Theoretical modeling for performance analysis of IEEE 1901 power-line communication networks in the multi-hop environment

As one of the most promising networking technologies, power-line communication (PLC) networks have been gradually used not only in home networking systems but also in industrial IoT (Internet of things). Current studies of PLC medium access control (MAC) protocol (i.e., IEEE 1901) only focus on the single-hop environment; however, in practical industrial IoT and home access communication systems, PLC networks generally utilize a multi-hop architecture. In addition, due to the difference in the MAC standard between 802 series and 1901, existing analytical models of multi-hop IEEE 802.11 wireless networks are not suitable for multi-hop IEEE 1901 PLC networks. In this paper, we propose a theoretical model for performance analysis of multi-hop IEEE 1901 PLC networks, where the impacts of traffic rate (containing relay traffic), buffer size, deferral counter process of 1901, hidden terminal problem and multi-hop environment are comprehensively considered. The modeling process is divided into two parts. In the local modeling part, we construct a brand-new Markov chain model to investigate the carrier sense multiple access with collision avoidance process of IEEE 1901 protocol under the multi-hop environment. In the coupling queueing modeling part, we employ queueing theory to analyze the packet transmission procedure between successive hops. On the basis, we derive the closed-form expressions of throughput, medium access delay, packet blocking probability, end-to-end successful transmission delay and goodput. Through extensive simulations, we verify that our theoretical model can accurately estimate the MAC performance of IEEE 1901 PLC networks in the multi-hop environment.

1. The carrier sensing region is assumed to be of circle shape.

2. Although both 2-D DMC and ML DMC models can describe 1901’s CSMA/CA mechanism, the detailed mathematical description for CSMA/CA process of 1901 protocol in multi-hop environment is different from that in single-hop environment.

3. The complexity of a Markov chain model is determined by the size of its corresponding balance equations, i.e., the size of state space. It should be stressed that this definition is not equal to the computational complexity of algorithm.

4. 1901 allows unlimited retransmission attempts.

5. This assumption relies on the fact that the 1901 standard does not use the request to send RTS frame, and the short inter-frame space SIFS is assumed to be zero.

6. In most network configurations, this probability is very small and negligible. Hence, we let it be zero so that we can derive the detailed expressions for \(T_s\) and \(T_c\).

7. \(Prb\{A\}\) denotes the probability that case A happens.

8. According to the 1901 standard, only the fully loaded station buffer can cause the drop packet problem.

9. We do not need to measure the end-to-end successful transmission delay \(Max\{T_{0H},T'_{0H}\}\), since there is a strict corresponding relation between G and \(Max\{T_{0H},T'_{0H}\}\).

10. Note that there is still a deviation between simulation results and numerical analysis results, since the DCP of 1901 can enhance the system jitter.

11. If a station triggers the DCP at backoff stage k, it has to jump to the next backoff stage (or reenters this backoff stage, if it is already at the last backoff stage).

The author would like to thank the editor and four anonymous reviewers for helpful comments that have improved the quality of the paper. This work is supported by the National Natural Science Foundation of China (No. 61772386).

Authors and Affiliations

1. School of Computer, Wuhan University, Wuhan, 430079, Hubei Province, People’s Republic of China

   Sheng Hao & Hu-yin Zhang

Corresponding author

Correspondence to Hu-yin Zhang.

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Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix 1: The derivation process of \(\tau _i\) for station i

Let \(\varPsi =\varSigma ^m_{k=1}b_{k,1}\cdot (1-p_i)(1-\mu ^i_0)+E(1-\nu _i)\), we first deduce the expressions of \(b_{1,r}\) and \(d_{1,g}\):

Based on the above derivation, we further let \(H^i_2=b_{1,1}p_i+d_{1,1}\), \(b_{2,r}\) and \(d_{2,g}\) be derived as follows:

By using recursion method, the generalized expressions of \(b_{k,r}\) and \(d_{k,g}\) (\(k\in [2,m-1]\)) can be written as:

where \(H_k\) is given by

To investigate the mathematical relation of \(\varPsi \) and \(H^i_k\) (\(k\in [2,m-1]\)), we first extend the expression of \(H^i_2\)

Similarly, we use recursion method to get the generalized expression of \(H^i_k\) (\(k\in [2,m-1]\))

Considering the reentrancy of the last backoff stage, let \(H^i_m=b_{m,1}p_i+d_{m,1}\). \(b_{m,r}\) and \(d_{m,g}\) be represented as:

Based on the definition of \(H^i_m\), we can build the relation of \(H^i_m\) and \(H^i_{m-1}\) as follows:

Solving Eq. 57, we have

In addition, E can be expressed as

Let \(F^i_k=H^i_k\cdot \varPsi ^{-1}\), then we can get Eq. 2 and finally write \(\tau _i=\sum ^m_{k=1}b_{k,1}\) as Eq. 1.

Appendix 2: The derivation process of Prb \(\{T^i_{mac}=(r-1)E[slot]+t_s\}\) (Eqs. 23–25)

\(Prb\{T_{mac}=(r-1)E[slot]+t_s\}\) denotes the probability that a successful transmission occurs at a backoff stage (\(k\in [1,m]\)) and just needs \(r-1\) slot times and a successful duration \(t_s\) (since the packet can be definitely successfully transmitted in the last time slot). Therefore, \(Prb\{T^i_{mac}=(r-1)E[slot]+t_s\}\) is determined by the probability that station i jumps from the initial state E to backoff stage k (\(k\in [1,m]\)), and the probability that the chosen backoff counter at backoff stage k is r (\(r\in [1,W_k]\)) without triggering the DCP.^{Footnote 11}

Firstly, we assume that the successful transmission occurs at backoff stage 1 and needs \((r-1)E[slot]+t_s\) slot times. Let \(P^i_{J_{(E\rightharpoonup k)}}\) be the probability that station i jumps from the empty state E to backoff stage k (\(k\in [1,m]\)), we have

As mentioned earlier (see Sect. 4.1), the probability that the chosen backoff counter at backoff stage k is r (note that if a successful transmission requires \((r-1)E[slot]+t_s\) slot times, we need to choose r as the initial backoff counter, since \(\lceil \frac{(r-1)E[slot]+t_s}{E[slot]}\rceil =r\), \(r\in [1,W_k]\)) without triggering the DCP can be, respectively, given by

Thus, for (\(k=1\)), \(Prb\{T^i_{mac}=(r-1)E[slot]+t_s\}\) can be expressed as Eq. 23.

Based on the 1901 standard, a station jumps to the next backoff stage through attempting an unsuccessful transmission or triggering the DCP. Thus, the probability \(P^i_{J_{(k\rightharpoonup k+1)}}\) that station i at backoff stage k jumps to backoff stage \(k+1\) can be written as

Because of the reentrancy of the last backoff stage m (caused by an unsuccessful transmission attempt or triggering the DCP), the probability that station i jumps from the empty state E to backoff stage k (\(k\in [2,m]\)) can be, respectively, expressed as follows:

For (\(k\in [2,m-1]\))

For (\(k=m\))

Through the above analysis, if the successful transmission occurs at backoff stage k (\(k\in [2,m]\)) and needs \((r-1)E[slot]+t_s\) slot times, \(Prb\{T^i_{mac}=(r-1)E[slot]+t_s\}\) can be, respectively, expressed as Eqs. 24 and 25.

Appendix 3: The computational complexity analysis about executing IEEE 1901 MAC protocol and solving the theoretical model

Firstly, we analyze the computational complexity of executing IEEE 1901 MAC protocol. In a multi-hop PLC network, each station independently executes the CSMA/CA process of IEEE 1901 protocol, and the pseudocode can be given as follows:

Analyzing this algorithm, it can be regarded as a modified CSMA/CA algorithm, and its computational complexity is determined by the network size and the expected number of backoff counters consumed by each station to occupy the medium.

As mentioned in Appendices (1 or 2), the IEEE 1901 protocol allows unlimited retransmission attempts; in other words, whether the expected number of backoff counters consumed by each station to occupy the medium is a constant should be examined. Recalling the derivation of medium access delay (Eqs. 23–26 and Appendix 2), the expected number of backoff counters consumed by each station to occupy the PLC medium C can be written as

Further reviewing Eq. 64 in Appendix 2, we can find that although unlimited retransmission attempts are allowed by IEEE 1901, the corresponding probability that a station transmits the packet at the last backoff stage m is still convergent. Accordingly, it is certain that C is a constant.

Thus, for the multi-hop PLC network, the computational complexity of executing IEEE 1901 protocol can be written as

where N represents the network size.

Secondly, we analyze the computational complexity of solving our theoretical model.

Our model provides a theoretical framework to analyze the MAC performance for multi-hop IEEE 1901 PLC networks. To solve the results of this model, we need to calculate \(p_i\), \(\mu ^i_0\) and \(\nu ^i\). Many numerical methods can be used to find the values of \(p_i\), \(\mu ^i_0\) and \(\nu ^i\). Here, fixed-point iteration (FPI) method is employed, and the pseudocode is as follows:

Clearly, if we set a reasonable \(\theta \), we can finally derive the results of the theoretical model in a finite number of iteration times (i.e., a constant) \(C_{iteration}\). Thus, the computational complexity of solving the theoretical model can be given by

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