Table 2 Pseudo-code for the proposed algorithm for calculating the reliability of a WSN S

Set all parameters (S = {s _i }, T = {t _j }, types of SNs, sink location, \( {\lambda}_s^i \), \( {\lambda}_t^i \), \( {\lambda}_p^i \) and \( {\lambda}_b^i \) for i = 1 , … , n and j = 1 , … , m)

Initialize R =Prob(π |s _i  ∈ S is in on mode ∀i = 1 , … , n)

Initialize Π ₁ = {(π |s _i  ∈ S is in on mode ∀i = 1 , … , n)}

Let k be the number of SNs in relay mode. Initialize k = 1.

Let \( {\mathcal{F}}_r^k \) be a k−combination of SNs in relay mode. Let \( {\boldsymbol{F}}_r^k \) be the set of k−combinations of SNs in relay mode that S can tolerate. Initialize\( \kern0.5em {\mathcal{F}}_r^k={\boldsymbol{F}}_r^k=\left\{\phi \right\} \).

For i = 1 , … , n

- Let s _i be in relay mode, i.e. \( {\mathcal{F}}_r^k=\left\{{s}_i\right\} \)

- Evaluate\( f\left(\boldsymbol{\pi} |\ {\mathcal{F}}_r^k\right) \) using (10)

- If \( f\left(\boldsymbol{\pi} |\ {\mathcal{F}}_r^k\right)=1\to {\boldsymbol{F}}_r^k={\boldsymbol{F}}_r^k\cup {\mathcal{F}}_r^k \)

End For loop

While \( {\boldsymbol{F}}_r^k\ne \left\{\phi \right\}\kern1em \to \kern0.75em k=k+1 \), Let \( {F}_{rl}^{k-1}\in {\boldsymbol{F}}_r^{k-1}, \)

\( \kern0.5em {\mathcal{F}}_r^k={\boldsymbol{F}}_r^k=\left\{\phi \right\} \)

For \( l=1,\dots, \left|\ {\boldsymbol{F}}_r^{k-1}\right| \) and i = 1 , … , n

- Let \( {\mathcal{F}}_{\mathrm{r}}^{\mathrm{k}}=\left\{\ {F}_{rl}^{k-1},{s}_i\kern0.5em \right\} \)

- Evaluate\( \mathrm{f}\left(\uppi |\ {\mathcal{F}}_{\mathrm{r}}^{\mathrm{k}}\right) \) using (10)

- If \( f\left(\pi |\ {\mathcal{F}}_r^k\right)=1\to {\boldsymbol{F}}_r^k={\boldsymbol{F}}_r^k\cup {\mathcal{F}}_r^k \)

End For loops, End While loop

Let k be the number of SNs in off mode. Initialize k = 1.

Let \( {\mathcal{F}}_o^k \) be a k−combination of SNs in off mode. Let \( {\boldsymbol{F}}_o^k \) be the set of k−combinations of SNs in off mode that S can tolerate. Initialize\( \kern0.50em {\mathcal{F}}_o^k={\boldsymbol{F}}_o^k=\left\{\phi \right\} \).

Repeat step 2.c. for off mode, i.e. \( {\mathcal{F}}_o^k=\left\{{s}_i\right\} \)

While \( {\boldsymbol{F}}_o^k\ne \left\{\phi \right\}\kern1em \to \kern0.75em k=k+1 \), Let \( {F}_{ol}^{k-1}\in {\boldsymbol{F}}_o^{k-1}, \)

\( \kern0.5em {\mathcal{F}}_o^k={\boldsymbol{F}}_o^k=\left\{\phi \right\} \)

Repeat 2.e. using \( {F}_{ol}^{k-1} \) and\( {\mathcal{F}}_o^k \) to get \( {\boldsymbol{F}}_o^k \)

End For loops, End While loop

Let ℱ _r and ℱ _o be a combination of SNs in relay and off modes respectively.

Let F _r and F _o be the sets of all combinations of SNs of in relay and off mode that that S can tolerate respectively.

Let \( {F}_{rl_r}\in {\boldsymbol{F}}_r \) and \( {F}_{ol_o}\in {\boldsymbol{F}}_o \)

For l _r  = 1 , … , |F _r | and l _o  = 1 , … , |F _o |

- Let \( {\mathcal{F}}_r={F}_{rl_r} \) and \( {\mathcal{F}}_o={F}_{ol_o} \)

- Evaluate f(π| ℱ _r , ℱ _o  ) using (10)

- If f(π| ℱ _r, ℱ _o ) = 1→ Π ₁ = Π ₁ ∪ π

End For loops

Let π _l  ∈ Π ₁

For l = 1 , … , |Π ₁|

- R(S) = R(S) + Prob(π _l )

End For loop

Output:  R(S)