# End-to-End Service Availability in Heterogeneous Multi-Tier Cloud-Fog-Edge Networks
## 4 Results
Availability, in terms of QoS, is often expressed as the fraction of time a service or system is operational and can be accessed, as expected by users or other systems. This is one of the key metrics in service-level agreements (SLAs) across various industries. Availability \(A\) for a given component is the probability that it is operational when required. It can be represented as:  
\[A=\frac{Uptime}{Uptime+Downtime} \tag{1}\]  
where:  
* _Uptime_ refers to the amount of time a system, service, or component is operational and available to perform its intended function. Uptime is essentially the period when a system is functioning without any interruptions. * _Downtime_ represents the duration when a system, service, or component is not operational due to planned maintenance, unplanned outages, or failures. It is the period when the system is not available for its intended purpose. For a multi-level system, if we assume that the service for the endpoint depends on each component working sequentially (like a series system in reliability theory), the availability of end-to-end service \(A_{s}\) is the product of the availabilities of each of the components:  
\[A_{s}=\prod_{i=1}^{n}A_{i} \tag{2}\]  
where \(n\) is the number of levels. At the apex of multi-tier hierarchy (level \(i=n\)) is the cloud tier. Quantifying cloud service availability depends on numerous architectural factors, including redundancy, failover mechanisms, and distributed deployment across geographically diverse data centers. Detailed analytical modeling of cloud reliability is an active research area, such as in [40]. However, in practice, cloud providers specify availability via service-level agreements (SLAs) with quantified uptime guarantees. Industry standards range from 99% uptime for routine, non-critical services up to \"five 9s\" (99.999%) or higher availability for mission-critical applications where even rare downtime has unacceptable consequences [41]. Thus, within our multi-tier availability model, the cloud tier availability \(A_{m}\) can be represented by a constant value dictated by the SLA tier purchased from the provider rather than requiring an elaborated availability model. This parameterized SLA approach allows flexibility in capturing a range of cloud reliability characteristics in the overall end-to-end quantification. We can represent the availability of the endpoint as a product of the availabilities along a single path from the top (cloud) to the endpoint. In essence, the availability of the endpoint becomes the product of the availabilities of all the individual connections leading to it, from the top to the bottom of the hierarchy (Figure 4). Endpoint availability is a crucial metric in multi-tier systems, providing insight into the overall system's ability to offer service without disruptions. For the multi-tier systems under study, determining endpoint availability can be determined using the following methodology. In the methodology described, availability is assessed step-by-step, factoring in the unique attributes of each hierarchy level and their interactions. The technique involves the next steps:  
1. Assessing the local availability between an endpoint device and its connected node at the lowest tier using a dedicated two-state Markov chain model representing up and down states. This local availability reflects reliability just for the specific endpoint's channel, disregarding other endpoints. 2. Iteratively applying the same Markov model at each higher tier to quantify the availability of nodes providing dedicated service to the tier below. Failures of other nodes at a given tier are assumed to not affect service to the endpoint. 3. Taking the product of the local availability values at each tier to derive the overall end-to-end availability for the selected endpoint and service. This combines reliability along the hierarchical service provisioning path. The model can be adapted to various multi-tier topologies by adjusting the number of tiers and Markov parameters.