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 . 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 .Opportunistic Large Array (OLA) is a form of physicallayer CT , 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 .The proposed model for a strip-shaped OLA networkin  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  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  studied the OLAline network with finite node density and modeled the nodelocations with Bernoulli point process. This model is thenextended in  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  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) , 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 . 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 . The model andits findings can also be used to set up an inter-vehicularcommunication system on the motorways  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