Analysis of the adaptive three-modes for PC+MPC+APC techniques using retransmission cycle mechanism

Abstract

This paper presents the new throughput analysis of the adaptive three modes for Packet Combining (PC), Modified Packet Combining (MPC) and Aggressive Packet Combining (APC) techniques. The three modes in this paper consist of X (low probability of packet error), Y (moderate probability of packet error) and Z (high probability of packet error) channel states. The three states are based on the number of received acknowledgement (either positive acknowledgement (ACK) or negative acknowledgement (NACK)) by Transmitter (Tx). The PC, the hybrid PC+MPC and the hybrid MPC+APC techniques are implemented in X, Y and Z channel states respectively for enhancing the fast error correction and also to reduce retransmission of duplicate copies. Thus, the proposed protocol is simulated using MATLAB and the results yield better performance over conventional PC, MPC and APC techniques in terms of throughput, the probability of packet error correction and energy consumption.

Keywords

PC MPC APC three states bit error rate

1. Introduction

The bit error rate of wireless channel is high. In uplink data communication, by characteristics the Tx uses retransmit for a particular packet multiple times before the Receiver (Rx) receives the correct one. Every retransmission consumes communication bandwidth and transmission energy. The bandwidth is a scarce and valuable resource because many portable computers may be sharing a limited free space spectrum. The energy is also a scarce and valuable resource because the portable computer is usually powered by battery. Therefore, to utilize the bandwidth and battery energy efficiently it is desirable to reduce the number of retransmissions. In conventional Backward Error Correction technique, the receiver discards the erroneous packet and requests the transmitter for a retransmission. However, an erroneous packet may contain both erroneous bits and correct bits and hence it may still contain useful information.

The receiver may be able to combine this information from multiple erroneous copies to recover the correct packet. Wicker [20] proposed the majority packet combining scheme. This scheme performs bit-by-bit majority voting on the erroneous copies and then performs error detection on the resulting combined packet. If the combined packet is found to be correct, the receiver accepts it; otherwise, the receiver requests the transmitter for a retransmission. Chakraborty [9–11] has also suggested a simple and elegant scheme based on Backward Error Correction technique, called Packet Combining (PC) technique. The technique is used to correct original copy by storing the received erroneous copies. If the packet is received erroneously, the receiver stores the erroneous copy and it requests for duplicate copy by sending NACK to the transmitter. If the duplicate copy is received without erroneous bits then the receiver request for next packet otherwise stores the duplicate copy and performs XOR operation (⊕) of two erroneous copies for locating the erroneous bit positions. Let the original packet is “10101010”. Assume the receiver receives on first transmission copy as “00101010” which is erroneous. Assume the receiver receives the copy on retransmission as “10101000” which is also erroneous. Now XOR operation is performed with two erroneous copies, i.e. $00101010 \oplus 10101000 = 10000010$ . Hence, error locations are identified at 1st and 7th positions from MSB. Bit inversion technique (converts 0 to 1 or 1 to 0) is performed on two erroneous copies at identified locations. Finally, it checks for error detection and hence it may get the original copy. The limitation of PC is that the technique fails when error(s) location(s) is/are at the same position of erroneous copies which is also called hidden error. The example is shown below:

Let an original copy: “00001111” and the receiver received the first and second erroneous copy as “00000111” and “00000111” i.e. $00000111 \oplus 00000111 = 00000000$ (PC technique fails to identify the erroneous bit location due to erroneous occurred at the same bit position). Thus, if the PC technique fails to recover the original copy then discard the erroneous copies and same procedure will be repeated.

Furthermore, Modified Packet Combining scheme [1] has been carried out for performing higher error correction and to enhance the error correction done by PCs. In this scheme, three erroneous copies are XORed pairwise in order to find the bit error location. The identified bit error locations are put in ascending order where the first minimum erroneous bit is used to perform first bit-by-bit inversion process followed by second and third if fails which same technique is done by PC. The three erroneous copies are assumed as Copy (C) C-1, C-2 and C-3 and then pairwise XOR operation is carried out among (C-2 ⊕ C-1), (C-3 ⊕ C-2) and (C-1 ⊕ C-3) to locate the erroneous bit position.

Assume an original packet “11100111” and it is received as three erroneous copies as first erroneous copy, C-1 = 11101111, second erroneous copy, C-2 = 11101011, and third erroneous copy, C-3 = 11100101. The erroneous copies are performed XOR between C-2 ⊕ C-1 = 00000100, C-3 ⊕ C-2 = 00001110 and C-1 ⊕ C-3 = 00001010. The ascending orders based on common erroneous bit position are indicated by XOR operation which is clearly shown in Table 1. Thus, error correction will start from C-1 and if necessary then C-3 and followed by C-2. However, if the above process fails to recover the original copy then the same procedure will be repeated after discarding all the erroneous copies.

Table 1
Modified Packet Combining scheme

Copies of erroneous pairs Erroneous bits (y) Erroneous detected position

[C-2 ⊕ C-1] 1 bit 6th position from MSB

[C-1 ⊕ C-3] 2 bits 5th and 7th positions from MSB

[C-3 ⊕ C-2] 3 bits 5th, 6th and 7th positions from MSB

Copies of erroneous pairs	Erroneous bits (y)	Erroneous detected position
[C-2 ⊕ C-1]	1 bit	6th position from MSB
[C-1 ⊕ C-3]	2 bits	5th and 7th positions from MSB
[C-3 ⊕ C-2]	3 bits	5th, 6th and 7th positions from MSB

Leung [12] has introduced an extension technique of Majority Packet Combining called Aggressive Packet Combining (APC) for performing higher error control in a wireless network. It consists of three steps and illustrated as follows:

Step 1: Let the three erroneous copies CP₁^err: 00100, CP₂^err: 00001 and CP₃^err: 00100 are buffered at the receiver and then it performs bit by bit majority voting and resultants bits are checked by error detection block whether it is erroneous bit exist or not. If the Step 1 recovers the original copy then receiver requests for next packet, if not; Step 2 and Step 3 will be performed.

Step 2: This step is performed to find the least reliable bits from Step 1. And the brief example is explained in Fig. 1. In this case 3rd and 5th bits from MSB are detected as the two least reliable bits.

Fig. 1.

Block diagram of conventional APC [12].

Step 3: This step is used to find the correct bit sequence for the least reliable bits. In this case, 00, 01, 10 are the possible correct bit pattern which can be obtained from $2^{l} - 1$ and ‘l’ is the number of least reliable bit. Thus, error detection block checks all the possible bit sequence whether it is correct copy or not. Finally, “00000” bit sequence is found as correct copy. If this technique fails to find the correct copy then discard the whole erroneous copies and request for next duplicate in order to repeat the same procedure.

Thus, these three techniques have discarded the whole erroneous copies if the techniques failed to recover the original copy and repeat the same procedure which drastically reduced the throughput efficiency. Several researchers [3–8,14–17] have conclusively established different techniques based on conventional PC, MPC and APC for higher error correction capability but lacked an analysis of the channel states and also discarded the half-corrected erroneous copies at the receiver. Since it was reported in the literature that MPC is an extension technique of PC, this paper focuses on utilizing the half-corrected copies of PC technique instead of discarding the two erroneous copies. Moreover, APC technique is also the modification of Majority Packet Combining. Thus, MPC and APC techniques require at least three erroneous copies and PC requires two erroneous copies in order to recover the original copy. It is also reported that PC does only one bit error correction in the condition of low bit error rate but the MPC and APC can correct two bits error which is more than PC but lower throughput efficiency. Thus, MPC and APC have the capability to adapt in higher bit error rate.

Thus, in order to re-use the resultant erroneous copies or half-corrected copies for having fast error correction and to reduce the retransmission of duplicate copies, the combined techniques of PC+MPC and MPC+APC are considered with the three channel states. Hence, the conventional PC is implemented in X mode followed by PC+MPC and MPC+APC on Y and Z states. The proposed protocol is simulated using MATLAB and the results show that they have a better performance in terms of throughput, the probability of packet error correction and energy consumption over conventional PC, APC and MPC.

The manuscript has been divided into four sections and is organized as follows: the proposed model is explained in Section 2. Section 3 discusses the numerical analysis and results of probability of packet error correction, throughput, and energy consumption. Finally the conclusion is discussed in Section 4.

2. Proposed model

The proposed protocol assumes that the channel state is in X mode if the Tx receives β consecutive ACK(s) and hence PC technique will be implemented. In X mode if the Tx receives α consecutive NACK(s), the channel switches to Y mode. In this mode, PC+MPC technique will be performed. If the Tx receives β consecutive ACK(s) in Y mode, the channel returns to X mode. If the Tx receives γ consecutive NACK(s) while in Y mode, the channel switches to Z mode, in which MPC+APC technique will be implemented. Similarly, the channel state returns to Y mode upon the receipt of δ consecutive ACKs and hence implementation of PC+MPC technique will be continued.

Fig. 2.

Markov two states model [18].

Fig. 3.

XX retransmission cycles [18].

Initially we assume that the channel state in order to adapt the proposed protocol into two states called Markov chain, as shown in Fig. 3. The chain goes into the state X0 (rectangle) upon successfully reception of ACK packets from Rx. X0 state ends the previous and starts the new retransmission cycle. Thus, it is called final or initial depending on ACK(s) or NACK(s) parameters. On the other hand, the state X1(ellipse) is called transient. This state is reached after NACK packets received from Rx. Thus, X0 and X1 belong to the same group of X.

The two pairs $S_{X} = [1 - P_{X}, T_{S X}]$ and $f_{X} = [P_{X}, T_{f X}]$ are assigned to the proposed model which leads into final/initial state and transient state, $P_{X}$ denotes the failure packet transmission. $T_{S X}$ and $T_{f X}$ denote the times required to complete the transition from some state into X0 and X1 respectively.

Fig. 4.

Transition diagram of the proposed work [18].

From Fig. 4, we can see the three states X, Y and Z and their transient states where $X_{i}$ , $i = 0, 1, \dots, β - 1$ . In states $Y_{l}^{'}$ , $l = 0, 1, \dots, γ - 1$ and $Y_{j}^{″}$ , $j = 0, 1, \dots, β - 1$ , and finally $Z_{i}$ , $i = 0, 1, \dots, δ - 1$ .

Due to the equal states probabilities, we can have $P (Y_{0}^{'}) = P (Y_{0}^{″}) ≜ P (Y_{0})$ where P(A) determines the probability of state A in stationary regime. Thus it enables us to express the probabilities of other states in terms of $P (Y_{0})$ . Now, we have got from the probability $\begin{array}{l} (1) & P (X_{0}) = \frac{{(1 - P_{Y})}^{β}}{P_{X}^{α}} P (Y_{0}), \\ (2) & P (Z_{i}) = {(1 - P_{Z})}^{i} \frac{P_{Y}^{γ}}{{(1 - P_{Z})}^{δ}} P (Y_{0}), i = 1, 2, \dots, δ - 1, \\ (3) & P (Y_{i}^{″}) = {(1 - P_{Y})}^{i} P (Y_{0}), i = 0, 1, 2, \dots, β - 1 \end{array}$ Now the three states X, Y and Z can be described by the following equations, $\begin{array}{l} (4) & X ≜ \frac{P (X_{0})}{P (Y_{0})} = \frac{{(1 - P_{Y})}^{β}}{P_{X}^{α}} \\ (5) & Y ≜ \frac{1}{P (Y_{0})} \sum_{i = 0}^{β - 1} P (Y_{i}^{″}) = \frac{1 - {(1 - P_{Y})}^{β}}{P_{Y}} \\ (6) & Z ≜ \frac{1}{P (Y_{0})} \sum_{i = 0}^{δ - 1} P (Z_{i}) = \frac{1 - {(1 - P_{Z})}^{δ - 1} P_{Y}^{γ}}{{(1 - P_{Z})}^{δ - 1} P_{Z}} \end{array}$ Figure 5 shows the block diagram of the proposed work. Initially, the Tx assumes that the channel state is X mode. Thus, the consecutive two copies (CP₁ and CP₂) are used to transmit from Tx to Rx. The Rx checks whether the received copies are correct or not. If one of the copies is found correct then discards the other erroneous copy and requests for next packet. If both the copies are found as erroneously then PC technique will be performed. If PC technique provides the correct copy then Rx requests for next packet otherwise requests for third copy (CP₃) and if the third copy is received as correct copy then discards the half-corrected copies. The Tx upon reception of NACK packet, the channel is assumed that the X state channel is switched to Y state. Thus, the hybrid PC+MPC technique will be performed. Similarly, if the PC+MPC technique fails to recover the correct copy then the hybrid MPC+APC technique will be performed by assuming that channel switches from Y to Z state. The main notion of the proposed scheme is used to reduce the retransmission of the duplicate copies by enhancing the probability of packet error correction.

Fig. 5.

Block diagram of proposed work.

The stepwise procedure for error recovery technique in the proposed scheme is given below:

The proposed protocol consists of PC, PC+MPC and MPC+APC blocks. In PC block, two erroneous copies CP₁^err and CP₂^err are performed XOR operation and it identifies the erroneous bit location. In PC+MPC block, the identified location from PC block performs $X^{1}$ operation where it converts the possible erroneous bit locations in CP₁^err in such a manner that if 0 then 1 or 1 then 0 but not both i.e. if $1 \to 0$ but $0 \to 0$ or if $0 \to 1$ but $1 \to 1$ . The $X^{2}$ operation converts the similar process of $X^{1}$ operation but in CP₂^err and then the resultant of $X^{1}$ and $X^{2}$ are performed pairwise XOR operation and finally, it checks whether it is an original copy or not.

Again in PC+MPC block, $Y^{1} \oplus Y^{2}$ , $Z^{1} \oplus Z^{2}$ are also performed from the resultant of $Y^{1} \oplus {CP}_{3}^{err}$ and $Z^{1} \oplus {CP}_{1}^{err}$ and finally checks for original copy.

Lastly, MPC+APC block performs the APC technique of $X^{1}$ , $Y^{1}$ and $Z^{1}$ or $X^{2}$ , $Y^{2}$ and $Z^{2}$ instead of CP₁^err, CP₂^err and CP₃^err in conventional APC technique.

Tables 2–7 show the examples of PC+MPC and MPC+APC techniques. Table 3 and Table 4 show the comparative examples of MPC+APC technique and conventional APC scheme. The resultant copies $X^{1}$ , $Y^{1}$ and $Z^{1}$ from PC+MPC block perform majority voting and it detects the number of least reliable bits ( $l = 3$ ) but in conventional APC the majority voting has been carried among the erroneous copies i.e. CP₁^err, CP₂^err and CP₃^err and found the value of $l = 4$ . Thus, we can see that the number of least reliable bits can be reduced by proposed scheme and hence the searching bit pattern can also be reduced to $2^{l - 1} - 1$ . Similarly, Table 6 and Table 7 also show another example and the value of l is also reduced.

Table 2

Example of proposed scheme under PC+MPC block with original copy, $O^{r} = 11111$

Table 3

Example of proposed scheme under MPC+APC block with original copy, $O^{r} = 11111$

MPC+APC block

Majority voting of $X^{1}$ , $Y^{1}$ and $Z^{1}$	Number of least reliable bits (l)	Number of searching correct bit patterns ( $2^{l} - 1$ )	Output
01111	3	000	11111 (correct copy)
11111		001
00011		010
$\dots \dots \dots$		100
01111 (incorrect copy)		101
		110
		111

Table 4

Example of conventional APC with original copy, $O^{r} = 11111$

Conventional APC scheme

Majority voting of CP₁^err = 01011, CP₂^err = 01101, CP₃^err = 10111	Number of least reliable bits (l)	Number of searching correct bit pattern ( $2^{l} - 1$ )		Output
01011	4	0000	1001	11111 (correct copy)
01101		0001	1010
10111		0010	1011
$\dots \dots \dots$		0100	1100
01111 (incorrect copy)		0101	1101
		0110	1110
		0111	1111
		1000

Table 5

Example of proposed scheme under PC and MPC blocks with original copy, $O^{r} = 00011$

Table 6

Proposed example of MPC+APC block with original copy, $O^{r} = 00011$

MPC+APC block

Majority voting of $X^{1}$ , $Y^{1}$ and $Z^{1}$	Number of least reliable bits (l)	Number of searching correct bit patterns ( $2^{l} - 1$ )		Output
00111	3	000	101	00011 (correct copy)
00011		001	110
11011		010	111
$\dots \dots \dots$		110
00011

Table 7

Example of conventional APC with original copy, $O^{r} = 00011$

Conventional APC scheme

Majority voting of CP₁^err = 11011, CP₂^err = 10011, CP₃^err = 00000	Number of least reliable bits	Number of searching correct bit patterns ( $2^{l} - 1$ )			Output
11011	4	0000	0101	1011	00011 (correct copy)
10011		0001	0110	1100
00000		0010	0111	1101
$\dots \dots \dots$		0011	1000	1110
10011 (correct copy)		0100	1001	1111
			1010

In any of the packet combining techniques, the retransmission is required when the first transmitted packet is received as an erroneous copy at the receiver. Thus, the probability that the first transmission of one copy from the N available number of duplicate copies of same packet is $\frac{1}{N}$ so, the mean service delay in the first transmission can be given by Eq. (7) [13] $\begin{matrix} (7) & T_{1} = \frac{1}{N} n [N + (N - 1) + \dots + 2 + 1] = \frac{n (N + 1)}{2} \end{matrix}$ Where N denotes the number of duplicate copies require for performing packet combining technique and n denotes the size of the packet.

The Tx retransmits the duplicate copies upon reception of NACK from Rx. Therefore, when the copies are retransmitted for r times, the equivalent retransmission delay for the PC technique can be given by Eq. (8) $\begin{array}{l} (8) & \begin{matrix} T_{r PC} & = n r + (r + 1) (T_{D} + 2 T_{P} + T_{A}) \\ = 2 n + 3 (T_{D} + 2 T_{P} + T_{A}); if r = 2 \end{matrix} \end{array}$ Where, $T_{D} = the transmission delay of n bits packet size$ , $T_{P} = propagation delay$ and $T_{A}$ = acknowledgement delay from Rx to Tx

Thus the total service delay time for the PC technique can be obtained from Eqs (7) and (8) and is given by Eq. (9) $\begin{matrix} (9) & {TS}_{PC} = T_{1} + T_{r PC} \end{matrix}$ Similarly, the total service delay time for the MPC and APC techniques can be given by Eqs (10) and (11) $\begin{array}{l} (10) & {TS}_{MPC} = T_{1} + T_{r MPC} \\ (11) & {TS}_{APC} = T_{1} + T_{r APC} \end{array}$ Where $T_{r MPC} = 3 n + 4 (T_{D} + 2 T_{P} + T_{A})$ and $T_{r APC} = 3 n + 4 (T_{D} + 2 T_{P} + T_{A})$ when $r = 3$ .

The objective of the manuscript is to determine the throughput that can be achieved during transmission of the packet as well as retransmission of the duplicate copies. The normalized throughput [18] retransmission cycle duration, $T_{R}$ . It is also noted that the retransmission cycle, $T_{R}$ can be concluded that the average time needed for successful packet transmission or retransmission for the conventional packet combining techniques.

Referring from Fig. 2, we can express for the mean duration of the retransmission cycle in a single mode combining techniques. $\begin{array}{l} T_{R} & = \sum_{i = 0}^{\infty} (i T_{f X} + T_{S X}) P_{X}^{i} (1 - P_{X}) \\ (12) & = \frac{P_{X}}{(1 - P_{X})} T_{f X} + T_{S X}, \end{array}$ from which we can obtain the throughput efficiency $\begin{array}{l} (13) & \begin{matrix} S_{eff} & = \frac{T_{E}}{T_{R}} = \frac{(1 - P_{X}) T_{E}}{T_{f X} P_{X} + T_{S X} (1 - P_{X})} \\ = \frac{(1 - P_{X})}{M_{X} P_{X} + K_{X} (1 - P_{X})} \end{matrix} \end{array}$ Where $M_{X} = \frac{T_{f X}}{T_{E}}$ and $K_{X} = \frac{T_{S X}}{T_{E}}$

In PC technique, the first copy is sent from Tx to Rx. If the first copy is successful transmission, then the successful transmission time is equal to the time required to emit the copy i.e. $K_{X} = 1$ . If the Tx upon reception of NACK from Rx then it is used to send a duplicate copy of the same packet in the next time slot. Thus due to the retransmission of the duplicate copies in PC, it assumes that $M_{X} = D_{PC}$ .

Let $p_{e}$ be the probability of bit error in wireless network having n bits packet size. If single bit 1 is sent from Tx to Rx and if Rx is received erroneously as 0. Then, the probability of error for single bit will be given by Eq. (14). $\begin{matrix} (14) & P = [1 - {(1 - p_{e})}^{n = 1}] \end{matrix}$ Then for PC technique [2] if the packet size is n bits, the probability of packet error can be given by Eq. (15) $\begin{matrix} (15) & P_{PC} = {[1 - {(1 - p_{e})}^{n}]}^{2} \end{matrix}$ Thus, for MPC and APC techniques, the probability of packet error with n bits packet size will be given by Eqs (16) and (17) $\begin{array}{l} (16) & P_{MPC} = {[1 - {(1 - p_{e})}^{n}]}^{3} \\ (17) & P_{APC} = {[1 - {(1 - p_{e})}^{n}]}^{3} \end{array}$ Thus, the normalized throughput for the PC technique can be given by Eq. (18) $\begin{matrix} (18) & S_{eff (PC)} = \frac{(1 - P_{PC})}{1 + P_{PC} (D_{PC} - 1)} \end{matrix}$ Similarly, for the MPC and APC techniques the retransmissions of duplicate copies are occurred and therefore, the normalized throughput can be given by Eqs (19) and (20). $\begin{array}{l} (19) & S_{eff (MPC)} = \frac{(1 - P_{MPC})}{1 + P_{MPC} (D_{MPC} - 1)} \\ (20) & S_{eff (APC)} = \frac{(1 - P_{APC})}{1 + P_{APC} (D_{APC} - 1)} \end{array}$ For the hybrid (PC+MPC), the throughput efficiency can be given by Eq. (21) $\begin{matrix} (21) & S_{eff (PC + MPC)} = S_{eff (PC)} \times S_{eff (MPC)} \end{matrix}$ Similarly, for the hybrid (MPC+APC), the throughput efficiency can be given by Eq. (22) $\begin{matrix} (22) & S_{eff (MPC + APC)} = S_{eff (MPC)} \times S_{eff (APC)} \end{matrix}$ Where $D_{PC}$ , $D_{MPC}$ and $D_{APC}$ denote the total number of received duplicate erroneous copies by Rx for performing PC, MPC and APC techniques.

It is also assumed that the probability of the three states is equal and the sums of these probabilities are equal to unity. Therefore, the throughput of the proposed scheme without considering the probability of error correction capability can be obtained from Eq. (23) $\begin{matrix} (23) & S_{prop}^{″} = P (X) \cdot S_{eff (PC)} + P (Y) \cdot S_{eff (PC + MPC)} + P (Z) \cdot S_{eff (MPC + APC)} \end{matrix}$ It is assumed that the three erroneous copies are received and stored in the buffer. Let the packet size of the erroneous copy is ‘n’ and ‘ $p_{e}$ ’ is the probability of BER. Let $k_{r}$ denotes the number of error bits in the rth corrupted packets and $p (k_{r})$ is the probability of having $k_{r}$ error bits at the rth corrupted packets, where $r ⩾ 2$ [19].

The probability of successfully getting at least one original copy by the PC, MPC and APC techniques if they perform independently at the receiver end and it is given by Eq. (24) $\begin{matrix} (24) & {PEC}_{prop} = \frac{(1 - {PE}_{x}) \cdot {PE}_{PC} \cdot {PE}_{MPC} \cdot {PE}_{APC}}{{PE}_{x}} \end{matrix}$ Where ${PE}_{x}$ is the probability of packet error (PPE) and $x = PC, MPC and APC$

The probability of packet error correction (PPEC) of the proposed scheme will be given by Eq. (25) by considering that all the three erroneous copies have no hidden errors. $\begin{matrix} (25) & p_{r}^{prop} (e c) = [1 - \sum_{i = 1}^{r - 1} p_{i}^{prop}] \cdot [{(1 - p_{e})}^{n} + {(1 - (1 - p_{e}))}^{n} \cdot (1 - h_{r})] \cdot [1 - {[1 - {(1 - p_{e})}^{n}]}^{3} \cdot (1 - h_{r})] \end{matrix}$ It is assumed that the probability of two corrupted copies consist of $k_{i}$ and $k_{j}$ error bits ( $i < j$ ), if it has a hidden error in the proposed scheme and it is given by Eq. (26) $\begin{matrix} (26) & p_{i, j} (h) = [1 - \frac{\sum_{k_{i} = 1}^{n} \sum_{k_{j} = k_{i}}^{n - k_{i}} (\binom{n}{k_{i}}) (\binom{n - k_{i}}{k_{j}})}{\sum_{k_{i} = 1}^{n} \sum_{k_{j} = k_{i}}^{n} (\binom{n}{k_{i}}) (\binom{n}{k_{j}})}] \end{matrix}$ But the occurrences of error bits in the corrupted packets are independent events. Thus, the proposed work provides the conditional probability ( $h_{r}$ ) when the ${CP}_{i}^{err}$ , $i = 1, 2 and 3$ arrives at the receiver. Thus, $h_{r}$ can be obtained by Eq. (27) $\begin{matrix} (27) & h_{r} = [\frac{1}{{[1 - {(1 - p_{e})}^{n}]}^{r} \prod_{l = 2}^{r - 1} h_{l} [\sum_{k_{1} = 1}^{n} \sum_{k_{2} = 1}^{n} \dots \sum_{k_{r} = 1}^{n} p (k_{1}) p (k_{2}) \dots p (k_{r}) \prod_{1 ⩽ i, j ⩽ r} p_{i, j} (h)]}] \end{matrix}$ The throughput of the proposed scheme with the probability of error correction capability can be obtained from Eq. (23) and (25), and is given by Eq. (28) $\begin{matrix} (28) & S_{prop} = [P (L) \cdot S_{eff (PC)} + P (Y) \cdot S_{eff (PC + MPC)} + P (Z) \cdot S_{eff (MPC + APC)}] \times [1 - p_{r}^{prop} (e c)] \end{matrix}$ Let the energy consumption is higher if the probability of packet error is higher. Therefore, the energy consumption (E) of the proposed scheme can be determined by Eq. (29) $\begin{matrix} (29) & E = \frac{1}{[1 - (1 - p_{r}^{prop} (e c))]} \end{matrix}$

Fig. 6.

Throughput of P(X) = 0.5, P(Y) = 0.25, P(Z) = 0.25 with packet size 1024 bits.

Fig. 7.

Throughput of P(X) = 0.2, P(Y) = 0.4, P(Z) = 0.4 with packet size 1024 bits.

Fig. 8.

Packet throughput of proposed scheme with different probabilities states.

Fig. 9.

PPEC of packet size 256 bits.

Fig. 10.

PPEC of packet size 512 bits.

Fig. 11.

PPEC of packet size 1024 bits.

Fig. 12.

Energy consumption of 256 bits packet size.

Fig. 13.

Energy consumption of 512 bits packet size.

Fig. 14.

Energy consumption of 1024 bits packet size.

3. Results and discussion

In the simulation, Fig. 6 portrays that when the probability of three states, P(X) = 0.5, P(Y) = 0.25 and P(Z) = 0.25 are taken without considering the exact values of α, β, γ and δ, the packet throughput is found 0.43 i.e. 43%. Similarly, Fig. 7 provides the packet throughput of the proposed scheme which can be reached up to 0.38 or 38% when the P(X) = 0.2, P(Y) = 0.4 and P(Z) = 0.4. Thus, the results prove that the proposed scheme still obtains better results even if the P(X) state is low; i.e. the P(Z) and P(Y) are higher. Figure 8 provides packet throughput of the proposed scheme with different probability states. It illustrates that BER is higher when the probability of P(Z) and P(Y) are increased but the proposed scheme can also be adapted even if the P(Z) and P(Y) are higher as the packet throughput is higher if the BER is increased. Figures 9–11 show the probability of packet error correction of the proposed scheme with packet size of 256, 512 and 1024 bits and provide higher probability of packet error correction. Finally, with the same packet size (Figs 12–14), the proposed scheme obtained less energy consumption over conventional PC, MPC and APC when retransmission cycle is considered.

4. Conclusion

The three states X, Y and Z are considered in the proposed protocol. After implementing the PC, the hybrid PC+MPC and hybrid MPC+APC techniques are in X (good state), Y (average state) and Z (bad state) states respectively. It is reported from literature that if conventional PC, MPC or APC techniques fail to recover the original copy then the whole erroneous copies are discarded and the same procedure is repeated, which consumes extra communication energy and also leads to a reduction in the throughput. In order to reduce retransmission of duplicate copies in a retransmission cycle, the proposed protocol re-used the half-corrected copies that failed from conventional PC or MPC technique instead of discarding all copies. The proposed protocol is simulated in MATLAB. The results yield better performance over conventional PC, MPC and APC in retransmission cycle in terms of packet throughput, the probability of packet error correction and energy consumption. Tables 2–7 also detail the example of proposed protocol that can reduce the least reliable bits (the value of l) and hence the number of searching correct bit pattern is reduced from $2^{l} - 1$ to $2^{l - 1} - 1$ . Hence, the proposed protocol enhances the performance of conventional PC, MPC and APC and consumes relatively less power when retransmission cycle is considered.

Footnotes

Acknowledgement

This research work under “Visvesvaraya Ph. D Scheme” is financially supported by the Ministry of Electronics & Information Technology (MeitY), Government of India.

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Analysis of the adaptive three-modes for PC+MPC+APC techniques using retransmission cycle mechanism

Abstract

Keywords

1. Introduction

Table 1 Modified Packet Combining scheme Copies of erroneous pairs Erroneous bits (y) Erroneous detected position [C-2 ⊕ C-1] 1 bit 6th position from MSB [C-1 ⊕ C-3] 2 bits 5th and 7th positions from MSB [C-3 ⊕ C-2] 3 bits 5th, 6th and 7th positions from MSB

4. Conclusion

Footnotes

Acknowledgement

References

Table 1
Modified Packet Combining scheme

Copies of erroneous pairs Erroneous bits (y) Erroneous detected position

[C-2 ⊕ C-1] 1 bit 6th position from MSB

[C-1 ⊕ C-3] 2 bits 5th and 7th positions from MSB

[C-3 ⊕ C-2] 3 bits 5th, 6th and 7th positions from MSB