A. MEMBERSHIP OF A NODEIt can be noticed that because of random channel charac-teristics, a node can be part of many levels in different CTsessions. The tendency of the nodes to be in the same hopor level in subsequent iterations of the CT is higher if thenodes are present around the center of a level and the tendencydecreases gradually for the nodes located near the boundary.The path loss is the main factor in governing such behaviorof the nodes. The nodes present near the boundary of a hopcan become part of the adjacent hop due to the lower pathloss as compared to the path loss of the nodes present aroundthe center of the hop. The membership probability of a nodethat it transmits in hopmis different for every other nodeof the hop as shown in Fig. 2. For instance, a node locatednear the boundary of hop,m−1, can become a member ofthe next hop,m, provided it has not transmitted the signalbefore and successfully decoded the signal in next time slot. Code Shoppy The nodes located at points1have almost equal probability ofbecoming a member of hopm−1 orm. A node has a non-zeroprobability of becoming a member of any level unless it hasnot transmitted before. The Kolmogorov-Smirnov (K-S) test is used to check the similarity of the membership proba-bility distribution to known distributions.Monte-Carlo simulations are used to collect the data forthe membership probability and K-S test is applied to find asimilar distribution. For the simulation purpose, we considera strip-shaped network of length 300 and width 8 in which M. Ahsenet al.: Propagation Modeling in Large-Scale Cooperative Multi-Hop Ad Hoc Networksnodes are distributed uniformly. A source node is placed atthe start of the network and it broadcasts the signal. Everyother node that receives the signal and able to successfullydecode it, based on threshold,τ=0.04, will transmit thesignal in the next time slot. The DF nodes that transmit thesignal at next time slot form level 1. The nodes with receivedpower greater thanτand which are not part of the previouslevel (or levels), form next level. This process continues tillthe signal is broadcasted to the entire network. We observe thedecoding pattern of the nodes in the subsequent levels formedat a later stage of the process.
In this section, we derive the distribution of the randomdistance between a pair of nodes in adjacent levels. Thenodes in one level communicate with the nodes in thenext level, where nodes in each level are modeled bytwo independent PPPs. A distance distribution between twonodes, which are part of two distinct PPPs, needs to bederived. In other words, a distance distribution between twonodes of two PPPs is required, which is different com-pared to the distance distribution between nodes of a singlePPP , . As shown in Fig. 1, the network has a fixedvertical length and extends in the horizontal dimension. Thenodes are distributed uniformly in the 2D network in bothdirections but the formation of levels changes the node distri-bution in horizontal direction with respect to a level or hop asshown in 2.It follows from the membership probability that at a level,the nodes in the horizontal direction are concentrated aroundthe center of the level and stretches in the outward direction,whereas vertical distribution of the nodes is not affectedand they follow uniform distribution. The candidate nodelocations within one of the levels is modeled with randomvariables (RV).