Performance Limits of Spatially Distributed Molecular Communications With Receiver Saturation

Molecular via diffusion are affected by unreliability and inter symbol interference in case of single transmitter. A promising solution to these issues appears to be the adoption of a swarm of randomly distributed transmitting nano-devices and a single spherical receiver. However, such a perspective assumes the receiver as fully absorbing (i.e., able to absorb an unlimited number of molecules per second). In this letter, we show that, if this assumption is relaxed, increasing the number of point transmitters leads to a saturation effect which makes communication impossible when digital transmission is considered. By means of a first and second order spatio-temporal stochastic model, we analytically derive the maximum allowed point transmitters density before saturation arises, as a function of parameters such as the diffusion coefficient, the maximum number of absorbed molecules, and the number of previously transmitted symbols. The analysis is validated via Monte Carlo simulation.

system is evaluated both in terms of received number of molecules and of bit error probability (BEP) when on-off keying (OOK) digital modulations is considered. These results have been extended in case of asynchronous transmissions, i.e., without assuming that all point transmitters emit molecules at the same time instants, in [10], [11], [12].
Two of the most relevant results regarding this kind of spatially distributed MCvD scenario can be summarized as follows: i) from a single transmitter MCvD scenario, the signal-to-noise ratio (SNR) saturates (i.e., when digital transmission is considered, the BEP shows a floor) as the number of emitted molecules increases; ii) both SNR and the signalto-noise-plus-interference ratio (SNIR) linearly increase (thus, the BEP decreases) as the spatial density of point transmitter increases. In other words, indefinitely increasing the density of point transmitters (rather than increasing the number of transmitted molecules for each point transmitter) appears as the key-solution to counteract both noise and inter-symbol interference (ISI) effects. However, this result clearly depends on the ideal assumption that the receiver can indefinitely absorb all molecules hitting its surface (i.e., fully absorbing receiver). But, in [13], it is shown that the absorbing rate is limited for chemical reasons and, in [14], the number of molecules absorbed within a certain interval (i.e., a bit interval) is thus considered bounded by a certain threshold. The problem of the receiver congestion when the fully absorbing assumption is removed has also been addressed in [15], where an algorithm is proposed to control molecules emission and avoid molecules wasting. However, [13], [14], [15] deal with a point-to-point molecular communication scenario.
Differently, in this work, we address a scenario with a swarm of point transmitters to study the effect of the receiver saturation in the terms of BEP. In addition, differently from [15], which is focused on an early congestion detection and a proper run-time molecules emission control in a pointto-point communication scenario, we analytically derive the maximum value of the point transmitters density before the congestion arises for a fixed emission rate.

II. SYSTEM MODEL
We consider a large-scale molecular communications scenario similar to that presented in [9] with a swarm of point transmitters and a spherical receiver (see Fig. 1). More specifically, the transmitters positions {x k } are a random set given by the outcomes of a spatial HPPP Π with intensity λ a restricted to Ω Ra \ Ω R , where Ω Ra and Ω R are spheres of radius R a and R, respectively. This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ Fig. 1.
A swarm of randomly placed point transmitters randomly emit molecules according to independent time-domain PPs with the same intensity function (t).
For the transmission process, we consider the asynchronous stochastic model proposed in [10], [11], [12]. The instants in which the molecules are emitted by the k-th point transmitter are given by the random set {τ (k ) l } constituted by the output of the (generally, non stationary) time domain PP Φ (k ) (i.e., τ (k ) l is the emission instant of the l-th molecule emitted by the k-th transmitter placed at x k ). The time domain PPs {Φ (k ) } are independent and have the same intensity function ρ(t). Such an asynchronous model describes a scenario where the point transmitters represents the randomly located points where spontaneous emissions happen in the fluid due to chemical reactions. Since the emissions are related to the collisions between the reagent molecules, it must be highlighted that the Brownian motion of the reagent molecules motivates the independence among the time domain PPs and ρ(t) takes the meaning of the propensity function as described by [16].
Note that, the aforementioned asynchronous transmissions (also known as timing transmissions) can be reduced to the synchronous case (also known as Poisson concentration transmission) by setting ρ(t) = N tx δ(t), where N tx is the average number of molecules instantaneously emitted by a Poisson concentration transmitter and δ(t) is the Dirac delta. In such a case, the model reduces to the synchronous scenario proposed in [9], where the point transmitters represent nano-machines which collaborate to transmit the messages.
The key difference with respect to the spatiotemporal model in [10] and [12] is at the receiver side. Here, the goal is to investigate the impact of the saturation effect on a spatially distributed molecular communication system. Thus, instead of assuming a fully absorbing receiver, we take into account the effect of saturation by means of the maximum absorbing rate f sat , indicating the number of molecules per second that the receiver can absorb. For the sake of simplicity, we assume that the maximum number of absorbed molecules for each symbol time results in N abs = f sat T b and that such a level is quickly resetted at each timestep. This is in accordance with the model proposed in [14] (establishing a more complex and accurate model for the reception mechanism, e.g., based on ligand-receptor interactions, is beyond the scope of the present letter). Thus, the random variable (RV) n abs denoting the total number of absorbed molecules is given by where n rx is the number of absorbed molecules (within T b ) when a fully absorbing spherical receiver of radius R is considered with hitting rate [3] being d the distance from the transmitter, D the diffusion coefficient, and t the time after the transmission of the unitary molecule. As known, f hit (d , t) represents the (infinitesimal) probability that a molecule emitted at the time origin at distance d is received (exactly) at time t. The probability that a molecule emitted at t = 0 is observed within the interval The time invariance of the Fick's law of diffusion allows to write that, if the emission happens at τ Due to the linearity of the Fick's law, the number of received molecules from all transmitters within the interval We now consider the digital transmission of j + 1 symbols, where the intensity function of all transmission processes is modeled by where a i is the i-th bit (which is 1 with probability p 1 and 0 with probability for the synchronous Poisson transmission case and g(t) = (N tx /T a ) rect(t/T a ) for the asynchronous transmission case, with T a the activity interval and T a /T b ≤ 1 the duty cycle. It has been shown in [10], [11], [12] that ρ(t) plays the role of the input of a linear time invariant (LTI) system, thus the superposition of effects holds. Focusing on the generic j-th instant (due to the causality, the molecules emitted after the j-th bit time provide no effect), the number of hitting molecules can be written as where n u is the useful component evaluated by (6) when Note that the randomness of the useful component n u represents the noise, while the ISI component n ISI is given by the molecules hitting the receiver in a bit interval different than the emission one.

A. BEP as a Function of the Threshold
Some analytical expressions derived for the fully absorbing receiver are here properly adapted to calculate the BEP taking the saturation effect into account. From (8), it follows that, if a j = 0 or a j = 1, the number n rx of hitting molecules results in n rx | 0 = n ISI and n rx | 1 = n u +n ISI , respectively. According to (1), the number of absorbed molecules in the two cases are thus given by n abs | 0 = min N abs , n ISI (9a) n abs | 1 = min N abs , n u + n ISI . (9b) The mean and the variance of n u and n ISI have been evaluated in [12]. The average BEP can instead be computed as: where, N th denoted the threshold, P e | 0 Prob{n abs | 0 > N th } and P e | 1 Prob{n abs | 1 < N th }, where n abs | 0 and n abs | 1 represents the number of absorbed molecules conditioned to a j = 0 and a j = 1, respectively. By making use of the total probability theorem, we can derive P e | 0 = Prob n abs | 0 > N th , n ISI ≤ N abs + Prob n abs | 0 > N th , n ISI > N abs (11a) P e | 1 = Prob n abs | 1 < N th , n u + n ISI ≤ N abs + Prob n abs | 1 < N th , n u + n ISI > N abs . (11b) From (9) and (11) one can readily note that the case N th > N abs makes the error event conditioned to a j = 0 an impossible event and the error event conditioned to a j = 1 a certain event. Moroever, the case N th < N abs can be easily computed by making use of (9). Thus, (11) becomes Denote as μ u and σ 2 u the mean and variance of the useful component n u . Clearly, the ISI component depends on the previously emitted symbols. However, if they are assumed independent and equiprobable, ISI mean μ ISI and variance σ 2 ISI can be calculated obtaining expressions independent on the particular symbols sequences [12]. In [17] it is proved that n ISI and n u + n ISI can be approximated as generalized Gaussian random variables with mean μ ISI and μ u + μ ISI and variance σ 2 ISI and σ 2 u + σ 2 ISI , respectively. We recall that the generalized Gaussian probability density function (PDF) is defined as where Γ(·) is the gamma function, μ is the mean, β is the shaping parameter (note that for β = 2 we obtain the normal distribution), and α is a scale parameter such that the variance results in α 2 Γ(3/β) Γ(1/β) . Thus, (12) becomes where γ(s, x ) . By substituting (13) in (10) we easily get the error probability as a function of the threshold.

B. Optimal Threshold
It is clear from (13) that the optimal threshold has to be lower than N abs . By deriving the expression of P e with respect N th we obtain, for (N th ≤ N abs ): where By setting (14) to zero we obtain: By denoting as N * th the solution of (16), the optimal threshold can be written as Thus, the error probability can be obtained through (10), by using (13), and (17), as well as the expressions of μ u , μ ISI , σ 2 u , and σ 2 ISI derived in [12] for both the synchronous and the asynchronous case.

B. Maximum Transmitter Points Density Evaluation
In the general case, by setting the threshold to (20), it is where SIR μ 2 u /σ 2 ISI , SINR μ 2 u /(σ 2 u + σ 2 ISI ), p fl (N abs ) is obtained from (10) by substituting N th = N abs in (13), and with ρ Γ(3/β)/4Γ(1/β). Now, we derive the maximum density of point transmitters before the saturation effect arises. To this aim, we substitute in (21) the expressions of μ u , μ ISI , SIR, and SINR as functions of λ a , derived in [12] for both the synchronous and the asynchronous case.
1) Synchronous Case: For Poisson concentration transmitters, the mean of useful and ISI components are where H all (t) is the so called collective impulse response resulting in H all (t) = 4λ a R √ πDt(2R + √ πDt). The SIR and SINR result as in [12, eq. (84)].
2) Asynchronous Case: For timing asynchronous transmitters with uniformly distributed emissions in an activity interval T a , the mean of useful and ISI components result in where and . The SIR and SINR are obtained as in [12, eq. (85)]. By substituting (22) and (23), after some algebra (21) becomes where λ M represents the maximum transmitters density, that, for Poisson concentration (synchronous) transmitters, is and, for timing (asynchronous) transmitters with uniformly distributed emissions in an activity interval T a , is Note that, since SIR and SINR are increasing functions of the point transmitters density λ a , λ M implies a floor value for the BEP. We also remark that, in both cases (synchronous and asynchronous), the maximum transmitters density λ M linearly increases with the saturation level N abs and decreases with the number of emitted molecules N tx , the diffusion coefficient D, and the square of the receiver radius R. However, since the maximum number of absorbed molecules depends on the number of receptors over the cell surface, it is reasonable to assume N abs proportional to R 2 (the number of receptors increases as the area increases). Hence, λ M should not substantially depend on the dimension of the receiver. It is clear that the dominant term at the denominator of (25) and (26) depends on the number j of the previously transmitted symbols which generate ISI. In Fig. 2 the maximum value of the transmitters density λ M before the saturation effect arises is plotted as a function of the total ISI duration (j + 1)T b , for both synchronous (25) and asynchronous (26) case, for different values of the saturation level N abs . Due to the lower efficiency of the asynchronous transmission (less molecules hit the receiver with respect to the synchronous case), the receiver saturation effect arises for higher values of point transmitters density (compared to the case of synchronous transmission).
Such a result establishes that, in the considered scenario, differently from the case in [12] related to the fully absorbing receiver, ISI cannot be counteracted by indefinitely increasing the density of point transmitters.

C. Analytical and Simulation Results
In this section, we show the impact of the saturation effect on the BEP and we validate the analytical expression (13) through Monte Carlo simulations. To this aim, we assume the validity of (4) (that has already been verified in [3] by particlebased simulations) and, we compute n rx according to (6) by using the superposition of effects (note that, according to (8), each contribution contains both useful and ISI terms). Then N abs is considered according to (9) (saturation effect), compared to the threshold N th and used (instead of n rx ) to count the error events. The resulting values for the bit error rate are finally averaged over 10 4 instantiations of the considered PPs. A receiver with radius R = 5μm, a bounded space of radius R a = 50μm, a diffusion coefficient D = 120μ m 2 / s, a bit time duration T b = 0.2s, an average number of emitted molecules per bit N tx = 80, and 4 ISI symbols are considered, according to [18], [19]. For what concerns the generalized Gaussian parameter, β ∈ [2,5] is considered [17].
In Fig. 3 the BEP is depicted as a function of the threshold N th for a point transmitters density λ a = 10 15 m −3 , according to (10) and (13). As expected, the BEP drops to 1/2 as soon as the threshold N th overcomes N abs . Both Gaussian (β = 2) and generalized Gaussian (β = 5) approximations are considered for the analysis and compared to Monte Carlo simulations results. The figure shows a good agreement for both the synchronous and the asynchronous case. More precisely, β = 2 well fits the synchronous case, while β = 5 is the best fitting for the asynchronous case.
In Fig. 4 the BEP is shown as a function of the point transmitters density λ a for the case of optimal threshold with different values of the saturation level N abs . Here, an intermediate value β = 3.5 is chosen for both the synchronous and the asynchronous case. It can be observed that, as expected, asynchronous transmission implies higher values of P e before the saturation effect arises.

V. CONCLUSION
A spatially distributed large-scale MCvD system is considered with a swarm of point transmitters and a not fully absorbing spherical receiver. An analytical model for the BEP in case of OOK modulation is obtained based on point processes theory for both synchronous and asynchronous transmitters. We show that, differently from the fully absorbing receiver case, if the density of the point transmitters increases over a given threshold, the BEP drops to a value close to 1/2 as a result of the saturation effect at the receiver side. The optimal values for point transmitters density are analytically determined and validated through Monte Carlo simulations.