
Slide2PLANT
OUTPUTProblem Domain: Queueing Output ProcessesDesired over long term:High ThroughputLow Variability
- Single Server
Queue
- Tandem
Queue
-
Re-Entrant
Line
Our focus:
for large T
Slide3The GI/G/1/K Queue
overflows
* Load:
* Squared coefficient of variation:
* Assume
Slide4Variance of Outputs
* Stationary
stable
M/M/1:
D(t) is
PoissonProcess
( ):
* Stationary
M/M/1/1
with
:
D(t) is
RenewalProcess
(
Erlang
(2, )):
* In general, for renewal process with
* The output process of most
queueing
systems is NOT renewal
Asymptotic Variance
Simple Examples:
Notes:
Slide5Asymptotic Variance for GI/G/1/K
What happens here?
Note: the figure assumes
B
alancing
R
educes
A
symptotic
V
ariance of
O
utputs
Slide6BRAVO Effect (for M/M/1/K)
Slide7BRAVO Effect (illustration for M/M/1)
More than a singular theoretic phenomenon
Slide8Balancing Reduces A
symptotic Variance of OutputsTheorem (Al Hanbali, Mandjes, N. , Whitt AAP 2011):For the GI/G/1 queue with , under some further technical conditions:Theorem (N. , Weiss
QUESTA 2008):
For the M/M/1/K queue with :
Conjecture (N. ,
Daely
, QUESTA To appear):
For the GI/G/1/K queue with , under further
technical conditions :
Slide9Additional Slides7/15/2011Valuetools 2008
9
Slide10The Basic Loss-Less Stable Queueing System
Q(t)
Slide110
1
K
K-1
Some (partial) intuition for M/M/1/K
Easy to see: