1. (unpublished, except as an NPS Technical Report)
ESTIMATING THE WORKLOAD CAPACITY OF A BLACK-BOX SIMULATED SERVICE SYSTEM
Mike Bailey
(unpublished, except as an NPS Technical Report)
with a huge assist from THE man who invented Standardized Time Series methods
Lee Schruben, now of VPI

2. DISCRETE EVENT SERVICE SYSTEMS
STUFF comes in through a central conduit at rate l
centralized, controllable, non-lattice process
completed work is ejected from the system
the system can process STUFF at m per unit time

3. l HOW BIG CAN l BE? completion rate Service Action
Motivation: Initiation-to-completion of
Basic Operational Subtask (BOST)
via radio transmissions

4. HOW BIG CAN l BE? TOO SMALL TOO LARGE
System handles the input with grace
Completion times largely unaffected by traffic
Unused capacity
TOO LARGE
Internal queues build up to unlimited size
Completion times grow indefinitely large

5. WORKLOAD ESTIMATION
Counterintuitive: we rarely observe constant input rates in real life!
Important: the maximum capacity of a service system (m-max)
Other motivating examples
Chip fabrication
Student homework

6. ALLOWABLE COMPLEXITY Failures Jammed Radios
Internal Queues of partially completed, prioritized work in progress
Movement into/out of range
Shift Voice/Digital
Change Nets

7. WORK-CONSERVING SYSTEMS
NO...
expiration of tasks
tasks creating other tasks
tasks splitting or combining
tasks that never finish

8. l RECIRCULATION l(t) is the RECIRCULATION rate of the system
completion rate
Service Action
l(t) is the RECIRCULATION rate of the system
Kelly, Walrand, Disney and Kiessler:

9. AND SO We re-circulate completed tasks
This starts out too fast or too slow
As the system stabilizes, l(t) becomes stationary
Solve this problem, and solve _____?

10. Brownian Motion
{X(t), t>=0} is a Brownian Motion stochastic process if
m is the drift of the Brownian Motion
s is the standard deviation of the “noise”
discovered by Brown in 1827 while observing
microscopic pollen in water
Einstein’s 1905 Nobel Prize in Physics
for characterizing Brownian Motion

11. Drift = 10, Sigma = 30

12. DRIFT = 0

13. Brownian Bridge
let Yi,j be the jth recirculation time of the ith independent replication
heavily correlated
for large enough j, Y i,j are identically distributed
let...

14. Scaling scale observation k to the interval [0,1] divide by (sn1/2)/k
result is {Ti(t), t in [0,1]} is a Brownian Bridge
a Brownian motion scaled to be 0 at each endpoint

15. Finale Ai is a Normal Random Variable independent for each i
Knowing its variance, we can construct F-statistic that rejects when the Y’s are NOT identically distributed
and doesn’t require estimate of s!

16. Applied Probability X is a random variable sums of X’s are Normal
“standardized” they become Z ~ Normal(0,1)
summed squared Z’s is chi-squared
ratios of such sums is has F-distribution

17. Procedure Keep moving backward until the F-statistic rejects
...re-circulation transient
BUILD F-STATISTIC
FOR ENDS OF
EACH REPL.
Keep moving backward until the F-statistic rejects

18. 20 independent replications sample-wise confidence intervals

19. F never leaves the critical region

20. Longer sample system service capacity (0.498, 0.504)