Title: The Twin Measure for System Unfairness and Predictability
1The Twin Measure for System Unfairness and
Predictability
- David Raz
- School of Computer Science, Tel Aviv University
- Jointly with
- Hanoch Levy, Tel Aviv University
- Network Seminar, November 2006
2What is Un-Predictability?
How long will I wait?
How well we can predictHow dispersed is the
real value around our prediction
- Prediction EW
- Un-Predictability VarW
3Importance of Predictability
- Uncertainty about waiting time influences overall
service satisfaction in a through way (Taylor
94). - if a consumer is uncertain about how long they
will have to wait before accessing the web site
of interest there will be negative affect, and
this negative affect will affect the evaluation
of the web site reading event. (Dellaert Kahn
1999) - Many other
4Information level
5Predictability Criterion (Wierman
Harchol-Balter Sigmetrics 05)
- A policy P is always predictable if for every
load ? every service requirement x - Motivation
- Growth rate of VarT(x) is O(x) for all work
conserving policies - Inequality holds for all work conserving policies
when x tends to infinity, equality holds for PS,
SRPT and more
6(No Transcript)
7Higher Information level
- Prediction EWx, state.
- Unpredictability Varx, state
- State can be
- Number of customers in queue
- Workload in the system
- More?
8Problem Black-Box systems
- In a black-box system customers are not aware of
the system structure or state - Web servers
- Call centers (if no queue position is given)
9Solution Active Prediction
Web Server
Internet
- Customers initiate probe messages to check the
system state - Customer can initiate more than one probe, or may
send two probes by accident
10Predictability in the Black-Box setting
Web Server
Internet
- Customer perception a customer experiencing
large difference will feel that the system is
unpredictable - Design perspective even with the same system
state the results are very different, so the
system cannot be predictable - We call identical and simultaneously arriving
jobs twins
11The Twin Measure
- Let C1 and C2 be identical customers, with
arrival times a and ae and service requirements
x and xd. - Random variable Z(x, e, d)d2-d1
- Y(x) lim e?0, d?0 Z(x, e, d)
- For service policy P and job size x
- TP(x)EY(x)
12Details, details
- In fact we define higher moments as well
- The limit must exist. In what way do e, d tend to
zero? - Customers must be indistinguishable egt0 ? d can
be negative. - All limits must be the same
13Policies whose Twin Measure equals zero
- Processor Sharing (PS)
- Least Attained Service (LAS)
- Longest Remaining Processing Time (LRPT)
14Twin Measure for Some Simple Policies
15NP-LCFS
time
C1
C2
Busy Period
Waiting
16P-LCFS
time
Busy Period
Busy Period
17Twin Measure for Some Simple Policies
18Shortest Job First (SJF)
- For service time probability density function
(pdf) b(x) - The load of customers of size x
- Then we can follow the same analysis
19Twin Measure for Some Simple Policies
- FCFS
- NP-LCFS
- P-LCFS
- P/NP-SJF
- P/NP-LJF
- x
- x/(1-?)
- x/(1-?)
- x/(1-?(x))
- x/(1-(?-?(x)))
20Shortest Remaining Processing Time (SRPT)
- Y(x)W(x)R(x)
- W(x) Waiting time of a customer of size x, until
first service - R(x) Service time of customer of size x
- Observe customers arriving in a period dt where
C1 has t remaining service - Customers with service req. ltt create busy period
of size - Customers arriving in this period with tltreqltx
are also served before C2, so the total load is
21Shortest Remaining Processing Time (SRPT)
- This creates a busy period for customers of size
ltx, so the total waiting created at the interval
dt is
22- Conclusion
- Well known
- And finally
23Twin Measure for Some Simple Policies
- FCFS
- NP-LCFS
- P-LCFS
- P/NP-SJF
- P/NP-LJF
- SRPT
- x
- x/(1-?)
- x/(1-?)
- x/(1-?(x))
- x/(1-(?-?(x)))
- x/(1-?(x))
FCFS lt SJFSRPT, LJF lt LCFS
24So which policies are predictable?
25So which policies are predictable?
26Conclusion
- Predictability is important
- The Twin Measure is a requirement for Good
Predictability - We can easily evaluate for many policies
- Results are important since they do not
necessarily agree with previous criterion
27So what are the questions?
- What are we measuring?
- Predictability?
- Fairness?
- Something else?
- Where do we go from here?
- Networks of queues
- Connection to measurement trains