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arXiv:1605.08311v2 [cs.IT] 3 Aug 2016

3D Stochastic Geometry Model for Large-Scale

Molecular Communication Systems

Yansha Deng∗, Adam Noel†, Weisi Guo‡, Arumugam Nallanathan∗, and Maged Elkashlan§

∗Department of Informatics, King’s College London, UK

†School of Electrical Engineering and Computer Science, University of Ottawa, Canada

‡School of Engineering, University of Warwick, UK

§School of Electronic Engineering and Computer Science, Queen Mary University of London, UK

Abstract—Information delivery using chemical molecules is an

integral part of biology at multiple distance scales and has

attracted recent interest in bioengineering and communication.

The collective signal strength at the receiver (i.e., the expected

number of observed molecules inside the receiver), resulting from

a large number of transmitters at random distances (e.g., due to

mobility), can have a major impact on the reliability and efﬁciency

of the molecular communication system. Modeling the collective

signal from multiple diffusion sources can be computationally

and analytically challenging. In this paper, we present the ﬁrst

tractable analytical model for the collective signal strength due

to randomly-placed transmitters, whose positions are modelled

as a homogeneous Poisson point process in three-dimensional

(3D) space. By applying stochastic geometry, we derive analytical

expressions for the expected number of observed molecules at

a fully absorbing receiver and a passive receiver. Our results

reveal that the collective signal strength at both types of receivers

increases proportionally with increasing transmitter density. The

proposed framework dramatically simpliﬁes the analysis of large-

scale molecular systems in both communication and biological

applications.

Index Terms—molecular communications, absorbing receiver,

passive receiver, stochastic geometry, interference modeling.

I. INTRODUCTION

Molecular communication via diffusion has attracted sig-

niﬁcant bioengineering and communication engineering re-

search interest in recent years [1]. Messages are delivered

via molecules undergoing random walks [2], a prevalent phe-

nomenon in biology [3]. In fact, molecular communication

exists in nature at both the nano- and macro-scales, offering

transmit energy and signal propagation advantages over wave-

based communications [4,5]. One application example is that

swarms of nano-robots can track speciﬁc targets, such as a

tumour cells, to perform operations such as targeted drug

delivery [6]. In order to do so, energy efﬁcient and tether-less

communications between the nano-robots must be established

in biological conditions [7], and possibly additional nano-bio-

interfaces need to be implemented [8].

Fundamentally, molecular communications involves modu-

lating information onto the property of a single or a group

of molecules (e.g., number, type, emission time). When mod-

ulating the number of molecules, each messenger node will

transmit information-bearing molecules via chemical pulses. In

a realistic environment with a swarm of robots (i.e., messenger

nodes) operating together, they are likely to transmit molecular

messages simultaneously. Due to limitations in transmitter de-

sign and molecule type availability, it is likely that many trans-

mitters will transmit the same type of information molecule.

Thus, it is important to model the collective signal strength due

to all transmitters with the same type of information molecule,

and to account for random transmitter locations due to mobility.

Existing works have largely focused on modeling: 1) the

signal strength of a point-to-point communication channel by

considering the self-interference that arises from adjacent sym-

bols (i.e., inter-symbol-interference (ISI)) at a passive receiver

[9], at a fully absorbing receiver [10], and at a reversible

adsorption receiver [11]; and 2) the collective signal strength of

a multi-access communication channel at the passive receiver

due to co-channel transmitters (i.e., transmitters emitting the

same type of molecule) with the given knowledge of their total

number and location [9].

The ﬁrst work to consider randomly distributed co-channel

transmitters in 3-D space according to a spatial homogeneous

Poisson process (HPPP) is [12], where the probability density

function (PDF) of the received signal at a point location was

derived based on the assumption of white Gaussian transmit

signals. Since the receiver size was negligible, the placement

of transmitters did not need to accommodate the receiver’s

location. More importantly, only the Monte Carlo simulation,

and not particle-based simulation, was performed to verify the

derived PDF.

From the perspective of receiver type, many works have

focused on the passive receiver, which can observe and count

the number of molecules inside the receiver without interfering

with the molecules [9,12]. In nature, receivers commonly

remove information molecules from the environment once they

bind to a receptor. One example is the fully absorbing receiver,

which absorbs all the molecules hitting its surface [10, 11].

However, no work has studied the channel characteristics and

the received signal at the fully absorbing receiver in a large-

scale molecular communication system, let alone its compari-

son with that at the passive receiver.

In this paper, we model the collective signal strength at the

passive receiver and fully absorbing receiver due to a swarm

of mobile point transmitters that simultaneously emit a given

number of information molecules. Unlike [12], which focused

on the statistics of the received signal at any point location,

we focus on examining and deriving exact expressions for

the expected number of molecules observed inside two types

of receiver for signal demodulation. This is achieved using

stochastic geometry, which has been extensively used to model

and provide simple and tractable results for wireless systems

[13]. Our contributions can be summarized as follows:

1) We use stochastic geometry to model the collective signal

at a receiver in a large-scale molecular communication

system, where the receiver is either passive or fully

absorbing. We distinguish between the desired signal due

to the nearest transmitter and the interfering signal due

to the other transmitters.

2) We derive a simple closed-form expression for the ex-

pected number of molecules absorbed at the fully absorb-

ing receiver, and a tractable expression for the expected

number of molecules observed inside the passive receiver

at any time instant.

3) We deﬁne and derive tractable analytical expressions for

the fraction of molecules due to the nearest transmitter

and the fraction of molecules due to the other transmit-

ters.

4) We verify our results using particle-based simulation and

Monte Carlo simulation, which prove that the expected

number of molecules observed at both types of receiver

increases linearly with increasing transmitter density.

II. SYSTEM MODEL

We consider a large-scale molecular communication system

with a single receiver in which a swarm of point transmitters are

spatially distributed outside the receiver in R3/VΩrraccording

to an independent and HPPP Φwith density λ, where VΩrr

is the volume of receiver Ωrr. This spatial distribution, which

was previously used to model wireless sensor networks [14],

cellular networks [13] and heterogenous cellular networks [15],

has also been applied to model bacterial colonies in [16] and

the interference sources in a molecular communication system

[12]. We consider a ﬂuid environment in the absence of ﬂow

currents: the extension for ﬂow currents will be treated in future

work.

At any given time instant, a number of transmitters will

be either silent or active. Thus, we deﬁne the activity prob-

ability of a transmitter that is triggered to transmit data as

ρa(0 < ρa<1). This activity probability is independent of the

receiver’slocation. Thus, the active point transmitters constitute

independent HPPPs Φawith intensities λa=λρa. Each

transmitter transmits molecular signal pulses with amplitude

NFA

tx (NPS

tx ) to the absorbing receiver (the passive receiver). We

assume the existence of a global clock such that all molecule

emissions can only occur at t= 0.

We consider two types of spherical receiver with radius rr:

1) Fully absorbing receiver [17], and 2) Passive receiver [4,

18]. To equivalently compare them, we assume both types of

receiver are capable of counting the number of information

molecules within the receiver volume at any time instant for

information decoding.

It is well known that the distance between the transmitter and

the receiver in molecular communication is the main contributor

Receiver

k th Point!

Transmitter

T

t

P

T

t

P

T

t

P

T

t

P

T

t

P

T

t

P

T

t

P

rr

Fig. 1. Illustration of a receptor receiving molecular pulse signals from point

transmitters at different distances.

to the degradation of the signal strength (i.e., the number of

molecules observed at the receiver). Instinctively, we assume

that the receiver is associated with the nearest transmitter to

obtain the strongest signal. Thus, the messenger molecules

transmitted by other active point transmitters act as interference,

which impairs the correct reception at the receiver. To measure

this impairment, we formulate the desired signal, the interfering

signal for the absorbing receiver and the passive receiver in the

following subsections.

A. Absorbing Receiver

In our proposed large-scale molecular communication sys-

tem, let us consider the center of an absorbing receiver located

at the origin. Using the Slivnyak-Mecke’s theorem [13], the

fraction FFA of molecules absorbed at the receiver until time

Tdue to an arbitrary point transmitter xat the location xwith

molecule emission occurring at t= 0 can be represented as

[17]

FFA ( Ωrr, T |kxk) = rr

kxkerfc nkxk − rr

√4DT o,(1)

where kxkis the distance between the point transmitter and the

center of the receiver where the transmitters follow a HPPP,

and Dis the constant diffusion coefﬁcient, which is usually

obtained via experiment as in [19, Ch. 5]. The fraction FFA

uof

molecules absorbed inside the receiver until time Tdue to a

single pulse emission by the nearest active transmitter can be

represented as

FFA

u(Ωrr, T | kx∗k) = rr

kx∗kerfc nkx∗k − rr

√4DT o,(2)

where kx∗kdenotes the distance between the receiver and the

nearest transmitter,

x∗= arg min

x∈Φakxk,(3)

x∗denotes the nearest point transmitter for the receiver, and

Φadenotes the set of active transmitters’ positions.

The fraction of molecules absorbed at the receiver until

time Tdue to single pulse emissions at each active interfering

transmitter FFA

Iand that due to a single pulse emission at each

active transmitter FFA

all is represented as

FFA

I(Ωrr, T | kxk) = X

Φa/x∗

rr

kxkerfc nkxk − rr

√4DT o,(4)

and

FFA

all (Ωrr, T | kxk) = X

Φa

rr

kxkerfc nkxk − rr

√4DT o,(5)

respectively.

The expected number of molecules absorbed at the receiver

by time Tdue to all active transmitters is equivalently the

expected number of molecules absorbed at the receiver until

time T, which can be calculated as

ENFA

all (Ωrr, T )=NFA

tx FFA

all (Ωrr, T | kxk)

=NFA

tx FFA

u(Ωrr, T | kx∗k)

|{z }

EFA

u

+NFA

tx FFA

I(Ωrr, T | kxk)

|{z }

EFA

I

,(6)

where FFA

all (Ωrr, T | kxk)is given in (5), EFA

uis the fraction of

absorbed molecules at the absorbing receiver until time Tdue

to the nearest transmitter, and EFA

Iis the fraction of absorbed

molecules at the absorbing receiver until time Tdue to the

other (interfering) transmitters.

B. Passive Receiver

In a point-to-point molecular communication system with a

single point transmitter located distance kxkaway from the

center of a passive receiver with radius rr, the local point

concentration at the center of the passive receiver at time T

due to a single pulse emission by the transmitter is given as

[20, Eq. (4.28)]

C( Ωrr, T |kxk) = 1

(4πD T )3/2 exp −kxk2

4DT .(7)

The fraction of information molecules observed inside the

passive receiver with volume VΩrrat time Tis denoted as

FPS (Ωrr, T | kxk) = Z

VΩrr

C(Ωrr, T | kxk)dVΩrr,(8)

where VΩrris the volume of the spherical passive receiver.

According to (8) and Theorem 2 in [21], the fraction FPS of

information molecules observed inside the passive receiver at

time Tdue to a single pulse emission by a transmitter at time

tis derived as

FPS ( Ωrr, T |kxk) = 1

2erfrr− kxk

2√DT + erfrr+kxk

2√DT

+√DT

√πkxk"exp −(rr+kxk)2

4DT −exp −(kxk − rr)2

4DT #,

(9)

which does not assume that the molecule concentration inside

the passive receiver is uniform. This is unlike the common as-

sumption that the concentration of molecule inside the passive

receiver is uniform. Although that assumption is commonly

applied, it relies on the receiver being sufﬁciently far from the

transmitter (see [21]), which we cannot guarantee here since

the transmitters are placed randomly.

In the large-scale molecular communication system with a

passive receiver centered at the origin, the expected number of

molecules observed inside the receiver at time Tdue to a single

pulse emission at all active transmitters at t= 0 is given as

ENPS

all (Ωrr, T )=E(X

Φa

NPS

tx FPS ( Ωrr, T |kxk))

=NPS

tx FPS

u(Ωrr, T | kx∗k)

|{z }

EPS

u

+NPS

tx FPS

I(Ωrr, T | kxk)

|{z }

EPS

I

,

(10)

where FPS ( Ωrr, T | kxk)is given in (9), EPS

uis the fraction

of molecules observed inside the receiver at time Tdue to the

nearest transmitter, and EPS

Iis the fraction of molecules ob-

served inside the receiver at time Tdue to the other (interfering)

transmitters.

III. RECEIVER OBSERVATIONS

In this section, we ﬁrst derive the distance distribution

between the receiver and the nearest point transmitter. By doing

so, we derive exact expressions for the expected number of

molecules observed inside the receiver due to the nearest point

transmitter and that due to the interfering transmitters.

A. Distance Distribution

Unlike the stochastic geometry modelling of wireless net-

works, where the transmitters are randomly located in the un-

bounded space, the point transmitters in a large-scale molecular

communication system can only be distributed outside the sur-

face of the spherical receiver. Taking into account the minimum

distance rrbetween point transmitters and the receiver center,

we derive the probability density function (PDF) of the shortest

distance between a point transmitter and the receiver in the

following proposition.

Proposition 1. The PDF of the shortest distance between any

point transmitter and the receiver in 3D space is given by

fkx∗k(x) = 4λaπx2e−λa(4

3πx3−4

3πrr

3),(11)

where λa=λρa.

Proof. See Appendix A.

Based on the proof of Proposition 1, we also derive the PDF

of the shortest distance between any point transmitter and the

receiver in 2D space in the following lemma.

Corollary 1. The PDF of the shortest distance between any

point transmitter and the receiver in 2D space is given by

fkx∗k(x) = 2λaπre−λa(πr 2−πrr

2),(12)

where λa=λρa.

B. Absorbing Receiver Observations

In this subsection, we derive a closed-form expression for the

expected number of molecules observed inside the absorbing

receiver in 3D space.

Using Campbell’s theorem, we derive the expected number

of absorbed molecules due to the nearest transmitter and that

due to the interfering transmitters until time tas

EFA

u=NFA

tx Z∞

rr

Πt

0(x) 4λaπx2e−λa(4

3πx3−4

3πrr

3)dx, (13)

and

EFA

I=NFA

tx (4πλa)2Z∞

rrZ∞

x

Πt

0(r)r2drx2e−λa(4

3πx3−4

3πrr

3)dx,

(14)

respectively.

Theorem 1. The expected net number of molecules absorbed

at the absorbing receiver in 3D space during any sampling

time interval [t, t +Tss ]is derived as

ENFA

all (Ωrr, t, t +Tss )

= 4NFA

tx √πλarrhD√πTss + 2√DrrpTss +t−√ti.

(15)

The expected number of molecules observed inside the fully

absorbing receiver in 3D space until time tis derived as

ENFA

all (Ωrr, t)= 4NFA

tx √πλarrhD√πt + 2rr√Dti.

(16)

Proof. See Appendix B.

From Theorem 1, we ﬁnd that the expected number of

molecules absorbed at the absorbing receiver at time tis

linearly proportional to the density of active transmitters,

and increases with increasing diffusion coefﬁcient or receiver

radius. As expected, we ﬁnd that the expected number of

molecules absorbed at the receiver until tis always increasing

with time t.

C. Passive Receiver Observations

In the following theorem, we derive the expected number of

molecules observed inside the passive receiver in 3D space.

Using Campbell’s theorem, we derive the expected number

of observed molecules due to the nearest transmitter and that

due to the interfering transmitters at time tas

EPS

u= 4λaπN PS

tx e4

3πrr

3λaZ∞

rr

Φ (x)x2exp n−4

3πx3λaodx,

(17)

and

EPS

I=(4πλa)2e4

3πrr

3λaNPS

tx Z∞

rrZ∞

x

Φ (r)r2dr

x2e−4

3πx3λadx, (18)

respectively. In (17) and (18), Φ (r) = FPS ( Ωrr, t|r).

Theorem 2. The expected net number of molecules observed

inside the passive receiver during any sampling time interval

[t, t +Tss ]in 3D space is derived as

ENPS

all ( Ωrr, t, t +Tss |kxk)= 4NPS

tx πλa

Z∞

rr

FPS (Ωrr, t +Tss |r)r2dr −Z∞

rr

FPS (Ωrr, t|r)r2dr,

(19)

where FPS (Ωrr, t|r)is given in (9).

The expected number of molecules observed inside the pas-

sive receiver at time tin 3D space is derived as

ENPS

all (Ωrr,0, t|kxk)=

4NPS

tx πλaZ∞

rr

FPS (Ωrr, t|r)r2dr. (20)

Proof. Analogous to Appendix B without solving the integrals.

In Theorem 2, we observe that the expected number of

molecules observed inside the passive receiver also increases

proportionately with the density of active transmitters.

IV. NUMERICAL AND SIMULATION RESULTS

In this section, we examine the expected number of

molecules observed at the absorbing receiver and the passive

receiver due to simultaneous single pulse emissions at all active

point transmitters. In all ﬁgures of this section, we set the

parameters as follows: rr= 5 µm and NFA

tx =NPS

tx = 104.

In all ﬁgures, the analytical curves of the expected number

of molecules absorbed at the absorbing receiver due to all

the transmitters, the nearest transmitter, and the interfering

transmitters are plotted using Eqs. (15), (13), and (14), and

are abbreviated as “Absorbing All”, “Absorbing Nearest”, and

“Absorbing Aggregate”, respectively. The analytical curves of

the expected number of molecules observed inside the passive

receiver due to all the transmitters, the nearest transmitter, the

interfering transmitters are plotted using (19), (17), and (18),

and are abbreviated as “Passive All”, “Passive Nearest”, and

“Passive Aggregate”, respectively.

A. Particle-Based and Pseudo Simulation Validation

In Fig. 2, we set D= 80 ×10−12 m2

s, and assume that the

transmitters are placed up to R= 50 µm from the center of the

receiver at a density of λa= 10−4transmitters per µm3(i.e.,

52 average number of transmitters, including the subtraction of

the receiver volume). The receiver takes samples every Tss =

0.01 s and calculates the net change in the number of observed

molecules between samples. The default simulation time step

is also 0.01 s. Unless otherwise noted, all simulation results

were averaged over 104transmitter location permutations, with

each permutation simulated at least 10 times.

In Fig. 2, we verify the analytical expressions for the

expected net number of molecules observed during [t, t +Tss ]

at the absorbing receiver in Eq. (13) and Eq. (14), and that

inside the passive receiver in Eq. (17) and Eq. (18) by com-

paring with the particle-based simulations and the Monte Carlo

10−1 100

0

0.2

0.4

0.6

0.8

1

0.01 0.02 0.03

0

0.2

0.4

0.6

0.8

1

Absorbing Nearest

Absorbing Aggregate

Absorbing Nearest

Absorbing Aggregate

Passive Aggregate

Passive Nearest

Analytical

Particle−Based Sim.

Monte Carlo Sim.

Time (s) Time (s)

Net Number of Observed Molecules

Fig. 2. Net number of observed molecules inside the receiver as a function of

time. All curves are scaled by the maximum value of the analytical curves in

the right subplot.

TABLE II

THE S IMUL ATI ON PARA METERS A ND SCA LING VALUE S APP LIED IN FIG. 2.

Transmitter Receiver Realizations Time Scaling

Step [s] Value

Nearest Passive 10410−2149.57

Nearest Active 10410−2354.52

Aggregate Passive 10410−29.252

Aggregate Active 10310−359.42

simulations. The particle-based simulations were performed by

tracking the progress of individual particles to obtain the net

number of observed molecules during [t, t +Tss]using the

AcCoRD simulator (Actor-based Communication via Reaction-

Diffusion) [22]. The pseudo simulations rely on the Monte

Carlo simulation method, which were performed by averaging

the expected number of observed molecules due to all active

transmitters with randomly-generated location, as calculated

from Eq. (1) and Eq. (9), over 104realizations.

In the right subplot of Fig. 2, we compare passive and absorb-

ing receivers and observe the expected net number of observed

molecules during [t, t +Tss ]due to the nearest transmitter and

due to the aggregation of the interfering transmitters. In the

left subplot of Fig. 2, we lower the simulation time step to

10−4s for the ﬁrst few samples of the two absorbing receiver

cases, in order to demonstrate the corresponding improvement

in accuracy. All curves in both subplots are scaled by the max-

imum value of the corresponding analytical curve in the right

subplot; the scaling values and other simulation parameters are

summarized in Table II.

1) Particle-Based Simulation Validation: Overall, there is

good agreement between the analytical curves and the particle-

based simulations in the right subplot of Fig. 2. The analytical

results for the net number of molecules observed inside the

passive receiver during [t, t+Tss]due to the nearest transmitter

is highly accurate, and even captures the net loss of molecules

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

0.5

1

1.5

2x 105

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

2000

4000

6000

8000

Analytical

Monte Carlo Sim.

Passive All

Absorbing All

Passive Nearest

Passive Aggregate

Absorbing Aggregate

Absorbing Nearest

Number of Observed Molecules

Time (s)

Fig. 3. Expected number of molecules observed inside the receiver as a

function of time.

observed after t= 0.1s. There is a slight deviation in the

particle-based simulation for the “passive aggregate” curve,

particularly as time approaches t= 1 s, which is primarily due

to the very low number of molecules observed at this time (note

that the scaling factor in this case is only 9.252; see Table II).

There is less agreement between the particle-based simula-

tions and the analytical expressions for the absorbing receiver,

and this is primarily due to the large simulation time step (even

though we used a smaller time step for the aggregate transmitter

case in the right subplot; see Table II). To demonstrate the

impact of the time step, the left subplot shows much better

agreement for the absorbing receiver model by lowering the

time step to 10−4s. This improvement is especially true in the

case of the nearest transmitter, as there is signiﬁcant devia-

tion between the particle-based simulation and the analytical

expression for very early times in the right subplot.

2) Monte Carlo Simulation Validation: There is a good

match between the analytical curves and the Monte Carlo

simulations for the net number of molecules observed at both

types of receiver during [t, t+Tss ]due to the nearest transmitter,

which can be attributed to the large number of molecules and

the shortest distance value compared with R= 50 µm (as

shown in Table II). There is slight deviation in the Monte Carlo

simulations for the expected number of molecules observed

inside both types of receiver due to the interfering transmitters,

and this is primarily due to the restricted placement of trans-

mitters to the maximum distance R= 50 µm. In Figs. 3, better

agreement between the analytical curves and pseudo simulation

is achieved by increasing the maximum placement distance R.

Due to the extensive computational demands to simulate

such large molecular communication environments, we assume

that the particle-based simulations have sufﬁciently veriﬁed the

analytical models. The remaining simulation results in Fig. 3

is only generated via Monte Carlo simulation.

B. Performance Evaluation

From Fig. 2 and the scaling values in Table II, we see

that the expected net number of molecules absorbed at the

absorbing receiver is much larger than that inside the passive

receiver, since every molecule arriving at the absorbing receiver

is permanently absorbed. We also notice that the expected net

number of absorbed molecules due to the nearest transmitter is

much larger than that due to the interfering transmitters, which

may be due to a relatively low transmitter density.

Interestingly, the concurrent single pulse transmission by the

transmitters at time t= 0 results in a longer and stronger

channel response at the absorbing receiver than that at the

passive receiver. If the demodulation is based on the number of

observed molecules during each bit interval, the longer channel

response at the absorbing receiver may contribute to higher ISI

than at a passive receiver for the same bit interval, whereas its

stronger channel response may beneﬁt signal detection.

In Fig. 3, we set the parameters: D= 120 ×10−12 m2

s,

R= 100 µm, and Tss = 0.1s. Fig. 3 plots the expected

number of molecules observed at the absorbing receiver and

the passive receiver at time t. We set the density of active

transmitters as λa= 10−3/µm3. As shown in the lower subplot

of Fig. 3, the channel responses of the receivers due to the

nearest transmitter in this large-scale molecular communication

system are consistent with those observed at the absorbing

receiver in [11, Fig. 4] and the passive receiver in [4, Fig. 2]

and [18, Fig. 1] for a point-to-point molecular communication

system.

In Fig. 3, we notice that the expected number of observed

molecules at time tdue to all the transmitters is dominated by

the interfering transmitters, rather than the nearest transmitter,

which is due to the higher density of transmitters. Furthermore,

as we might expect, the expected number of molecules observed

inside the passive receiver at time tstabilizes after t= 0.8s,

whereas that at the absorbing receiver increases linearly with

increasing time. This reveals the potential differences in optimal

demodulation and interference cancellation design for these two

types of receiver.

V. CONCLUSIONS AND FUTURE WORK

In this paper, we provided a general model for the transmitter

modelling in a large-scale molecular communication system

using stochastic geometry. The collective signal strength at a

fully absorbing receiver and a passive receiver are modelled

and examined. We derived tractable expressions for the ex-

pected number of observed molecules at the fully absorbing

receiver and the passive receiver, which were shown to increase

with transmitter density. Our analytical results were validated

through particle-based simulation and Monte Carlo simulation.

The analytical model presented in this paper can also be applied

for the performance evaluation of other types of receiver

(e.g., partially absorbing, reversible adsorption receiver, ligand-

binding receiver) in large-scale molecular communication sys-

tems by substituting its corresponding channel response.

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