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Molecular communication is a novel nanoscale communication method. It can use ions, biochemical molecules or other information carriers to transmit information. However, due to molecules are easily accumulated in the channel to produce strong internal symbol interference, the information transmission in the channel is vulnerable to a low reliability. Therefore, reliability has become an important research issue in the field of molecular communication. At present, the existing reliability model of molecular communication does not take into account the drift velocity of the medium. Nevertheless, in some scenarios, it is often necessary to consider the effect of the drift velocity of the medium on the reliability of molecular communication. In this paper, we introduce the drift velocity of the medium and propose a reliability model of molecular communication based on drift diffusion (MCD2) in different topologies. Furthermore, in the case of transmission failure, a retransmission mechanism is used to ensure reliable transmission of information. Finally, we also compare the reliability performance of molecular communication between reliability model of MCD2 and reliability model of molecular communication based on free diffusion. The simulation results indicate that the proposed reliability model is superior to the existing reliability model of molecular communication based on free diffusion in analyzing the reliability of MCD2.

In recent years, with the rapid development of the Internet and 5G communication technologies, people increasingly depend on network communication technologies. At the level of micro-network communication outside the traditional communication network, nanocommunication technologies with different mechanisms have received extensive attention from researchers.

There are two main methods for implementing nanonetworks, namely bio-inspired molecular communication and electromagnetic nanocommunication. Bioinspired molecular communication generally uses chemical signals, cells, organelles or DNA as carriers. As for electromagnetic nanocommunication generally uses carbon nanotubes as a carrier. With the development of nanonetwork research, multiple nano-sensors can be deployed in the human body to monitor the substances in the human body, such as glucose, sodium and cholesterol. Then these sensors can transmit detection information to mobile phones or computers to dynamically display the health of the human body. Nano-sensors can also detect the specific parts of human infections to prevent human diseases. To achieve the above functions, it must rely on tiny devices with bionic functions, such as nanomachine. Nanomachine can be made up of nanoscale components that perform tasks such as calculation, storage, sensing, and driving. At present, there are three different ways to make nanomachine [

However, due to the limited task types and space scope of a single nanomachine, in order to complete more complex tasks in a larger scope, it is necessary to connect multiple nanomachines to form a nanonetwork for task collaboration and information sharing. To the best of our knowledge, there are four ways to achieve communication between nanomachines, nanomechanical communication, acoustic communication, electromagnetic communication, and molecular communication. Among the above four communication methods, researchers have found that molecular communication is the most likely way to achieve communication between nanomachines. This mainly depends on bio-inspired molecular communication, which generally uses chemical signals, cells or organelles, and DNA as vectors to transmit information to adapt to functional characteristics of nanomachine.

At present, the main goal in the field of molecular communication is to enable nanomachine can communicate with each other in a biological environment. In this new mode of communication, information can be encoded as a carrier. such as a molecule or ion for propagation. Examples of using molecular communication to solve problems including calcium ion signals [

Molecular communication is a new communication technology. It can use bioinformatics molecules or ions to encode different information for transmission. However, the controllability and reliability of molecular communication are low due to the interference between various symbolic molecules and the random walk of molecules. Therefore, the reliability of molecular communication has become an important research direction in the field of molecular communication. Nevertheless, the reliability research of existing molecular communication has mainly involved the reliability analysis of molecular communication in free diffusion channels, without considering the influence of the drift velocity of the medium on the reliability of molecular communication. In this paper, a reliability model of molecular communication based on drift diffusion is proposed on the premise of drift velocity.

In the reliability research of molecular communication, Frank et al. [

The motivation of this paper is to analyze the reliability of molecular communication on the premise of drift velocity. Based on this goal, we propose a reliability model of molecular communication based on drift diffusion under different topologies. In the case of a transmission failure, a retransmission mechanism is used to ensure reliable transmission of information.

1) Our Contribution

• According to the existing research of molecular communication reliability, we introduced the drift velocity of the medium to propose a reliability model of molecular communication based on drift diffusion under different topologies.

• In the case of transmission failure, a retransmission mechanism is used to ensure reliable transmission of information.

• In addition, we also analyze the influence of different model parameters and different topologies on the reliability of MCD2, which can guide us to select optimal parameter and topology to improve the reliability of molecular communication.

2) Paper Outline

The rest of this paper is organized as follows. Section 2 a system model can be described. In Section 3, we proposed the reliability model of MCD2 in the single link, single path and multipath. Simulation results and analysis are presented in Section 4. Finally, we summarize this paper in Section 5.

In this section, as shown in

• Transmitter nanomachine (TN)

The transmitter nanomachine can continuously generate the same molecules to transmit information. Assuming that the TN can precisely control the release time of the molecule, and once these molecules are released into the channel by the TN, they will no longer be affected by the TN.

• Transmission medium

In a fluid medium, information molecules propagate information between a TN and a receiver nanomachine.

• Receiver nanomachine (RN)

When these molecules arrive at the RN, the RN can decode the information from the TN. Then, these molecules can be completely removed from the current channel by the RN.

Here, it is supposed that the TN and the RN are highly synchronized at all times and that the information molecules only fully elastically collide with the RN surface in the drift diffused channel. At the same time, the diffusion motion of the information molecules released by the TN, which can be attributed to the one-dimensional Brownian motion in the forward direction of the drift velocity to simplify the analysis process.

MCD2 has great application prospects in the field of biomedicine. For example, in the human body, cell to cell communication can through the diffusion of some information molecules or ions to transmit information. This process can be abstracted as a communication process between two nanomachines. The communication system model can be described as

As shown in

It can be seen from

It is assumed that the transmission of information between the TN and the RN is mainly based on the binary sequence. In order to avoid mutual interference of the same type of molecules, we use two different types of molecules A and B to represent “1” and “0”, respectively. Then, we use the array a[i] to represent the information transmitted each time. And the encode of the information by the TN can be formally defined as follows:

a [ i ] = { 1 , TN → M A 0 , TN → M B

Since the motion of the information molecules in the channel is affected by the Brownian motion, the diffusion process is random, and the transmission time of these molecules to the RN is also random. Without considering the drift velocity, the information molecules propagate in the form of Brownian motion in the channel. According to Fick’s second law, the partial derivative of the concentration of information molecules with respect to time is expressed as follows [

∂ c ( t ) ∂ t = D × ∂ 2 c ( t ) ∂ x 2 (1)

Therefore, under the molecular communication channel based on free diffusion, for any information molecule released by the TN at time t = 0, the probability of the molecule at different position x can be used P ( x , t ) to calculate, which its expression is as follows:

P ( x , t ) = 1 4 π D t exp ( − x 2 4 D t ) (2)

In the molecular communication channel based on drift diffusion, it is

necessary to consider the influence of the drift velocity of the medium on the impulse response of the system. We suppose that the drift velocity only consists of the positive drift velocity in the horizontal direction. In this paper, we only consider the propagation of information molecules in the drift diffusion channel in the one-dimensional case. Therefore, according to Fick’s second law, under the drift diffusion channel, the partial derivative relationship of the concentration c ( t ) of the information molecule with respect to time t is as follows:

∂ c ( t ) ∂ t = D × ∂ 2 c ( t ) ∂ x 2 − v x ∂ c ( t ) ∂ x (3)

In this paper, we suppose that the RN does not have an absorption boundary under the flow medium, ranging from negative infinity to positive infinity. For any information molecule released by the TN at time t = 0, the probability of the molecule at different position x can be calculated by using the position distribution function g ( x , t ) [

g ( x , t ) = 1 4 π D t exp ( − ( x − v x t ) 2 4 D t ) (4)

In one dimensional environment, the position of any information molecule changes with time, which obeys the position probability density function g ( x , t ) . Using this distribution, we can get that the time of any information molecule released by the TN reaches the RN after the moving d distance obeys the probability density function f ( t ) . Here we make the horizontal drift velocity ( v x = v ), and its function expression f ( t ) is as follows:

f ( t ) = d 4 π D t 3 exp ( − ( d − v t ) 2 4 D t ) , t > 0 (5)

In the above Equation (5), D represents the diffusion coefficient of medium and v represents the drift velocity of the medium. According to the function expression of f ( t ) , the cumulative distribution function F ( t ) can be obtained, which represents the probability any information molecules generated by the TN reaches the RN before time t. Then, the calculation expression of F ( t ) is as follows:

F ( t ) = 1 2 ( 1 + erf ( d 4 D t ( v t d − 1 ) ) ) + 1 2 exp ( v d D ) ( 1 + erf ( − d 4 D t ( v t d + 1 ) ) ) . (6)

In this section, we will give a mathematical model of reliability of MCD2 in single link, and then we also extend this reliability model into single path and multipath.

If there is no other relay node between the TN and the RN, this transmission path is defined as a single link. As shown in

P ( s A = 1 ) + P ( s A = 0 ) = 1 (7)

We suppose that TN transmits bit information “1” or “0” with the same channel transmission probability of λ, the above Formula (7) can be reduced to 2 λ = 1 . Namely, λ = 0.5 .

When time slot k ∈ { 1 , 2 , 3 , ⋯ , n − 1 } , TN transmits the information molecules with the channel transmission probability of λ without being received by RN, the probability that this molecule reaches the RN in the n^{th} time slot is recorded as γ i j ( k , n ) , and its calculation expression is as follows:

γ i j ( k , n ) = λ ∫ ( n − k ) τ ( n − k + 1 ) τ f ( t ) d t = λ [ F ( ( n − k + 1 ) τ ) − F ( ( n − k ) τ ) ] = 1 2 [ F ( ( n − k + 1 ) τ ) − F ( ( n − k ) τ ) ] , n > k (8)

where F ( τ ) is given in Equation (6). And when n = k , γ i j ( n , n ) can be calculated by γ i j ( n , n ) = 0 .5 F ( τ ) .

According to the above analysis, the probability that M molecules of type A or B released by the TN are not received by the RN in the n^{th} time slot is represented by β i j , and the calculation expression is as follows:

β i j = ∏ k = 1 n ( 1 − 1 2 [ F ( ( n − k + 1 ) τ ) − F ( ( n − k ) τ ) ] ) M (9)

The reliability of MCD2 under a single link is defined as the probability that at least one information molecule released by the TN is successfully received by the RN before time T, which is recorded as P i j . In summary, the calculation expression of the reliability of MCD2 in a single link is as follows:

P i j = 1 − ∏ n = 1 m ∏ k = 1 n ( 1 − 1 2 ( F ( ( n − k + 1 ) τ ) − F ( ( n − k ) τ ) ) ) M (10)

where M represents the total number of A type or B type information molecules released by the TN in each time slot, and m represents the number of time slots.

In the case of transmission failure, we use the automatic repeat request mechanism (ARQ) to ensure the reliable transmission of information. The working mechanism is as shown in

and RN will generate M acknowledgment information molecules for transmission to the TN. Therefore, when the TN receives the confirmation molecule transmitted from the RN, the transmission of one bit information is successfully completed. Here, it is assumed that information molecules and confirmation information molecules are different, but they have the characteristics of common diffusion.

As shown in _{ij} during the retransmission process. Therefore, after using the automatic retransmission request mechanism, the probability that the RN receives at least one molecule is denoted as P ′ i j , and its calculation expression is as follows:

P ′ i j = P i j + ( 1 − P i j ) P i j + ( 1 − P i j ) 2 P i j + ( 1 − P i j ) 3 P i j + ⋯ + ( 1 − P i j ) C i j P i j = ∑ n = 0 C i j ( 1 − P i j ) n P i j . (11)

A single path with two links is described in

As can be seen from

This communication process can be thought of as two processes. In the first stage, TN releases these information molecules to the relay nanomachine, and from TN to relay nanomachine any molecule experienced by the time t obey f_{1}(t), it represents the probability density function (PDF), it can be with the Equation (5), then f_{1}(t) can be described as the following Equation (12):

f 1 ( t ) = d 1 4 π D t 3 exp ( − ( d 1 − v t ) 2 4 D t ) , t > 0 (12)

The CDF is denoted by F_{1}(t) which could be connected with the probability density function f_{1}(t) in Equation (12) as follows Equation (13):

F 1 ( t ) = 1 2 ( 1 + erf ( d 1 4 D t ( v t d 1 − 1 ) ) ) + 1 2 exp ( v d 1 D ) ( 1 + erf ( − d 1 4 D t ( v t d 1 + 1 ) ) ) (13)

According to Equation (12), (13), then we can use P_{1} to represent the reliability of communication between the TN and the relay nanomachine. Then P_{1} can be calculated as follows Equation (14):

P 1 = 1 − ∏ n = 1 m ∏ k = 1 n ( 1 − 1 2 ( F 1 ( ( n − k + 1 ) τ ) − F 1 ( ( n − k ) τ ) ) ) M (14)

Similarly, when these information molecules are released by the relay nanomachine first hit the RN. Then we could consider that the time t experienced by any molecule from relay nanomachine to RN obeys the f_{2}(t), which could be associated with Equation (5), then f_{2}(t) is described as follows Equation (15):

f 2 ( t ) = d 2 4 π D t 3 exp ( − ( d 2 − v t ) 2 4 D t ) , t > 0 (15)

At the same time, the CDF is denoted by F 2 ( t ) which could be connected with the probability density function f 2 ( t ) in (15) as follows Equation (16):

F 2 ( t ) = 1 2 ( 1+erf ( d 2 4 D t ( v t d 2 − 1 ) ) ) + 1 2 exp ( v d 2 D ) ( 1+erf ( − d 2 4 D t ( v t d 2 + 1 ) ) ) (16)

According to Equations (15), (16), then we can use P_{2} to represent the reliability of communication between relay nanomachine and the RN. Then P_{2} can be computed as follows Equation (17):

P 2 = 1 − ∏ n = 1 m ∏ k = 1 n ( 1 − 1 2 ( F 2 ( ( n − k + 1 ) τ ) − F 2 ( ( n − k ) τ ) ) ) M (17)

As for the process of the TN releases information molecules to relay nanomachine and then relay nanomachine simultaneously forward these information molecules to RN. We consider that no link failure from TN to RN in the single path. Then we get the reliability of the single path P_{3} can be denoted as follows Equation (18):

P 3 = P 1 P 2 . (18)

In this section, we will investigate the reliability of MCD2 in the multipath. As you can see from _{1}, RN_{2} or RN_{n} before time T. There are 2n paths in the S which include TN-M_{1}-N_{1}-RN_{1}, TN-M_{2}-N_{2}-RN_{1}, TN-M_{3}-N_{3}-RN_{2}, TN-M_{4}-N_{4}-RN_{2}, TN-M_{i}-N_{i}-RN_{n} and TN-M_{n}-N_{n}-RN_{n}. These paths are labeled as s 1 , s 2 , s 3 , s 4 , s i , ⋯ , s 2 n , which are all disjoint and independent. On the one hand, we define S as the set of the multipath and each path s k has 3 hops. Then set S can be assigned as S = { s 1 , s 2 , s 3 , s 4 , s i , ⋯ , s 2 n } . On the other hand, we also define a set of corresponding link reliabilities P s k = [ P i j , P j l , P l m , ⋯ , P m 2 n ] , where P i j is the reliability between node i and j.

From

RN_{1}. These information molecules can also arrive at the destination nanomachine RN_{2} by passing along the path s 3 or s 4 . In addition, the TN releases some molecules can reach the RN_{n} through s i or s 2 n before one time slot. Therefore, the reliability of the multipath can be defined as P m u l which means the probability that at least one molecule from TN can be successfully received by RN_{1}, RN_{2}, RN_{3}, ∙∙∙, RN_{i} or RN_{n} before time T.

Firstly, we consider that the reliability of the case at least one information molecule from TN can be successfully received by RN_{1} passing along the path s 1 or s 2 can be defined as P S 1 , 2 . In this case, we can get the reliability P S 1 , 2 for the two paths from the TN to the RN_{1} as follows:

P S 1 , 2 = 1 − ∏ s k ⊆ S ( 1 − P s k ) , k ∈ [ 1 , 2 ] , k ∈ Z (19)

P s k = ∏ s k ⊆ S P i j (20)

where P i j can be calculated by the Equation (10).

Then we can consider the reliability of the case that at least one information molecule from TN can be successfully received by RN_{2} passing along the path s 3 or s 4 can be defined as P S 3 , 4 . In this case, we can get the reliability P S 3 , 4 for the two paths from the TN to the RN_{2} as follows:

P S 3 , 4 = 1 − ∏ s k ⊆ S ( 1 − P s k ) , k ∈ [ 3 , 4 ] , k ∈ Z (21)

where P s k can be calculated by the Equation (20).

Finally, we compute the reliability of the case that at least one information molecule from TN can be successfully received by RN_{n} passing along the path s i or s 2 n can be represented by P S i , 2 n . Base on this case, we can get the reliability P S i , 2 n for the two paths from the TN to the RN_{n} as follows:

P S i , 2 n = 1 − ∏ s k ⊆ S ( 1 − P s k ) , k ∈ [ i , 2 n ] , k ∈ Z (22)

where P s k can be calculated by the Equation (20).

Therefore, we can calculate the reliability of the case that at least one information molecule from TN can be successfully received by RN_{1}, RN_{2}, RN_{3}, ∙∙∙, RN_{i}_{ }or RN_{n}. According to our previous definition about the reliability of the multipath, P m u l can be calculated as follows Equation (23):

P m u l = P S 1 , 2 + P S 3 , 4 + P S 4 , 5 + ⋯ + P S i , 2 n . (23)

In this section, we will via numerical analysis to obtain simulation results. This experiment runs on a Windows 10 (64-bit) operating system, a PC with a memory size of 8GB, and a CPU of i7-9700. Then, we use MATLAB software to do some simulation experiments. Firstly, we investigate the influence of different model parameters for the reliability of MCD2 in the single link. These parameters are given in

Symbol | Description | Unit |
---|---|---|

v | the drift velocity of medium | um/s |

d | distance between two nanomachines | um |

D | the diffusion coefficient of environment | um^{2}/s |

m | number of time slot | - |

M | number of molecules each time slot | - |

on free diffusion. Furthermore, we also compare the reliability of MCD2 between multipath and single path.

We consider that drift velocity of medium, the distance from TN to RN, diffusion coefficient, time slot numbers have effect on the reliability of MCD2 in the single link. Therefore, it is necessary to investigate how these parameters affect the reliability of MCD2.

Here, we suppose that the diffusion coefficient D = 0.8 um^{2}/s, time slot number m = 10, diffusion distance d = 8 um, and then we can use the reliability model of MCD2 to analyze the influence of different drift velocity on the reliability of MCD2, Such as v = 0.8 um/s, v = 0.85 um/s and v = 0.9 um/s, Therefore, we can see that the reliability of MCD2 varies with different drift velocity from

As can be seen from

It is supposed that the diffusion coefficient D = 0.8 um^{2}/s, drift velocity v = 0.8 um/s, and time slot number m = 10. Then, we can use the reliability model of MCD2 to analyze the influence of different diffusion distance on the reliability. Such as d = 7.5 um, d = 8 um and d = 8.5 um. Therefore, we can see that the reliability of MCD2 varies with diffusion distance from

It can be seen from

molecules diffuse in the channel, information molecules are prone to decay in the long time irregular motion. It makes the concentration of information molecules on the RN surface to gradually decrease, which leads to the problem of information decode error during the RN decodes information. Thus, the reliability of MCD2 is reduced to some extent.

It is assumed that drift velocity v = 0.8 um/s, d = 8 um, and time slot number m = 10. Then we use the reliability model of molecular communication to study the influence of different diffusion coefficient on the reliability of MCD2. Such as D = 0.59 um^{2}/s, D = 0.6 um^{2}/s, D = 0.64 um^{2}/s. Therefore, we can see that the reliability of MCD2 varies with different diffusion coefficient from

From

We consider that drift velocity v = 0.8 um/s, diffusion coefficient D = 0.8 um^{2}/s, d = 0.8 um. Then, we can analyze the impact of different time slot length on the reliability of single link. For example, we take time slot τ = 1 s , τ = 2 s , and τ = 3 s , respectively.

From

We presume that the diffusion coefficient of the environment D = 0.8 um^{2}/s, d = 8 um, drift velocity v = 0.8 um/s, τ = 1, the number of time slot m = 10. In the

process of transmission failure, a retransmission mechanism is used to ensure reliable transmission of information. Such as C_{ij} = 0, C_{ij} = 1, C_{ij} = 2, C_{ij} = 3. Therefore, we can see that the reliability of MCD2 varies with different retransmission times from

From

In the single link, the reliability model of MCD2 is compared with the existing model of molecular communication reliability based on free diffusion [^{2}/s, d = 5 um, τ = 1, drift velocity v = 0.6 um/s, the number of time slot m = 10 to compare the performance of reliability in different molecular communication reliability model. Therefore, we can analyze the performance of the reliability between reliability model of MCD2 and the existing model of molecular communication reliability based on free diffusion from

From

In this part, we mainly focus on comparing the reliability of MCD2 between multipath and single path. Firstly, we assume that diffusion coefficient D = 1 um^{2}/s, v = 0.5 um/s and the number of time slot m = 10 in the multipath and single path. Secondly, we take n = 3 to indicate that there are three receiver nanomachines in the multipath. What’s more, we also assume that the distance between the corresponding two nanomachines is equal and independent. The distance of each hop between two adjacent nanomachines is d 1 = 4 um , d 2 = 5 um , d 3 = 6 um , d 4 = 7 um , d 5 = 8 um , d 6 = 9 um in six paths. As for the single path, we also suppose that d 1 = 4 um , d 2 = 5 um . Then we can use Equation (18) and Equation (23) to analyze the reliability of MCD2 in the single path and multipath.

In this paper, we propose a reliability model of MCD2 to investigate the reliability of different topologies between TN and RN. In the case of transmission failure, a retransmission mechanism is used to ensure the reliability of information transmission. Furthermore, it can be concluded by numerical analysis that with the increase of drift velocity, diffusion coefficient, retransmission times, the length of time slot, and the decrease of diffusion distance, the reliability of MCD2

can be improved. At the same time, we also find that our proposed model of reliability is superior to the existing reliability model of molecular communication based on free diffusion in analyzing the reliability of MCD2.

The authors declare no conflicts of interest regarding the publication of this paper.

Wang, X.L. and Jia, Z. (2020) Reliability Analysis of Molecular Communication Based on Drift Diffusion in Different Topologies. Journal of Computer and Communications, 8, 71-89. https://doi.org/10.4236/jcc.2020.81005