In this paper, a strip-shaped multi-hop ad hoc network is analyzed
using a spatial Poisson pointprocess (PPP) and stochastic geometry. The
decode-and-forwardprotocol is considered for transmission overthe
multi-hop network where cooperative communications is employed at each
hop. An analytical expressionfor the probability density function of the
received power at an arbitrary node is derived, given a set of
nodestransmits in the previous hop, which is further used to
characterize the coverage performance of the network.The received power
at a node becomes a doubly stochastic process owing to random path loss
and a Rayleighfading channel. The notions of one-hop success probability
and coverage range are analyzed for variousnetwork parameters. An
algorithm for conserving energy is also proposed by considering PPP
thinning andits performance in terms of the fraction of energy saved is
quantified. It is shown that the proposed algorithmis more energy
efficient as compared with an independent thinning algorithm.
Future wireless networks pose critical challenges in terms
ofreliability and seamless coverage. Whereas future networkswill be an
amalgamation of sophisticated techniques underthe umbrella of fifth
generation (5G), an integral part of5G communications will largely be
composed of Internet ofThings (IoT).
Sensor and ad hoc networks constitute a majorportion of IoT and
communications between various entitiesof these networks plays a vital
role for their successful oper-ation. Cooperative transmission (CT) is
one of the relayingtechniques for wireless sensor and ad hoc networks
usedprimarily to enhance the reliability of the received signals.The
nodes cooperate to form a virtual antenna array andtransmit the same
signal towards the nodes of the next level orhop, thereby providing
spatial diversity gain [1]. A CT multi-hop mechanism provides an
efficient method for reaching adistant destination as the transmit
powers of the nodes can bereduced without compromising the reliability
[2].Opportunistic Large Array (OLA) is a form of physicallayer CT [3],
where multiple nodes in a hop transmit the samemessage, without any
coordination among each other andwithout any addressing scheme. A
promising characteristicof this technique is that it does not require
any prior infor-mation of the number of cooperating nodes or their
locations,which makes it scalable and suitable for transmission
withoutany cluster head. In an OLA transmission, a source nodebroadcasts
its message and all nodes in the vicinity thatcan decode this message,
become part of level 1, which areknown as decode-and-forward (DF) nodes.
In the next timeslot, these DF nodes transmit the same message
concurrentlyin the forward direction using cooperation and the
processcontinues until the data is reached at the destination or
broad-casted to the entire network. However, the modeling of
signaltransmission over this multi-hop network is not very
straightforward and the foremost model was proposed in [4].The proposed
model for a strip-shaped OLA networkin [4] assumes infinite number of
nodes transmitting con-stant power per unit area, which guaranteed
infinite signalpropagation over the network. This assumption
ofcontinuumof nodes confined its application to networks with very high
node density. However, it was shown in [5] that a finitenode density
cannot lead to an infinite broadcast and thatthe path loss exponent
plays a major role in controlling thebroadcast region. Hassan and Ingram
[6] studied the OLAline network with finite node density and modeled
the nodelocations with Bernoulli point process. This model is
thenextended in [7] to a strip-shaped network with deterministichop
boundaries. Specifically, the authors model an ad hocnetwork where the
number of nodes in each hop is knowna priori. Moreover, fixed hop
boundaries were assumed and aMarkov chain model was derived to study the
characteristicsof multi-hop transmissions over the network. We extend
themodel in [8] with random number of nodes per hop and fixedhop
boundaries.In this paper, we study the coverage of a more generalsetup,
where the number of nodes per hop as well as hopboundaries are kept
random. The transmission model resem-bles a typical OLA, where the
transmission of the signal froma source to a distant destination forms
irregular levels or hopswith random number of nodes in each hop. We
derive thecoverage probability of this network using the distributionof
the received power at a node, which is subject to channelimpairments
that include independent Rayleigh fading andpath loss.Our stochastic
model is based on the theory of Poissonpoint process (PPP) [9], where
the nodes in each hop areindependent and distributed according to a
Poisson randomvariable (RV). The analytical tractability of the PPP
modelmakes it a suitable candidate to analyze the random numberof nodes
in each hop; unlike fixed number of nodes, whichcan be generally modeled
using Markov chains [10]. Oncemodeled, the void probability of PPP is
used to computevarious network performance metrics such asm-hop
successprobability, coverage range (CR) and required node densityto
achieve a particular CR under a quality of service (QoS)constraint.The
proposed stochastic model helps in determining theCR of a pure OLA
network given the node density of thenetwork for various hop distances.
It provides useful insightsin designing a network in terms of one-hop
success prob-ability,m-hop success probability and fraction of
energysaved (FES). Its applications include, but not limited to,
smartgrid communication system or fault recognition system
fortransmission lines, and structural health monitoring systemfor the
overhead bridges and tunnels [11]. The model andits findings can also be
used to set up an inter-vehicularcommunication system on the motorways
[12] or a vehicularad hoc network (VANET) to monitor highway activity
bydistributing the nodes along the highway.The geometric complexity of
the system increases withrandom node locations and irregular hop
boundaries, mak-ing path loss a random process. The path loss is
dependentupon the Euclidean distance between the nodes, which
whencombined with fading provides the notion of signal-to-noiseratio
(SNR) for a single link. In CT, multiple single-inputsingle-output
(SISO) links are averaged over a PPP to analyzevirtual multiple-input
single-output (MISO) links. Our maincontributions in this paper are the
followings.•Derivation of the distribution of the Euclidean
distancebetween a pair of nodes distributed randomly in
adjacentoverlapping levels without any hypothetical boundary inbetween
them.•It is shown with the help of some statistical approachessuch as
the moment matching method that the distribu-tion of the distance raised
to a positive power can be wellapproximated by a Weibull
distribution.•We derive the distribution of the received power for
avirtual MISO link, which is the random sum over a PPPof the ratio of an
exponential random variable (RV) anda Weibull RV.•We derive the
coverage range of a 2-dimensional (2D)strip-shaped multiple hop OLA
network.•We devise a thinning of PPP to conserve energy for thefinite
node density OLA networks with random nodeplacements by allowing only a
subset of nodes to trans-mit and quantify its performance. View More
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