# End-to-End Service Availability in Heterogeneous Multi-Tier Cloud-Fog-Edge Networks
## 4 Results
It provides a probabilistic methodology to assess service availability spanning interconnected heterogeneous network resources. The Markov model is the key element in the described methodology. Let us develop the specified Markov model for one level of the architecture under consideration and determine the availability of the dedicative service for the endpoint element in this model. The main interest in this study is the definition of availability of service \(A_{i}\) provided to the endpoint node as a dynamic object at the \(i\) (\(i=1,\ldots n-1\)) level. In practice, when organizing access to user applications, it is of interest to determine the availability of service for a separate dedicated node (marked service in Figure 4). In this case, the user is not interested in the availability of service at all other nodes or IoT devices  
Figure 4: Endpoint path as individual connections of device from IoT level to cloud level (from the top to the bottom of the architecture hierarchy). and applications provided for other users. Let us study the availability \(A_{j}\) of a dedicated endpoint service (DES) at the \(j\)\((i=j,\ldots n-1)\) level for one selected IoT device. For a practical important system with \(l=1\) repair bodies and \(k\) nodes at the level of hierarchy architecture under study, the behavior of the examined system is described by the Markov Chain state transition diagram (Figure 5), where \(H_{i}\) represents the state with \(i\) failed nodes, but the dedicated user service is fault-free, and \(H_{if}\) represents the state with \(i\) failed nodes unused by the dedicated user and one failed DES. Figure 5 illustrates the Markov model for analyzing the availability of a dedicated endpoint service (DES) within a tier of the multi-tier architecture at each level of hierarchy architecture. The model consists of state \(H_{j}\) representing \(j\) nodes failed but the DES still working, and state \(H_{if}\) denotes \(j\) nodes failed along with failure of the DES. The system can be in state \(H_{x}\) with probability \(p_{x}(t)|\forall x\). The transitions between states are governed by the failure rate \(\lambda\) and repair rate \(\mu\). Both failure and repair transitions are shown between the working and failed DES states. From working state \(H_{j}\), failure of non-DES nodes lead to a state with a higher index, \(j\). Repairs from state \(H_{if}\) bring the DES back to working state \(H_{j}\). Solving this continuous-time Markov chain provides the steady-state probabilities of being in each state. The availability of the DES is computed by summing the probabilities of being in operational state \(H_{j}\). This elegantly captures the reliability characteristics of the DES within its tier environment. Extending this to multiple tiers gives an end-to-end multi-tier availability model. The flexibility to specify failure and repair rates enables modeling a range of environments from reliable to unstable. This is a core strength of the Markov approach to availability analysis. On the basis of the diagram (Figure 5), the Chapman-Kolmogorov system of differential equations can be writing in accordance with the general rules [39]:  
\[ \begin{array}{c}p_{0}^{\prime}(t)=-k\lambda p_{0}(t)+\mu p_{1}(t)+\mu p_{0f}( t)\\ p_{1}^{\prime}(t)=(k-1)\lambda p_{0}(t)-[(k-1)\lambda+\mu]p_{1}(t)+\mu p_{2}(t)+ \frac{1}{2}\mu p_{1f}(t)\\ \ldots\\ p_{k-1}^{\prime}(t)=\lambda p_{k-2}(t)-(\lambda+\mu)p_{k-1}(t)+\frac{1}{k}\mu p _{k-1,f}(t)\\ p_{0f}^{\prime}(t)=\lambda p_{0}(t)-[(k-1)\lambda+\mu]p_{0f}(t)+\frac{1}{2}\mu p _{1f}(t)\\ \end{array}  \tag{3}\]  
\[ \begin{array}{c}p_{1f}^{\prime}(t)=\lambda p_{1}(t)+(k-1)\lambda p_{0f}(t)- [(k-2)\lambda+\mu]p_{1f}(t)+\frac{2}{3}\mu p_{2f}(t)\\ \ldots\\ p_{k-1}^{\prime}(t)=\lambda p_{k-1}(t)+\lambda p_{k-2f}(t)-\mu p_{k-1,f}(t)\\ \end{array} \]  
Figure 5: Markov Chain state transition diagram.