Proportional Fair Coding for Wireless Mesh Networks

1.4 MB·English

by  Premkumar Karumbu

1
Proportional Fair Coding for Wireless Mesh
Networks
K. Premkumar, Xiaomin Chen, and Douglas J. Leith
Abstract—We consider multi–hop wireless networks carrying 4
unicast flows for multiple users. Each flow has a specified
3 Optimal
delay deadline, and the lossy wireless links are modelled as Classical
2
binarysymmetricchannels(BSCs).Sincetransmissiontime,also
1
called airtime, on the links is shared amongst flows, increasing flow f
2 the airtime for one flow comes at the cost of reducing the flow f1 3 Utility 0
1 airtime available to other flows sharing the same link. We −1
0 derive the joint allocation of flow airtimes and coding rates that −2
2 achievestheproportionallyfairthroughputallocation.Thisutility flow f2 −3
optimisation problem is non–convex, and one of the technical
an contributions of this paper is to show that the proportional −40 10 20 De3la0y thres4h0old of5 d0elay−s6e0nsitive7 f0low 80 90 100
J fair utility optimisation can nevertheless be decomposed into (a) Topolopgy (b) OptimumUtilityvsClassicalUtility
a sequence of convex optimisation problems. The solution to
8
1 this sequence of convex problems is the unique solution to the Fig. 1. An illustration of a single cell wireless LAN with 3 flows. The
original non–convex optimisation. Surprisingly, this solution can centralnoderepresentstheAccessPoint(AP),andtheothernodesrepresent

be written in an explicit form that yields considerable insight thewirelessSTAtions.
]
I into the nature of the proportional fair joint airtime/coding rate
N allocation.Toourknowledge,thisisthefirsttimethattheutility
fair joint allocation of airtime/coding rate has been analysed,
.
s and also, one of the first times that utility fairness with delay jointlyoptimisetheflowairtimesandcodingrates.Further,we
c
deadlines has been considered.
[ show that the resulting allocation of airtime and coding rates

IndexTerms—Binarysymmetricchannels,coderateselection, is qualitatively different from classical results. For example,
1
cross–layer optimisation, optimal packet size, network utility consider a single hop wireless network carrying three flows,
v
maximisation, resource allocation, scheduling
3 see Figure 1. Flow f1 is a delay–sensitive flow (e.g. video)
2 while flows f2 and f3 are delay–insensitive flows (e.g. TCP
7 I. INTRODUCTION data). Transmissions are scheduled in a TDMA manner, and
3 the delay deadline for flow f is one schedule period, while
. In this paper, we consider wireless mesh networks with 1
1 the delay deadline for flows f and f is infinite. The channel
lossy links and flow delay deadlines. Packets which are 2 3
0 symbol error rate is 10−2 for all flows, and flows use MDS
decodedafteradelaydeadlinearetreatedaslosses.Wederive
2
codes for error correction. The proportionally fair airtime
1 the joint allocation of flow airtimes and coding rates that
and coding rate allocation that we show in this paper (see
: achieves the proportionally fair throughput allocation. To our
v Eqns. (21) and (22)) results in the allocation of 41% of the
knowledge, this is the first time that the utility fair joint
i
X allocation of airtime/coding rate has been analysed, and also, airtime for flow f1 while flow f2 and flow f3 each receive
29.5%. Observe that the proportionally fair allocation assigns
r one of the first times that utility fairness with delay deadlines
a unequal airtimes to the flows, which is a notable departure
has been considered (also, see [1], [2]).
from the usual equal–airtime property of the proportional fair
In the special cases where all links in a network are loss–
allocation when selection of delay deadlines and coding rate
free or all flow delay deadlines are infinite, we show that
are not included, e.g. see [3]. The optimal coding rate is 0.62
the proportionally fair utility optimisation decomposes into
for flow f and 0.97 for flows f and f . The coding rate
decoupled airtime and coding rate allocation tasks. That is, 1 2 3
for flow f is much lower than for flows f and f , since
alayeredapproachthatseparatesMACschedulingandpacket 1 2 3
a smaller block size must be used by flow f (and more
coding rate selection is optimal. This corresponds to the 1
redundant symbols for error–recovery) in order to respect the
currentpractice,andthesetaskscanbesolvedseparatelyusing
delaydeadline.Duetothedelaydeadline,theseoptimalcoding
a wealth of classical techniques.
rates yield non–zero loss rates. For flow f , the packet loss
However,weshowthatnosuchdecompositionoccurswhen 1
rate at the receiver, after decoding, is 20%, whereas flows f
one or more links are lossy or, one or more flows have finite 2
andf arelossfree.Thishighlightsanimportantfeatureofthe
delay deadlines. Instead, in such cases, it is necessary to 3
jointairtimeandcodingrateutilityoptimisation.Namely,that
The authors are with the Hamilton Institute, National University of it allows the throughput/loss/delay trade–off amongst flows
Ireland Maynooth, Ireland. E–mail: {Premkumar.Karumbu, Xiaomin.Chen, sharing network resources to be performed in a principled,
Doug.Leith}@nuim.ie
fair manner. Without consideration of coding rate, the trade–
This work is supported by Science Foundation Ireland under Grant No.
07/IN.1/I901. off between throughput and loss cannot be fully understood
2
or optimally managed. Without consideration of airtime, the The problem of Network Utility Maximisation (NUM) has
contentionbetweenflowsforsharednetworkresourcescannot been studied in various contexts, with NUM as a network
be fully captured. layering tool introduced in [4].
Proportionalfairnesscanbeformulatedasautilitymaximi- Much of the work on NUM is concerned with the flow
sation task, with the utility being the sum of log flow rates. schedulingandthroughputallocationthatachievesthenetwork
Figure 1(b) compares the optimal network utility with that stability region. This work focuses on throughput and largely
obtained with a classical type of approach where all flows are ignores delay constraints. Resource allocation problems from
allocated equal airtime and the coding rates are chosen based the viewpoint of network control and stability is studied by
on the channel error probabilities alone (this corresponds to Georgiadis et al. in [5]. Network flow scheduling problems
ignoring the delay deadline of flow f ). It can be seen that are studied in a utility optimal framework by Shakkottai and
1
the optimal approach that we present in this paper potentially Srikantin[6].Inalltheseworksandthereferencestherein,the
offerssignificantperformancebenefitsoverclassicalmethods. emphasis is on the MAC layers and above. In [5], an energy
We note that one of the reasons why the joint selection of optimal scheduling problem is studied in which the PHYsical
airtime/coding rate has not been previously studied is that the layer is also considered.
proportionalfairutilityoptimisationisnon–convex,andhence, Some recent work explicitly includes delay constraints in
powerful tools from convex optimisation cannot be applied the utility optimisation. In [1], Li and Eryilmaz studied the
directly. Also, the study of the throughput performance by problemofend–to–enddelayconstrainedschedulinginmulti–
jointly considering the coding and the MAC has not been hop networks. They propose algorithms based on Lyapunov
performed before. One of the technical contributions of this drift minimisation and pricing, and show that by dynamically
paper is to show that the proportional fair utility optimisation selecting service disciplines, the proposed algorithms sig-
can nevertheless be decomposed into a sequence of convex nificantly outperform existing throughput–optimal scheduling
optimisationproblems.Thesolutiontothissequenceofconvex algorithms.In[2],JaramilloandSrikantstudiedaresourceal-
problems is the unique solution to the original non–convex locationproblem inad hocnetworks withelastic andinelastic
optimisation. Moreover, this solution can be written in an traffic with deadlines for packet reception, and obtained joint
explicit form thereby yielding considerable insight into the congestion control and scheduling algorithm that maximises
nature of the proportional fair airtime/coding rate allocation. a network utility. In this work the focus is on congestion
The rest of the paper is organised as follows. The related control and scheduling, with the PHYsical layer considered
literature on utility optimal resource allocation is discussed in to be error–free.
SectionII.SectionIIIdefinesthenetworkmodel;inparticular, Ashort,preliminaryversionoftheworkinthecurrentpaper
we describe the mesh network architecture, the traffic model, was presented in [7].
and the channel model. We also discuss the transmission
scheduling model, decoding delay deadline, and the network III. NETWORKMODEL
constraints. In Section IV, we obtain a measure for the end– A. Cellular Mesh Architecture
to–end packet decoding error, and describe the throughput of
We consider networks consisting of a set of C ≥ 1 cells,
thenetwork.InSectionV,weformulateanetworkutilitymax-
C ={1,2,··· ,C}whichdefinethe“interferencedomains”in
imisation problem subject to constraints on the transmission
thenetwork.Weallowintra–cellinterference(i.etransmissions
schedule lengths, and discuss the optimization framework. In
by nodes within the same cell interfere) but assume that
Section VI, we discuss two special cases of networks: delay–
there is no inter–cell interference. This captures, for example,
insensitive and loss–free networks, and show that the tasks of
commonnetworkarchitectureswherenodeswithinagivencell
obtaining optimal airtimes and coding rates decouple in these
use the same radio channel while neighbouring cells using
special cases. We discuss the optimal airtime/coding solution
orthogonal radio channels. Within each cell, any two nodes
with some examples in Section VII. Finally we conclude
are within the decoding range of each other, and hence, can
in Section VIII. The proofs of Lemmas and Theorems are
communicatewitheachother.Thecellsareinterconnectedus-
provided in the Appendix.
ing multi–radio bridging nodes to create a multi–hop wireless
network. A multi–radio bridging node i connecting the set of
II. RELATEDWORK cells B(i) = {c1,..,cn} ⊂ C can be thought of as a set of n
singleradionodes,oneineachcell,interconnectedbyahigh–
We consider a multi–hop Network Utility Maximisation
speed, loss–free wired backplane. See, for example, Figure 2.
(NUM)problemwithdeadlineconstraintsandwithapractical
modelforthePHYlayer.Bymeansofchannelcoding,wetry
to recover a packet from the channel errors. Having a low
B. Unicast Flows
coding rate helps in recovering the packets, but at the cost of
asmallfractionofpayload,andatthecostofthetransmission Dataistransmittedacrossthismulti–hopnetworkasasetF
airtimes of other flows. Thus, we consider the problem of = {1,2,··· ,F}, F ≥ 1 of unicast flows. The route of each
resource allocation that answers the following question: how flow f ∈ F is given by C = {c (f),c (f),··· ,c (f)},
f 1 2 (cid:96)f
to allocate throughput across competing flows with each flow where the source node s(f)∈c (f) and the destination node
1
seeing different channel conditions and respecting the delay d(f)∈c (f). We assume loop–free flows (i.e., no two cells
(cid:96)f
deadline. in C are same).
f
3
flow f2 constraintthat(cid:80) T ≤T whereT istheperiodofthe
f:c∈Cf f,c c c
flow f d schedule in cell c. We consider a periodic scheduling strategy
1
in which, in each cell c, service is given to the flows in a
round robin fashion, and that each flow f in cell c gets a time
slice of T units in every schedule.
f,c
E. Flow Decoding Delay Deadline
At the source node s(f) for flow f, we assume that k
f
symbols arrive in each time slot, which allows us to sim-
4 plify the analysis by ignoring queueing. Information symbols
are formed into blocks of D k symbols, where D ∈
a f f f
{1,2,3,···} is the number of time slots that the block may
b c
span. Each block of D k information symbols is encoded
f f
into a block of D n coded symbols, where n = k /r
f f f f f
symbols, with coding rate 0 < r ≤ 1. Here, n is the
flow f3 f f
number of encoded symbols transmitted in one slot i.e. the
transmitted packet size. The code employed for encoding is
Fig.2. Anillustrationofawirelessmeshnetworkwith4cells.Cellsa,
b,c,andduseorthogonalchannelsCH1,CH2,CH3,andCH4 respectively. discussedinSectionIV.ThequantityDf isauseroroperator
Nodes 3, 5, and 6 are bridge nodes. The bridge node 3 (resp. 5 and 6) is suppliedqualityofserviceparameter.Itspecifiesthedecoding
provided a time slice of each of the channels CH1 & CH2 (resp. CH2 & delay deadline for flow f, since after the flow destination has
CH4 for node 5 and CH2& CH3& CH4 for node 6). Three flows f1,f2,
andf3 areconsidered.Inthisexample,Cf1 ={a,b},Cf2 ={d,b,a},and collectedatmostDf successivecodedpacketsitmustattempt
Cf3 ={c,d}. to decode the encoded information symbols.
C. Binary Symmetric Channels F. Network Constraints on Coding Rate
We associate a binaryrandom variable E [b] with the b’th For flow f in cell c, let w be the rate of transmission
f,c f,c
bit transmitted by flow f in cell c. E [b] = 0 indicates that in symbols/second, which is determined by the modulation
f,c
the bit is received correctly, and E [k] = 1 indicates that and spectral bandwidth used for signal transmission and the
f,c
the bit is received incorrectly, i.e., the bit is “flipped”. We within-cellFECused.Eachcellc∈C alongtherouteofflow
f
assume that E [1],E [2],··· are independent and identi- f allocates an airtime of at least nf in order to transmit the
cally distributefd,c(iid), afn,cd P{Ef,c[b] = 1} = αf,c ∈ [0,0.5). packets of flow f. Let Fc := {wff,c∈ F : c ∈ Cf} be the
That is, we have a binary symmetric channel with cross-over set of flows that are routed through cell c. We recall that the
probability αf,c. A transmitted bit may be “flipped” multiple transmissions in any cell c are scheduled in a TDMA fashion,
times as it travels along the route of flow f, and is received and hence, the total time required for transmitting packets for
incorrectly at the flow destination only if there is an odd allflowsincellcisgivenby(cid:80) nf .Since,forcellc,the
number of such flips. The end-to-end cross-over probability transmissionscheduleintervalisf∈TFcuwnfit,csoftime,theencoded
c
along the route of flow f is therefore given by packet size n must satisfy the schedulability constraint
f
αf ={xc∈{0,1},c∈(cid:88)Cf:c∈(cid:80)Cfxc isodd} c(cid:89)∈Cfαfx,cc (1−αf,c)1−xc. f(cid:88)∈Fc wnff,c (cid:54)Tc
Notethatsincewerequiresufficienttransmittimeateachcell
Note that we can accommodate transmission of symbols from
any2m =M–aryalphabet(i.e.notjusttransmissionofbinary alongrouteCf toallownf codedsymbolstobetransmittedin
every schedule period, there is no queueing at the cells along
symbols) by associating m channel uses of the BSC for every
the route of a flow.
transmitted symbol. The symbol error probability (for any
m≥1) is then given by β =1−(1−α )m.
f f
In this channel model, the channel processes across time IV. PACKETERRORPROBABILITY
are independent copies of the BSCs. In practice this can be Each transmitted symbol of flow f reaches the destination
realised by means of an interleaver of sufficient depth (after node erroneously with probability β . Hence, to help protect
f
the channel encoder), which randomly shuffles the encoded against errors when recovering the information symbols, we
symbols, combined with a de-interleaver (before the channel encodeinformationsymbolsatthesourcenodesusingablock
decoder) at the receiver. This interleaving and de–interleaving code (we note here that a convolutional code with zero–
randomly mixes any channel fades, which can then be mod- padding is also a block code). An (n,k,d) block code has
elled as independent channel processes across time. the following properties. The encoder takes a sequence of
k information symbols as input, and generates a sequence
D. Flow Transmission Scheduling of n ≥ k coded symbols as output. The decoder takes a
A scheduler assigns a time slice of duration T >0 time sequenceofncodedsymbolsasinput,andoutputsasequence
f,c
units to each flow f that flows through cell c, subject to the of k information symbols. These information symbols will
4
be error–free provided no more than (cid:98)d−1(cid:99) of the coded and x := 1−rf, B(x) is the Bernoulli distribution with
2 f 2
symbols are corrupted. The Singleton bound [8] tells us that parameterx,H(P)istheentropyofprobabilitymassfunction
d(cid:54)n−k+1, with equality for maximum–distance separable (pmf) P, and I(P(cid:107)Q) is the information divergence between
(MDS) codes. Thus, an MDS code can correct up to the pmfs P and Q.
(cid:22) (cid:23) (cid:22) (cid:23)
d−1 n−k Proof: See Appendix B.
= (1)
2 2
errors.ExamplesforMDScodesincludeReed–Solomoncodes B. Tightness of Bounds
[8], and MDS–convolutional codes [9]. In [9], the authors It can be verified that
show the existence of MDS–convolutional codes for any
code rate. Hereafter, we will make use of Eqn. (1), and so, IEf[1](xf;θf)=θfxf −ln(cid:0)1−βf +βfeθf(cid:1)
confine consideration to MDS codes. However, the analysis Since θ >0 is a free parameter, we can select the value that
f
can be readily extended to other types of code provided a maximises I (x ;θ ) and so provides the tightest upper
corresponding bound on d is available. bound. It caEnf[b1e] vefrififed (e.g. by inspection of the second
Consider a coded block of flow f and let i ∈ derivative) that I (x ;θ ) is concave in θ and so the
{1,2,··· ,D n }indexthesymbolsintheblock.LetE [i]be Ef[1] f f f
f f f KKT conditions are necessary and sufficient for an optimum.
a binary random variable which equals 0 when the i’th coded
The KKT condition here is
symbol is received correctly and which equals 1 when it is
received corrupted. P{E [i]=1}=β and P{E [i]=0}= ∂IEf[1](xf;θf) =x − βfeθf =0
1 − β . From Eqn. (1),fthe probabilifty of the bflock being ∂θf f 1−βf +βfeθf
f
decoded incorrectly is given by which is solved by
  (cid:18) (cid:19) (cid:18) (cid:19)
e =PD(cid:88)fnfE [i]> Dfnf −Dfkf θf∗ =ln xβf −ln 11−−xβf .
(cid:101)f f 2 f f
 
i=1 provided x >β . Substituting for θ∗,
f f f
BTherenosyumllibroalnderormorsvaErifa[b1l]e,s,Eanfd[2]s,o,·t·h·e,(cid:80)EDif=f[1DnffEnff][i]aries ai.ib.di-. mθfa>x0IEf[1](xf;θf)=IEf[1](cid:0)xf;θf∗(cid:1)
nomial random variable. Hence, the probability of a decoding (cid:18)x (cid:19) (cid:18)1−x (cid:19)
=x ln f +(1−x )ln f
error can be computed exactly. However, the exact expression f β f 1−β
f f
is combinatorial in nature, and is not tractable for further
=I(B(x )(cid:107)B(β ))
analysis. We therefore proceed by obtaining upper and lower f f
bounds on the error probability, and show that the bounds are and by Lemmas 1 and 2, the probability e of a decoding
(cid:101)f
the same up to a prefactor, and that the prefactor decreases error satisfies
as the block size D n increases. Hence, we pose the NUM
f f
based on the upper bound on the error probability. Also, we
relax the following constraints: n ∈ Z and k ∈ Z , and Γe−DfnfI(B(xf)(cid:107)B(βf)) ≤e(cid:101)f ≤e−DfnfI(B(xf)(cid:107)B(βf)).
f + f +
allow them to take positive real values, i.e., nf ∈ R+ and It can be seen that the upper and lower bounds are the same
kf ∈R+. to within prefactor Γ, and the gap between these bounds
decreases exponentially as the block size D n increases.
f f
A. Upper and Lower Bounds
Lemma 1 (Upper Bound). The end–to–end probability e of V. NETWORKUTILITYOPTIMISATION
(cid:101)f
a decoding error for flow f satisfies We are interested in the fair allocation of flow airtimes
(cid:0) (cid:1) and coding rates amongst flows in the network. Other things
e ≤ exp −D n I (x ;θ ) (2)
(cid:101)f f f Ef[1] f f beingequal,weexpectthatdecreasingthecodingraterf (i.e.,
=: ef(θf,nf,rf). increasing the number Dfnf −Dfkf of redundant symbols
where x := 1−rf, r =k /n is the coding rate, θ >0 is transmitted) for flow f will decrease the error probability ef,
f 2 f f f f and so increases the flow throughput. However, decreasing
the Chernoff–bound parameter and the function I (x;θ) :=
Z
θx−ln(E(cid:2)eθZ(cid:3))iscalledtheratefunctioninlargedeviations the coding rate increases the coded packet size Dfnf, and
so increases the airtime used by flow f. Since the network
theory.
capacity is limited and shared by other flows, this generally
Proof: See Appendix A. decreasestheairtimeavailabletootherflowsandsodecreases
their throughout. Similarly, increasing the packet size D k
Lemma 2 (Lower Bound). The end–to–end probability e of f f
(cid:101)f
of flow f increases its throughput but at the cost of increased
a decoding error for flow f satisfies
airtime and a reduction in the throughput of other flows. We
e(cid:101)f ≥Γexp(−DfnfI(B(xf)(cid:107)B(βf))) (3) formulate this tradeoff as a utility fair optimisation problem.
Inparticular,wefocusontheproportionalfairallocationsince
where
β it is of wide interest and, as we will see, is tractable, despite
Γ= f exp(−D n H(B(x )))
1−β f f f the non–convex nature of the optimisation.
f
5
The utility fair optimisation problem is Lemma 3. . For convex sets Y and Z, and for a function
f : Y ×Z → R that is concave in y ∈ Y and in z ∈ Z,
max U(θ,(n,x)) (4) but not jointly in (y,z), the solution to the joint optimisation
θ,(n,x)
subject to (cid:88) nf ≤T , ∀c∈C (5) problem
w c
f:c∈Cf f,c max f(y,z) (9)
y∈Y,z∈Z
θf >0, ∀f ∈F (6) is unique, and is the same as the solution to
x ≤λ ∀f ∈F (7)
f f maxmaxf(y,z), (10)
x ≥λ ∀f ∈F (8) z∈Z y∈Y
f f
with θ := [θ ] the vector of Chernoff parameters, n :=
f f∈F
[n ] the vector of flow packet sizes, and x := [x ] iff(y∗(z),z)isaconcavefunctionofz,whereforeachz ∈Z,
f f∈F f f∈F
the vector of flow coding rates (where we recall that x = y∗(z):=argmaxf(y,z).
f
(1−r )/2).Eqn.(5)enforcesthenetworkcapacity(ortheflow y∈Y
f
schedulability)constraints,Eqn.(6)thepositivityconstrainton Proof: See Appendix C.
the Chernoff parameters, and the constraints Eqns. (7)–(8) are This lemma establishes conditions under which we can
introducedfortechnicalreasonsthatwillbediscussedinmore transform a non–convex optimisation into a sequence of con-
detail shortly. vex optimisations. Roughly speaking, we proceed by optimis-
For proportional fairness, we select the sum of the log of ing over each variable in turn and substituting the optimal
the flow throughputs as our network utility U. For flow f variable value that is found back into the objective function.
the expected throughput is D k (1−e ) symbols in every This creates a sequence of objective functions. Provided each
f f (cid:101)f
time interval of duration D T (we recall that d(f) is the member of this sequence is concave in the variable being
f d(f)
destination cell of flow f), which is the same as k (1−e ) optimised (but not necessary jointly concave in all variables),
f (cid:101)f
symbols every time interval of duration T , where D k is thesolutiontothesequenceofconvexoptimisationscoincides
d(f) f f
the information packet size and e the packet decoding error withthesolutiontotheoriginalnon–convexoptimisation.Ev-
(cid:101)f
probability. As the exact expression of e is intractable, we idently, the condition that concavity holds for every objective
(cid:101)f
use the upper bound for e which is e . Thus, the objective function in this sequence is extremely strong. Remarkably,
(cid:101)f f
function is given by however, we show that it is satisfied in our present network
utility optimisation.
(cid:88)
U(θ,(n,x)):= ln(k (1−e (θ ,n ,x )))
f f f f f
f∈F C. Optimal θ∗(x )
(cid:88) f
= ln(n r (1−e (θ ,n ,x )))
f f f f f f Takingasequentialoptimisationapproach,webeginbyfirst
f∈F solving the optimisation
(cid:88)
= ln(n (1−2x )(1−e (θ ,n ,x )))
f f f f f f
max U(θ,(n,x))
f∈F θ
(cid:88) (cid:88)
= ln(nf)+ ln(1−2xf) subject to θf >0, ∀f ∈F
f∈F f∈F given packet sizes n ∈ ZF and coding rates x ∈ [λ ,λ ]F.
(cid:88) + f f
+ ln(1−ef(θf,nf,xf)). The objective function is separable and concave in the θfs.
f∈F The partial derivative of U(θ,(n,x)) with respect to θf is
given by
The optimisation problem yields the proportional fair flow
coding rates and coded packet size nf. Since the PHY trans- ∂U(θ,(n,x)) = efDfnf (cid:20)x − βfeθf (cid:21) (11)
mission rates wf,c are known parameters, the coded packet ∂θf 1−ef f 1−βf +βfeθf
size is proportional to the airtime used by a flow (i.e., the
Setting this derivative equal to zero, provided x >β this is
airtime is given by n /w ). f f
f f,c
solved by
A. Non–Convexity βfeθf∗ =x
The objective function U(θ,(n,x)) is separable in 1−βf +βfeθf∗ f
x 1−β
(θf,(nf,xf)) for each flow f. However, it can be readily or,eθf∗ = f f
verified that ln(1−ef(θf,nf,xf)) is not jointly concave in βf 1−xf
(cid:18) (cid:19) (cid:18) (cid:19)
(θ ,(n ,x )),andso,theoptimisationisnon–convex.Hence, x 1−x
f f f or,θ∗ =ln f −ln f . (12)
thenetworkutilitymaximisationproblemdefinedinEqns.(4)– f β 1−β
f f
(8) is not in the standard convex optimisation framework.
Observe that in fact θ∗ is function only of x and not both
f f
n andx .Therequirementforx >β ensuresthatθ∗ >0.
f f f f f
B. Reformulation as Sequential Optimisations When x ≤ β , the derivative (Eqn. (11)) is negative for all
f f
θ > 0. In this case, the optimum θ∗ is zero which yields
We proceed by making the following key observation. f f
an error probability e of one. Thus, for error recovery we
f
6
require x >β i.e. the coding rate r <1−2β , and for a Lemma 4.
f f f f
non–empty feasible region in the NUM problem formulation (cid:16) (cid:17) (cid:88) (cid:88) (cid:16) (cid:17)
in Eqns. (4)–(8) the constraints on x should satisfy the U θ∗(I˜),(n˜,I˜) = n˜f + ln 1−2xf(I(cid:101)f)
f
f∈F f∈F
following:λ <1−2β andλ >0.Wenotethatthecapacity
region for afBSC havinfg a crofss–over probability α with an + (cid:88)ln(cid:16)1−e−Dfen˜f+I˜f(cid:17).
f
m–ary signalling is (0,m(1−H(αf))), and the coding rate f∈F
1−2β lies in the capacity region.
f is jointly concave in n(cid:101)f and I(cid:101)f.
Proof: See Appendix D.
D. Optimal (n∗,x∗) Hence, we have the following convex optimisation problem
f f
(cid:16) (cid:17)
The next step in our sequential optimisation approach is to max U θ∗(I˜),(n˜,I˜) (17)
solve (n˜,I˜)
(cid:88) en(cid:101)f
max U(θ∗(x),(n,x)) subject to ≤T , ∀c∈C (18)
w c
(n,x) f,c
subject to (cid:88) wnff,c ≤Tc, ∀c∈C fI˜f:c∈≤Cfλ˜f ∀f ∈F (19)
f:c∈Cf I˜ ≥λ˜ ∀f ∈F (20)
x ≤λ ∀f ∈F f f
f f
x ≥λ ∀f ∈F We solve the above maximisation problem using the La-
f f
grangian relaxation approach. The Lagrangian function of the
That is, we substitute into the objective function for the problem is given by
(cid:16) (cid:17)
optimalθ∗ foundinSectionV-C.DefiningI =x ln xf + (cid:16) (cid:17)
f (cid:16) (cid:17) f f βf L n(cid:101),I(cid:101),p,ν,ν
(1−xf)ln 11−−βxff , (cid:88) (cid:88) (cid:88) en(cid:101)f 
U(θ∗(x),(n,x))= (cid:88)ln(nf)+ (cid:88)ln(1−2xf) :=f∈FUf(θf∗(I(cid:101)f),(n(cid:101)f,I(cid:101)f))−c∈Cpcf∈Fcwf,c −Tc
+f∈(cid:88)F ln(cid:0)1−ef−∈FDfnfIf(cid:1). − (cid:88)νf(cid:16)I˜f −λ˜f(cid:17)+ (cid:88)νf(cid:16)I˜f −λ˜f(cid:17)
f∈F f∈F
f∈F
where p ≥ 0, ν ≥ 0, and ν ≥ 0 are Lagrangian multipliers
ItcanbeverifiedthatU(θ∗(x),(n,x))isnotjointlyconcave correspondingtotheconstraintsgiveninEqns.(18)–(20).The
in (n,x). To proceed, we therefore rewrite the objective in channel error probabilities β s are strictly positive, and the
f
terms of the log–transformed variables n = ln(n ) and channel coding rates are always assumed to be in the interior
(cid:101)f f
I(cid:101)f = ln(If). Observe that the mapping from nf to n(cid:101)f is ofthefeasibilityregion.Hence,theconstraintsforthechannel
invertible and similarly the mapping from xf to I(cid:101)f. Since I(cid:101)f coding rate given in Eqns. (19), (20) are not active at the
is a monotone increasing function of xf (this can be verified optimal point, and the Lagrangian costs νfs and νfs are zero.
by inspection of the first derivative), the inverse mapping Thus,theshadowcostscorrespondingtotheseconstraintswill
from I(cid:101)f to xf exists and is one-to-one. With the obvious not appear in the Lagrangian relaxation.
abuse of notation, we write inverse map as xf(I(cid:101)f). In terms Since the optimisation problem falls within convex optimi-
of these log–transformed co-ordinates, the objective function sation framework, and the Slater condition is satisfied, strong
(cid:16) (cid:17)
is U θ∗(I˜),(n˜,I˜) . We note that the problem defined in duality holds. Hence, the KKT conditions are necessary and
sufficient for optimality. Differentiating the Lagrangian with
Eqns. (4)–(8) is equivalent to the problem,
respect to n at n =n∗, and setting equal to zero yields the
(cid:16) (cid:17) (cid:101)f (cid:101)f (cid:101)f
max U θ,(n˜,I˜) KKT condition
θ,(n˜,I˜) (cid:88) en˜f 1+ Dfexp(n(cid:101)∗f +I(cid:101)f∗)e−Dfexp(n(cid:101)∗f+I(cid:101)f∗) = (cid:88) pcen(cid:101)∗f
subject to wf,c ≤Tc, ∀c∈C (13) 1−e−Dfexp(n(cid:101)∗f+I(cid:101)f∗) c∈Cf wf,c
fθ:fc∈>Cf0, ∀f ∈F (14) or, 1+ Dfn∗fIf(x∗f)e−Dfn∗fIf(x∗f) = (cid:88) pcn∗f (21)
I˜f ≤λ˜f ∀f ∈F (15) 1−e−Dfn∗fIf(x∗f) c∈Cf wf,c
I˜f ≥λ˜f ∀f ∈F (16) Similarly, the KKT condition for I(cid:101)f∗ is
and hence, by Lemma 3, the solution to the log–transformed
problem is the same as that of the problem defined in Dfexp(n(cid:101)∗f +I(cid:101)f∗)e−Dfexp(n(cid:101)∗f+I(cid:101)f∗) = 2 If∗
Eqns. (4)–(8). We solve the maximisation problem by convex 1−e−Dfexp(n(cid:101)∗f+I(cid:101)f∗) 1−2x∗f θf∗
optimisation method. We show that the objective function is
jointly concave in (cid:16)n˜,I˜(cid:17) in the following Lemma. or, Dfn∗fIf(x∗f)e−Dfn∗fIf(x∗f) = 2 If(x∗f) (22)
1−e−Dfn∗fIf(x∗f) 1−2x∗f θf∗(x∗f)
7
Combining eqns. (21) and (22), yields and the projected subgradient descent update is
c(cid:88)∈Cf pwcfn,c∗f −1= 1−22x∗f θIff∗((xx∗f∗f)). (23) pc(i+1)=pc(i)−γ·Tc−f(cid:88)∈Fcn∗fw(pf,(ci))+
Observe that the LHS is a function of n∗f and the RHS is a where γ > 0 is a sufficiently small stepsize, and [f(x)]+ :=
function of x∗. Thus, the choice of packet size parameter n∗
f f max{f(x),0}ensuresthattheLagrangemultipliernevergoes
and coding rate parameter x∗ are in general coupled.
f negative (see [10]).
The subgradient updates can carried out locally by each
E. Distributed Algorithm for Solving Optimisation cell c since the update of pc only requires knowledge of the
packet sizes n∗(p(i)) of flows f ∈F traversing cell c. Thus,
GiventhevaluesoftheLagrangemultipliersp∗,thesolution f c
at the beginning of each iteration i, the flow source nodes
to Eqn. (23) specifies the optimal packet size and coding rate.
choose their packet sizes as D n∗(p(i)) and the coding rates
To complete the solution to the optimisation it therefore re- f f
as 1−2x∗(p(i)), and each cell computes its cost based on
mainstocalculatethemultipliersp∗.Thesecannotbeobtained f
the packet sizes (or equivalently the rates) of flows through
in closed form since their values reflect the network topology
it. The updated costs along the route of each flow are then
anddetailsofflowrouting.However,theycanbereadilyfound
fed back to the source nodes to compute the packet size and
inadistributedmannerusingastandardsubgradientapproach.
coding rate for the next iteration.
We proceed as follows. The dual problem for the primal
Observe that the Lagrange multiplier p can be interpreted
c
problem defined in Eqn. (17) is given by
as the cost of transmitting traffic through cell c. The amount
of service time that is available is given by ∆ = T −
min D(p), c
p≥0 (cid:80) n∗f(p(i)). When ∆ is positive and large, then the La-
where the dual function D(p) is given by f∈Fc wf,c
grangian cost p decreases rapidly (because the dual function
c
(cid:88) D(·) is convex), and when ∆ is negative, then the Lagrangian
D(p)=max Uf(θf∗(I(cid:101)f),(n(cid:101)f,I(cid:101)f)) cost p increases rapidly to make ∆ ≥ 0. We note that the
(n(cid:101),I(cid:101)) f∈F c
increase or decrease of p between successive iterations is
  c
(cid:88) (cid:88) en(cid:101)f proportionalto∆,theamountofservicetimeavailable.Thus,
+ pcTc− w  (24) thesub–gradientprocedureprovidesadynamiccontrolscheme
f,c
c∈C f∈Fc to balance the network load.
(cid:88) (cid:16) (cid:17)
= Uf θf∗(I(cid:101)f(p)),(n(cid:101)∗f(p),I(cid:101)f∗(p)) The resulting distributed implementation of the joint air-
time/codingrateoptimisationtaskissummarisedinAlgorithm
f∈F
  1.
(cid:88) (cid:88) en(cid:101)∗f(p)
+ pcTc− w . Algorithm 1 Distributed implementation of joint air-
f,c
c∈C f∈Fc time/coding rate optimisation.
Each cell c runs:
From Eqn. (24), for any (n(cid:101),I(cid:101)),
loop
  (cid:34) (cid:32) (cid:33)(cid:35)+
D(p)≥f(cid:88)∈FUf(θf∗(I(cid:101)f),(n(cid:101)f,I(cid:101)f))+(cid:88)c∈CpcTc−f(cid:88)∈Fcwen(cid:101)ff,c, en1d.lpoco(pi+1)= pc(i)−γ· Tc−f(cid:80)∈Fcn∗fw(pf,(ci))
and in particular, the dual function D(p) is greater than that The source for each flow f runs:
for I(cid:101)f =I(cid:101)f∗(p(cid:101)) for some arbitrary p(cid:101), i.e., loop
1. Measure (cid:80) pc , the aggregate cost of using the
D(p)≥ (cid:88) Uf(cid:16)θf∗(I(cid:101)f∗(p(cid:101))),(n(cid:101)∗f(p(cid:101)),I(cid:101)f∗(p(cid:101)))(cid:17) cells along thecr∈oCuftewfo,fc flow f. E.g. if each cell updates
f∈F theheaderoftransmittedpacketstoreflectthissum,itcan
 
+(cid:88)pcTc− (cid:88) ewn(cid:101)∗f(p(cid:101)) t2h.eFninbdetehcehouendiqbuaeckpatcoktehtessizoeurnc∗feabnydthceodflionwgrdaetsetixn∗fatitohnat.
f,c solve (23). Since there are only two variables, a simple
c∈C f∈Fc
  numerical search can be used.
(cid:88) (cid:88) en(cid:101)∗f(p(cid:101)) end loop
=D(p(cid:101))+ (pc−p(cid:101)c)Tc− w  (25)
f,c
c∈C f∈Fc
Thus, a sub–gradient of D(·) at any p is given by the vector
(cid:101) VI. TWOSPECIALCASES
 
Tc− (cid:88) nw∗ff(,p(cid:101)c) , A.SDupelpaoys–eIntsheensdietilvaeyNdeetawdolirnkes Df → ∞ for all flows. For
f∈Fc c∈C any positive bounded n∗I∗, i.e., 0<n∗I∗ <∞, the LHS of
f f f f
8
Eqn. (22) can be written as B. Loss-Free Networks
Dfn∗fIf∗e−Dfn∗fIf∗ = Dfn∗fIf∗ FroSmupEpoqsne. t(h1e2)c,hwanenoeblsseyrmvebothlaetrrorrateβf =0forallflows.
1−e−Dfn∗fIf∗ eDfn∗fIf∗ −1
= (cid:80)∞Df(nD∗ffInf∗∗fIf∗)j βlfi→m0θf∗ =∞, (31)
j=1 j! and this yields e = 0 for all flows. The objective function
f
1 (cid:80)
= in Eqn. (17) degenerates to f∈Fln(nf(1−2xf)). We note
1+(cid:80)∞j=2 (Dfn∗fjI!f∗)j−1 that for any x∗f > βf > 0, as βf ↓ 0, If(x∗f) → ∞. Hence,
the LHS of Eqn. (22) becomes,
∴Dfli→m∞Df1n−∗fIef∗−eD−fDnf∗fnIf∗∗fIf∗ =0. (26) lim Dfn∗fIf(x∗f)e−Dfn∗fIf(x∗f) =0. (32)
Thus, the asymptotic optimal coding rate x∗ as the delay βf↓0 1−e−Dfn∗fIf(x∗f)
f
deadline requirement D →∞ is the solution to In the same was as in Eqn. (28), this limit can be achieved
f
by x∗ =0 (i.e. r∗ =1). Similarly, the optimal packet size is
1−22x∗f Iθf∗((xx∗f∗f)) =0. (27) nth∗fat=ffor(cid:80)dce∈laCfy1–picn/sweffn,cs.itiTvheinseotwptoimrkasl,speaecEkeqtn.si(z2e9)i,sainddenitticcaanlbtoe
Since βf <xf <1/2, it is sufficient to find the solution to verifiedthatinfactitcorrespondstotheclassicalproportional
I (x∗) fair rate allocation for loss–free networks, as expected.
f f =0.
θ∗(x∗)
f VII. EXAMPLES
I (x∗)
Note that lim f f =0, A. Single cell
x∗f→βf θ∗(x∗f)
We begin by considering network examples consisting of a
and hence, lim x∗ =β . (28)
f f single cell carrying multiple flows. The network topology is
Df→∞
illustratedschematicallyinFigure1andmightcorrespond,for
Since this is the limiting solution and x∗ > β , one can use
f f example, to a WLAN.
x∗ =β +(cid:15) for some arbitrarily small (cid:15)>0. Similarly, from
f f 1) Mix of delay–sensitive and delay–insensitive flows:
Eqn. (21), the asymptotic optimal packet size n∗ as D →∞
f f Supposetheflowsinthenetworkbelongtotwoclasses,oneof
is
which is delay–sensitive and has a delay–deadline D whereas
1
n∗ = (29) theotherisdelay–insensitivei.e.hasaninfinitedelaydeadline.
f (cid:80) p /w
c∈Cf c f,c These classes might correspond, for example, to video and
wherethemultipliersp areobtained,asbefore,bysubgradient data traffic. Figure 3(a) plots the optimal airtime allocation
c
descent as the delay deadline D is varied. In this example, there is
  + a single delay–sensitive flow and two delay–insensitive flows,
pc(i+1)=pc(i)−γ·Tc− (cid:88) n∗fw(p(i)) (30) aflnodwtahnedafiortrimoneeaollfotchaetiodnelaisy–sihnosewnnsitfiovre tflhoewdse(lbaoyt–hsernescietiivvee
f,c
f∈Fc the same airtime allocation). As expected, it can be seen
Observe that the optimal coding rate x∗ = β +(cid:15) which is that the airtime allocations of the delay–sensitive and delay–
f f
given by the solution of Eqn. (28) is determined solely by insensitive flows approach each other as the delay deadline
the channel error rate β of flow f. It is therefore completely D is increased. However, it is notable that they approach
f
independent of the other network properties. In particular, it each other fairly slowly, and when the delay deadline is low
is independent of the packet size n∗ used, of the other flows the airtime allocated to the delay–sensitive flow is almost
f
sharing the network and of the network topology. Conversely, 50% greater than that allocated to a delay–insensitive flow.
observe that the optimal packet size n∗ in Eqn. (29) and This behaviour is qualitatively different from the classical
f
Eqn. (30) is dependent on the network topology and flow proportional fair allocation neglecting coding rate and delay–
routes, but is completely independent of the error rate β deadlines, which would allocate equal airtimes to all flows.
f
and coding rate x . That is, in delay–insensitive networks, By taking coding rate and delay deadlines into account, our
f
the joint airtime/coding rate optimisation task breaks into approachallowstheresourceallocationtoflowswithdifferent
separate optimal airtime allocation and optimal coding rate qualityofservicerequirementstobecarriedoutinaprincipled
allocation tasks which are completely decoupled. Our optimi- and fair manner.
sationthereforeyieldsaMAC/PHYlayering,wherebyairtime Figure 3(b) plots the optimal airtime allocation as the
allocation/transmission scheduling is handled by the MAC number N of delay–insensitive flows is varied. It can be seen
whereas coding rate selection is handled by the PHY, with no that the airtime allocated to each flow decreases as N is
cross-layer communication. It is important to note, however, increased, as expected since the number of flows sharing the
that this layering does not occur in networks where one or network is increasing. Interestingly, observe that the airtime
more flows have finite delay–deadlines; see Section VII for a allocated to the delay–sensitive flow is a roughly constant
more detailed discussion. marginabovethatallocatedtothedelay–insensitiveflows.The
9
0.45 0.56
Delay−sensitive flow Flow 1
Delay−insensitive flows 0.55 Other flows
me (fraction of schedule period)0.03.54 me (fraction of schedule period)0000....05555.12345
mal Airti 0.3 mal Airti0.49
Opti Opti0.48
0.47
0.25 0.46
0 10 20 30 40 50 60 70 80 90 100 10−5 10−4 10−3 10−2 10−1
Delay threshold of delay−sensitive flow Symbol error rate for flow 1
(a) OptimalairtimeallocationvsdelaydeadlineD,N=2. (a) Optimalairtimeallocationvschannelsymbolerrorratefor
flow1,symbolerrorrateforflow2isheldfixedat10−2.
0.7
Delay−sensitive flow
Delay−insensitive flows 0.58
Optimal Airtime (fraction of schedule period)000000......123456 Optimal Airtime (fraction of schedule period)00000.....044555.682465 FOltohwe r1 flows
0.44
0
2 4 6 8 10 12 14 16 18 20
Number N of delay−insensitive flows 0.42
0 5 10 15 20 25 30 35 40 45 50
PHY rate for flow 1 (symbols/schedule)
(b) OptimalairtimeallocationvsN.D=1.
(b) OptimalairtimeallocationvsPHYrateofflow1,PHYrate
Fig.3. SingleWLANwithonedelay–sensitiveflowandNdelay–insensitive forflow2isheldfixedat10symbols/schedule.
flows.DelaysensitiveflowhasdelaydeadlineD,delay–insensitiveflowshave
infinitedelaydeadlines.Rawchannelsymbolerrorrateis10−2forallflows, Fig. 4. Single WLAN with two delay–sensitive flows, both with delay
PHY rate for all flows is 10 symbols per schedule period. Optimal airtimes deadline D = 1. In upper figure, PHY rate for both flows is 10 symbols
aregivenasaproportionofthescheduleperiod. per schedule period and channel symbol error rate for flow 1 is varied. In
lowerfigure,channelsymbolerrorrateforbothflowsis10−2,andPHYrate
forflow1isvaried.
delay–sensitive flow is therefore “protected” from the delay–
insensitive flows. However, in contrast to ad hoc approaches, Conversely,asthePHYrateisdecreased,theairtimeallocation
this protection is carried out in a principled and fair manner. for flow 1 increases. Again, note that this is qualitatively dif-
ferentfromtheclassicalproportionalfairallocationneglecting
2) Mix of near and far stations: Consider now a situation
coding rate and delay–deadlines which would allocate equal
whereallflowshavethesamedelaydeadlineD,butwherefor
airtimes to all flows.
someflowsthesourcesarelocatedclosetothedestinationand
3) Unequal Airtimes: The basic observations in these ex-
forotherflowsthesourcesarefurtheraway.Wethereforehave
amples apply more generally. In particular, as noted above,
two classes of flows, one with a higher channel symbol error
in a loss-free, delay–insensitive single-cell network the pro-
rate than the other when both use the same PHY rate. Figure
portional fair allocation is to assign equal air–time to all
4(a) plots the optimal airtime allocation for a flow in each
flows ([3] and Section VI-B). However, when delay deadlines
classasthechannelerrorrateforoneclassisvaried.Whenthe
channel error rates for both classes is the same (β =10−2), are introduced and/or links are lossy, we see an interesting
f
phenomenon.
it can be seen that the airtime allocation is the same. As the
channel error rate decreases, the airtime allocated to flow 1 Lemma 5. The optimum rate allocation x∗ (or equivalently
decreases. Conversely, as the channel error rate increases, the r∗) is not equivalent to an equal air–time allocation.
airtime allocated to flow 1 increases.
Proof: See Appendix E
Figure 4(b) plots the optimal airtime allocation when flows
In particular, flows that see a better channel get less air–
in both classes have the same channel error rate but different
times than flows that see a worse channel.
PHY rates i.e. where the PHY modulation has been adjusted
to equalise the channel error rates. When the PHY rates
B. Multiple cells
are the same (w = 10 symbols per schedule period), the
f,c
airtime allocation is the same to both classes. As the PHY We now consider a mesh network consisting of N cells
rate is increased, the airtime allocation for flow 1 decreases. carryingN+1flowsinthewell-studiedParkingLottopology.
10
1.15
D1=D2=1
flow 1 D1=1,D2=1e5
D1=1e5,D2=1
2
flow 2 flow 3 flow 4 flow N+1 ofclass 1.1
me
1 2 3 N-1 N 1/airti
Fig. 5. A linear Parking Lot network with N cells and N+1 flows (one class 1.05
of
multi-hopflowandN single-hopflows). me
Airti
1
ThenetworktopologyisillustratedinFig.5.Theflowsinthis 2 3 4 5 6 7 8
Numberofcells
networkcanbeassignedtotwoclasses:class1consistsofthe
N-hop flow, and class 2 consists of the single–hop flows 2, Fig.6. Ratioofairtimesvs.numberN ofcellsinParkingLottopologyof
3, ···, N +1. Each cell has the same schedule period, i.e. Fig.5.They-axisistheratiooftheairtimeallocatedtotheN-hopflowto
thatallocatedtoasinglehopflow;notethattheairtimeoftheN-hopflowis
T =T,∀c∈C.
c thesumofallocatedairtimeineachcellalongtheflowsroute.Dataisshown
1) Impact of number of hops: Suppose that both classes forthreedifferentdelaydeadlinerequirements,asindicatedinthelegend.All
flowshavethesamePHYrate.
of flow use the same symbol transmission PHY rate and
experience the same loss rate in each cell. Then the N-hop
12000
flow will experience a higher end-to-end symbol error rate class1
class2
that the single hop flows, and the loss rate will increase with 10000 class3
N. Fig. 6 plots the ratio of optimal airtimes allocated to each mbols
class of flow versus N. Results are shown for three delay sy 8000
1b(dwDeeiiats1hdse=lddieenenl1elaa)yrtyhe–wqaidhntueisilaireenednmlcsitlinheateinesvsteDsfi:2wr1bsioht=stihdlceDaecslcl2aealy,as=–sswsien1sh2s;eoercfnieslsflaisdbotsieowvlte1ahay(ri–Dsecsldda2esnee=sllsaaeityys1i––v0hsse5eea.)vnn;Iesstciilttcatiiahvvsneees Codedpacketlengthin 46000000
2000
same delay deadline, the ratio of airtimes is larger than 1.
This is in accordance with the previous observation that flows 0
1 2 3 4 5 6 7 8
withpoorerchannelconditionsareallocatedmoreairtimethan w/w,w=54Mbps
3
flowswithbetterchannelconditions.Inthesecondcase,where
class 2 is delay–insensitive (D = 105), additional airtime Fig. 7. Coded packet size vs. ratio of PHY rates w3/w for Parking Lot
2 topology of Fig. 5 with N =3 cells. Class 1 consists of multi-hop flow 1,
is allocated to class 1, the delay–sensitive flow, which also class 2 consists of single-hop flows 2 and 4, class 3 consists of single-hop
corresponds with the single cell analysis. In the third case, flow3;class1and2flowsusePHYratew bit/sec,theclass3flowusesa
where class 1 is delay–insensitive (D = 105) and class 2 is PHYrateofw3 bit/sec.AllflowshavedelaydeadlineD=1.
1
delay–sensitive, it can be seen that class 2 flows are allocated
slightly more airtime that the class 1 flow. Interestingly,
seenthatvaryingthePHYrateforthesingle-hopflowinclass
however, observe that the airtime allocated to the class 1 flow
3 does not affect the optimal coded packet sizes of flows in
is insensitive to the number N of hops. This contrasts with
class 1 and class 2, and hence the airtime of class 1 and class
the behaviour when the class 1 flow is delay–sensitive.
2 flows remains the same as w is varied. The coded packet
2) Impact of different flow PHY rates: Now consider a 3
size of the class 3 flow increases linearly with w /w, and so
situation where the number of cells N = 3 and all flows 3
the airtime of the class 4 flow remains invariant as well.
have the same delay deadline D =D =1. Flow 2 and flow
1 2
4 have symbol error rate 10−4, and flow 1 and flow 3 have
symbol error rate 2.5×10−1. We classify the flows into three VIII. CONCLUSIONS
sets: class 1 consists of multi-hop flow 1, class 2 consists of
single-hop flows 2 and 4, class 3 consists of single-hop flow In this paper, we posed a utility fair problem that yields
3. Let w denote the PHY rate used used by class 1 and class the optimum airtime and the coding rate across flows in
2flows,andw denotethePHYrateusedbytheclass3flow. a capacity constrained multi-hop network with delay dead-
3
Fig. 7 plots the optimal coded packet size versus the ratio lines. We showed that the problem is highly non–convex.
w /w. We begin by observing that when w /w =1, all flows Nevertheless, we demonstrate that the global network utility
3 3
havethesamePHYrateanditcanbeseenthatflowsinclasses optimisationproblemcanbesolved.Weobtainedtheoptimum
2 and 3 are allocated the same packet sizes (and so the same airtime/packet size, channel coding rate, and analysed its
airtime). Hence, although the flow in class 3 crosses a much properties. We also analysed some simple networks based on
morelossylinkthantheflowsinclass2,theoptimalallocation theutilityoptimumframeworkweproposed.Tothebestofour
ensures that all of the single-hop flows have the same airtime. knowledge, this is the first work on cross–layer optimisation
Themulti-hopflowinclass1isallocatedasmallerpacketsize thatstudiesoptimumcodingacrossflowswhicharecompeting
(and so less airtime) than the single hop flows. It can also be for network resources and have delay–deadline constraints.