# End-to-End Service Availability in Heterogeneous Multi-Tier Cloud-Fog-Edge Networks
## 4 Results
The normalizing condition is:  
\[\sum_{i\in z}p_{i}(t)=1,\ \ \ \ \ z=\overline{1,k-1},\ \overline{0_{f},(k-1)_{f}} \tag{4}\]  
For stationary conditions of operation, the system of differential Equations (3) and (4) is transformed into a linear system of equations, in which \(p_{i}^{\prime}(t)=0,\ \forall i\) and \(p_{i}(t)=P_{i},\ \forall i\). Steady-state probabilities can be determined in this case in accordance with the general rules [39]:  
\[P_{i}=\frac{(k-1)!}{(k-i-1)!}\gamma^{i}P_{0},\ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i\leq k-1\]  
\[P_{i,f}=\frac{(k-1)!(i+1)}{(k-i-1)!}\gamma^{i+1}P_{0},\ \ \ \ 0\leq i\leq k-1\]  
The value of \(P_{0}\) can be obtained by replacing \(P_{i},\ \overline{1,k-1}\) and \(P_{i,f},\ \overline{0,k-1}\) in the normalizing Equation (4):  
\[P_{0}=\left[1+\sum_{i=1}^{k-1}\frac{(k-1)!\gamma^{i}}{(k-i-1)!}+\sum_{i=0}^{k- 1}\frac{(k-1)!(i+1)\gamma^{i+1}}{(k-i-1)!}\right]^{-1}\]  
Considering the obtained expressions for steady-state probabilities, the availability of the DES at the one hierarchy level is obtained as  
\[A_{1}=1-\sum_{\forall i,f}P_{if}=\frac{a_{1}}{a_{1}+a_{2}}, \tag{5}\]  
where:  
\[a_{1}=(k-1)!\sum_{i=0}^{k-1}\frac{\gamma^{i}}{(k-i-1)!},\]  
\[a_{2}=(k-1)!\sum_{i=0}^{k-1}\frac{(i+1)\gamma^{i+1}}{(k-i-1)!},\]  
\[\gamma=\lambda/\mu\]  
The resulting Equation (5) is generalized for determining the availability of a dedicated endpoint service for other hierarchical levels in the architecture under consideration \(A_{i}\ (i=\overline{1,\ n-1})\). By consistently determining the availability of service at all levels in accordance with the reliability indicators of their nodes, we can obtain the required service availability in the system by substituting the obtained availability values into Equation (1). Let us investigate the availability of dedicated user service at the \(i\) level in the network in accordance with Equation (5). Figure 6 shows the availability of a dedicated endpoint service at a single tier in the multi-tier architecture as a function of the number of nodes at that tier. Two cases with different node reliabilities are presented: 1 represents \(\gamma=0.01\) and 2 represents \(\gamma=0.001\), where \(\gamma\) is the ratio of failure to repair rate. As observed, the availability of the endpoint service decreases as the number of nodes increases, for both reliability cases. The decreasing trend is logical since more nodes means more potential failure points, reducing overall service availability. The case with the higher \(\gamma\) value exhibits lower availability overall because of the poorer inherent reliability (higher failure rate). This demonstrates how the Markov model can quantify the availability of a single tier based on node count and reliability. This tier availability then contributes to the overall multi-tier availability based on the end-to-end path. The model enables exploring tradeoffs between tier size and reliability targets to meet desired availability thresholds. This is valuable in designing multi-tier systems and validating service-level agreements. Figure 7 shows the unavailability parameter of unavailability \(U=1-A\) as a function of the number of nodes at a given tier for two different node reliability cases: (a) \(\gamma=0.01\) and (b) \(\gamma=0.001\). Higher unavailability corresponds to lower availability. As observed in both Figure 7a,b, the unavailability increases as the number of nodes increases, demonstrating the reduced availability. The trends illustrate how adding more potential failure points (nodes) reduces overall service availability, seen through increasing unavailability. Comparing Figure 7a,b also shows that better node reliability (lower \(\gamma\)) yields lower unavailability, as expected. This analysis demonstrates using the Markov model to quantify node relationships between tier size, node reliability, and unavailability/availability metrics. The model provides a tool to explore tradeoffs between dimension tiers and set reliability requirements to achieve target availability thresholds.