Molecular communication in nature can incorporate a large number of nano-things in nanonetworks as well as demonstrate how nano-things communicate. This paper presents molecular communication where transmit nanomachines deliver information molecules to a receive nanomachine over an anomalous diffusion channel. By considering a random molecule concentration in a space-time fractional diffusion channel, an analytical expression is derived for the first passage time (FPT) of the molecules. Then, the bit error rate of the ℓth nearest molecular communication with timing binary modulation is derived in terms of Fox's H-function. In the presence of interfering molecules, the mean and variance of the number of the arrived interfering molecules in a given time interval are presented. Using these statistics, a simple mitigation scheme for timing modulation is provided. The results in this paper provide the network performance on the error probability by averaging over a set of random distances between the communicating links as well as a set of random FPTs caused by the anomalous diffusion of molecules.This result will help in designing and developing molecular communication systems for various design purposes. ). 3 [25], and references therein). Subdiffusion is used to explain the divergence property of waiting time with finite moments of the jump length distribution of the particles. It has been found in various contexts-e.g., the movement of lipids in membranes, cytoplasmic macromolecules in living cells, proteins in the nucleoplasm, and the translocation of polymers [14], [15]-and the mean squared displacement of molecules scales slower than a linear relation in time. For a finite mean waiting time and divergent jump length variance of particles, superdiffusion (also known as Lévy flights) has been explored in [16], which can be observed in turbulent flows or bacterial motions [17], [18]. The mean square displacement of superdiffusing molecules increases more rapidly in time than for normal diffusion.In the context of molecular communication, anomalous diffusion can appear when the concentration of molecules is very high since the collisions between molecules lead to anomalous movement of the molecules in a given medium. For example, calcium signaling based molecular communication [26], [27] cannot avoid anomalous diffusion since calcium ions interact with each other due to the electrostatic forces. Furthermore, experimental studies of molecular communication showed that the channel response is nonlinear and does not obey theoretical results from previous works [28]. These results motivate the use of extraordinary diffusion in molecular communication for many applications [23]- [25]. Since the molecular system can consist of a vast number of molecules, it is difficult to characterize the dynamic behavior of the system analytically. Specifically, the modeling of a dynamic concentration (density) of molecules that undergo absorption, reaction, elastic collision
The random propagation of molecules in a fluid medium is characterized by the spontaneous diffusion law as well as the interaction between the environment and molecules. In this paper, we embody the anomalous diffusion theory for modeling and analysis in molecular communication. We employ H-diffusion to model a non-Fickian behavior of molecules in diffusive channels. H-diffusion enables us to model anomalous diffusion as the subordinate relationship between self-similar parent and directing processes and their corresponding probability density functions with two H-variates in a unified fashion. In addition, we introduce standard H-diffusion to make a bridge of normal diffusion across wellknown anomalous diffusions such as space-time fractional diffusion, Erdélyi-Kober fractional diffusion, grey Brownian motion, fractional Brownian motion, and Brownian motion. We then characterize the statistical properties of uncertainty of the random propagation time of a molecule governed by Hdiffusion laws by introducing a general class of molecular noise-called H-noise. Since H-noise can be an algebraic tailed process, we provide a concept of H-noise power using finite logarithm moments based on zero-order statistics. Finally, we develop a unifying framework for error probability analysis in a timing-based molecular communication system with a concept of signal-to-noise power ratio. ). 2 3 for timing and amplitude modulations. This work further extended to a connectivity problem with a random time constraint in a one-dimensional nanonetwork, where the random locations of molecules at the initial time are modeled by poisson point process [20]. The Cox process has been considered in [21] to capture the dynamic variation of the molecule concentration arising from the mobility of anomalously diffusive molecules, and the spatial ordering of the molecular communication performance has been characterized in terms of the error rate in the presence of interfering molecules. However, there is no comprehensive study on the modeling of anomalous diffusion channels in the context of molecular communication. Various anomalous diffusion processes typically can be modeled numerous ways including continuous random walk (CTRW), generalized diffusion equation, generalized master equation, fractional Brownian motion, and fractional kinetic equation (fractional diffusion equation) [15], [23]-[25]. 3 In particular, the CTRW simply describes diffusion of molecules in the medium with arbitrary distributions of jump lengths and waiting times. 4 In addition, the combination of a stochastic operational time-a directing process-and the self-similar parent process is equivalent to the subordination integral mechanism for the product of two random variables in the context of subordinated processes [28]-[31]. 5 This subordination law generates the solution of the fractional diffusion equation in purely analytical ways using the machinery offered by convolution properties of the Mellin transform. 6In this paper, we embody anomalous diffusion according to th...
The spatial randomness of nanomachines and propagation time of molecules play an essential role for determining the quality of molecular communication between nanomachines. In this study, we introduce a connectivity model in which the connection between a transmit nanomachine (TN), which is randomly distributed in space, and a receive nanomachine (RN) is achieved when a molecule emitted from the TN arrives at the RN within a time constraint. In particular, this time constraint is modeled as a random lifetime to explain the dissipation phenomenon of molecules in a medium or the random arrival time of interfering molecules. Then, we characterize the local connectivity of the RN in terms of the in-degree by averaging over the spatial randomness of nanomachines and the random first passage time of molecules, which is governed by an anomalous diffusion law.
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