# On Maximizing the Probability of Achieving Deadlines in Communication Networks
## 2 Related Work
It employs dedicated paths for real-time traffic, calculated using a metric that incorporates the initial energy, expected transmission count, inverse expected transmission count, and minimum loss. Combinations of multiple approaches have also been proposed. For instance, opportunistic routing (OR) has been integrated with RL for live video streaming over multi-hop wireless networks [21]. Techniques from GR and OR have been combined in a QoS-aware scheme for wireless sensor networks [22]. [PERSON] et al. [21] introduced a distributed RL-based OR scheme for video streaming, where a novel metric was proposed to estimate end-to-end (E2E) delays. All of the above approaches focus on individual metrics, predominantly velocity or hop count, to ensure the timely delivery of data. However, the importance of modeling link latencies directly as probability distributions instead of using individual metrics has been highlighted in works such as [4; 23]. Stochastic networks are characterized by aspects with uncertain behavior, and the parameters or conditions that affect network operations are subject to variability or randomness. Several works have considered deadline-aware routing for such networks. The reliable shortest path problem, introduced by [PERSON] in [24], considers the joint latency of individual paths to select the optimal path for a specific deadline. While [24] is a contribution from operations research, the reliable shortest path problem has also received some attention in the domain of wireless multi-hop networks in works such as [4; 23]. In contrast to the reliable shortest path problem, which focuses on predefined paths, the stochastic on-time arrival problem [25] allows for optimal decisions made on the fly. The stochastic on-time arrival problem has received little attention from researchers studying communication networks. It has been considered to provision soft QoS guarantees in mobile ad hoc multimedia networks [26] or for accomplishing timely data delivery in a practical bus network [27]. Routing algebra [28] can be used as a holistic mathematical modeling approach for designing and analyzing routing metrics. The key property of the algebra, isotonicity, is necessary and sufficient to prove that the routing produces loop-free paths. In [29], the authors extended the algebra to OR. Routing algebra can also be applied for combined metrics, such as a combination of a link quality indicator and the remaining energy [30]. In contrast, the optimal paths found by the stochastic on-time arrival problem may include loops [25]. This violates isotonicity, and therefore routing algebra is not sufficient to describe the stochastic on-time arrival problem. Numerous studies have focused on the subject of deadline-aware routing in recent years, as indicated by our literature review. However, these works predominantly offer solutions grounded in either single or combined metrics. There is a clear absence of algorithms created through analytical methods that are based on the distribution of link delays. In this paper, we use an analytical approach based on the stochastic on-time arrival problem in order to find the achievable deadline performance in communication networks and thereby provide a tool for the design of routing protocols that maximize the probability of achieving deadlines. We elaborate on this approach next.