Abstract
Energy enhancement is a crucial factor while designing routing models in wireless sensor networks (WSNs). Many energy efficiency routing schemes are implemented to exchange various forms of gathered data by sensors in an optimal routing path through the network to increase its lifespan and maintain high scalability of the WSN. In this paper, a stochastic energy enhancement routing model is proposed to reduce the resource usage by nodes during the routing process. We aim to adapt the stochastic formalism based on the hidden Markov models (HMM) to learn from existing sensor networks, and design a new optimal routing mechanism that significantly exploits the available resources. Meanwhile, the proposed stochastic routing algorithm performs overall energy reduction in the network and permits optimal data transmission. The experimental results show that the proposed technique is efficient in terms of energy consumption, overall resource enhancement, and permits to increase network lifetime at least 19.05% compared to other existing methods.
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Appendices
Appendix 1: Stochastic routing parameters calculation
In order to illustrate our methodology, we used the numerical values reported in the sample graph of Fig. 2 computed with Eqs. (2), (9), and (10), which stand for initial probabilities, transition probabilities, and emission probabilities, respectively.
The \(\Pi \) vector defines in which state the system should be at the beginning, the A matrix reflects the connection between each state, and finally, B describes the probabilities that the system observes symbols L, M, H being in a given state.
We launch observation sequence steps Y randomly for the time \(T = 10\). Each entity \(Y_t\) represents the random path decision made at that time t and show as follows :
\(Y = \{Y_1 = v_1, Y_2 = v_2, Y_3 = v_3, Y_4 = v_3, Y_5 = v_2, Y_6 = v_1, Y_7 = v_1, Y_8 = v_2, Y_9 = v_3 , Y_{10} = v_1 \}\)
We suppose that we observe the short sequence below:
Y sequence represents a random observation selected symbols where each symbol specifies the level of the energies in the system. The probability to observe the sequence given the model \(\lambda \) is estimated in Forward algorithm. Symbols L, M, and H correspond to energy Low, Medium, and High, respectively. Each symbol V illustrates the energy level in which each state can be: \(V = {v_1 = Low(L), v_2 = Medium(M), v_3 = High(H) }.\)
Fully connected HMM trellis
In Fig. 8, the transitions coefficients and initial vector for the proposed model are illustrated. The graph is interconnected by the fact that all transition coefficients between hidden states are not null.
The transition coefficients between states are provided in Table 8.
Table 7 deals with transition between hidden states. Hidden states stand for the energy consumed by nodes corresponding in the shortest path. \(E_1\) for energy in shortest path 1 or state 1, \(E_2\) for energy consumed by nodes in shortest path 2, and \(E_3\) for the energy consumed by nodes in state 3. This part is a completion of the Sect. 4.4.1 where we describe our Fully Connected Hidden Markov Model. These coefficients are then obtained based on the equation proposed to estimate the initial parameters. The K=3 shortest paths between source and destination nodes with a best minimal energy according to the network graph (Figs. 2, 3).
Compute \(\alpha \) process for the observation sequence. This evaluation is based on the sequence of the symbols observed in the sequence \(Y_t\)
Appendix 2: Forward calculation
The main steps to compute forward probabilities consist of an initialization, induction, and termination as follows:
1.1 Initialization
\(\alpha _1(E_1) = \Pi _{E_1}.b_{E_1}(Y_1 = v_1(L)) = 0.332 \times 0.668 = 0.304776\)
\(\alpha _1(E_2) = \Pi _{E_2}.b_{E_2}(Y_1 = v_1(L)) = 0.333 \times 0.082 = 0.027306\)
\(\alpha _1(E_3) = \Pi _{E_3}.b_{E_3}(Y_1 = v_1(L)) = 0.335 \times 0.250 = 0.08375\)
1.2 Induction
\(\alpha _2(j) = b_j({Y_2}).\sum _{i=1}^{N=3} \alpha _1(i).a_{ij}\)
\(\alpha _2(E_1) = b_{E_1}({Y_2=v_2(M)}).[ \alpha _1(E_1).a_{11} + \alpha _1(E_2).a_{21} + \alpha _1(E_3).a_{31}] \)
\(\alpha _2(E_2) = b_{E_1}({Y_2=v_2(M)}).[ \alpha _1(E_1).a_{12} + \alpha _1(E_2).a_{22} + \alpha _1(E_3).a_{32}] \)
\(\alpha _2(E_3) = b_{E_1}({Y_2=v_2(M)}).[ \alpha _1(E_1).a_{13} + \alpha _1(E_2).a_{23} + \alpha _1(E_3).a_{33}] \)
\(\quad ...\)
\(\alpha _{10}(j) = b_j(Y_{10}).\sum _{i=10}^{10} \alpha _9(i).a_{ij}\)
1.3 Termination
The termination process gives likelihood probability to get the best observe sequence. Equation (23) is used.
At each time (t), we observe a symbol. That means the system can have an energy level depending on a probability. At the time \(T=10\), a summation is made to get the probability to observe our random sequence given the initial model \(\lambda \).
In Table 9, we present the results of the forward process around the time. In the table, the results of Forward algorithm, the \(\alpha \) (see Fig. 9), are the probabilities to observe a symbol \(v_k\) in each hidden state based on the initial random observation sequence. It is important to show that because these numerical values are used to define the Best Routing Probability (BRP) of the stochastic model.
In Fig. 9, we compute the maximal score to get the best path corresponding to the observation sequence Y. That means, find the best sequence in the model \(\lambda \) that maximize \(P(X, Y / \lambda )\) (see Eq. 13). Based on that process and other intermediate one, we forecast that the best hidden state that produced the initial sequence Y with a high score is that observed in Table 7.
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Affane, A.R., Satori, H., Sanhaji, F. et al. Energy enhancement of routing protocol with hidden Markov model in wireless sensor networks. Neural Comput & Applic 35, 5381–5393 (2023). https://doi.org/10.1007/s00521-022-07970-3
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DOI: https://doi.org/10.1007/s00521-022-07970-3