Delay-Guaranteed Cross-Layer Scheduling in Multi-Hop Wireless Networks

0.25 MB·English

by  Dongyue Xue

Delay-Guaranteed Cross-Layer Scheduling in
Multi-Hop Wireless Networks
Dongyue Xue, Eylem Ekici
Department of Electrical and Computer Engineering
Ohio State University, USA
Email: {xued, ekici}@ece.osu.edu
1 Abstract—In this paper, we propose a cross-layer scheduling O(1) in the delay bound, where ǫ denotes the distance from
1 ǫ
algorithm that achieves a throughput “ǫ-close” to the optimal the optimal throughput.
0
throughput in multi-hop wireless networks with a tradeoff of
2 O(1) in delay guarantees. The algorithm aims to solve a joint Our main algorithm is further extended: (1) to a set of
ǫ low-complexitysuboptimalalgorithms;(2)fromamodelwith
n congestion control, routing, and scheduling problem in a multi-
a hop wireless network while satisfying per-flow average end-to- constantly-backlogged sources to a model with sources of
J end delay guarantees and minimum data rate requirements. arbitrary input rates at transport layer; (3) to an algorithm

5 This problem has been solved for both backlogged as well as employing delayed queue information; and (4) from a node-
2 arbitrary arrival rate systems. Moreover, we discuss the design exclusivemodelwith constantlink capacities to a modelwith
of a class of low-complexity suboptimal algorithms, effects of
] delayed feedback on the optimal algorithm, and extensions of arbitrary link capacities and interference models over fading
T the proposed algorithm to different interference models with channels.
I arbitrary link capacities. The rest of the paper is organized as follows: Section
.
s II discusses the related work. In Section III, the network
[c I. INTRODUCTION modelispresented,followedbycorrespondingapproachesfor
the considered multi-hop wireless networks. In Section IV,

2 Cross-layer design of congestion control, routing and the optimal cross-layer control and scheduling algorithm is
v scheduling algorithms with Quality of Service (QoS) guaran- described, and its performance analyzed. In Section V, we
4
teesisoneofthemostchallengingtopicsinwirelessnetwork- provide a class of feasible suboptimal algorithms, consider
5
9 ing. The back-pressure algorithm first proposed in [1] and its sources with arbitrary arrival rates at transport layer, employ
4 extensionshavebeenwidelyemployedindevelopingthrough- delayed queue information in the scheduling algorithm, and
. put optimaldynamic resource allocation and scheduling algo- extend the model to arbitrary link capacities and interference
9
0 rithms for wireless systems. Back-pressure-based scheduling models over fading channels. We present numerical results in
0 algorithms have also been employed in wireless networks Section VI. Finally, we conclude our work in Section VII.
1 with time-varying channels [2][3][4]. Congestion controllers
v: at the transport layer have assisted the cross-layer design of II. RELATEDWORK
i schedulingalgorithmsin[5][6][7],so thattheadmittedarrival Delay issues in single-hop wireless networks have been
X
rate is guaranteed to lie within the network capacity region. addressedin[13]-[21].Especially,theschedulingalgorithmin
r
a Low-complexitydistributedalgorithmshavebeen proposedin [18]providesathroughput-utilitythatisinverselyproportional
[8][9][10][11].Algorithmsadaptedtoclusterednetworkshave to the delay guarantee. Authors of [19] have obtained delay
been proposed in [12] to reduce the number of queues main- bounds for two classes of scheduling policies. A random
tained in the network. However, delay-related investigations access algorithm is proposed in [20] for lattice and torus
are not included in these works. interference graphs, which is shown to achieve order-optimal
Inthispaper,weproposeacross-layeralgorithmtoachieve delay in a distributed manner with optimal throughput. But
guaranteedthroughput while satisfying network QoS require- these works are not readily extendable to multi-hop wireless
ments. Specifically, we construct two virtual queues, i.e., a networks, where additional arrivals from neighboring nodes
virtual queue at transport layer and a virtual delay queue, and routing must be considered. Delay analysis for multi-
to guaranteeaverage end-to-enddelay bounds.Moreover,we hop networks with fixed-routing is provided in [22]. Delay-
construct a virtual service queue to guarantee the minimum related scheduling in multi-hop wireless networks have been
dataraterequiredbyindividualnetworkflows.Ourcross-layer proposedin[23][24][25][26][27].However,noneoftheabove-
designincludesacongestioncontrollerfortheinputratetothe mentionedworksprovideexplicitend-to-enddelayguarantees.
virtual queue at transport layer, as well as a joint policy for There are several works aiming to address end-to-end
packet admission, routing, and resource scheduling. We show delay or buffer occupancy guarantees in multi-hop wireless
that our algorithm can achieve a throughput arbitrarily close networks. Worst-case delay is guaranteed in [28] with a
totheoptimal.Inaddition,thealgorithmexhibitsatradeoffof packet dropping mechanism. However, dropped packets are
2
not compensated or retransmitted with the algorithm of [28], ∀(m,n) ∈ L. We also assume that µc (t) is bounded
s(c)b(c)
whichmayleadtorestrictionsinitspracticalimplementations. above by a constant µ ≥1:
M
A low-complexity cross-layer fixed-routing algorithm is de-
0≤µc (t)≤µ , ∀c∈F,∀t, (1)
velopedin[29]toguaranteeorder-optimalaverageend-to-end s(c)b(c) M
delay, but only for half of the capacity region. A scheduling
i.e., a source node can receive at most µ packets from the
M
algorithm for finite-buffer multi-hop wireless networks with
transport layer in any time slot. To simplify the analysis,
fixed routing is proposed in [30] and is extended to adaptive-
we prevent looping back to the source, i.e., we impose the
routing with congestion controller in [31]. Specifically, the
following constraints
algorithm in [31] guarantees O(1)-scaling in buffer size with
ǫ
aǫ-lossinthroughput-utility,butthisisachievedattheexpense (µc (t))=0 ∀c∈F,∀t. (2)
X mb(c)
ofthe bufferoccupancyofthesourcenodes,wherean infinite
m∈N
buffer size in the network layer is assumed in each source
Weemploythenode-exclusivemodelinouranalysis,i.e.,each
node. This leads to large average end-to-end delay since the
node can communicatewith at most one other node in a time
networkstabilityisachievedbasedonqueuebacklogsatthese
slot. Note that our model is extended to arbitrary interference
source nodes.
models with arbitrary link capacities and fading channels in
Compared to the above works, the algorithm presented in
Section V.D.
thispaperdevelopsandincorporatesnovelvirtualqueuestruc-
We now specify the QoS requirements associated with
tures. Different from traditional back-pressure-based algo-
each flow. The network imposes an average end-to-end delay
rithms, where the network stability is achieved at the expense
threshold ρ for each flow c. The end-to-end delay period
oflargepacketqueuebacklogs,inouralgorithm,“theburden” c
of a packet starts when the packet is admitted to the source
of actual packet queue backlogs is shared by our proposed
node from the transport layer and ends when it reaches its
virtual queues, in an attempt to guarantee specific delay per-
destination. Note that the delay threshold is a time-averaged
formances.Specifically, we design a congestion controllerfor
upper-bound,not a deterministic one.In addition,each flow c
avirtualinputrateandassignweightsintheschedulingpolicy
requires a minimum data rate of a packets per time slot.
as a productof actualpacketqueue backlogand the weighted c
backlogofa designedvirtualqueue,whichwillbe introduced
B. Network Constraints and Approaches
in detail in Section IV. As such, the network stabilization is
For convenience of analysis, we define Lc , L ∪
achieved with the help of virtual queue structures that do not
{(s(c),b(c))},wherethepair(s(c),b(c))canbeconsideredas
contribute to delay in the network. Since all packet queues
avirtuallinkfromtransportlayertothesourcenode.We now
in the network, including those in source nodes, have finite
model queue dynamics and network constraints in the multi-
sizes, all average end-to-end delays are bounded independent
hop network. Let Uc(t) be the backlog of the total amount
of length or multiplicity of paths. n
of flow c packets waiting for transmission at node n. For a
flow c, if n = d(c) then Uc(t) = 0 ∀t; Otherwise, the queue
III. NETWORK MODEL n
dynamics is as follows:
A. Network Elements
Uc(t+1)≤[Uc(t)− µc (t)]+
We consider a time-slotted multi-hop wireless network n n X ni
i:(n,i)∈L
consisting of N nodes and K flows. Denote by (m,n)∈L a (3)
link from node m to node n, where L is the set of directed + X µcjn(t), if n∈N\d(c),
links in the network. Denoting the set of flows by F and j:(j,n)∈Lc
the set of nodes by N, we formulate the network topology
where the operator [x]+ is defined as [x]+ = max{x,0}.
G=(N,L). Note thatwe consideradaptiveroutingscenario,
Note that in (3), we ensure that the actual number of packets
i.e.,theroutesofeachflowarenotdeterminedapriori,which
transmitted for flow c from node n does not exceed its
is more general than fixed-routing scenario. In addition, we
queue backlog, since a feasible scheduling algorithm may
denote the source node and the destination node of a flow
not depend on the information on queue backlogs. The terms
c∈F as b(c) and d(c), respectively.
µc (t) and µc (t) represent, respec-
We assume that the source node for flow c is always Pi:(n,i)∈L ni Pj:(j,n)∈Lc jn
tively, the scheduled departure rate from node n and the
backloggedatthetransportlayer.Lettheschedulingparameter
scheduledarrivalrateintonodenbytheschedulingalgorithm
µc (t) denote the link rate assignment of flow c for link
mn with respect to flow c. Note that (3) is an inequality since the
(m,n) at time slot t according to scheduling decisions and
arrivalratesfromneighbornodesmaybe less than µc (t)
let µc (t) denote the admitted rate of flow c from the Pj jn
s(c)b(c) if some neighbor node does not have sufficient number of
transportlayerof flow to the sourcenode,where s(c) denotes
packets to transmit. Since we employ the node-exclusive
the source at the transport layer of flow c. It is clear that
model, we have
in any time slot t, µc (t) = 0 ∀n 6= b(c). For simplicity
s(c)n
of analysis, we assume only one packet can be transmitted 0≤ [ µc (t)+ µc (t)]≤1, ∀n∈N. (4)
X X ni X jn
over a link in one slot, so (µcmn(t)) takes values in {0,1} c∈F i:(n,i)∈L j:(j,n)∈L
3
From (1)(2), we also have average end-to-end delay constraint and minimum data rate
requirementareachievedifqueuesUc(t)andthethreevirtual
µc (t)≤µ , if n=b(c), (5) n
X jn M queues are stable for any node and flow, i.e.,
j:(j,n)∈Lc
if it is ensured that no packets will be looped back to the 1 t−1
limsup E{X (τ)}<∞, ∀c;
source. t X c
t→∞
Now we construct three kinds of virtual queues, namely, τ=0
virtual queue Uc (t) at transport layer, virtual service queue t−1
s(c) 1
Z (t) at sources,andvirtualdelayqueueX (t), tolater assist limsup E{Uc(τ)}<∞, ∀n∈N ∪{s(c):c∈F};
c c t X n
the developmentof our algorithm: t→∞ τ=0
(1) For each flow c at transport layer, we construct a virtual
t−1
qpuroepuoeseUdsci(nc)(tht)e nwehxitcshecwtioilnl.bWeeedmenpoloteyetdheivnirttuhael ainlgpourtitrhamte limt→s∞up1t XE{Zc(τ)}<∞, ∀c.
τ=0
to the queue as R (t) at the end of time slot t and we upper-
c Nowwe definethe capacityregionofthe consideredmulti-
boundR (t) byµ . Letr denotethe time-averageofR (t).
c M c c hop network. An arrival rate vector (z ) is called admissible
We update the virtual queue as follows: c
if there exists some scheduling algorithm (withoutcongestion
Uc (t+1)=[Uc (t)−µc (t)]++R (t), (6) control) under which the node queue backlogs (not including
s(c) s(c) s(c)b(c) c
virtual queues) are stable. We denote Λ to be the capacity
where the initial Uc (0) = 0. Considering the admitted rate
µc (t) as the ss(ecr)vice rate, if the virtual queue Uc (t) region consisting of all admissible (zc), i.e., Λ consists of
s(c)b(c) s(c) all feasible rates stabilizable by some scheduling algorithm
is stable, then the time-average admitted rate µ of flow c
c without considering QoS requirements (i.e., delay constraints
satisfies:
and minimum data rate constraints). To assist the analysis in
1 t−1 1 t−1 thefollowingsections,welet(r∗ )denotethesolutionstothe
µ , lim µc (τ)≥r , lim R (τ). (7) ǫ,c
c t→∞ t X s(c)b(c) c t→∞ t X c following optimization problem:
τ=0 τ=0
max r
(2)Tosatisfytheminimumdatarateconstraints,weconstruct (rc):(rc+ǫ)∈ΛcX∈F c
a virtual queue Z (t) associated with flow c as follows:
c
s.t. r ≥a , ∀c∈F.
Z (t+1)=[Z (t)−R (t)]++a , (8) c c
c c c c
where ǫ is a positive number which can be chosen arbitrarily
where the initial Z (0)=0. Consideringa as the arrivalrate
c c small.Forsimplicityofanalysis,weassumethat(a )isinthe
c
andR (t)astheservicerate,ifqueueZ (t)isstable,wehave:
c c interior of Λ and without loss of generality, we assume that
r ≥ a . Additionally, if Uc (t) is stable, then according to
(7c), thecminimum data ratesf(oc)r flow c is achieved. thereexistsǫ′ >0 such thatrǫ∗,c ≥ac+ǫ′ ∀c∈F. According
to [32], we have
(3) To satisfy the end-to-end delay constraints, we construct
a virtual delay queue Xc(t) for any given flow c as follows: ǫli→m0Xrǫ∗,c = Xrc∗,
X (t+1)=[X (t)−ρ R (t)]++ Uc(t) (9) c∈F c∈F
c c c c X n where (r∗) is the solution to the following optimization:
n∈N c
where the initial X (0) = 0. Considering the packets kept
c max r
in the network in time slot t, i.e., Pn∈N Unc(t), as the arrival (rc):(rc)∈ΛcX∈F c
rateandρ R (t)astheservicerate,andaccordingtoqueueing
c c s.t. r ≥a , ∀c∈F.
c c
theory, if queue X (t) is stable, we have
c
IV. CONTROL SCHEDULING ALGORITHMFORMULTI-HOP
t−1 t−1
lim 1 Uc(τ)≤ρ lim 1 R (τ)=ρ r . WIRELESS NETWORKS
t→∞ t X X n ct→∞ t X c c c
τ=0n∈N τ=0 Now we propose a control and scheduling algorithm ALG
Furthermore, if Uc (t) is stable, then according to (7), we for the introduced multi-hop model so that ALG stabilizes
s(c) the network and satisfies the delay constraint and minimum
have:
t−1 data rate constraint. Given ǫ, the proposed ALG can achieve
1 1
µc tl→im∞ t X X Unc(τ)≤ρc. (10) a throughput arbitrarily close to Pc∈Frǫ∗,c, under certain
τ=0n∈N conditionsrelatedtodelayconstraintswhichwillbelatergiven
Inaddition,byLittle’sTheorem,(10) ensuresthattheaverage in Theorem 1.
end-to-enddelayofflowcislessthanorequaltothethreshold TheoptimalalgorithmALGconsistsoftwoparts:aconges-
ρ with probability (w.p.) 1. tion controllerof R (t), and a jointpacketadmission, routing
c c
From the above description, we know that the network and scheduling policy. We propose and analyze the algorithm
is stable (i.e., each queue at all nodes is stable) and the in the following subsections.
4
A. Algorithm Description and Analysis To analyze the performance of the algorithm, we first
introduce the following proposition.
Let q ≥ µ be a control parameter for queue length.
M M
Proposition 1: EmployingALG,eachqueuebackloginthe
We first propose a congestion controller for the input rate of
network has a deterministic worst-case bound:
virtual queues at transport layer:
1) Congestion Controller of R (t): Uc(t)≤q , ∀t,∀n∈N,∀c∈F. (14)
c n M
(q −µ )Uc (t) Proof:Weusemathematicalinductionontimeslotinthe
M M s(c)
0≤Rmc(ti)n≤µMRc(t)( qM −Xc(t)ρc−Zc(t)−(1V1)) hpyropooft.hWesihse,nwte=su0p,poUsnce(0in) =tim0e≤sloqtMt∀wne,ch.aIvnetUhenc(itn)du≤ctqioMn
where V > 0 is a control parameter. Specifically, when ∀n,c. In the induction step, for any given n∈N and c∈F,
qM−µMUc (t) − X (t)ρ − Z (t) − V > 0, R (t) is set we consider two cases as follows:
toqzMero; Ost(hc)erwise, Rc (t)c=µ c. c (1) We first consider the case when n = b(c), i.e., when n
c M
After performing the congestion control, we perform the is the source node of flow c. Since Unc(t) ≤ qM from the
following joint policy for packet admission, routing and induction hypothesis, we further consider two subcases:
scheduling (abbreviated as scheduling policy): • Inthefirstsubcase,Ubc(c)(t)≤qM−µM.Thenaccording
2)SchedulingPolicy:Ineachtimeslot,withtheconstraints tothequeuedynamics(3)andtheinequality(5),Uc (t+
b(c)
of the underlying interference model as described in Section 1)≤Ubc(c)(t)+µM ≤qM;
III including (1)(2)(4), the network solves the following opti- • In the second subcase, qM − µM < Ubc(c)(t) ≤ qM.
mization problem: According to the weight assignment (13), we have
wc (t) < 0 which leads to µc (t) = 0. Hence,
s(c)b(c) s(c)b(c)
(µmcmna(xt))Xm,nµcm∗mnn(t)(t)wmn(t) (12) (2)InUtbch(ce)(stec+on1d)≤casUeb,c(nc)(6=t)b≤(c)q,Mi.eb.,yn(2is)(n3o).tthesourcenode
of flow c. Similar to the first case, we further consider the
s.t. µc (t)=0 ∀c6=c∗ (t), ∀(m,n)∈Lc,
mn mn following two subcases:
µc (t)=0 if n=s(c), ∀c∈F, • Inthefirst subcase,Unc(t)<qM. Then,since we employ
mn node-exclusive model, Uc(t+1) ≤ Uc(t)+1 ≤ q by
n n M
where c∗ (t) and w (t) are defined as follows: (3)(4).
mn mn
c∗ (t)=argmaxwc (t), • In the second subcase, Unc(t) = qM. According to the
mn mn weight assignment (13) we have wc (t) ≤ 0 ∀m :
c∈F mn
(m,n) ∈ L. Now, for any given node m : (m,n) ∈ L,
w (t)=[maxwc (t)]+,
mn mn we have:
c∈F
(i) If c6=c∗ (t), then by (12), µc (t)=0;
with weight assignment as follows mn mn
(ii) Otherwise, c = c∗ (t), which induces w (t) =
mn mn
Uc (t) [wc (t)]+ = 0 and by the scheduling policy, µc (t) =
wmc n(t)=Usscqq((ccM))(t)[[UqMmc(−t)µ−MU−nc(Ut)bc](,c)i(ft)(]m, ,n)∈L, (13) U0H.necmn(tcn)e=µcmqMn(tb)y=th0e ∀qumeu:e(dmy,nnam) ∈icsL(,3)a.nd Unc(tm+n1) ≤
M The above analysis holds for any given n ∈ N and c ∈ F.
0, oifth(emrw,nis)e.=(s(c),b(c)), wThhNeicroehwfocrowemethppelreeitnesesdnutthcteoioupnrromsoteafpi.nhroelsdusl,tsi.ein.,UThnce(ot+rem1)1≤. qM ∀n,c,
In addition,when w (t)=0, withoutloss of optimality,we
mn Theorem 1: Given that
set µc (t)=0 ∀c∈F to maximize (12).
mn 2N −1+µ2 Nq
Note that L∪{(s(c),b(c)):c∈F} forms the (m,n) pairs qM > 2ǫ M +µM and ρc > r∗M ∀c∈F, (15)
in (µc (t)) over which the optimization (12) is performed. ǫ,c
mn
Thus,theoptimizationisatypicalMaximumWeightMatching ALG can achieve a throughput
(MWM) problem.We first decoupleflow schedulingfrom the t−1
1 B
MWM. Specifically, for each pair (m,n), the flow c∗mn(t) is litm→∞inf t XXE{Rc(τ)}≥ Xrǫ∗,c− V , (16)
fixed as the candidate for transmission. We then assign the τ=0c∈F c∈F
fwoerimghtofasthwemwne(itg)h.tNaostseiganlmsoenttha(t13a)lthhoauvgehbseiemnilaurtilpizreodduicnt w1KheNre2Bq2,+12N1KKµq2Mµ+M1K+KqMq−MµaM2.µ2M + 12µ2MPc∈Fρ2c+
[30][31], no virtual queues are involved there. Whereas in 2 M 2 M 2 Pc∈F c
In addition, ALG ensures that the virtual queues have a
ALG,weassignweightsasaproductofweightedvirtualqueue
time-averaged bound:
Uc (t)
backlog ( s(c) ) and the actual back-pressure, in an aim to
qM 1 t−1 B′
shift the burden of the actual queue backlog to the virtual limsup E{Uc (τ)+X (τ)+Z (τ)}≤ , (17)
backlog. t→∞ t XX s(c) c c δ
τ=0c∈F
5
where B′ , B + VB , with B and δ constant positive put arbitrarily close to the optimal value with distributed
R R
numbers given in the next subsection. implementation, we can employ random access techniques
Remark 1 (Network Stability): The inequalities (14) from [37][38] in the scheduling policy with fugacities [39] chosen
Proposition1and(17)fromTheorem1indicatethatALGsta- as exp{αU¯sc(c)(t)[Umc(t)−Unc(t)]+} for each link (m,n) ∈ L,
bilizes the actual and virtual queues. As an immediate result, where U¯c (t) isqaMlocal estimate (e.g., delayed information)
ALG stabilizes the network and satisfies the average end-to- s(c)
of Uc (t) and α a positive weight. It can be shown that the
end delay constraint and the minimum data rate requirement. s(c)
distributed algorithm can still achieve an average end-to-end
In addition, Proposition 1 states that the actual queues are
delayoforderO(1)withthetime-scaleseparationassumption
deterministically bounded by q , which ensures finite buffer ǫ
M [20][36][37].1 A variation of such distributed implementation
sizes for all queues in the network, including those in source
in single-hop networks can be found in our recent work [40].
nodes.
We prove Theorem 1 in the following subsection.
Remark 2 (Optimal Utility and Delay Analysis): Since
(Uc (t)) are stable, the inequality (16) gives a lower-bound B. Proof of Theorem 1
s(c)
on the throughput that ALG can achieve. Given some ǫ > 0, Beforeweproceed,wepresentthefollowinglemmaswhich
since B is independentof V, (16) also ensures that ALG can will assist us in proving Theorem 1.
achieve a throughput arbitrarily close to r∗ . When ǫ Lemma 1: For nonnegative numbers A ,A ,A ,Q ∈ R
c∈F ǫ,c 1 2 3
P
tends to 0, ALG can achieve a throughput arbitrarily close such that Q ≤ [A −A ]++A , we have Q2 ≤ A2+A2+
1 2 3 1 2
to the optimal value r∗ with the tradeoff in queue A2−2A (A −A ).
Pc∈F c 3 1 2 3
backlog upper-boundq and the delay constraints (ρ ), both The proof of Lemma 1 is trivial and omitted. We will later
M c
of which are lower-bounded by the reciprocal terms of ǫ as use Lemma 1 to simplify virtual queue dynamics.
shown in (15) in Theorem 1. In other words, the average Lemma 2: For any feasible rate vector (θ ) ∈ Λ with
c
end-to-end delay bound is of order O(1). We note that in θ ≥ a ∀c ∈ F, there exists a stationary randomized
ǫ c c
ALG,thecontrolparameterV,whichistypicallychosentobe algorithm STAT that stabilizes the network with input rate
large, does not affect the actual queue backlog upper-bound vector (µSTAT (t)) and scheduling parameters (µc,STAT(t))
s(c)b(c) mn
or the average end-to-end delay bound, but only affects the independentof queue backlogs, such that the expectedadmit-
upper-bound of the virtual queue backlogs (shown in (17)). ted rates are:
In comparison, in the algorithm proposed in [31], the authors
E{µc,STAT(t)}=θ ,∀t,∀c∈F.
show that the internalbuffersize is deterministicallybounded s(c)b(c) c
with order O(1), but at the expense of the buffer occupancy In addition, ∀t, ∀n∈N,∀c, the flow constraint is satisfied:
ǫ
at source nodes which is of order O(V), where V has to
E{ µc,STAT(t)− µc,STAT(t)}=0.
be large enough for their algorithm to approach r∗ . X ni X jn
c∈F ǫ,c
ThisdesignassumesaninfinitebuffersizeatsourcePnodesand i:(n,i)∈L j:(j,n)∈Lc
typically results in congestion at the source nodes as shown
Note that it is not necessary for the randomized algorithm
in the simulation results in [31], which further induces an
STAT to satisfy the average end-to-end delay constraints.
unguaranteed and large average end-to-end delay. Moreover,
SimilarformulationsofSTATandtheirproofshavebeengiven
one can expect that there are no buffer-size guarantees for
in [5] and [6], so we omit the proof of Lemma 2 for brevity.
single-hop flows by employing the algorithm in [31]. In
Remark 4: According to the STAT algorithm in Lemma
contrast, in our proposed ALG, we shift “the burden of V”
2, we assign the input rates of the virtual queues at trans-
from actual queues to virtual queues and ensure that the port layer as RSTAT(t) = µc,STAT(t). Thus, we also have
average end-to-end delay constraints are satisfied with finite E{RSTAT(t)} =c θ . Accordisn(cg)bt(oc)the update equation (6),
buffer sizes for all actual packet queues. c c
it is easy to show that the virtual queues under STAT are
Remark 3 (Implementation Issues): To update the virtual
bounded above by µ and the time-average of RSTAT(t)
queue Xc(t) and perform the Rc(t) congestion controller at satisfies: rSTAT =θ .MNote that(θ ) can take valuescas (r∗ )
the transport layer, the queue backlog information of flow c c c ǫ,c
or (r∗ +ǫ) or (r∗ − 1ǫ′), where we recall (r∗ +ǫ) ∈ Λ
c is crucial. This information can be collected back to the ǫ,c ǫ,c 2 ǫ,c
and r∗ ≥a +ǫ′ ∀c∈F.
source node by piggy-backingit on ACK from each node. In ǫ,c c
To prove Theorem 1, we first let Q(t) =
ordertoaccountforsuchdelayofqueuebackloginformation,
((Uc(t)),(Uc (t)),(X (t)),(Z (t))) and define the
the R (t) congestion controller (11) of the algorithm can n s(c) c c
c Lyapunov function L(Q(t)) as follows:
employ delayed queue backlog of X (t). Similarly, delayed
c
qwueeiguhetbaascskiglongmiennfto(r1m3a)tioofnthoefUscsch(ec)d(utl)incganpobleiceym.Tphloeymedodatifithede L(Q(t))= 12{X qMq−MµMUsc(c)(t)2+XXc(t)2
c∈F c∈F
(18)
algorithm and its validity are further discussed in Section 1
+ Z (t)2+ Uc(t)2Uc (t)}.
V.C. By employing delayed queue backlog information, we X c X X q n s(c)
M
can extend the algorithm to distributed implementation in c∈F c∈Fn∈N
much the same way as in [8][11] to achieve a fraction
1Notethattherandomaccessworkscitedaboveeitherdonotprovidedelay
of the optimal throughput. In order to achieve a through- guarantees orarenotreadily extended tomulti-hopsettings.
6
Itis obviousthat L(Q(0))=0. We denotethe Lyapunovdrift From the virtual queue dynamics (9), we have:
by
1
(X (t+1)2−X (t)2)
∆(t)=E{L(Q(t+1))−L(Q(t))|Q(t)}. (19) 2 X c c
c∈F
1
≤ (ρ2R (t)2+( Uc(t))2
From the queue dynamics (3)(6), we have: 2 X c c X n
c∈F n∈N
X X q1 Unc(t+1)2Usc(c)(t+1) −2Xc(t)(ρcRc(t)−nX∈NUnc(t))) (23)
c∈Fn∈N M ≤1µ2 ρ2+ 1KN2q2
≤X q1 (Rc(t)+Usc(c)(t)) X Unc(t+1)2 2 McX∈F c 2 M
M
c∈F n∈N − X (t)ρ R (t)+Nq X (t).
1 X c c c M X c
≤µMqMNK+X q Usc(c)(t) X{Unc(t)2 (20) c∈F c∈F
M
c∈F n∈N From the virtual queue dynamics (8), we have:
+( µc (t))2+( µc (t))2
X ni X jn 1
i:(n,i)∈L j:(j,n)∈Lc 2 X(Zc(t+1)2−Zc(t)2)
−2Uc(t)( µc (t)− µc (t))}, c∈F
n X ni X jn 1
i j ≤2 X(Rc(t)2+a2c −2Zc(t)(Rc(t)−ac)) (24)
c∈F
where we recall that R (t) ≤ µ and we employ Lemma 1 1 1
c M ≤ Kµ2 + a2− Z (t)R (t)+ a Z (t).
to deduce the second inequality. 2 M 2 X c X c c X c c
c∈F c∈F c∈F
From (20), we have
Substituting(21)(22)(23)(24)into theLyapunovdrift(19) and
subtracting V E{R (t)|Q(t)} from both sides, we then
1 1 c c
( (Uc(t+1)2Uc (t+1) have: P
2 X X q n s(c)
M
c∈Fn∈N
∆(t)−V E{R (t)|Q(t)}
−Uc(t)2Uc (t))) X c
n s(c)
c∈F
≤12 X (2N −1+qMµ2M)Usc(c)(t) + 21NKqMµM (21) ≤B+XE{Rc(t)((qM −µqM)Usc(c)(t)
c∈F M
c∈F
Uc(t)Uc (t)
− n s(c) −Xc(t)ρc−Zc(t)−V)|Q(t)}
X X q
c∈Fn∈N M +NqM XXc(t)+XacZc(t)
( µc (t)− µc (t)), c∈F c∈F
j:(nX,j)∈L nj i:(iX,n)∈Lc in 1 (2N −1+µ2M)Usc(c)(t) (25)
+
2 X q
M
c∈F
where we employ the fact deduced from (4)(5) that
q −µ
Piµµcnci((tt)) ≤≤ 1µandwhPenjµncjn=(t)b≤(c).1Nwohteenthnat6=web(ucs)eatnhde −E{ MqM M cX∈FUsc(c)(t)µcs(c)b(c)(t)
j jn M
sPummation index i and j interchangeably for convenience of Uc(t)Uc (t)
n s(c)
+
analysis. X X q
M
c∈Fn∈N
From the queue length dynamics (6) and by employing
( µc (t)− µc (t))|Q(t)}.
Lemma 1, we have: X nj X in
j:(n,j)∈L i:(i,n)∈Lc
1 qM −µM(Uc (t+1)2−Uc (t)2) WecanrewritethelasttermofRHSof(25)bysimplealgebra
2 X qM s(c) s(c) as
c∈F
≤21 X qMq−MµM(µcs(c)b(c)(t)2+Rc(t)2 −E{X X µcmn(t)Uscq(c)(t)(Umc(t)−Unc(t))
c∈F M
c∈F(m,n)∈L
−2Usc(c)(t)(µcs(c)b(c)(t)−Rc(t))) (22) Uc (t) (26)
q −µ + µc (t) s(c) (q −µ −Uc (t))|Q(t)}.
≤K M Mµ2 X s(c)b(c) q M M b(c)
q M c∈F M
M
q −µ
− M M Uc (t)(µc (t)−R (t)). Then, the second term and the last term of the RHS of
qM X s(c) s(c)b(c) c (25) are minimized by the congestion controller (11) and the
c∈F
7
scheduling policy (12), respectively, over a set of feasible al- By taking liminf of t on both sides of (29), we have
gorithmsincludingthestationaryrandomizedalgorithmSTAT
t−1
introduced in Lemma 2 and Remark 4. We can substitute liminf 1 E{R (τ)}
into the second term of RHS of (25) a stationary randomized t→∞ t XX c
τ=0c∈F
algorithm with admitted arrival rate vector (r∗ ) and into the
ǫ,c δ 1 t−1
lasttermwithastationaryrandomizedalgorithmwithadmitted ≥ liminf E{X (τ)+Uc (τ)+Z (τ)} (31)
arrival rate vector (r∗ +ǫ). Thus, we have: V t→∞ t XX c s(c) c
ǫ,c τ=0c∈F
B
− + r∗ ,
V X ǫ,c
∆(t)−V E{R (t)|Q(t)} c∈F
X c
c∈F which proves (16) since the first term of the RHS of (31) is
≤B−V r∗ nonnegative.
X ǫ,c
c∈F
Uc (t) 2N −1+µ2 (27) V. FURTHERDISCUSSIONS
−X sq(c) (ǫ(qM −µM)− 2 M) A. Suboptimal Algorithms
M
c∈F
Solving MWM optimization problem can be NP-hard de-
− (r∗ −a )Z (t)− (ρ r∗ −Nq )X (t).
X ǫ,c c c X c ǫ,c M c pending on the underlying interference model as indicated
c∈F c∈F in [33]. In this section, we introduce a group of suboptimal
When (15) holds, we can find ǫ >0 such that ǫ ≤ρ r∗ − algorithms that aim to achieve at least a γ fraction of the
1 1 c ǫ,c
optimal throughput. We denote the scheduling parameters of
NisqdMefin∀ecd∈suFchatnhdatǫr1ǫ∗,≤c ≥ǫ(aqMc+−µǫM′q)∀−Mc2∈N−F12+.µT2Mh.usR,ewcaellhathvaet: ǫ′ asulsboodpetinmoateltahlegoscrihthedmuslinbgyp(aµrcma,mSnUeBte(rts)o)f.AFLorGcboynv(eµncmi,OennPcTe,(tw))e.
Algorithmsarecalledsuboptimaliftheschedulingparameters
∆(t)−V E{R (t)|Q(t)} (µc,SUB(t)) satisfy the following:
X c mn
c∈F (28) µc∗mn(t),SUB(t)w (t)≥γ µc∗mn(t),OPT(t)w (t),
≤B−δ (X (t)+Uc (t)+Z (t))−V r∗ , X mn mn X mn mn
X c s(c) c X ǫ,c m,n m,n
c∈F c∈F (32)
where γ ∈ (0,1) is constant and we recall that c∗ (t) and
where δ ,min{ǫ ,ǫ′}. mn
1 w (t)aredefinedinSectionIV.A.Inaddition,thecongestion
mn
We take the expectation with respect to the distribution of
controllerofsuboptimalalgorithmsisthesameasthatofALG
Q on both sides of (28) and take the time average on τ =
(11).
0,...,t−1, which leads to
FollowingthesameanalysisofALG,Proposition1holdsfor
suboptimal algorithms, i.e., the queue backlogs are bounded
t−1
1 V
E{L(Q(t))}− E{R (τ)} above by qM, and we derive the following theorem:
t t XX c Theorem 2: Given that
τ=0c∈F
≤B−V cX∈Frǫ∗,c (29) qM > 2N −21γǫ+µ2M +µM and ρc > Nγrqǫ∗M,c ∀c∈F, (33)
− δ t−1 E{X (τ)+Uc (τ)+Z (τ)}. ∃ǫ2 >0 s.t. γrǫ∗,c ≥ac+ǫ2 ∀c∈F,
t XX c s(c) c
a suboptimal algorithm ensures that the virtual queues have a
τ=0c∈F
time-averaged bound:
Sincelimsup 1 t−1 E{R (τ)}isboundedabove
(say,bya constatn→t∞BRt Pwiτth=0BPRc≤KµcM) andE{L(Q(t))} is limsup1 t−1 E{Uc (τ)+X (τ)+Z (τ)}≤ B¯, (34)
nonnegative, by taking limsup of t on both sides of (29), we t→∞ t XX s(c) c c δ
τ=0c∈F
have:
where B¯ , B+γVB . In addition, a suboptimal algorithm
R
1 t−1 can achieve a throughput
limsup E{X (τ)+Uc (τ)+Z (τ)}
t XX c s(c) c
t→∞ t−1
τ=0c∈F 1 B
liminf E{R (τ)}≥γ r∗ − . (35)
≤B + V [limsup1 t−1 E{R (τ)}− r∗ ] (30) t→∞ t τX=0cX∈F c cX∈F ǫ,c V
δ δ t XX c X ǫ,c
t→∞
τ=0c∈F c∈F
Proof: The proof is provided in Appendix A.
B′
≤ , Remark 5: From Theorem 2, given an arbitrarily small ǫ,
δ
we show that a suboptimal algorithm can at least achieve
which proves (17). a throughput arbitrarily close to a fraction γ of the optimal
8
results r∗ . Suboptimal algorithms include the well- Specifically, when ηY (t) − V ≥ 0, v (t) is set to zero;
known PGrce∈eFdyǫ,Mc aximal Matching (GMM) algorithm [34] Otherwise, v (t) = µc . When qM−µMUc c (t) −ηY (t)−
with γ = 1 as well as the solutions to the maximum X (t)ρ − Zc (t) ≥ M0, R (t) isqMset tos(zc)ero; Othecrwise,
2 c c c c
weighted independentset (MWIS) optimization problemsuch R (t)=min{L (t)+A (t),µ }.
c c c M
as GWMAX and GWMIN proposed in [35] with γ = 1, 2)SchedulingPolicy:Theschedulingalgorithmisthesame
∆
where ∆ is the maximum degree of the network topology G. as that of ALG provided in Section IV.B, except for the up-
The delay bound and throughput tradeoff in Theorem 1 still datedconstraints:0≤µc (t)≤min{L (t)+A (t),µ }.
s(c)b(c) c c M
hold in Theorem 2. Since the scheduling policy is not changed, Proposition 1
still holds. And we present a theorem below for the perfor-
B. Arbitrary Arrival Rates at Transport Layer
mance of the algorithm:
Note that in the previous model description, we assumed Theorem 3: Given that
that the flow sources are constantly backlogged, that is, the
2N −1+µ2 Nq
congestion controller (11) can always guarantee Rc(t)=µM qM > 2ǫ M +µM and ρc > r∗M ∀c∈F,
when qM−µMUc (t)− X (t)ρ −Z (t)− V ≤ 0. In this ǫ,c
subsectioqnM, we ps(rce)sent an ocptimcal algocrithm when the flows the algorithm ensures that the virtual queues have a time-
have arbitrary arrival rates at the transport layer. averaged bound:
Let Ac(t) denote the arrival rate of flow c packets at 1 t−1 B
the beginning of the time slot t at the transport layer. We limsup E{Uc (τ)+X (τ)+Z (τ)+Y (τ)}≤ 2,
t XX s(c) c c c δ′
assume thatAc(t) is i.i.d.with respectto t with mean λc. For t→∞ τ=0c∈F
simplicity ofanalysis, we assume (λc) to be in the exteriorof where B , B+Kηµ2 +VB and δ′ is constant positive
2 M R
thecapacityregionΛsothatacongestioncontrollerisneeded
number. In addition, the algorithm can achieve a throughput
and we assume that A (t) is boundedabove by µ ∀c∈F.2
c M
t−1
Let Lc(t) denote the backlog of flow c data at the transport liminf 1 E{v (τ)}≥ r∗ − B1,
layer which is updated as follows: t→∞ t XX c X ǫ,c V
τ=0c∈F c∈F
Lc(t+1)=min{[Lc(t)+Ac(t)−µcs(c)b(c)(t)]+,LM}, (36) where B1 ,B+Kηµ2M.
Proof: The proof is provided in Appendix B.
where L ≥ 0 is the buffer size for flow c at the transport
M
Theorem 3 shows that optimality is preserved and O(1)
layer. Note that we have LM = 0 and Lc(t) = 0 if there is ǫ
delay scaling is kept.
no buffer for flow c at the transport layer.
Following the idea introducedin [5], we constructa virtual C. Employing Delayed Queue Backlog Information
queue Y (t) and an auxiliary variable v (t) for each virtual
c c Recallthatin ALG,congestioncontroller(11) isperformed
input rate R (t), with queue dynamics for Y (t) as follows
c c at the transport layer and link weight assignment in (13) is
Y (t+1)=[Y (t)−R (t)]++v (t), (37) performedlocallyateachlink.Thus,inordertoaccountforthe
c c c c
propagation delay of queue information, we employ delayed
where initially we have Y (0)=0. The intuition is that v (t)
c c queue backlog of (X (t)) in (11) and employ delayed queue
c
serves as the function of Rc(t) in congestion controller (11) backlog of (Uc (t)) for links in L in (13). Specifically, we
andwenotethatwhenY (t)isstable,wehaver ≥v ,where s(c)
c c c rewrite (11) in ALG as:
v is the time average rate for v (t), recalling that r is the
c c c (q −µ )Uc (t)
timeaveragerateforR (t).Thus,whenY (t)andUc (t)are M M s(c)
c c s(c) minRc(t)( −Xc(t−T)ρc−Zc(t)−V),
stotatbhlee,oipftiwmealcvaanlueensurert∗he, tvhaelunesoPisctvhceitshraorubgithrapruilty cloµse qM (40)
Pc ǫ,c Pc c whereT isanintegernumberthatislargerthanthemaximum
since µ ≥r ≥v .
c c c propagationdelayfromasourcetoanode,andwerewrite(13)
Now we provide the optimal algorithm for arbitrary arrival
as:
rates at the transport layer:
Uc (t−T)
1) Congestion Controller:  s(c) [Uc(t)−Uc(t)],
wRmhci(entr)eRηsc.(tt.>)(q0Misq00−≤Ma≤vµcmw(MRti)enc≤Ui(gµtsch)M(tc≤)va(ctsm)(sto−i)cn(iη{ηaLYtYecccd(((tttw)))−+−ithVAXt)chc,((ett))v,ρiµcrMt−ua}Zl cq(u(te3)u)8e)(39)wmc n(t)=0U,scq(cM)q(Mt)[qM −oµiimfftMh((emm−rw,,Unnis))bec(n.∈=c)(L(ts),(],c),b(c)), (41)
Proposition 1 still holds and we present a theorem for the
Y (t). Note that (38) and (39) can be solved independently.
c scheduling algorithm using delayed queue backlog informa-
tion, which maintains the throughput optimality and O(1)
2Note that our analysis also works for the case when Ac(t) is bounded ǫ
abovebysomeconstant AM ∀c∈F,whereAM ≥µM. scaling in delay bound:
9
Theorem 4: Given that
A E
2N −1+µ2 Nq
q > M +µ and ρ > M ∀c∈F,
M 2ǫ M c r∗ B F
ǫ,c
the algorithm ensures that the virtual queues have a time- D
H
averaged bound:
C G
t−1
1 B
limsup E{Uc (τ)+X (τ)+Z (τ)}≤ 4,
t XX s(c) c c δ
t→∞
τ=0c∈F
Fig.1. Networktopology forsimulations
where B , B + VB and B , B + KNµ T +
4 3 R 3 M
Nq Tµ ρ + Kρ2µ2 T. In addition, the algorithm can
M M c c M
achieve a throughput VI. NUMERICAL RESULTS
In this section, we present the simulation results for the
t−1
1 B
liminf E{R (τ)}≥ r∗ − 3. proposed optimal algorithm ALG. Simulations are run in
t→∞ t XX c X ǫ,c V Matlab 2009A with results averaged over 105 time slots. In
τ=0c∈F c∈F
the network topology illustrated in Figure 1, there are three
Proof: The proof is provided in Appendix C. source-destinationpairs(A,G),(D,E)and(F,H)withsame
On employing delayed queue backlogs, we can extend the Poissonarrivalratesandµ =2.Therequiredminimumdata
M
centralizedoptimizationproblem(12)todistributedimplemen- rateforthethreeflowsareallsetto0.1.We denotebyBPthe
tations with methods introduced in Remark 3. back-pressure scheduling algorithm in [1] with a congestion
controller in [5], and denote by Finite Buffer the cross-layer
D. ArbitraryLinkCapacitiesandArbitraryInterferenceMod-
algorithmdevelopedin[31]withbuffersizeequaltothequeue
els with Fading Channels
length limit q . Note that it is shown in simulation results in
M
Recall that in the model description in Section III, the link [31]thatFiniteBufferalgorithmensuresmuchsmallerinternal
capacity is assumed constant (one packet per slot) and node- queue length (of nodes excluding the source node) than BP
exclusivemodelisemployed.Inthissubsection,weextendthe algorithm (and queue length is related to delay performance).
model to arbitrary link capacities and arbitrary interference We set the controlparameterV =1000,wherein simulations
models with fading channels of finite channel states. Thus, wefindthatahigherV cannotfurtherimprovethethroughput.
instead of (4), we have (µc (t)) ∈ I(t), where I(t) We first illustrate in Table I the throughput optimality of
mn (m,n)∈L
is the feasible activation set for time slot t determined by the ALG when the sources are constantly backlogged. We loosen
underlyinginterferencemodelandcurrentchannelstates,with thedelayconstraintasρ =30q .Asweincreasethecontrol
c M
link capacity constraints µc (t) ≤ l , where l is parameterq ,theALGachievesathroughputapproachingthe
c∈F mn mn mn M
P
the arbitrarily chosen link capacity for a link (m,n) ∈ L. throughputofBPalgorithmwhichisknowntobeoptimal.We
We define l ,max µc (t). alsonotethatthisapproximationinthroughputresultsinworse
n (µcmn(t))∈I(t)Pc∈FPm:(m,n)∈L mn
Note that it is clear that l ≤ l . Then we average end-to-end delay performance, which complies with
n Pm:(m,n)∈L mn
can update the optimization (12) and weight assignment (13), Remark 1.
respectively, as follows: Wethenillustratethethroughputanddelaytradeoffforboth
(µmc a(xt))Xµmc∗mnn(t)(t)wmn(t) tFhieguArLeG2faonrdthitesccaosrereosfpaornbdiitnragrysuabroripvtailmraalteGsMatMtraanlsgpoorirtthlmayienr
mn m,n withL =0,wherewesetq =5andρ =50foreachflow
M M c
s.t. (µcmn(t))(m,n)∈L ∈I(t) and µs(c)b(c)(t)≤µM ∀c∈L. c. Note that this pair of qM and ρc shows that the bound in
(15)isactuallyquiteloose,andthusouralgorithmcanachieve
Uc (t)
 s(c) [Uc(t)−Uc(t)−l ], if (m,n)∈L, better delay performance than stated in (15). Figure 2 shows
wmc n(t)=Uscqq(cM)(t)[qMm−µM −nUbc(c)(nt)], tFthhieantitcetohneBsutarfvafeeinrratag(leρgcoern=idth-mt5o0s-)e.nTadnhdedetllhoarwyoeuurgnhtdhpeaurntAothLfaGAtLuisGndweisrelclBloPbseelaontwod
M
0, oifth(emrw,nis)e.=(s(c),b(c)), (araarlrtteihvoaislugr0ah.t3el,oswAaeLrerGtshmaacnahl)liet(hv≤aets0oa.f3)tthh.reSopouepgcthiipmfiucatall1lBy0,P%wahlmgeoonrrietthhetmhaawnrrhitvheanel
It is not difficult to check that Proposition 1 still holds GMM algorithm and 9.0% less than BP algorithm, with an
with q ≥ max{max l ,µ } and Theorem 1 holds average end-to-end delay 35.2% less than the BP algorithm.
M n∈N n M
with a different definition of constant B. The above modified Inthelarge-input-rate-region(>0.3),wealsoobservethatthe
algorithm can be further extended to solve power allocation delay in both the BP and Finite Buffer algorithm violates the
problems,wherewereferinterestedreaderstoourrecentwork delayconstraints.Inaddition,intheaboveillustratedscenarios
[41]. with backlogged and arbitrary arrival rates, the minimum
10
TABLEI
THROUGHPUTPERFORMANCEOFALGWHENSOURCESAREBACKLOGGEDATTHETRANSPORTLAYER
ALG(ρc=150) ALG(ρc=300) ALG(ρc=3000) ALG(ρc=30000) BP
Throughput(sumforthreeflows) 0.9368 1.1834 1.2007 1.2305 1.2315
End-to-enddelay(averaged overthreeflows) 45.76 131.47 1.514×103 1.3687×104 3.753×104
arrival rates and average end-to-end delay requirements are [2] M. Neely and E. Modiano, “Dynamic power allocation and routing for
satisfied for individual flows under ALG. As a side note, the time varying wireless networks”, in IEEE Journal on Selected Area in
Communications, vol.23,no.1,pp.89-103,March2005.
average end-to-end delay in all four algorithms in Figure 2
[3] A. Eryilmaz, R. Srikant and J. Perkins, “Stable scheduling policies for
firstdecreases,whichcanbeexplainedbytheintuitionthatall fading wireless channels”, in IEEE/ACM Transactions on Networking,
thealgorithmsarebasedonback-pressureoflinks(i.e.,queue vol.13,no.2,pp.411-424,April2005.
[4] M. Lotfinezhad, B. Liang and E. Sousa, “On stability region and delay
backlog difference of links) and the queue backlog difference
performance of linear-memory randomized scheduling for time-varing
tendstobelargerforeachhopwithalargerarrivalrate.When networks”, in IEEE/ACM Transactions on Networking, vol. 17, no. 6,
arrivalrate furtherincreases, congestionlevelbecomeshigher pp.1860-1873, December2009.
[5] L.Georgiadis,M.NeelyandL.Tassiulas,“Resourceallocationandcross-
since more packets are admitted into the network.
Layer control in wireless networks”, in Foundations and Trends in
Networking, pp.1-149,2006.
[6] M. Neely, “Energy Optimal Control for Time Varying Wireless Net-
1.2 works”, in IEEE Transactions on Information Theory, vol. 52, no. 7,
ALG pp.2915-2934, July2006.
1.1 103 BP [7] A.EryilmazandR.Srikant,“Jointcongestioncontrol,routingandMAC
GMM
1 ay FiniteBuffer for stability and fairness in wireless networks”, in IEEE Journal on
0.9 del Delayconstraint SelectedAreasinCommunications,vol.24,no.8,pp.1514-1524,August
throughput000...678 eend-to-end102 [[98]] PgMX2.r0.eo0CebW6dih.lyuae,pCosRcro.khmaeSrdpr,uuikltKi.a,nn.gvtKoalaa.nlrgd6o,aJrnni.tdohP.meS6rs.k,iSpninpasr.,kw5“a9Sirr,5ce-hl“6eeT0sdhs5ur,loinJnueuggtnwheepofu2rfikt0cs0ig”e7,un.acirynanoItfeEeEdsiEstthrTirboruautngesdh.
0.5 ABPLG verag mCoanxfiemreanlcsechoenduClionmgminunwiciraeslteisosnsn,etCwoonrtkrso”l,ainndPCroocm.4p3urtidngA,n2n0u0a5l.Allerton
0.4 GMM a [10] A.Eryilmaz,A.Ozdaglar,D.ShahandE.Modiano,“Distributedcross-
0.3 FiniteBuffer layeralgorithmsfortheoptimalcontrolofmulti-hopwirelessnetworks”,
toappearinIEEE/ACMTransactions onNetworking.
0.20 .1 0.15 0.2 0.25ar0r.3iv0a.3l5r0a.4te0.45 0.5 0.55 0.6 1001 .1 0.15 0.2 0.25ar0r.3iv0a.3l5r0a.4te0.45 0.5 0.55 0.6 [11] X.LinandS.Rasool,“Adistributedjointchannel-assignment,scheduling
and routing algorithm for multi-channel ad hoc wireless networks”, in
Proc.IEEEINFOCOM07,May2007.
Fig. 2. Throughput and delay tradeoff under Alg. with performances [12] L. Ying, R. Srikant and D. Towsley, “Cluster-based back-pressure
compared to Finite Buffer algorithm and BP algorithm, with varying arrival routingalgorithm”, inProc.IEEEINFOCOM08,April2008.
rates atthetransport layer. [13] G. Gupta and N. Shroff, “Delay analysis for wireless networks with
single hop traffic and general interference constraints”, in IEEE/ACM
Transactions onNetworking, 2010.
VII. CONCLUSIONS AND FUTURE WORKS [14] I.HouandP.Kumar,“UtilitymaximizationfordelayconstraintedQoS
inwireless”, inProc.IEEEINFOCOM2010,March2010.
In this paper, we proposed a cross-layer framework which [15] I.Hou andP.Kumar, “Scheduling heterogeneous real-time traffic over
fadingwirelesschannels”,inProc.IEEEINFOCOM2010,March2010.
approachesthe optimalthroughputarbitrarily close for a gen-
[16] I. Hou and P. Kumar, “A theory of QoS for wireless”, in Proc. IEEE
eral multi-hop wireless network. We show a tradeoff between INFOCOM09,April2009.
the throughput and average end-to-end delay bound while [17] I. Hou and P. Kumar, “Admission control and scheduling for QoS
guarantees for variable-bit-rate applications on wireless channels”, in
satisfying the minimum data rate requirements for individual
Proc.ACMMobiHoc, pp.175-184,2009.
flows. [18] M. Neely, “Delay-based network utility maximization”, in Proc. IEEE
Ourworkaimsatabetterunderstandingofthefundamental INFOCOM2010,March2010.
[19] K.Kar,X.Luo,andS.Sarkar,“Delayguaranteesforthroughput-optimal
properties and performance limits of QoS-constrained multi-
wireless linkscheduling”, inProc.IEEEINFOCOM09,April2009.
hop wireless networks. While we show an O(1) delay bound [20] M. Lotfinezhad and P. Marbach, “Throughput-optimal random access
ǫ
with ǫ-loss in throughput, how small the actual delay can withorder-optimal delay”, toappear inIEEEINFOCOM’11.
[21] H.Xiong,R.Li,A.Eryilmaz,andE.Ekici,“Delay-AwareCross-Layer
become still remains elusive, which is dependent on specific
Design for Network Utility Maximization in Multi-hop Networks”, to
networktopologies.Inourfuturework,wewillinvestigatethe appearinIEEEJournalonSelected AreasinCommunications, 2011.
capacity region under end-to-end delay constraints applied to [22] G. Gupta and N. Shroff, “Delay analysis for multi-hop wireless net-
works”,inProc.IEEEINFOCOM09,April2009.
differentnetworktopologies.Ourfutureworkwillalsoinvolve
[23] V.Venkataramanan, X.Lin,L.YingandS.Shakkottai, “Onscheduling
power management in the scheduling policies. forminimizingend-to-endbufferusageovermultihopwirelessnetworks”,
inProc.IEEEINFOCOM2010,March2010.
REFERENCES [24] J.Ryu,L.YingandS.Shakkottai, “Back-pressure routingforintermit-
tently connected networks”, in IEEE INFOCOM2010 Mini-Conference,
[1] L. Tassiulas and A. Ephremides, “Stability properties of constrained March2010.
queueing systems and scheduling policies for maximum throughput in [25] L.Bui, R.Srikant andA.Stolyar, “Novel architectures andalgorithms
multihop radio networks,” in IEEE Trans. Autom. Control, vol. 37, no. for delay reduction in back-pressure scheduling and routing”, in IEEE
12,pp.1936-1948, Dec.1992. INFOCOM2009Mini-Conference, April2009.