Opportunistic Buffered Decode-Wait-and-Forward (OBDWF) Protocol for Mobile Wireless Relay Networks

0.31 MB·English

by  Rui Wang

TOBEAPPEAREDINIEEETRANS.WIRELESSCOMMUN. 1
Opportunistic Buffered Decode-Wait-and-Forward
(OBDWF) Protocol for Mobile Wireless Relay
Networks
Rui Wang, Vincent K. N. Lau, Senior Member, IEEE, and Huang Huang, Student Member, IEEE
Abstract—In thispaper, weproposeanopportunisticbuffered works have focused on the physical layer performance (such
1 decode-wait-and-forward(OBDWF)protocoltoexploitbothrelay asthroughput)andfailedtoexploitthebufferdynamicsinthe
1 buffering and relay mobility to enhance the system throughput
relay.Furthermore,they have assumedall the relaysare static
0 and the end-to-end packet delay under bursty arrivals. We con-
2 sider a point-to-point communication link assisted by K mobile and have ignored the potential benefit introduced by mobility
relays. We illustrate that the OBDWF protocol could achieve a in the network.On the other hand,there are also some papers
n
betterthroughputanddelayperformancecomparedwithexisting focusingonstudyingthe macroscopicbehaviorofcooperative
a
J baseline systems such as the conventional dynamic decode-and- ad-hocnetworks.Forexample,the scalinglaw of the wireless
forward (DDF) and amplified-and-forward (AF) protocol. In
6 ad-hocnetworkisderivedin[8]–[11]anditisshownthateach
additiontosimulationperformance, wealsoderived closed-form
2 node can achieve the throughput of the order O( 1 )1
asymptotic throughput and delay expressions of the OBDWF √Klog2K
protocol. Specifically, the proposed OBDWF protocol achieves whenK fixednodesarerandomlydistributedoveraunitarea.
]
T an asymptotic throughput Θ(log2K) with Θ(1) total transmit Theseresultsimplythatthethroughputofeachnodeconverges
power in the relay network. This is a significant gain compared
I to zero when the number of nodes increases. Nevertheless, it
. with the best known performance in conventional protocols
s (Θ(log K) throughput with Θ(K) total transmit power). With is found in [12] that the per-node throughput can arbitrarily
c 2
[ bursty arrivals, we show that both the stability region and close to constant by hierarchical cooperation. In [13], [14],
average delay of the proposed OBDWF protocol can achieve it’s shown that the source-destination throughput can scale

1 order-wiseperformance gainΘ(K) comparedwithconventional as Θ(log K) when all the relays amplify and forward the
v DDF protocol. received p2acket to the destination cooperatively with Θ(K)
9
8 Index Terms—relay networks, mobile relays, opportunistic total transmission power.In [15], the authorshave shown that
9 buffered decode-wait-and-forward (OBDWF) protocol, queueing a per link throughputof Θ(1) can be achievedat the expense
theory
4 ofpotentiallylargedelaywhenthenodesaremobile.Allthese
.
1 works have suggested that there are potential advantage of
0 I. INTRODUCTION relay buffering and relay mobility. However, there are also
1 In wireless communication networks, cooperative relaying various technical challenges to be addressed before we could
1
not only extends the coverage but also contributes the spatial better understand the potential benefits.
:
v diversity.Asaresult,cooperativerelayisoneofthecoretech- • Low Complexity Relay Protocol Design Exploiting
i
X nology components in the next generation wireless systems Relay Buffering and Relay Mobility: Although the
r suchasIEEE802.16mandLTE-A.Thereareextensivestudies idea of utilizing the mobility has been studied in [15],
a on the theory and algorithm design of cooperative relay, and [16] in the study of ad-hoc network throughput analysis
they can be roughly classified as focusing on the microscopic (scaling laws), there is notmuch work that addressesthe
and macroscopic aspects. Examples of microscopic studies microscopicdetails(suchasprotocoldesign)oftheprob-
include the decode-and-forward (DF), amplify-and-forward lem. For example, most of the existing relay protocols
(AF)andcompress-and-forward(CF) protocolsforsingle hop have focused entirely on the physical layer performance
[1]–[3]aswellasmulti-hopcooperativerelaynetworks[4].In (information theoretical capacity or Degrees-of-Freedom
[5], the authors demonstrated that AF could achieve optimal (DoF)) and they did not fully exploit the potential of
end-to-end DoF for the MIMO point-to-point system with relay buffering. In fact, it is quite challenging to design
multiple relay nodes. In [6], it is shown that a variant of the lowcomplexityrelayprotocolthatcouldexploitboththe
CFrelayingachievesthecapacityofanygeneralmulti-antenna relaybufferingandrelaymobility.Furthermore,theissue
Gaussianrelaynetworkwithinaconstantnumberofbit.In[7], is further complicated by the bursty source arrival and
relay selection protocolis shownto achievehigherbandwidth randomly coupled queue dynamics in the systems.
efficiency while guaranteeing the same diversity order as that • Performance Analysis: It is very important to have
of the conventional cooperative protocols. However, all these closed form performance analysis to obtain insights to
The authors are with the Department of Electronic and Computer 1f(K) = O(g(K)) means that there exists a constant C such that
Engineering (ECE), The Hong Kong University of Science and Tech-
f(K) ≤ Cg(K) for sufficiently large K, f(K) = o(g(K)) means that
nology (HKUST), Hong Kong. (email: [email protected], eekn- lim f(K) = 0, and f(K) = Θ(g(K)) means that f(K) = O(g(K))
[email protected],[email protected]). This work was supported by the Research K→∞g(K)
GrantsCounciloftheHongKongGovernmentthroughthegrantRGC615609. andg(K)=O(f(K)).
TOBEAPPEAREDINIEEETRANS.WIRELESSCOMMUN. 2
understand the fundamentaltradeoff between throughput
gain and delay penalty in cooperativesystems. However,
it is very challenging to analyze closed form tradeoff
between the throughput, stability region and end-to-end
delay. For instance, most of the existing papers studying
delay and throughput scaling laws in ad-hoc network
[15], [16] are focused on the macroscopic aspects of the
systems and they have ignored the microscopic details
such as the random bursty arrivals and queue dynamics
in the systems. When these dynamicsare taken into con-
sideration, the problem involves both information theory
(to modelthe physicallayer dynamics)and the queueing
theory (to model the queue dynamics), which is highly
non-trivial.
In this paper, we shall propose an opportunistic buffered
decode-wait-and-forward (OBDWF) protocol to exploit both
relay buffering and relay mobility to enhance the system
throughput and the end-to-end packet delay under bursty
arrivals. We consider a point-to-point communication link
assisted byK mobilerelays.Byexploitingtherelaybuffering
and relay mobility in the phase I andphase II of the proposed
OBDWF, we shall illustrate that the OBDWF could achieve Fig. 1. System and random walk mobility model for the point-to-point
communication system with a half-duplexing mobile relay network. The
a better throughput and delay performance compared with
parameter q determines the mobility of the network (larger q means higher
existing baseline systems such as the conventional Dynamic mobility).
Decode-and-Forward (DDF) [3] and AF protocol [17]. In
addition to simulation performance, we also derived closed-
form asymptotic throughput and delay expressions of the 1. The source and the destination nodes are fixed at two
OBDWF protocol. Under random bursty arrivals and queue ends of a diameter, and the disk is divided horizontally into
dynamics,theproposedOBDWFprotocolhaslowcomplexity M equal-area regions along the source-destination diameter.
with only O(K) communication overhead and Θ(1) total Theseregionsaredenotedasregion1,region2,...,andregion
transmit power in the relay network. It achieves a throughput M, from the source to the destination. As illustrated in Fig.
Θ(log K), which is a significant gain compared with the 1, the movement of each relay is modeled as a random walk
2
bestknownperformanceinconventionalprotocols(Θ(log K) (Markov chain) over these regions:
2
throughput with Θ(K) total transmit power). Furthermore, • At the beginning, each relay is uniformly distributed on
both the stability region and average delay can achieve order- the disk. Movementsof relayscan only occur in discrete
wise performance gain Θ(K) compared with conventional frames with time index t.
DDF protocol. • Let Xk(t) denote the region index of the k-th relay in
the t-th frame, X (t)t=1,...+ is a Markovchain
II. SYSTEMMODEL { k | ∞}
with the following transition matrix
In this section, we shall describe the system model for the
point-to-point communication system with a half-duplexing Pr Xk(t+1)Xk(t) =
{ | }
mobile relay network. Specifically, there are K mobile relay q Xk(t+1)=Xk(t) 1
±
nodes between the source node and the destination node, as 1 2q Xk(t+1)=Xk(t)=2,...,M 1
 − −
snhoodwesnhinasFaign. i1n.fiEnaitcehloenfgtthhebsuofufrecre. TnoodfeacainlidtattehethKe rreellaayy  10−q Xokth(etr+wi1s)e=Xk(t)=1 or M
sischfeudrtuhleinrgd,itvriadnesdmiinstsoiotnhriseepatyrtpiteisonoefdsilnottosfdreafimneesd.Easacfholflroawmse: • When one relay moves into a region, its actual locati(o1n)
in this region is uniformly distributed.
• ChannelEstimationSlotisusedbyrelaysforestimating
the channel gains with the source and destination nodes. Remark 1 (Interpretation of Parameter q): The region
• Control Slot is used by relays for distributive control transition probability q ∈ (0,12] measures how likely one
signaling of the OBDWF protocol. The details is given relay will move into another region in the next frame, and
in the protocol description. therefore, it is related to the average speed of the relays.
• Transmission Slot is used for data transmission, and it
last τ seconds.
B. Physical Layer Model
A. Relay Mobility Model Let H and d be the small scale fading gain and
s,j s,j
Following [8], [14], we assume that the K relays are the distance between the source node and the j-th relay
distributed on a disk with radius R as illustrated in Fig. respectively, and let H and d be the small scale fading
j,d j,d
TOBEAPPEAREDINIEEETRANS.WIRELESSCOMMUN. 3
gain and the distance between the j-th relay and destination
node respectively.
Assumption 1 (Assumption on the Channel Model): We
assume that H ,H are quasi static in each frame.
s,j j,d
{ }
Furthermore, H ,H are i.i.d over frames according to
s,j j,d
{ }
a general distribution Pr H and independent between each
{ }
link.
The relay network shares a common spectrum with band-
width WHz, and each node transmits at a peak power P.
Let x be the transmitted symbol from the source node,
s
the received signal at the j-th relay is given by: y =
j
H /dα x +z , where α is the path loss exponent, and
s,j s,j s j
zqj isthe i.i.d (0,1)noise.Theachievabledata rate between
N
the source node and the j-th relay is given by:
C =W log (1+PξH /dα ), (2)
s,j 2 s,j s,j
Fig. 2. Frame sequences for conventional DDF protocl and opportunistic
where ξ (0,1] is a constant can be used to model both the buffered decode-wait-and-forward (OBDWF)protocol. Inconventional DDF
codedand∈uncodedsystems.Similarly,theachievabledatarate protocol,thephaseIandthephaseIIofthesamepacketappearasinseparable
atomic actions. On the other hand, the proposed OBDWF protocol exploits
between j-th relay and the destination node is given by buffersintherelaystocreatetheflexibilitytoschedulephaseIandphaseIIof
thesamepacketbasedontheinstantaneouschannelstate.Coupledwithrelay
C =W log (1+PξH /dα ), (3) mobility,theproposedOBDWFprotocolallowstherelaytobufferthepacket
j,d 2 j,d j,d
andwaitforgoodopportunity(whentherelaysisclosetothedestination)to
All the packets are transmitted at data rate R = Wlog β deliver thepacket.
2
for some constant β. The j-th relay could correctly decode
the packets transmitted from the source node if R C ,
s,j
≤
and the destination node could correctly decode the packets packet from its buffer if there is an ACK from at least
transmitted from the j-th relay if R C . For easy one of the relays or the destination node.
j,d
≤
discussion, we shall denote a link as a connected link if its
Note that the OBDWF protocol has only O(K) commu-
achievable data rate is larger than R, and otherwise a broken nication overhead with Θ(1) total transmit power in the
link.
relay network, which is the same as the conventional DDF
and conventional AF protocol, elaborated in Table I. Unlike
III. THE OBDWFPROTOCOL conventionalDDFprotocolwherethephaseI(sourcetorelay)
and the phase II (relay to destination) of the same packet
In this section, we shall first describe the proposed op-
appear as inseparable atomic actions, the proposed OBDWF
portunisticbuffereddecode-wait-and-forward(OBDWF)relay
protocol exploits buffers in the relays to create the flexibility
protocol for mobile relays.
to schedule phase I and phase II of the same packet based on
Protocol 1 (OBDWF Protocol for Mobile Relays):
theinstantaneouschannelstateasillustratedinFig.2.Coupled
1. Each relay measures the current states connected, bro-
withrelaymobility,theproposedOBDWFprotocolallowsthe
{
ken of its linkswith the destinationnode in the channel
relaytobufferthepacketandwaitforgoodopportunity(when
}
estimation slot.
the relays is close to the destination)to deliver the packet. As
2. Thecontrolslotisdividedintomini-slots.Ifthebufferin
aresult,relaymobilityallowsthesystemtooperateatahigher
a relay is not empty and the link state to the destination
throughput at the expense of larger delay. We shall quantify
is connected, it will submit a request in a control mini-
such tradeoff in Section IV and V.
slot. Using standard contention mechanism, one active
relay is selected to transmit its packet from its buffer to
the destination node2. The selected relay as well as all
IV. THROUGHPUT PERFORMANCE WITHINFINITE
the other relays will dequeue the same packet from their
BACKLOG
buffers.
3. IfnoneoftheK relaysattemptstocompeteforaccessto
In this section, we shall first discuss the average system
the destination node in the control slot, the source node
throughput of the proposed OBDWF protocol with infinite
will broadcast a new packet to the K relays and the
backlog at the source buffer. We first define the average
destination node using a fixed data rate R = Wlog β
2 throughputbelow.
for some constant β. The source node will dequeue the
Definition 1 (Average End-to-End System Throughput):
LetT bethenumberofinformationbitssuccessfullyreceived
2 The algorithm can be extended to consider spatial combining from i
multiple relays. However, the performance gain associated with that will be by the destination node at the i-th frame. The average end-
quitelimitedduetothepathlosseffects.Forinstance,thereisverylowchance to-end system throughput between the source node and the
ofhavingmultiplerelaysnearthesourceormultiplerelays(havingthesame destination node T is defined as T =lim Pκi=1Ti.
commonpacket) nearthedestination. κ→∞ κτ
TOBEAPPEAREDINIEEETRANS.WIRELESSCOMMUN. 4
A. Throughput Performance of the OBDWF Protocol Therefore, we have the following corollary on the perfor-
mance gain of the OBDWF protocol:
Note that for a fixed number of relay nodes K, when the
Corollary 1: (Comparison of the Average System Through-
source node increases the data rate R, the associate radio
put): The throughput gain of the OBDWF protocol is
coverageandthenumberofrelayswhocandecodethesource
npuacmkbeetrboefcoremlaeyssswmhaollecra.nOdnecthoedeotthheersohuanrcde, pfoarckfiexteidncRre,atshees T∗OBDWF = Θ log2l(oKg2qKM−1) if Kl→im∞KqM−1 =∞
as K increases. We shall quantifysuch scaling relationship in T∗DDF  Θ(cid:16)(KqlMog−21K)1/M(cid:17) otherwise
Lemma 1. (7)
(cid:16) (cid:17)
Lemma 1 (Scaling Relationship of Connected Link): Remark 3(Interpretation of Corollary 1): Note that when
Denote the transmit data rate R = W log2β and γ = βα2 the system mobility is low (q = (logK2K)M1−1), there is an
for some constant β. For sufficiently large K, the following order-wisethroughputgainachievedbytheOBDWFprotocol.
statements are true:
• I. If lim γ = 0, the probability that there are Θ(K)
relaysK→ha∞vinKg connected links with the source node (γor V. STABILITY AND DELAYPERFORMANCE WITH BURSTY
the destination node) tends to 1. ARRIVALS
• II. If lim γ > 0, the probability that there are Θ(1) Inthissection,weshallfocusonthestabilityregionandthe
relaysKh→av∞inKg connected links with the source node (or delayperformanceanalysisof theproposedOBDWFprotocol
the destination node) is Θ(K). under bursty packet arrivals. We shall first define the busty
γ
source model, followed by the analysis on the stability region
Proof: Please refer to Appendix A.
and average delay performance.
Using Lemma 1, we obtain the closed-form asymptotic
averagesystemthroughputunderinfinitebacklogatthesource
buffer. A. Bursty Source Model
Theorem 1: (ThroughputPerformance of the OBDWF Pro- Let A (t) representsthe numberof new packets arrivingat
s
tocol): For sufficiently large K and infinite backlog at the the source node at the beginning of the t-th frame.
source buffer, the maximal average system throughput of the Definition 2 (Bursty Source Model): We assume that the
proposed OBDWF protocol under the random walk mobility arrival process A (t) is i.i.d over the frame index t according
s
model in (1) is given by toageneraldistributionPr A .Eachnewpackethasafixed
s
{ }
T∗OBDWF =Θ(log2K) (4) narurmivbaelrporofcbeistss aNre. Tdehneofitersdtbayndλse=conEd[Aord]earnmdoλm(2e)n=tsEof[Ath2e]
s s s s
This order-wise throughput is achieved when γ = Θ(Kσ) respectively.
for some constant σ ∈(0,1]. Furthermore, T∗OBDWF is upper- Let Qs(t) be the number of packets in the source buffer
bounded by but infinitely close to Wαlog K. at the t-th frame. The dynamics of the source buffer state is
4 2
Proof: Please refer to Appendix B. given by:
Remark 2 (Interpretation of Theorem 1): Since there are Q (t+1)=A (t+1)+[Q (t) U (t)]+ (8)
s s s s
infinitely large buffers at the relay nodes and the random- −
walk transition probability q is positive, the average system where Us(t) 0,1 is the number of packets transmitted to
∈{ }
throughput is R/2 as long as there are always relays having the relay networkatt-th frame.Furthermore,let Qk(t) be the
connected links to the source and destination node (which is number of packets in the the k-th relay node’s buffer at the
presented mathematically as lim γ =0 by Lemma 1). t-thframe.Thedynamicsofthe relaybufferstate isgivenby:
K→∞K
Q (t+1)=A (t+1)+[Q (t) U (t)]+ (9)
k k k k
−
B. Comparison with the Conventional DDF Protocol
where A (t) 0,1 is the number of packets received by
k
Similarly, we shall summarize the closed-form asymptotic the k-th relay∈no{de fr}om the source node at the beginning of
averagesystem throughputforthe conventionalDDFprotocol the t-th frame, and U (t) 0,1 is the number of packets
k
( elaborated in Table I) below. dequeued from the k-th rel∈ay{node}at the t-th frame.
Lemma 2: (ThroughputPerformanceof ConventionalDDF
Protocol): For sufficiently large K and infinite backlog at the
B. Stability Performance
source buffer, the maximal average system throughput of the
In this section, we shall derive the stability region of the
conventional DDF protocol under the random walk mobility
OBDWF protocol and the conventional DDF protocols. We
model in (1) is given by:
first define the notion of queue stability [18], [19] below.
T∗ =min Θ((KqM−1)1/M),Θ log (KqM−1) (5) Definition 3 (Stability of the Queueing System): The
DDF 2
queueing system is stable, if
This order-wisenthroughput is achieved(cid:0)when (cid:1)o
lim Pr Q(t)<x =F(x) and lim F(x)=1 (10)
γ =max Θ(1),Θ( KqM−1) (6) t→∞ { } x→∞
where Q(t) is the queue state in the queueing system at the
Proof: Please refernto the Apppendix C. o t-th frame.
TOBEAPPEAREDINIEEETRANS.WIRELESSCOMMUN. 5
Using Definition 3, we have the following Theoremfor the
2
OBDWF protocol.
Theorem 2 (Stability Region of the OBDWF Protocol): 1.8 q=Θ(K−2)
For sufficiently large K, the system of queues
1.6
Q (t),Q (t), ,Q (t) under the proposed OBDWF
s 1 K
{protocolare sta·b·l·e if and o}nly if λs < ζ1, where ζ is givenby K)1.4 q=Θ(K−3/2)
ζ =max{Θ(1),Θ(γ/K)} (11) D)/log(S 1.12 q=Θ(K−1)
SimPirloaorlfy:,PltehaesestraebfielritytorAegpipoenndoixf Eth.e conventional DDF D:log(S 0.8
protocol is given by: 0.6
Lemma 3 (Stability Region of the Conventional DDF Protocol): 0.4 q=Θ(K−1/3)
For sufficiently large K, the system of queues
Qs(t),Q1(t), ,QK(t) underconventionalDDFprotocol 0.2 q=Θ(1)
{ ··· }
are stable if and only if λs < 1 , where DS is given by 0
DS 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
γ:log(γ)/log(K)
D =
S
max Θ(1),
 Θ (K(cid:8)qγM2−1)M1 If Kl→im∞Kγ =0 and Kl→im∞qKγ2 >0 xF-iagx.i3s.isTlhoeg(aγs)y,mapntdotitchereyl-aatixoinssihsiplobge(tDwSe)en. DS,γ andq.Specifically, the
 ΘΘ((cid:2)γK2γ)/qM−1 M1(cid:3)(cid:9) IIff Kll→iimm∞Kγγ =>00 aanndd Kll→iimm∞q1Kγ2>=00 log(K) log(K)
 ΘPr(ohγ(cid:0)o)f: Using(cid:1)LeimmaIf5KKiln→→im∞∞ApKKγpe>nd0ixaCnd, tKKhle→→im∞∞resqqu1γγlt(s=12i0n) tDathote-e=tnhsdoleiumrpcdaeecsktneinotadPtdeiκieo=lna1anyIdir3n−otIodhiafee. tihr-eethsprepeclaatciyvkeeltnye.stuwTcochreekssfiausvleldyraegfireenceedeivnedads-
Lemma 3 follows using similar argument as in Appendix E. κ→∞ κ
The following Theorem summarizes the average delay per-
formance of the proposed OBDWF protocol.
The following Corollary summarizes the performance gain
Theorem 3: (Average End-to-End Packet Delay of the OB-
of the OBDWF protocol in stability region.
DWFProtocol):ForsufficientlylargeK andλ inthestability
Corollary 2 (Stability Region Comparison): Let s
region in Theorem 2, the average end-to-end packet delay of
λ∗(OBDWF) and λ∗(DDF) be the maximum source
s s the OBDWF protocol is given by:
arrival rate the system can support and maintain queue
stability using OBDWF and conventional DDF protocol
λ(2)ζ
respectively.For sufficientlylarge K, the gain on the stability D =max Θ s ,Θ(D ) (14)
OBDWF S
region is given by: ( λs(1 λsζ)! )
−
λ∗sλ(∗sO(BDmDDaWFx)F) Θ=(1), wherPerζooisf:gPivleeansebyre(f1er1)toanAdpDpeSndiisxgFiv.en by (12).
 Θ (K(cid:8)qγM2−1)M1 If Kl→im∞Kγ =0 and Kl→im∞qKγ2 >0 Remark 5 (Interpretation of Theorem 3): The first term in
 ΘΘ((cid:2)γKK2)(γq)1−MM (cid:3)(cid:9) IIff KKll→→iimm∞∞KKγγ =>00 aanndd KKll→→iimm∞∞qq1Kγγ2>=00 tfbhryaemthReesH)Sstohuaotrfcthe(e1a4pr)aricvikaselttsmhseotadyaevsle,irnia.tgehe.,eswλosauirtacinnedgbuλtif(smf2e)er..IT(tnhiusemasfbfeeecrcotneoddf
ReΘma((cid:16)Krk)4 (Interpr(cid:17)etationIfofKlt→ihme∞CKγoro>lla0ryan2d):KlN→imo∞teq1γ (=t1h3a0)t sptetaarcmykseDtinSsitz=heeΘNre(lDaaynRd)n,etwthwehoemrrkeo.DbTilRhitiysisfotahfcettoharveednreaepgtewenotdirsmkoe(ntqh)be.opFthaigct.khe3et
whenthepacketsize N islargesuchthatγ >Θ(ψ), thenthe further illustrates the asymptotic relationship between DS, γ
OBDWFprotocolcanobtainanorder-wisegain.Forexample, andq.Specifically,thex-axisis logγ,andthey-axisis logDS.
logK logK
when γ = Θ(K) and q = Θ(1), then λ∗sλ(∗O(BDDDWF)F) = Θ(K). Observe that DS is an increasing function of γ and 1/q.
s Similarly, the delay performance of the conventional DDF
protocol is summarized in the following Lemma:
Lemma 4: (Average End-to-End Packet Delay of the Con-
C. Delay Performance
ventional DDF Protocol): For sufficiently large K and λ in
s
In this section, we shall compare the average end-to-end
thestabilityregioninLemma3,theaverageend-to-endpacket
packet delay performance. The average end-to-end packet
delay of the relay network is defined below.
3Note that it is implicitly assumed that the system of queues are stable
Definition 4 (Average End-to-End Packet Delay): Let Ia
i when we discuss average delay because otherwise, the probability measure
and Ir be the frame indices of the i-th packet arrival at behindtheexpectation isnotdefined.
i
TOBEAPPEAREDINIEEETRANS.WIRELESSCOMMUN. 6
delay under the conventional DDF protocol is given by: versusthenumberofrelaysK atdifferentnetworkmobilityq
under the bursty source model. Similar significant gains over
λ(2)D
D =Θ s S (15) the baselines can be observed. Moreover, it can be observed
DDF
λs(1 λsDS)! in these two figuresthat the simulation results match with the
−
theoretical results derived in Section IV and V.
where D is given by (12).
S Fig.6illustratestheaverageend-to-endpacketdelayversus
Proof: Please refer to the Appendix G.
thenumberofrelaysKatdifferentmobilityqwithfinitebuffer
ThefollowingCorollarysummarizestheaveragedelaygain
length of 25 packets for all the nodes. Note that, the delay
of the OBDWF protocol.
performanceis an increasing functionof 1/q for all protocols
Corollary 3 (Average Delay Comparison): For sufficiently
and there is also a significant gain of the proposed OBDWF
largeK andλ inthestabilityregioninLemma3,theaverage
s protocol.
delaygainoftheOBDWFandtheconventionalDDFprotocols
Fig.7illustratestheaverageend-to-endpacketdelayversus
is given by:
the average arrival rate λ with infinite buffer length for all
s
DDDF = the nodes. Note that the baseline 2 and 3 are not stable under
DOBDWmFin Θ λ(s2) ,Θ DS(1−λsζ) (16) the operating regime considered. The delay performance of
λs(1−λsDS) ζ(1−λsDS) baseline 4 quickly blows up at λs = 0.08, which is the
where ζ is givnen b(cid:16)y (11) and Θ(cid:17)(D )(cid:16)is given by(cid:17)(o12). maximal stability input rate for this baseline. On the other
S
hand,thedelayperformanceoftheproposedOBDWFprotocol
Remark 6 (Interpretation of the Corollary 3): There are
significantlyout-performsallthebaselineovertheentirerange
several scenarios that the OBDWF protocol will have
of traffic loading. Fig. 8 illustrates the average end-to-end
significant order-wise gain on the delay performance.
system throughput T versus the number of relays K under
For example, when λ is close to the service rate
s
1/D , i.e., λ = 1 ǫ where ǫ = o( 1 ), we have the Random Waypoint Model, which is also widely used in
S s
DS − DS [20]–[22]. Similar performance gains can be observed.
DDDF = Θ( 1 ) > Θ(1). On the other hand, even if
λDOBDisWFnot so cǫlDosSe to 1 , i.e., 1 λ D = Θ(1), there
s DS − s S VII. CONCLUSIONS
will still be order-wise gain as long as λs = o(1) and
1 = o(1). Specifically, if Pr A (t) =λ(s2)Θ(K) = ν, Inthispaper,we proposean opportunisticbuffereddecode-
PDrS A (t)=0 =1 ν, where ν{=Θs ( 1 ), γ =Θ}(K) and wait-and-forward(OBDWF)protocolforapoint-to-pointcom-
{ s } − K2 municationsystemwithK mobilerelays.Unlikeconventional
q =Θ(1), then we can obtain DDDF =Θ(K) by (16). DDF protocol, the proposed OBDWF protocol exploits both
DOBDWF
the relay buffering and relay mobility in the systems. We
VI. SIMULATION RESULTSAND DISCUSSION derive closed-form expressions on the asymptotic system
In this section, we shall compare the performance of the throughputunder infinite backlog as well as the average end-
proposed OBDWF protocol with various baseline schemes. to-enddelayunderageneralburstyarrivalmodel,Basedonthe
Baseline 1 refers to the conventionalDDF protocol[3], base- analysis,wefoundthatthereexistsathroughputdelaytradeoff
line 2 refers to the conventional AF protocol [17]. Baseline inthebufferedrelaynetwork.Thesystemcanachieveahigher
3 and 4 are extensions of Baseline 1 and 2 respectively with throughput Θ(log2K) using the proposed OBDWF protocol
spatial combining from multiple relays, which are elaborated at the expense of extra delay. The system mobility affects the
inTableIWeconsiderasystemwiththesourcenodeat(0,0) tradeoff as below:
and the destinationnodeat(5,0). The K relaysare randomly • Effect on the Throughput/Stability Region Perfor-
distributed between the source node and the destination node, mance: According to Theorem 1, the maximal average
asshowninFig.1.Themovementofeachrelayisgivenbythe system throughput of the proposed OBDWF protocol
randomwalkmobilitymodelin(1),wherethenumberofrelay Θ(log K) is not influenced by the relay mobility.
2
mobilityregionsisM =5.Thesmallscalefadinggainfollows • Effect on the Delay Performance: If the movement of
complex Guassian with unit variance. The pass loss exponent the mobility is fast (large region transition probability
α= 4, and the transmit SNR P = 20dB. For bursty arrivals, q), the chance one relay with source’s packet gets close
we assume Pr A =15 =0.001 and Pr A =0 =0.999. to the destination is high, leading to small delay in the
s s
{ } { }
This corresponds to an arrival rate λ = 0.015. The packet relaynetwork,andviceversa.Thiscanbeobservedfrom
s
size N = Wτlog K, the frame duration τ = 5 ms and the Theorem 3.
2
bandwidth W = 1MHz. Using Lemma 2, the physical data Finally, theoretical and numerical results demonstrate the
rate at the source node is set to be R = W log2K, which is significantperformancegainsofthe proposedOBDWF proto-
the optimal choice for conventionalDDF. col against various baseline references.
Fig. 4 illustrates the average end-to-end system throughput
T versus the number of relays K at different mobility q. Ob-
APPENDIX A:THEPROOF OF LEMMA1
serve that the proposed OBDWF protocolhas significant gain
compared with the baselines. Furthermore, the performance Without loss of generality, we only study the number of
of the OBDWF protocol is insensitive to the mobility of the connected links with the source. The connection with the
networkq. Fig. 5 illustrates the maximalstable arrivalrate λ destination can follow the same approach.
s
TOBEAPPEAREDINIEEETRANS.WIRELESSCOMMUN. 7
TABLEI
ELABORATIONOFTHEBASELINES.SPECIFICALLY,BASELINE1REFERSTOconventional DDFPROTOCOL[3],BASELINE2REFERSTOconventional AF
PROTOCOL[17],BASELINE3REFERSTOAFwithspatialcombiningANDBASELINE4REFERSTODFwithSpatial Combining.
Baseline Name Description
Baseline 1 •Thesourcenodebroadcasts apacketfromthebufferatthebeginningoftheframeuntilatleastonerelay
(Conventional DDF) ordestination nodereturns withanACK.
•Ifthedestination nodereturnswithanACK,thesourcenodestarttobroadcast anewpacket; Iftherelay
nodereturnswithanACK,thesourcenodestopsbroadcasting andtherelaynodeforwardthepacket tothe
destination nodeinthenextframe.
Baseline 2 •Thesourcenodebroadcasts apacket fromthebufferatthebeginning ofaframe.
(Conventional AF) •Alltherelayslistenandstorethereceivedsamplesfromthesourceduringthelisteningphaseandtherelay
with the largest metric (PSPs,2jS+sP,jSSjj,,dd+1, where Ss,j = Hdαss,,jj and Sj,d = Hdαjj,,dd ) is selected to amplify
andforwardtothedestination nodeinthenextframe.
Baseline 3 •Thesourcenodebroadcasts apacket fromthebufferatthebeginning ofaframe.
(AF with Spatial • All the relays listen and store the received samples from the source during the listening phase and NA
Combining) relays with the largest metric (PSPs,2jS+sP,jSSjj,,dd+1, where Ss,j = Hdαss,,jj and Sj,d = Hdαjj,,dd ) are selected to
amplifyandforwardtothedestination nodeinthenextframe.
Baseline 4 •Thesourcenodebroadcasts apacketfromthebufferatthebeginningoftheframeuntilatleastND relay
(DF with Spatial nodesordestination nodereturnwithanACK.
Combining) •Ifthedestination nodereturns withanACK,thesourcenodestarts tobroadcast anew packet; Ifatleast
ND relay nodes return with anACK, the source node stops broadcasting andall the relay nodes that have
decodedthepacket fromthesourcenodewillforwardthepacket tothedestination nodeinthenextframe.
4
Average end−to−end system throughput (b/s/Hz)0123....1235555 (AF wBiathsBA eslanpiOnsaaeelBtyislaitDnilc Weca ol3F mR pebrsionutiolntcsgo ()[Cl Θo(nlovge2nKtBi)o]ansa(eCl lDoinn(DeDv Fe1F np BwtriaooitstnhoeaB cslliao npAslea)Fe t2il aipn lr eco ot4omcboiln)ing) λMaximal stable arrival rate (packets/frame)s00000.....0000001234.....5555512345 (AF withOB sBapDsaeWtliianFle (cpD o3rFom (C( tCwobocioinntohnivBln vesgaenpB)sntaeiatBotiloisinaanenalse laceli nl4oAl ieDn mF eD2 bAp F1irnn opiantrlogyoct)tiooclca)ol lR)esults [ Θ(B1a) s]elines
0 0
50 60 70 80 90 100 110 120 130 50 60 70 80 90 100 110 120 130
Number of relays K Number of relays K
(a) q=1/10 (a) q=1/10
4
e end−to−end system throughput (b/s/Hz)123...123555 (AF wBitahAsB sneapalisanlyetetilaisicnla ecl o3Rmebsiunlitnsg ([O) CΘBo(DnlovWge2nFKt Bi)po](arnCosatoeol nlcDi(nvoDDeelF nFB1 t wiaposirntoehatBl olis nacApesoaF el2t) ilp ainrlo ect oo4mcobl)ining) λmal stable arrival rate (pakets/frame)s0000....2345 (DF wiOthBB saDpsWaetliFian lep c rB4oo((C mCatosoobcennionlvvilnieneengnBst)tBiaiooasnnesaaellil nl AiDne(Fe DA2 pFF1 Ar opwntriaoothlctyBo otscailcp)osaale)lt liRianlee cs o3um ltsb i[n Θin(g1)) ]
Averag0.5 Maxi 0.1
0 0
50 60 70 80 90 100 110 120 130 50 60 70 80 90 100 110 120 130
Number of relays K Number of relays K
(b) q=1/5 (b) q=1/5
Fig.4. Averageend-to-endsystemthroughputT versusthenumberofrelays Fig.5. Themaximalstablearrival rateλs versusthenumberofrelays K
tKo Latemdimffeare2n,t(mNoAbi,lNityDq).T=he(5p,h5y)siicnalthdeatabarsaeteliniseR3=andW4loregs2peKctiavceclyo.rdTinhge (a5t,d5if)feirnentthemboabsileiltiyneq.3Tahnedp4ackreestpseiczteivieslyN. T=heWmaτrlkosg“2oK,x,,+(,N(cid:3),A⋄,”NdDen)o=te
marks“o,x,+,(cid:3),⋄”denotesimulationresultsandthesolidlinesrepresentthe simulation results and the solid lines represent the analytical results for
analytical results forOBDWFandbaseline 1(conventional DDF)protocol. OBDWFandbaseline 1(conventional DDF)protocol.
TOBEAPPEAREDINIEEETRANS.WIRELESSCOMMUN. 8
1200 350
Average end−to−end packet delay (frames)1246800000000000 BBaasseelilninee s3 (AF wBiaths es(lDpinaFet iwa2l i t(chCoB osmapnsbaveietnilaniinnlt( iegcoC )on4omanlvb AeinFnin tBpiogran)osateol lcDinoDel) F1 protocol) Average end−to−end packet delay (frames)11223050505000000 (DF withB sapsaetilainl ec o4mbining)OBDWF protocol
OBDWF protocol 0
0.06 0.065 0.07 0.075 0.08
050 60 70 80 90 100 110 120 130 Average arrival rate λs (packets/frame)
Number of relays K
(a) q=1/10 Fig.7. Theaverageend-to-endpacketdelayversustheaveragearrivalrate
λs. Note that the other baselines are not stable under the operating regime
considered. The packet size is N = Wτlog K, K = 110, q = 1/5 and
2
900 ND =5 in the baseline 4. The source arrival model is given byPr{As =
Baseline 2 (Conventional AF protocol) 1}=λs,andPr{As=0}=1−λs.
800
mes)700
Average end−to−end packet delay (fra234560000000000 Base(DlinFe swithB sapsaetilainl ec o4m(bCinoinngve)ntBioansael( lADinFDe Fw1 ipthroB staopscaeotillai)nl ec o3mbining) −end system throughput (b/s/Hz)123...234555 (ConventBioansael lDin(DeA FF1 pwriothOtoB BscapDosalW)etilaiFnl ecp or3omtobcino(ilnDgF) withB sapsaetilainl ec o4mbining)
100 OBDWF protocol d−to
en 1 Baseline 2
0 e (Conventional AF protocol)
50 60 70 80 90 100 110 120 130 ag
Number of relays K Aver0.5
(b) q=1/5 0
50 60 70 80 90 100 110 120 130
Number of relays K
Fig.6. Theaverageend-to-endpacketdelayversusthenumberofrelaysK
atdifferentmobilityqwithfinitebufferlengthof25packetsforallthenodes.
aTnhde4parcekseptecstiizveeliys.NTh=e sWouτrcleogar2riKva,l(mNoAd,elNiDs)gi=ven(5b,y5)Pirn{Athes b=as1el5in}e=3 FKigu.n8d.erAthveerraagnedeonmd-wtoa-yenpdoinsytsmteomdethlr[o2u0g]–h[p2u2t]T.Svpeercsiufisctahlley,nuamnboedreocfhroeolasyess
arandomdestinationdistributeduniformlyinthecoverageareaandmovesin
0.001andPr{As=0}=0.999.
thatdirectionwitharandomspeedchosenuniformlyintheinterval[0.1,0.6]
(per frame). On reaching the destination the node pauses for some random
timechosenuniformlyintheset{0,1,···,10}(frames)andtheprocessrepeats
ForagivensmallscalefadingrealizationHs,j,thecoverage itself. Thephysical datarate isR=Wlog2K,(NA,ND)=(5,5)inthe
baseline 3and4respectively.
radius of the source within which the relays have connected
links to the source is given by d = (PξHs,j)α1/(β 1)α1.
−
Therefore, the area of the source’s coverage can be approx-
simouartceed’sacsovπe2dr2a,geanids tdh2e.pTraokbianbgilcitoynsoindeeraretiloanyoffaltlhseirnatnodothme nuWmbheern, wKle→imh∞avKγe = 0, i.e., Kl→im∞φK = ∞, by law of large
2R2
realizationofH , theprobabilitythatonerelayfallsintothe
s,j Pr[φ o(φ)<X/K <φ+o(φ)] 1 when K ,
source’s coverage is given by − → →∞
(18)
∞ d2 1 where lim o(φ) = 0. Therefore, the Pr X = Θ(φK) =
φ=Z0 2R2f(Hs,j)dHs,j =Θ(γ) (17) Θ(Kγ)}Kt→en∞ds tφo 1 when K → ∞. This fin{ishes the proof of
statement I.
where f(H ) is the pdf of the small scale fading gain H .
s,j s,j When lim γ > 0, since the probability that one relay
Suppose there are X relays having connected links with K
K→∞
the source, therefore, Pr(X = x) = K (φ)x(1 φ)K−x. has the connected link with source node is φ = Θ(1), the
x − γ
X can be treated as one binomial random variable generated probability that at least one relay can receive the transmitted
from Bernoulli trail where it has value(cid:0)1 (cid:1)with probability φ. packetis givenby 1 (1 φ)K =Θ(K/γ). This finishesthe
− −
In other words, X Binomial(K,φ). proof of statement II.
∼
TOBEAPPEAREDINIEEETRANS.WIRELESSCOMMUN. 9
APPENDIX B: THEPROOF OF THEOREM1 Note that DS is an increase function of γ by Lemma 5.
henWceheγn=γΘ=(1Θ)(i1s)n,oitttihseoobpvtiimouaslcthhaoticTe.OWBDhWeFn=liΘm(1γ),=and0 nSuinmceeraTtoDrD,Fw=e hΘav(eloDgD2SγS)(γa∗n)d=duΘe(t1o),thwehleorge fγu∗ncitsiotnheinopthtie-
and γ > Θ(1), the coverage radius of theKs→ou∞rcKe and mal value that maximizes TDDF, i.e., γ∗ = argmaxγTDDF.
According to the delay expression in (20), when γ∗ =
destinationnodescan bemadesufficientlysmallgivena fixed
max Θ(1),Θ( KqM−1) , we have D (γ∗) = Θ(1). As
transmitpowerP.Onlytherelaysclosetothesourcenode(in { } S
a result, if KqM−1 = o(1), the optimal value is γ∗ =
region 1) can decode the packet transmitted from the source Θ(1), leading pto T∗ = Θ(KqM−1)1/M . Otherwise, the
nodeandonlytherelaysclosetothedestination(inregionM) optimal valupe is γ∗DD=F Θ( KqM−1), leading to T∗ =
can forward the packet to the destination node. Furthermore, DDF
Θ log (KqM−1) .
there are (with probability 1) some relays in region 1 can 2 p
decode the source’s packets by Lemma 1. Similarly, there are
(cid:0) APPEND(cid:1)IX D:THE PROOF OF LEMMA 5
(with probability 1) some relays in region M can forward
We provide the proof in two scenarios, lim γ = 0 and
packets to the destination, and hence the average system K
throughput is given by: T =. R = W log β, where lim γ >0, respectively. K→∞
. OBDWF 2 2 2 K→∞K
= denotesthe equalitywith high probability(with probability
1 1 as in [9]). A. lim γ =0 Scenario
−WKhen increasing the β such that lim γ >0, The source K→∞K
should continue to transmit Θ(γ)Ksl→ot∞s bKefore Θ(1) relays canWdheecnotdheetshoiusrcpeacbkreotadwciathstsparopbaacbkielitt,yth1erbeyarLeeΘm(mKγa)1r,elaanyds
K
can decode the packet by Lemma 1, and hence the average henceρ=Θ(1).Themovementofrelayswiththesaidpacket
system throughput is TOBDWF = Θ Kloγg2β . Since the log can be dividedin to two stage: (1) un-balanced,which means
function is in the numerator, increasing the order of β will the order of relays with the said packet in each region is not
reduce the order of Klog2β. As a(cid:0)result, th(cid:1)e optimal γ is the same; (2) balanced, which means the order of relays with
γ
γ∗ = O(K). γ∗ should be infinitely close to Θ(K) from the said packet in each region is the same. Obviously, after
below to make that there are always relays in the connected Θ(1/q) frames, the system is balanced, i.e., there are Θ(K)
region of the source/destination node ( lim γ = 0). The relays with the said packet in each of the M regions. γ
K→∞K
correspondingmaximalaveragesystemthroughputisinfinitely Whentheconnectedlinkmainlyhappensintheun-balanced
close to Wαlog K, when K . stage, after η = O(1/q) frames, the number of relays with
4 2 →∞ the said packet in the region M is N = K(qη)M−1.
M γ
APPENDIX C: THEPROOF OF LEMMA2 The chance that these relays have connected link with the
destination is min 1,Θ(NM) . It can be obtainedfollowing
Obviously, under the conventionalDDF protocol, the aver- γ
age system throughputfor a given data rate R=W log β is the same approachnas Lemmao1. Therefore, after W(η) =
2
given by max 1,Θ( γ ) frames, there is at least one relay with
NM
the said packethaving connectedlink to the destination node.
T =Θ(log β/D )=Θ(log γ/D ) (19) n o
DDF 2 S 2 S Increasing the order of η, the number of relays with the said
where D is the average service time (number of frames) for packet in the region M increases, but W(η) decrease. The
S
a packet, i.e., the average time spent for the source node to actual delay should satisfy: η =Θ(W(η)), which leads to
transmit a packet to the destination node. DS can be divided η =max Θ(1),Θ ( γ2 )1/M (21)
into two part, i.e., DS = ρ+η, where ρ is the average time KqM−1
spent for the source node to transmit a packet to the relay (cid:26) (cid:20) (cid:21)(cid:27)
As a result, the requirement of η =O(1/q) is satisfied when
network,andη istheaveragetimespentfortherelaynetwork
lim K >0.
to forward the said packet to the destination. Specifically, we qγ2
K→∞
When the connected link mainly happens in the balanced
have following Lemma for the average service time D .
S stage,thusη >Θ(1/q), i.e., lim qη = .Thisrequirement
Lemma 5: (Average Service Time of Conventional DDF
K→∞ ∞
Protocol): The average service time for a packet under con- issatisfiedwhen lim K =0.Inthiscase,theaveragedelay
qγ2
ventional DDF protocol is given by: DS = ρ+η = Θ(η), is mainly due toKth→e∞waiting time after the relays’ movement
where ρ=max Θ(1),Θ(Kγ) , and is balanced, given by η =Θ(γ2).
K
η = (cid:8) (cid:9)
B. lim γ >0 Scenario
 mΘa(xK(cid:8)qΘγM2(−11)),M1 If Kl→im∞Kγ =0 and Kl→im∞qKγ2 >0 WK→he∞n KKl→im∞Kγ >0, the source should continue to transmit
 ΘΘ((cid:2)γK2γ)/qM−1 M1(cid:3)(cid:9) IIff Kll→iimm∞Kγγ =>00 aanndd Kll→iimm∞q1Kγ2>=00 ΘLwei(tmhKγmt)haes1lpo,atics.kee.b,teρfion=rtehΘeΘr(e(Kγ1la))y.rnTeelhatweyroecrfkoanrrea,dthtehecerordteheaanrtehΘeΘ(p(K1ac))kareestlatbhyyes
K→∞K K→∞qγ γ
 ΘPr(ohγ(cid:0)o)f: please(cid:1)refier toIAfpKpl→iemn∞dixKγD>. 0 and Kl→im∞q1γ (=200) sKthhl→ieomw∞abnoKγtvhea=tsut0hbessecacevtineoarnarigboey. dFreeolplalloyawciisinngggiΘvteh(neKγbs)iymwLiiletahmrΘmapa(p1r5)o.,aicthcaansbine
TOBEAPPEAREDINIEEETRANS.WIRELESSCOMMUN. 10
APPENDIX E:THE PROOF OF THETHEOREM2 where P0 is the probability that no packets arrives. Λ(z) =
In this proof, we shall first study the stability region of ∞As=0Pr{As}zAs is the p.g.f of the bursty arrival As. The
the source buffer Q (t), and then prove that under the same p.g.f A(z) of the number a arriving within a service time is
s P
stabilityregion,therelaybuffers Q1, ,QK arestabletoo. A(z) = ∞ Pr a za
From [18], [19], the queueing{syste·m··is stab}le, if and only = ∞a=0Pr{X}=l ∞ Pr aX =l za (24)
if the average arrival rate λs is smaller than the service rate = Pl∞=1Pr{X =l}[Λ(az=)0]l { | }
1/b,i.e.,λ b<1,wherebistheaverageservicetimetoserver Pl=1 { }P
s
a queueing packet out of the queueing buffer. For the source Similarly, thePp.g.f A(z) of packet arrival a is given by
i
buffer Qs, the average service time is the average number of ∞
frames for the source node to transmit a packet to the relay A(z)= l=1Pr{eX =l}[Λ(z)]l−1 =Be(Λ(z))/Λ(z) (25)
nfoeltlwoworikn,gdtewnootsecdeansarζio.sF.rom Lemma 1, we can discuss in the wheere B(zX) = ∞l=1Pr{X = l}zl is the p.g.f of the service
time. From (22)-(25), the p.g.f P (z) of the number in the
If lim γ =0, when the source broadcastsa packet,there P n
K→∞K system immediately after the service completion instants is
are Θ(K) relays can decode this packet with probability 1. [23]
γ
Therefore, ζ =Θ(1). (1 λ b)[1 Λ(z)]B(Λ(z))
If lim γ > 0, the source should continue to transmit Pn(z)= −λ Λs(z)[B−(Λ(z)) z] (26)
K→∞K s −
Θ(γ)slotsbeforeΘ(1)relayscandecodethepacket.There-
K Itis shownin [24] thatthep.g.fP(z)forthepacketsinthe
fore, ζ =Θ(γ).
K queueingsystemimmediatelyafteranarbitraryframeisgiven
Note that the K queues Q (t), ,Q (t) are statisti-
1 K by
{ ··· }
cally identical, and they are either all stable or all unsta-
ble. Consider a fictitious queueing system with Q (t) = P(z)=P (z)λs(1−z)Λ(z) = (1−λsb)(1−z)B(Λ(z)) (27)
r n 1−Λ(z) [B(Λ(z))−z]
K Q (t) with the average arrival rate λ and service
k=1 k r Therefore, by Little’s law [23], the average time a packet
rate 1/b . Obviously, Q (t) Q (t), k and hence, all the
P r k ≤ r ∀ spends in the buffer will be
relayqueues Q (t),...,Q (t) aredominatedbythefictitious
1 k
{ }
λque=uemQirn(tλ).T,1he,aavnedragtheeararviveraalgreatneuomfbtheerfiocftfitriaomusesqufoeruethies DQ = λ2sb(2)−λ2sb−λsb+λ(s2)b +b (28)
r { s ζ} 2λs(1 λsb)
fictitious system to deliver a packet to the destination b <ζ. −
r
Therefore,ifλ < 1,thenλ < 1.Inotherwords,thequeues Note that, to make the system stable, the arrival rate λs
s ζ r br should be smaller than the service rate 1/b, i.e., λ b < 1. It
of the relay nodesare also stable if the queue Q (t) is stable. s
s
agrees with the stability condition given in [18], [19].
APPENDIX F:THEPROOF OF THETHEOREM3 If lim γ =0, whenthe sourcebroadcastsa packet, there
K→∞K
Note that DOBDWF =DQ+DR, where DQ is the average areΘ(K)relayscan decodethispacketwith probability1 by
γ
queueing delay in the source buffer before transmitted to the Lemma 1. Therefore, b=Θ(1) and b(2) =Θ(1).
relay network, and DR is the average waiting time in the If lim γ > 0, the first and second order moments of
relay network. Following the same approach as the proof of K→∞K
the service time for a packet to enter the relay network is
Lemma 5, it is easy to verify that D =Θ(η) given in (20).
R b=Θ(γ) and b(2) =Θ(γ2) respectively from Lemma 1.
The remaining task is to find D , which is discussed below. K K2
Q Let ζ = max Θ(1),Θ(γ) , and note that λs = O(1).
traLnsemt iXtabpeactkheetninutmobtheer roeflafyranmeetwso(rnka,maenldydseenrovtiecebt=imEe)[Xto] The average end(cid:8)-to-end pacKket(cid:9)delay is given byλ(s2)
andb(2) =E[X2]respectively.Furthermore,letn denotesthe
i DOBDWF =DQ+DR
number of packets in the source buffer Qs immediately after = max Θ λ(s2)ζ ,Θ(D ) (29)
transmitting the i-th packet to the relay network. ai is the λs(1−λsζ) S
number of packets arriving during the service time of the i- n (cid:16) (cid:17) o
th packet, a is the number of packets arriving in one frame APPENDIX G:THE PROOF OF THELEMMA 4
given that a 1, i.e., Pr a =Pr As =aa 1 , and ai+1 From Appendix C, the average service time to server
≥ { } { | ≥ }
is the number of packets arriving during the service time of a packet to the destination node from the source node is
the(i+1)-thpacketminusoneframe(Thenumberofarreivals b = DS = E[DS] = Θ(DS). In the followings, we shall
duringthelastm 1framesiftheservicetimeofthe(i+1)-th provethattheseconderordermomentofservicetime isgiven
−
packet is m frames). Then ni will form a Markov chain with by b(2) =Θ (DS)2 . Specifically,
the following transitions.
b(2) (cid:0)= E[D(cid:1)2]= Pr D D2
ni+1 = nai−11++aai+i+11 iofthneirw=is0e (22) = Θ (SDS)2P+DfS−(K{)+S}f+S(K) (30)
(cid:26) −
Specifically, the probability geenerating function (p.g.f) A(z) where f−(K) (in t(cid:2)erms of K) is contributed(cid:3)by the pos-
sibilities that the service time D = o(D ), and f+(K)
of a is given by S S
is contributed by the possibilities that the service time
A(z) = a∞=1Pr{As =a|a≥1}za (23) DS > Θ(DS). Clearly, f−(K) is neglectable as f−(K) =
= ∞ Pr{As=a}za = Λ(z)−P0 o (D )2 , i.e, b(2) =Θ (D )2+f+(K) .
Pa=1 1−P0 1−P0 S S
P (cid:0) (cid:1) (cid:2) (cid:3)