analysis of a multiservice and an elastic traffic model on a cdma link ioannis koukoutsidis...
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Analysis of a Multiservice and an Elastic Traffic
Model on a CDMA link
Ioannis KoukoutsidisPost-Doctoral Fellow, INRIA
Projet MAESTRO
2
Traffic Demand in a Multiservice Network
Real-time traffic: strict QoS requirements
duration, bit rate
(conversational traffic: audio, video, streaming traffic)
Performance metric: blocking probability
Non real-time traffic: Elastic
transmission rate is freely adjusted
(documents, web pages, downloadable audio/video)
Performance metric: transfer time
3
Traffic Analysis of CDMA networks
Evaluation of capacity is more difficult than FDMA, TDMA or wireline networks
- interference-limited capacity
- different problem and parameters in uplink, downlink
- traffic and transmission power strongly coupled through power control
Need to consider various services and classes of traffic
(variable bit rates, traffic characteristics)
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Modeling Capacity and Throughput on a CDMA Link
W
R
N
EIC
o
b
o
b
N
E: energy per bit to noise density
: processing gain R
W
s
os E
WN
sf
sR
)(1
)(
Uplink:
Capacity is expressed as a function of the number of users the CDMA cell can theoretically sustain without the total power going to infinity
s
s
IC
ICfs
1
)1()(
f : ratio of intercell to intracell interference
5
s
os E
WN
saFa
sR
)(
)(
Downlink: s
s
ICa
ICFas
1
)()(
F : ratio of received intercell to intracell power
a : fraction of received own cell power experienced as intracell interference due to multipath fading
Notes
• Δ(s) is an increasing function of Rs
• Eb/No requirements are higher on the downlink
• DL: power used up for SCH and CCH channels
DL is the bottleneck, even on a symmetric link (despite the use of orthogonal signaling on the downlink!)
6
Objectives of Analysis
Solution of a multiservice model with RT and NRT calls Integration of RT and NRT with “interactive use of resources”
use of QBD process theory for numerical solution resource sharing trade-offs, admission control policies
Solution of an elastic traffic model with only NRT calls Processor-sharing for NRT traffic
application of a GPS model access-control policies
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Multiservice traffic model
RT traffic has priority over the system resources GoS control: more RT calls with degraded transmission rates
NRT traffic employs processor sharing a portion of the total capacity, LNRT is reserved use of whatever capacity is left-over from RT traffic
maxminmaxmin ,, RR
max RTRT LN (number of calls with max rate)
minmax RTRT LM (max number of RT calls)
RTRTRTRTRT
RTRTRT MMNforML
NMforM
max
max
,
,)(
NRTRT LL ,
otherwiseL
NMifMMC
NRT
RTRTRTRT ,
,)( max
8
Two models for NRT capacity usage
HSDPA, HSUPA: High-speed downlink (uplink) packet access (WCDMA) total capacity assigned to a single mobile for a very short time
Total throughput (downlink)
Processor-sharing (standard CDMA) capacity used simultaneously by the number of mobiles present
Total throughput (downlink)
s
o
RT
RTRT
NRTtotal E
WN
MCaFa
MCMR
)(
)()(
s
o
RTNRT
RTNRTNRTRT
NRTtotal E
WN
MCaFaM
MCMMMR
)()(
)(),(
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QuasiBirthDeath Analysis
Departure rate of NRT calls:
QBD process with
for level
),(),( RTNRTNRTtotalNRTRTNRT MMRMMv
),1(),0(
),(,),1,(),0,()( , maxRTMiiiii phases 1max
RTM
00
0
0
00
02
122
011
12
AAA
AAA
AB
Q
o
)(0 NRTdiagA )0 );,(( max2
RTi MjjivdiagA ),(],[
]1,[
]1,[
1
1
1
jivjjjA
jjjA
jjA
NRTRTRTi
RTi
RTi
• HSDPA: Homogeneous QBD process• PS: Non-homogeneous QBD process (LDQBD)
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Numerical Results
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Ergodicity of the LDQBD process
For a homogeneous QBD process, a necessary and sufficient ergodicity condition is
e Aπe Aπ 02
NRTNRTtotalNRT RE ][
What is an ergodicity condition in the LDQBD case?
We observe that the total throughput reaches a limit in both the UL and DL cases, i.e. the sub-matrices of the LDQBD process converge to level-independent submatrices
Theorem: If the homogeneous QBD process is ergodic, the LDQBD process alsois. Conversely, if the homogeneous QBD is not ergodic with positive expected drift,d=πQ0e- πQ2e>0, the LDQBD process is also not ergodic
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Proof sketch
• Denote )( ),( tXtX the LDQBD and QBD processes respectively
• It holds that 2
)(22
22
12 i.e. , AAAAA k
• Then we can show that )()( tXtX st , from which the forward part
of the proof follows
• In the reverse part, we show that there exists a modified QBD process
)(tX which is not ergodic and for which holds )()( tXtX Lst
• Then )(tX L is not ergodic, from which we can establish that the
original LDQBD is not ergodic
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Elastic Traffic Model
s
otot E
WN
Cfn
CnnRUplink
)1()( :
s
otot E
WN
CaFan
CnnRDownlink
)()( :
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Generalized Processor Sharing (GPS)
The GPS model, defined and studied by Cohen (1979), applies here:
)()(
)( ,)(0each of rate service )(
issionfor transm requests
nRnfn
nfnnfnf
n
tot
)(limmin nRR totn
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Poisson arrivals model
- K different groups of flows
- (arrival process)k~ Poisson(λk), service requirement
-
)(kF with mean ][ kE
][: kkk E
0 )(
)(Pr
z
z
n
zg
ngnN
n
k k ifiwhere1i
)(:g(n) ,: ,
- ][][ NENE kk
- Mean transfer times can be derived by Little’s law
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Theorem P1: The stochastic process of the number of flows in the system is ergodic if and only if minR
Theorem P2: The mean sojourn time of a flow whose service requirement is deterministic, c, is given by:
][
][)]([
ETE
ccTE
where E[T] is the mean sojourn time in a corresponding single class system with the same total load and maximum number of admitted flows (in loss systems) and with mean service requirement E[σ]
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Engset-like model
Both service rate reduction and blocking finite population of Mk sources for each class k, total max. no. of flows S
K
k Kzk
k
k
Szzz
M
z
M
z
M
z
Knk
K
kk
k
nnn
zzzz
M
nnnn
M
npk
K
K
K
k
K
1 210 0 0
211
,,,
)(
)(
)(
21
1
1
2
2
21
for .][
][: ,21
k
kkK E
ESnnn
1
1)(:)(
n
iifn
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Theorems
Theorem E1: The blocking probability of a class-m source is given by
K
mkk K
zm
zk
m
m
k
k
Szzz
M
z
M
z
M
z
Snnn Knm
nk
K
mkk
m
m
k
k
mB
zzzz
M
z
M
nnnn
M
n
M
Pmk
K
K
K
K
mk
1 210 0 0
211
)(1
)(1
21
1
1
2
2
21
Theorem E2: The sojourn time of a class-m source is given by
)( 1
)( 121
21
21
),,(
),,()(][
][
Sxxx Km
Sxxx KKm
km
K
K
xxpx
xxpxxxfxE
TE
19
Proof (E2): Consider the countable state space of the system, S.
In a processor-sharing system that is ergodic, the arrival rate must equal the
departure rate, since flows are not queued. Then it suffices to show that
is the departure rate of class-m flows, defined as:
This is straightforward if we consider the regenerative process structure of
Cohen (extended to K classes, viz. that the process
is regenerative), since then the time average equals the mean of the limiting
distribution.
SK
m
Km xxpE
xxfx),,(
][
)(1
1
.][
))()(()(1lim
0
1
d
E
xxfx
t
t
m
Km
t
),,,( 21tK
ttt xxx x
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Insensitivity and truncation properties
Insensitivity properties apply to all GPS examined models In loss systems, truncation principle applies
We can prove insensitivity by an easier and more general method
(Burman’s restricted flow equations, Schassberger’s method of clocks) Truncation principle then follows since the associated Markov
process of the system is reversible
Extend results to other access models (dedicated access, fully
shared access, or other strategies in between)
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Examples
Poisson arrivals, 2 classes, separate limits M1, M2, common limit M (M<M1+M2)
)(!!
)(!!
),(
212
2
1
10 0
212
2
1
1
21 21
21
1
1
2
2
21
zzzz
xxxx
xxp zz
Mzz
M
z
M
z
xx
Engset-like system, 2 classes, source populations M1, M2, separate limits S1,
S2.
1
1
2
2
21
21
0 0 21212
2
1
1
21212
2
1
1
21
)(
)(
),(S
z
S
z
zz
xx
zzz
M
z
M
xxx
M
x
M
xxp
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Graphs
Blocking probabilities in a 2-class,Engset-like system with a common constraint (S=20, M1=15, M2=8)Total load ρ=1000
Blocking probabilities in a 2-class,Engset-like system with separate constraints (S1=10, S2=5, M1=15, M2=8). Total load ρ=1000
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Other Research Directions
Capacity model compare with Shannon’s capacity include spatial density of mobiles
Combine different access techniques (e.g. CDMA and WiFi) study resource sharing and scheduling techniques for different
traffic models