1. The Variance of Production Counts over a Long Time Horizon
Yoni Nazarathy*
EURANDOM, TU/e
Contains joint work with Ahmad Al-Hanbali, Yoav Kerner, Michel Mandjes, Gideon Weiss and Ward Whitt
Workshop on Stochastic Models of Manufacturing Systems
Eindhoven, June 2010
*Supported by NWO-VIDI Grant of Erjen Lefeber

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

3. Asymptotic Variance Rate of Outputs For Renewal Processes:
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, )):
Asymptotic Variance Rate of Outputs
For Renewal Processes:

4. Asymptotic Variance Rate
M/M/1
Non-Stop Service
Burkes Theorem

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

6. Our main focus: Overloaded and critically loaded systems

7. GI/G/1
Non-Stop Service

8. Queues in Tandem (with 1 bottleneck)
Bottleneck Server
Just as simple…

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

10. Overloaded case --> Infinite Supply Re-entrant Line
Result:

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

12. Shocking result* coming up…
* at least for me

13. Back to Single Server (GI/G/1/K)
What happens here?
Balancing Reduces Asymptotic Variance of Outputs
Note: the figure assumes

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

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

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

17. Questions?