Reliability Analysis of Molecular Communication Based on Drift Diffusion

Molecular communication is a novel nanoscale communication method. The information can be encoded by using different molecules for transmission. However, as a result of various symbol molecules are easy to interfere with each other and the stochastic behaviour of the molecules, molecular communication is easily susceptible to a low reliability. Therefore, reliability is a vital issue in the field of molecular communication. At present, the reliability analysis of existing molecular communication has not taken the drift velocity into consideration. In this paper, we introduce the drift velocity of molecules and propose a reliability model of molecular communication based on drift diffusion in the single link and single path. Furthermore, retransmission mechanism is introduced on the condition of the transmission failure. Finally, our simulation experiments show how the parameters in the model affect the reliability of molecular communication based on drift diffusion, which can guide us how to improve the reliability of molecular communication based on drift diffusion in the future.


Introduction
In recent years, with the rapid development of the Internet and 5G communica-Journal of Computer and Communications 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. [7] used virus particle as information carrier to study the reliability and delay of multi-hop molecular communication. Balasubramaniam et al. [4] analyzed the characteristics of multi-hop nanonetworks, such as the reliability of information transmission under different topologies. Cheng et al. [8] have studied the reliability and delay of multicast topology under binary communication. Einolghozati et al. [9] have studied the reliability problem between two nodes in a bacterial colony molecular communication network. Tepekule et al. [10] have proposed two different types messenger molecules to reduce the impact of ISI to improve the reliability of molecular communication. Shih et al. [11] used new channel code techniques to improve the reliability of molecular communication. Leeson et al. have used error correction code to reduce the decode error rate and improve the reliability of the communication link [12]. Based on the above reliability analysis of molecular communication, these researches have not mentioned the influence of the drift velocity on the reliability of molecular communication.
Nevertheless, in our previous laboratory, Lu et al. [13] [14] considered the drift velocity of medium, they proposed the molecular communication model in the vertical direction and its test bed, but no further involves the reliability of molecular communication.
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.

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.

System Model
In this section, as shown in Figure 1, a system of molecular communication based on drift diffusion is described as consisting of the following subsystems.  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 Figure 1. As shown in Figure 1, when the TN releases the information molecules into the channel, after a period of time, the information molecules reach the RN by the drift diffusion, and the RN will decode the received information molecules to obtain the original information.
It can be seen from Figure 2, under the channel of MCD2, the communication process mainly includes the five processes, which are modulation, send, transmission, receive and demodulation, respectively.
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: 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 [15]: 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: In the molecular communication channel based on drift diffusion, it is 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 [16], which is expressed as follows: 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 , and its function expression ( ) f t is as follows: 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:

Reliability Analysis of MCD2
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.

The Analysis of Reliability in Single Link
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 Figure 1, we present a system model When the TN transmits bit information "1" in a time slot, the probability of the event is recorded as ( ) Similarly, when the TN transmits the bit information "0" in a time slot, the probability of the event is recorded as ( ) Then the following Equation can be established: 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 1 λ = . Namely, , 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 ( ) , ij k n γ , and its calculation expression is as follows: 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 ij β , and the calculation expression is as follows: 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 ij P . In summary, the calculation expression of the reliability of MCD2 in a single link is as follows: 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 Figure 3, that is, when the M information molecules  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 Figure 3, it is assumed that the maximum number of retransmissions of single link is C 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 ij P′ , and its calculation expression is as follows:

The Analysis of Reliability in Single Path
A single path with two links is described in Figure 4. There is only one relay nanomachine between the TN and the RN. As can be seen from Figure 4, the communication system consists of TN, relay nanomachine and RN. At the same time, TN, relay nanomachine and RN are placed in a row. Then in single path communication, we use the same type of molecule to transfer information from TN to RN. In this process, we believe that TN can release M molecules in the channel to transmit a bit information to the relay nanomachine. Then, these information molecules arrive at relay nanomachine which can continue to replicate M molecules by forwarding these information molecules to RN using the same type of molecule. Then, these information molecules can bind to the corresponding receptors from RN, and the information molecules are eventually decoded by RN, while the decoded molecules are removed from the current environment. This means that the communication is successfully completed in the single path.
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)  ( ) ( ) 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): 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): 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): At the same time, the CDF is denoted by ( ) 2 F t which could be connected with the probability density function ( ) 2 f t in (15) as follows Equation (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) 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 Journal of Computer and Communications path. Then we get the reliability of the single path P 3 can be denoted as follows Equation (18):

The Analysis of Reliability in Multipath
In this section, we will investigate the reliability of MCD2 in the multipath. As you can see from Figure 5, the multipath consists of one TN, 4n relay nanomachines and n Receiver nanomachines. As shown from Figure  , where ij P is the reliability between node i and j.
From Figure 5, we can see that the information molecules originate from TN can pass along the paths 1 s or 2 s arriving at the destination nanomachine

Simulation Experiment and Result Analysis
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 Table 1. Then we compare the reliability performance of different models between MCD2 and the existing molecular communication based

Single Link
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.

The Effect of Drift Velocity on the Reliability
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 Figure 6. As can be seen from Figure 6, the reliability of MCD2 varies with drift velocity. When the drift velocity of the medium is gradually increased, the reliability of MCD2 is significantly improved. This mainly depends on the drift velocity of the medium, which makes the information molecules fast transmission in the channel. These information molecules are less likely to accumulate in the channel and cause symbol interference, which reduces the ISI noise in the channel to some extent. Therefore, it can be inferred that when some information molecules are sensitive to delay, the drift velocity can be faster, the system delay is reduced, and the reliability of MCD2 also has been improved.

The Effect of Distance on the Reliability
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 Figure 7.
It can be seen from Figure 7 that as the diffusion distance between TN and RN increases, the reliability of molecular communication gradually decreases. This is mainly because the longer the diffusion distance, the longer the information molecules released by the TN arrive at the RN. When this information  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.

The Effect of Diffusion Coefficient for the Reliability
It is assumed that drift velocity v = 0.8 um/s, d = 8 um From Figure 8, it can be found that the reliability of the single link gradually increases as the number of molecules released in each time slot gradually increases. When the number of molecules increases to a certain upper threshold, the reliability of the link approaches 1. In addition, when the diffusion coefficient of the environment is gradually increased, the reliability of molecular communication under a single link is maximized in a shorter time. The reason is that the larger the diffusion coefficient, the faster the information molecules released by the TN move in the channel, and the greater the probability of reaching the RN in the same time.

The Effect of Time Slot for the Reliability
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 1 s τ = , 2 s τ = , and 3 s τ = , respectively.
From Figure 9, it can be seen that the reliability of molecular communication differs at different time slot length. Moreover, it can be concluded that as the length of time slot is larger, the reliability of molecular communication is higher. Therefore, we consider to extend the time slot properly within the acceptable delay range of both sides of the communication link. In this way, the influence of residual molecules in the channel on the decode of another symbol bit in the next time slot can be reduced, and the reliability of MCD2 can be further improved.

The Effect of Retransmission Times on Reliability
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 Figure 10.

Performance Comparison of Different Reliability Model
In the single link, the reliability model of MCD2 is compared with the existing model of molecular communication reliability based on free diffusion [8]. Under the condition that the parameters are set to the same value, we take the diffusion coefficient D = 0.59 um 2 /s, d = 5 um, τ = 1, drift velocity v = 0.6 um/s, the num-   From Figure 11, it can be seen that the reliability of molecular communication increases with the increase of the number of molecules. When the number of molecules increases to a certain extent, the reliability of link will no longer increase and tend to 1. Moreover, we also find that the reliability of MCD2 obviously higher than existing molecular communication based on free diffusion, which is mainly due to drift velocity accelerates information molecules reaching the RN, thereby increasing the possibility that the RN receives the information molecules. Based on the above analysis, if we continue to use the existing free diffusion-based molecular communication reliability model to analyze the reliability of MCD2, the analysis of the reliability of the communication link will produce a higher error.

The Comparison of Reliability between Multipath and Single Path
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 1 4 um d = , 2 5 um d = , 3 6 um d = , 4 7 um d = , 5 8 um d = , 6 9 um d = in six paths. As for the single path, we also suppose that 1 4 um d = , 2 5 um d = .
Then we can use Equation (18) and Equation (23) to analyze the reliability of MCD2 in the single path and multipath. Figure 12 shows that the reliability performance of MCD2 between multipath and single path. We can find that the reliability of the multipath obviously higher than the single path. This is mainly because the information molecules can easily reach the RN through multiple paths in the multipath, and the more information molecules can be received by RN. Therefore, the reliability of information transmission in the multipath is higher than that of single path.

Conclusion
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 Figure 12. The comparison of reliability between multipath and single path.
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.