Reliable networking in Ethernet ring mesh networks using regular topologies

Abstract

Reliable networking is an important factor in Ethernet ring mesh networks (ERMs) with ITU-T G.8032 Ethernet ring protection recommendation or with the IEC 62439-3 high-availability seamless redundancy protocol. However, there are two major challenges for this purpose: (1) Hitherto, irregular topologies are used in ERMs that it causes difficulty in analyzing and improving reliability. (2) A topology is appropriate for practical implementation, if it is low cost, simple to implement, and simple to develop for large scale-systems (i.e. scalable). However, irregular topologies are extremely difficult to implement and develop. To deal with these challenges, this paper introduces two regular topologies called chordal ring and k-cube networks, for the first time in the area of ERMs. In addition, the proposed topologies are carefully analyzed in terms of reliability. These analyzes prove that the proposed regular topologies outperform existing irregular multiple-ring networks namely shared link, shared node, complex shared link, redundant link, and 3-connected network.

Appendix

Here, how to use the decomposition method (DM) for calculating the all-terminal reliability equations in Sect. 4.1 will be described in detail.

• (1) Shared link topology reliability (Eq. 4): First, we examine Eq. (4), which is related to the reliability of shared link topology. How to choose DM key components is shown appropriately by a hierarchical graph in Fig. 6. In this figure, the key components are labeled by $k_{1}$, $k_{2}$, $k_{3}$, and $k_{4}$, in the order they were selected. Two states corresponding to each key component (healthy or failed) is also shown in Fig. 6. For example, $\overline{{k_{1} }}$ is indicative of the failure of first key component.

  Fig. 6

  

  The hierarchy of the key components in the decomposition method for shared link topology

Using Fig. 6, we can calculate the network reliability related to different network states according to different states of key components that are shown by a sequence of key components and their states:

$$ \overline{{k_{1} }} = 6e^{ - 5\lambda t} - 5e^{ - 6\lambda t} $$

(33)

$$ k_{1} \overline{{k_{2} }} = e^{ - 2\lambda t} \left( {3e^{ - 2\lambda t} - 2e^{ - 3\lambda t} } \right) $$

(34)

$$ k_{1} k_{2} \overline{{k_{3} }} = e^{ - \lambda t} \left( {3e^{ - 2\lambda t} - 2e^{ - 3\lambda t} } \right) $$

(35)

$$ k_{1} k_{2} k_{3} \overline{{k_{4} }} = 3e^{ - 2\lambda t} - 2e^{ - 3\lambda t} $$

(36)

$$ k_{1} k_{2} k_{3} k_{4} = 3e^{ - 2\lambda t} - 2e^{ - 3\lambda t} $$

(37)

Now, according to Eqs. (33)–(37), the reliability of the entire network can be calculated as follows (i.e. equal to the Eq. (4) in the previous section):

$$ \begin{aligned} R_{shared link} \left( t \right) & = \left( {1 - e^{ - \lambda t} } \right)\left( {6e^{ - 5\lambda t} - 5e^{ - 6\lambda t} } \right) \\ & \quad + {\kern 1pt} e^{ - \lambda t} \left( {e^{ - 2\lambda t} \left( {3e^{ - 2\lambda t} - 2e^{ - 3\lambda t} } \right)\left( {1 - e^{ - \lambda t} } \right)} \right. \\ & \quad + {\kern 1pt} e^{ - \lambda t} \left( {e^{ - \lambda t} \left( {3e^{ - 2\lambda t} - 2e^{ - 3\lambda t} } \right)\left( {1 - e^{ - \lambda t} } \right)} \right. \\ & \quad + {\kern 1pt} e^{ - \lambda t} \left. \left. \left( \left( {3e^{ - 2\lambda t} - 2e^{ - 3\lambda t} } \right)\left( {1 - e^{ - \lambda t} } \right) \right.\right.\right.\\ &\quad + e^{ - \lambda t}\left. \left.\left. \left( {3e^{ - 2\lambda t} - 2e^{ - 3\lambda t} } \right) \right) \right) \right) \\ & = 15e^{ - 5\lambda t} - 23e^{ - 6\lambda t} + 9e^{ - 7\lambda t} \\ \end{aligned} $$

(38)

• (2) Shared node topology (Eq. 5): The hierarchy diagram of the key components selection for shared node topology is shown in Fig. 7. Considering this figure, we have:

  Fig. 7

  

  The hierarchy of the key components in the decomposition method for shared node topology

  $$ \overline{{k_{1} }} = e^{ - 3\lambda t} \left( {4e^{ - 3\lambda t} - 3e^{ - 4\lambda t} } \right) $$

  (39)
  $$ k_{1} \overline{{k_{2} }} = e^{ - 2\lambda t} \left( {4e^{ - 3\lambda t} - 3e^{ - 4\lambda t} } \right) $$

  (40)
  $$ k_{1} k_{2} \overline{{k_{3} }} = e^{ - \lambda t} \left( {4e^{ - 3\lambda t} - 3e^{ - 4\lambda t} } \right) $$

  (41)
  $$ k_{1} k_{2} k_{3} \overline{{k_{4} }} = 4e^{ - 3\lambda t} - 3e^{ - 4\lambda t} $$

  (42)
  $$ k_{1} k_{2} k_{3} k_{4} = 4e^{ - 3\lambda t} - 3e^{ - 4\lambda t} $$

  (43)

As a result, according to Eqs. (39)–(43), the reliability of the entire network is given by (i.e. equal to the Eq. 5):

$$ \begin{aligned} R_{shared node} \left( t \right) & = \left( {1 - e^{ - \lambda t} } \right)\left( {e^{ - 3\lambda t} \left( {4e^{ - 3\lambda t} - 3e^{ - 4\lambda t} } \right)} \right) \\ & \quad + {\kern 1pt} e^{ - \lambda t} \left( {e^{ - 2\lambda t} \left( {4e^{ - 3\lambda t} - 3e^{ - 4\lambda t} } \right)\left( {1 - e^{ - \lambda t} } \right)} \right. \\ & \quad + {\kern 1pt} e^{ - \lambda t} \left( {e^{ - \lambda t} \left( {4e^{ - 3\lambda t} - 3e^{ - 4\lambda t} } \right)\left( {1 - e^{ - \lambda t} } \right)} \right. \\ & \quad + {\kern 1pt} e^{ - \lambda t} \left. \left. \left( \left( {4e^{ - 3\lambda t} - 3e^{ - 4\lambda t} } \right)\left( {1 - e^{ - \lambda t} } \right) \right.\right.\right.\\ &\quad + e^{ - \lambda t} \left.\left.\left. \left( {4e^{ - 3\lambda t} - 3e^{ - 4\lambda t} } \right) \right) \right) \right) \\ & = 16e^{ - 6\lambda t} - 24e^{ - 7\lambda t} + 9e^{ - 8\lambda t} \\ \end{aligned} $$

(44)

• (3) Complex shared link topology (Eq. 6): The hierarchy diagram of the key components selection for complex shared link topology is shown in Fig. 8. Considering this figure, network reliability for different modes is as follows:

  Fig. 8

  

  The hierarchy of the key components in the decomposition method for complex shared link topology

  $$ \overline{{k_{1} }} = e^{ - 2\lambda t} \left( {15e^{ - 5\lambda t} - 23e^{ - 6\lambda t} + 9e^{ - 7\lambda t} } \right) $$

  (45)
  $$ k_{1} \overline{{k_{2} }} = e^{ - \lambda t} \left( {15e^{ - 5\lambda t} - 23e^{ - 6\lambda t} + 9e^{ - 7\lambda t} } \right) $$

  (46)
  $$ k_{1} k_{2} \overline{{k_{3} }} = 15e^{ - 5\lambda t} - 23e^{ - 6\lambda t} + 9e^{ - 7\lambda t} $$

  (47)
  $$ \begin{aligned} k_{1} k_{2} k_{3} & = \left( {1 - e^{ - \lambda t} } \right)\left( {e^{ - \lambda t} \left( {1 - e^{ - \lambda t} } \right)\left( {4e^{ - 3\lambda t} - 3e^{ - 4\lambda t} } \right)} \right. \\ & \quad \left. +\, {\kern 1pt} e^{ - \lambda t} \left( \left( {1 - e^{ - \lambda t} } \right)\left( {4e^{ - 3\lambda t} - 3e^{ - 4\lambda t} } \right) \right.\right.\\ &\quad + e^{ - \lambda t}\left.\left. \left( {4e^{ - 2\lambda t} - 4e^{ - 3\lambda t} + e^{ - 4\lambda t} } \right) \right) \right) \\ & \quad + {\kern 1pt} e^{ - \lambda t} \left( \left( {1 - e^{ - \lambda t} } \right)\left( {4e^{ - 3\lambda t} - 3e^{ - 4\lambda t} } \right) \right. \\ &\quad + e^{ - \lambda t} \left. \left( {4e^{ - 2\lambda t} - 4e^{ - 3\lambda t} + e^{ - 4\lambda t} } \right) \right) \\ & = 20e^{ - 4\lambda t} - 41e^{ - 5\lambda t} + 29e^{ - 6\lambda t} - 7e^{ - 7\lambda t} \\ \end{aligned} $$

  (48)

Therefore, according to the Eqs. (45)–(48), we have:

$$ \begin{aligned} & R_{complex shared link} \left( t \right) \\ & \quad = \left( {1 - e^{ - \lambda t} } \right)\left( {e^{ - 2\lambda t} \left( {15e^{ - 5\lambda t} - 23e^{ - 6\lambda t} + 9e^{ - 7\lambda t} } \right)} \right) \\ & \qquad + {\kern 1pt} e^{ - \lambda t} \left( {\left( {15e^{ - 5\lambda t} - 23e^{ - 6\lambda t} + 9e^{ - 7\lambda t} } \right)\left( {1 - e^{ - \lambda t} } \right)} \right. \\ & \qquad + {\kern 1pt} e^{ - \lambda t} \left( {\left( {15e^{ - 5\lambda t} - 23e^{ - 6\lambda t} + 9e^{ - 7\lambda t} } \right)\left( {1 - e^{ - \lambda t} } \right)} \right. \\ & \qquad + {\kern 1pt} e^{ - \lambda t} \left( {\left( {1 - e^{ - \lambda t} } \right)\left( {e^{ - \lambda t} \left( {1 - e^{ - \lambda t} } \right)\left( {4e^{ - 3\lambda t} - 3e^{ - 4\lambda t} } \right)} \right.} \right. \\ & \qquad + {\kern 1pt} e^{ - \lambda t} \left( \left( {1 - e^{ - \lambda t} } \right)\left( {4e^{ - 3\lambda t} - 3e^{ - 4\lambda t} } \right) \right. \\ &\qquad +\, e^{ - \lambda t} \left. \left. {\left. {\left( {4e^{ - 2\lambda t} - 4e^{ - 3\lambda t} + e^{ - 4\lambda t} } \right)} \right)} \right) \right. \\ & \qquad +\, {\kern 1pt} e^{ - \lambda t} \left( \left( {1 - e^{ - \lambda t} } \right)\left( {4e^{ - 3\lambda t} - 3e^{ - 4\lambda t} } \right) \right.\\ &\qquad + e^{ - \lambda t} \left. \right.\left. {\left. {\left. {\left. {\left( {4e^{ - 2\lambda t} - 4e^{ - 3\lambda t} + e^{ - 4\lambda t} } \right)} \right)} \right)} \right)} \right) \\ & \quad = 65e^{ - 7\lambda t} - 155e^{ - 8\lambda t} + 125e^{ - 9\lambda t} - 34e^{ - 10\lambda t} \\ \end{aligned} $$

(49)

• (4) Redundant link topology (Eq. 7): Based on the hierarchy of the key components selection for redundant topology that is shown in Fig. 9, network reliability related to different network states according to different states of key components are given by:

  Fig. 9

  

  The hierarchy of the key components in the decomposition method for redundant link topology

  $$ \overline{{k_{1} }} = 15e^{ - 5\lambda t} - 23e^{ - 6\lambda t} + 9e^{ - 7\lambda t} $$

  (50)
  $$ k_{1} \overline{{k_{2} }} = e^{ - 2\lambda t} \left( {3e^{ - 2\lambda t} - 2e^{ - 3\lambda t} } \right) $$

  (51)
  $$ k_{1} k_{2} \overline{{k_{3} }} = e^{ - \lambda t} \left( {3e^{ - 2\lambda t} - 2e^{ - 3\lambda t} } \right) $$

  (52)
  $$ k_{1} k_{2} k_{3} \overline{{k_{4} }} = 3e^{ - 2\lambda t} - 2e^{ - 3\lambda t} $$

  (53)
  $$ k_{1} k_{2} k_{3} k_{4} = 3e^{ - 2\lambda t} - 2e^{ - 3\lambda t} $$

  (54)

According to the above equations, the reliability of the entire network is calculated as follows:

$$ \begin{aligned} & R_{redundant link} \left( t \right) \\ & \quad = \left( {1 - e^{ - \lambda t} } \right)\left( {15e^{ - 5\lambda t} - 23e^{ - 6\lambda t} + 9e^{ - 7\lambda t} } \right) \\ & \quad\quad\quad + e^{ - \lambda t} \left( {e^{ - 2\lambda t} \left( {3e^{ - 2\lambda t} - 2e^{ - 3\lambda t} } \right)\left( {1 - e^{ - \lambda t} } \right)} \right. \\ & \quad\quad\quad + e^{ - \lambda t} \left( {e^{ - \lambda t} \left( {3e^{ - 2\lambda t} - 2e^{ - 3\lambda t} } \right)\left( {1 - e^{ - \lambda t} } \right)} \right. \\ & \left. \left. \quad\quad\quad +\, e^{ - \lambda t} \left( \left( {3e^{ - 2\lambda t} - 2e^{ - 3\lambda t} } \right)\left( {1 - e^{ - \lambda t} } \right) \right.\right.\right.\\ &\quad\quad\quad \left.\left.\left. +\, e^{ - \lambda t} \left( {3e^{ - 2\lambda t} - 2e^{ - 3\lambda t} } \right) \right) \right) \right) \\ & \quad = 24e^{ - 5\lambda t} - 50e^{ - 6\lambda t} + 36e^{ - 7\lambda t} - 9e^{ - 8\lambda t} \\ \end{aligned} $$

(55)

• (5) 3-Connected network (Eq. 8): How to choose key components in decomposition method for the 3-connected network is shown in Fig. 10. Based on this figure, network reliability for different modes of key components is given by:

  Fig. 10

  

  The hierarchy of the key components in the decomposition method for 3-connected network

  $$ \overline{{k_{1} }} \overline{{k_{2} }} = E_{1} = 56e^{ - 7\lambda t} - 130e^{ - 8\lambda t} + 102e^{ - 9\lambda t} - 27e^{ - 10\lambda t} $$

  (56)
  $$ \overline{{k_{1} }} k_{2} \overline{{k_{3} }} = E_{2} = 40e^{ - 6\lambda t} - 90e^{ - 7\lambda t} + 69e^{ - 8\lambda t} - 18e^{ - 9\lambda t} $$

  (57)
  $$ \overline{{k_{1} }} k_{2} k_{3} \overline{{k_{4} }} = E_{3} = 32e^{ - 5\lambda t} - 68e^{ - 6\lambda t} + 49e^{ - 7\lambda t} - 12e^{ - 8\lambda t} $$

  (58)
  $$ \begin{aligned} \overline{{k_{1} }} k_{2} k_{3} k_{4} & = E_{4} = 32e^{ - 4\lambda t} - 84e^{ - 5\lambda t} \\ &\quad + 86e^{ - 6\lambda t} - 40e^{ - 7\lambda t} + 7e^{ - 8\lambda t} \end{aligned} $$

  (59)
  $$ k_{1} \overline{{k_{2} }} \overline{{k_{4} }} = E_{2} = 40e^{ - 6\lambda t} - 90e^{ - 7\lambda t} + 69e^{ - 8\lambda t} - 18e^{ - 9\lambda t} $$

  (60)
  $$ k_{1} \overline{{k_{2} }} k_{4} \overline{{k_{3} }} = E_{3} = 32e^{ - 5\lambda t} - 68e^{ - 6\lambda t} + 49e^{ - 7\lambda t} - 12e^{ - 8\lambda t} $$

  (61)
  $$\begin{aligned} k_{1} \overline{{k_{2} }} k_{4} k_{3} & = E_{4} = 32e^{ - 4\lambda t} - 84e^{ - 5\lambda t} \\ &\quad + 86e^{ - 6\lambda t} - 40e^{ - 7\lambda t} + 7e^{ - 8\lambda t} \end{aligned} $$

  (62)
  $$\begin{aligned} k_{1} k_{2} & = E_{5} = 112e^{ - 5\lambda t} - 398e^{ - 6\lambda t} + 582e^{ - 7\lambda t} \\ &\quad - 434e^{ - 8\lambda t} + 164e^{ - 9\lambda t} - 25e^{ - 10\lambda t} \end{aligned} $$

  (63)

Considering the Eqs. (56)–(63), the reliability of the 3-connected network is calculated as Eq. (64):

$$ \begin{aligned} & R_{3 - connected} \left( t \right) \\ & \quad = \left( {1 - e^{ - \lambda t} } \right)\left( \left( {1 - e^{ - \lambda t} } \right)\left( {E_{1} } \right) +\, e^{ - \lambda t} \left( \left( {1 - e^{ - \lambda t} } \right)\left( {E_{2} } \right) \right.\right.\\ &\qquad + e^{ - \lambda t} \left.\left. \left( {\left( {1 - e^{ - \lambda t} } \right)\left( {E_{3} } \right) + e^{ - \lambda t} \left( {E_{4} } \right)} \right) \right) \right) \\ & \qquad + e^{ - \lambda t} \left( \left( {1 - e^{ - \lambda t} } \right)\left( \left( {1 - e^{ - \lambda t} } \right)\left( {E_{2} } \right) \right.\right.\\ &\qquad + e^{ - \lambda t} \left.\left.\left( {\left( {1 - e^{ - \lambda t} } \right)\left( {E_{3} } \right) + e^{ - \lambda t} \left( {E_{4} } \right)} \right) \right) +\, e^{ - \lambda t} \left( {E_{5} } \right) \right) \\ & \quad = 376e^{ - 7\lambda t} - 1476e^{ - 8\lambda t} + 2352e^{ - 9\lambda t} \\ &\qquad - 1895e^{ - 10\lambda t} + 770e^{ - 11\lambda t} - 126e^{ - 12\lambda t} \\ \end{aligned} $$

(64)

• (6) Chordal rings (Eqs. (25)–(28)): To calculate the reliability of the chordal rings of $CHR4\left( {8; 2} \right)$ and $CHR4\left( {16; 2} \right)$, two cases have been considered: When all the chord links have failed (lower bound) and when all the chord links are active and healthy (upper bound). In case of lower bound for a $CHR4\left( {p; q} \right)$, network reliability is the reliability of a $p$-node single-ring network. Reliability block diagram (RBD) related to the lower bound is shown in Fig. 11(a). In addition, in the event of upper bound for a $CHR4\left( {p; q} \right)$, network reliability is equal to the reliability of a parallel system contains $p$ components in parallel from the standpoint of reliability. Figure 11(b) shows the RBD related to this case.

  Fig. 11

  

  a Lower bound RBD of $CHR4\left( {p; q} \right)$, b upper bound RBD of $CHR4\left( {p; q} \right)$

Therefore, according to Fig. 11a, the lower bound reliability for $CHR4\left( {8; 2} \right)$ and $CHR4\left( {16; 2} \right)$ is given by:

$$ R_{{CHR4\left( {8; 2} \right)}} \left( t \right) = 8e^{ - 7\lambda t} - 7e^{ - 8\lambda t} $$

(65)

$$ R_{{CHR4\left( {16; 2} \right)}} \left( t \right) = 16e^{ - 15\lambda t} - 15e^{ - 16\lambda t} $$

(66)

Also, according to Fig. 11b, the upper bound reliability of $CHR4\left( {8; 2} \right)$ and $CHR4\left( {16; 2} \right)$ is calculated as follows:

$$ \begin{aligned} & R_{{CHR4\left( {8; 2} \right)}} \left( t \right) \\ & \quad = 1 - \left( {1 - e^{ - \lambda t} } \right)^{8} \\ & \quad = 8e^{ - \lambda t} - 28e^{ - 2\lambda t} + 56e^{ - 3\lambda t} - 70e^{ - 4\lambda t} \\ & \qquad + 56e^{ - 5\lambda t} - 28e^{ - 6\lambda t} + 8e^{ - 7\lambda t} - e^{ - 8\lambda t} \\ \end{aligned} $$

(67)

$$ \begin{aligned} & R_{{CHR4\left( {16; 2} \right)}} \left( t \right) = 1 - \left( {1 - e^{ - \lambda t} } \right)^{16} \\ & \quad = {\kern 1pt} 16e^{ - \lambda t} - 120e^{ - 2\lambda t} + 560e^{ - 3\lambda t} - 1820e^{ - 4\lambda t}\\ & \qquad + 4368e^{ - 5\lambda t} - 8008e^{ - 6\lambda t} \\ & \qquad + {\kern 1pt} 11440e^{ - 7\lambda t} - 12870e^{ - 8\lambda t} + 11440e^{ - 9\lambda t} \\ & \qquad - 8008e^{ - 10\lambda t} + 4368e^{ - 11\lambda t} - 1820e^{ - 12\lambda t} \\ & \qquad + {\kern 1pt} {\kern 1pt} 560e^{ - 13\lambda t} - 120e^{ - 14\lambda t} + 16e^{ - 15\lambda t} - e^{ - 16\lambda t} \\ \end{aligned} $$

(68)

• (7) k-cube networks (Eqs. 25–28): 3-cube network structure is topologically equivalent to the structure of a 3-connected network. Therefore, the reliability of the 3-cube is equal to 3-connected reliability. In case of 4-cube, two modes are intended for calculation of reliability: When all the links between the two 3-cubes are failed, except for one of the links because working at least one of them is necessary for working the entire network (lower bound). In this case, reliability of the network is equal to the reliability of two 3-connected networks that are connected in series. Second mode, when all the links between the two 3-cubes are active and healthy (upper bound). In this case, the network can be seen as a 3-cube network with redundant links.

Therefore, according to the above description, the 3-cube reliability can be calculated as follows:

$$ \begin{aligned} R_{3 - cube} \left( t \right) & = 376e^{ - 7\lambda t} - 1476e^{ - 8\lambda t} + 2352e^{ - 9\lambda t} \\ & \quad - 1895e^{ - 10\lambda t} + 770e^{ - 11\lambda t} - 126e^{ - 12\lambda t} \\ \end{aligned} $$

(69)

Also, according to the above description, the 4-cube lower bound reliability is given by:

$$ R_{4 - cube} \left( t \right) = \left( {R_{3 - cube} \left( t \right)} \right)\left( {R_{3 - cube} \left( t \right)} \right) $$

(70)

Moreover, the 4-cube upper bound reliability can be defined as the a 3-cube network reliability with the difference that its link failure rate ($\lambda_{p}$) is equal to the failure rate of a parallel system includes two 3-cube links with failure rate $\lambda$:

$$ \begin{aligned} R_{4 - cube} \left( t \right) & = 376e^{{ - 7\lambda_{p} t}} - 1476e^{{ - 8\lambda_{p} t}} + 2352e^{{ - 9\lambda_{p} t}} \\ & \quad - 1895e^{{ - 10\lambda_{p} t}} + 770e^{{ - 11\lambda_{p} t}} - 126e^{{ - 12\lambda_{p} t}} \\ \end{aligned} $$

(71)

In Eq. (71), $\lambda_{p}$ is equal to:

$$ \begin{aligned} & \lambda_{p} = \frac{1}{{1 - \left( {1 - e^{ - \lambda t} } \right)^{2} }}\left( { - \frac{{d\left( {1 - \left( {1 - e^{ - \lambda t} } \right)^{2} } \right)}}{dt}} \right) \\ & \quad = \frac{{2\lambda \left( { - \frac{1}{{\varvec{e}^{t\lambda } }} + 1} \right)}}{{ - \frac{1}{{\varvec{e}^{t\lambda } }} + 2}} \\ \end{aligned} $$

(72)