distributed admission control and congestion pricing peter key peterkey@microsoft.com
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Distributed Admission Control and Congestion Pricing
Peter Keypeterkey@microsoft.comhttp://research.microsoft.com/network/disgame.htm
Subplot …
Can guarantees be provided using pricing alone?
Refs: F.P. Kelly (Stats Lab, Cambridge Uni.), P.B Key , S.Zachary (Heriot-Watt Uni.), Distributed Admission Control, preprint
Outline
Introduction Congestion Pricing
Adaptive Admission Control Mathematical Framework Examples
Binomial ModelVirtual Queue markingCritical Timescales
DiscussionCommodity markets and Futures
Resource system (‘network’)
Resource j
Capacity Cj
User /route r Ajr links users to resources
Basic Idea
Users generate load (packets)Network sends back signals (load
dependent)Signals : proportional to load
Act as feedback indicators Represent pricing signals
marginal incremental costs (derivatives …)congestion costsreal money or virtual / distributed mint
Optimisation Framework (for fairness)
CAxts
xUMaxr
r
..
U System optimum
U is utility
rr txxUMax User optimum
rr
jr
jj
rr
xAy
yCxUMax
U C is cost function,eg
C y Cj j c h
Solution
Consistent set of taxes (prices) and load exist s.t.
rr
jj
jj
rrj
jr
txU
dy
dC
onstraineddC
d
xxU
nedunconstrai
case c
0 if
U
Eg Network chooses taxes, user chooses load,solution is network, user and System optimal.But dependent on Utility function, so ….
Matching prices to load
For bounded prices, have to match price load to capacity
ie, require maximum amount users prepared to pay < maximum network can charge
Eg, if xr satisfies then require
'( ) 1r rU x
rr
x C
Admission Control
Send a number of probe packets through the network
Enter the network if none of these packets are marked
Assume: Poisson arrivals, rate Let a(mj) be probability accepted at
node jindependently
User policy
User /route r
M probepackets
Enter if less than m probePackets marked
Product form distribution
1
0 1!
jr mnr
jr R j J kr
n a kn
Equilibrium distribution for the number of calls in progress
n-1 n+1n
v a(n-1) v a(n)
n+1n
Fixed Point Approximation
Define stationary acceptance probability for J={j}, R={r}
Then fixed point approximation for network has unique solution
1, ( )
defr
j j jnr r
E nA n n
{ }:
( , )j j j j
j r ii r jr j r
A A
A
Acceptance ProbabilitiesExample 1
Eg , let 1-aj(mj) be probability any of a number of probe packets are marked
Eg for a burst-scale model, where there is long-range dependence
Pr ( , )j j j ja m Binomial m p
Rejection Probabilities & PDFs
20 40 60 80 100n
0.2
0.4
0.6
0.8
1
1- aHnLRejection probabilities Equilibrium CDF
Setup: =50, thresholds 10, 20
n20 40 60 80 100
n
0.2
0.4
0.6
0.8
1
PDFs
Setup: =50, thresholds 10, 20
n
20 40 60 80 100n
0.025
0.05
0.075
0.1
0.125
0.15
0.175
20 40 60 80 100n
0.05
0.1
0.15
0.2
Setup: =100, thresholds 10, 20
Shadow Prices (buffered)
Max Queue Length Q
Time
Qu
eu
e L
en
gth
packets
Q
Fixed ServiceRate
Mark all packets from start of ‘busy period’ until last packet loss.
Virtual Queue Marking
Put arrivals into a virtual queue, and mark on this
Capacity capacity cv c, eg c
Buffer size bv b , eg b
Virtual Queue Example
Suppose we want to track derivative of queue , (or suppose cost=P[exceed thresh)
M/M/1 (can use other SRD processes)
Equate derivate to a VQ with reduced rate
For virtual queue, rate , thresh K-1, put
[buffer exceeds ] KP K
111 K
K
VQ Thresholds
20000 40000 60000 80000 100000
20
40
60
80
100
120
VQ treshold , Price 0.2 , tcrit =0.1 , H=0.8
Timescales
Connection
Reaction (RTT)
Packet Level
averagerate
Seconds
line rate
ms
s
Application Network
ms
s
Critical timescale
Critical Timescales
Large deviation approach (many sources asymptotic – Courcoubetis and Weber)
1lim log Pr , , supinf , ( )N st
Nc Nb Nn stn s t s b ctN
where 0,1, log sX ts t E e
st
t* and s* are extremals, t* is the critical timescale.
If mark as shadow price, is typical marking time
Critical timescales for VQ
Eg for a Gaussian process, arrival rate , Hurst parameter H
*(1 ) 1
b Ht
c H
1* *
1(1 ) 1vq
b Ht t
Hc
11
2 214
1 1log
2
b cP
n
2 1H
Critical timescales
Example 1Critical Timescales
Threshold Loading Leased line LAN Backbone
1 0.2 0.005 0 0.0000210 0.89 0.27 0.06 0.00145 0.984 5.1 1 0.017
180 0.997 81 17 0.27
Critical timescales
Example 2 BUT, need to have critical timescale less
than time between arrivals (for decisions to be independent)
This is (mean holding time)/(number of calls) in equilibrium 0 as n
Hence, keep virtual queue small, just for cell
scale
Congestion Prices (Timescales)
60 secs LAN)
1 Sec (Backbone)
Example 2 – packet marking
Mark packets if size of Virtual Queue exceeds threshold
IfM probe packets sent
where is mean packet service time(if connections generate packets at rate r,
service rate is c, then =r/c, and 1/ represents “capacity” of queue )
1{0,1 ( ) }Kj j ja m Max K m
Rejection Probabilities & PDFs, VQ marking
Rejection probabilities Equilibrium CDF
Setup: =50, thresholds 5,10
n
20 40 60 80 100n
0.2
0.4
0.6
0.8
1
1- a
20 40 60 80 100n
0.2
0.4
0.6
0.8
1
1- a
PDFs
Setup: =50, thresholds 5,10
n
Setup: =100, thresholds 5,10
20 40 60 80 100n
0.05
0.1
0.15
0.2
1- a
20 40 60 80 100n
0.02
0.04
0.06
0.08
0.1
0.12
1- a
Blocking vs. Marking (price)
50 75 100 125 150 175 200n
0.1
0.2
0.3
0.4
0.5
0.6
Pr
50 75 100 125 150 175 200n
0.1
0.2
0.3
0.4
0.5
0.6
Pr
VQ marking, threshold K=10, capacity (1/)=100
1 Probe packet 5 Probe packets
Blocking
marking
Mixing adaptive and non-adaptive traffic
Simple model: two types of traffic Non-adaptive traffic, requires unit
bandwidth Adaptive traffic: reacts to signals can
halve its bandwidth requirementSuppose price (congestion marking
probability) not to go above 0.2Gives acceptance boundaries
Price regions
Acceptance Boundaries
5 10 15 20 25 30 35n1
2.5
5
7.5
10
12.5
15
n2 8Target Excedeence Prob 0.010 , C= 25.54<
Underload
Overload
To give a price blocking of 10-4
5 10 15 20 25 30
0.2
0.4
0.6
0.8
1
Capacity =25.5 (eg LAN with voice, PCM coding)
Prop. Of adaptive traffic required
Total arrival rate
Discussion
Congestion pricing works well for adaptive applications
We have constructed a model for streams/flows where decisions made by end-systems
System is robust, and can be analysed /engineered
Facilitators
Critical timescales (of marking) small compared to interarrival times, (comparable to RTTs?)
Small buffers in Virtual Queue (compared to transmission delay) to detect quickly
Target loads below 100% …Simple feedback signal, eg ECN bit/byteSignal reflects costsPrices need to match demandUser interface simple (risk apportionment)
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