1. Departure Process Variability of Queues and Queueing Networks
Yoni Nazarathy
Swinburne University of Technology
Contains joint work with Ahmad Al-Hanbali, Yoav Kerner, Michel Mandjes, Gideon Weiss and Ward Whitt
ANIZAM11 Conference
Adelaide, February 2011

2. Outline
Problem domain: Departure process variability, Asymptotic Variance Rate
Surprising results for single server queues
Clean results for complicated re-entrant lines

3. Problem Domain: Departure Counting Processes
PLANT
DEPARTURES
Models:
- Detailed Monte Carlo Sims - Single Server Queues
- Re-Entrant Lines
Desired over long term:
High Throughput
Low Variability
Our focus:
for large T

4. “Reminder”: Queueing Departure Processes
A Single Server Queue:
Buffer
Server …
State: 1 2 3 4 5 6
e.g. M/M/1 Queue:
Poisson Arrivals:
Exponential Service times:
Continuous Time Markov Chain
The Classic Theorem on M/M/1 Outputs: Burke’s Theorem (50’s):
Output process of stationary version is
Departure Process:
So for M/M/1 with :

5. More on Variance Curves
Example: Stationary stable M/M/1, D(t) is PoissonProcess( ):
Example: Stationary M/M/1/1 with D(t) is RenewalProcess(Erlang(2, )):
Example: M/M/1/K queues with K> D(t) is not a renewal process. No explicit Var(D(t)) formula
Asymptotic Variance Rate
For Renewal Processes:

6. More Generally: Any Loss-Less Stable Queueing System
Q(t)

7. Surprising Results for Single Server Queues

8. General Single Server Queue (GI/G/1)
What happens here?
Balancing Reduces Asymptotic Variance of Outputs
Note: the figure assumes

9. BRAVO Effect (for GI/G/1/K)

10. Clean Results for Complicated Re-Entrant Lines

11. Re-entrant Line
bottleneck
In the stable case:

12. Overloaded case --> Infinite Supply Re-entrant Line

13. Overloaded case --> Infinite Supply Re-entrant Line1 6 8 1 2 3 5 6 4 8 7 9

14. References
Nazarathy Y., “The Variance of Departure Processes: Puzzling Behaviour and Open Problems.”, Queueing Systems, to appear.
Al–Hanbali A., Mandjes M., Nazarathy Y., Whitt W., “The Asymptotic Variance of Departures in Critically Loaded Queues.”, Advances in Applied Probability, to appear, March 2011.
Nazarathy Y. and Weiss G., “Positive Harris Recurrence and Diffusion Scale Analysis of a Push Pull Queueing Network.”, Performance Evaluation, 67(4), pp. 201–217, (2010).
Nazarathy Y. and Weiss G., “The Asymptotic Variance Rate of the Output Process of Finite Capacity Birth–Death Queues.”, Queueing Systems, 59, pp. 135–156, (2008).
Kerner Y. and Nazarathy Y., “The Linear Asymptote of M/G/1 and GI/M/1 Departure Variance.”, in preparation. Preprint available.

15. Some (partial) intuition for M/M/1/K
Believe me: 1 K
K-1

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