1. Departure Process Variability 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
IFORS 2011, Melbourne,
July 15, 2011.

2. Problem Domain: Queueing Output Processes
PLANT
OUTPUT
- Single Server Queue
- Tandem Queue
- Re-Entrant Line
Desired over long term:
High Throughput
Low Variability
Our focus:
for large T

3. The GI/G/1/K Queue overflows * Assume * Load:
* Squared coefficient of variation:

4. Variance of Outputs Asymptotic Variance Simple Examples:
* Stationary stable M/M/1: D(t) is PoissonProcess( ):
* Stationary M/M/1/1 with : D(t) is RenewalProcess(Erlang(2, )):
Notes:
* In general, for renewal process with
* The output process of most queueing systems is NOT renewal

5. Asymptotic Variance for GI/G/1/K
What happens here?
Balancing Reduces Asymptotic Variance of Outputs
Note: the figure assumes

6. BRAVO Effect (for M/M/1/K)

7. BRAVO Effect (illustration for M/M/1)
More than a singular theoretic phenomenon

8. Balancing Reduces Asymptotic Variance of Outputs
Theorem (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 :

9. Additional Slides
4/28/2019Valuetools 2008

10. The Basic Loss-Less Stable Queueing System
Q(t)

11. Some (partial) intuition for M/M/1/K
Easy to see: 1 K
K-1