## 1 Introduction  
Wireless Sensor Networks (WSNs) have drawn intensive attention from the industrial community recently due to their flexibility, low cost, and powerful networking ability [1; 2; 3]. They could be applied for long-term surveillance, smart manufacturing, process control, and so on [4; 5; 6]. In industries, mobile elements are extensively involved in WSNs and play an increasingly important role [7]. On one hand, inspectors or patrol robots would be required to collect sensing data from various deployed industrial sensors (e.g., radiation sensor, gas sensor, temperature sensor) for decision-making. On the other hand, exploiting mobility of mobile entities in WSNs could improve the energy efficiency, load balance and network connectivity. Consequently, data collection in WSNs with mobile sinks (also referred to as mobile base stations or mobile data collectors) has become a significant issue [8]. There exist three sink mobility patterns [9]: (1) _random mobility_ where the sink roams in the sensing field randomly [10; 11; 12; 13; 14]; (2) _path-controllable mobility_ where the trajectory and the sink speed can be controlled to improve the network performance [15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26]; (3) _path-fixed mobility_ where the sink moves on a fixed path and under strict constraint its motion parameters such as the speed and pause time are also uncontrollable [27; 28; 29; 30; 31; 32; 33; 34; 35; 36]. In the past several years, extensive research has been conducted on efficient data delivery in WSNs with above sink mobility patterns and could be classified into three categories from the perspective of the routing scheme [21]: _single-hop routing, immediate multi-hop routing_, and _rendezvous-based routing_. In the single-hop routing, mobile sinks visit each sensor node and gather data via one-hop communication, resulting in low energy cost but extremely high delivery latency [10; 17; 27; 28; 29; 30]. With respect to the immediate multi-hop routing, all nodes immediately deliver packets to the mobile sink along dynamic routes with multiple hops, enabling real-time data collection at the cost of tremendous energy consumption [11; 12; 18; 29; 31; 19]. Rendezvous-based routing strikes a balance between aforementioned two methods, wherein data packets originated from deployed nodes will first be transferred to specified rendezvous nodes and then uploaded to the sink when it arrives [13; 16; 21; 22; 23; 24; 25; 32; 33]. This paper focuses on delay-constrained energy-efficient data delivery towards a path-fixed mobile sink. This mobility pattern is common in industrial scenarios due to restrictions on accessible areas and the requirements of performing assigned tasks (e.g., robots making regular inspections, people walking along roads, and transport vehicles). Several works adopting the rendezvous-based scheme have been proposed for the path-fixed mobility [32; 33]. However, they are dedicated to improving the network throughput or energy efficiency while neglecting the delay requirement. Actually, unlike in the situation that the sink mobility can be controlled, the traditional rendezvous-based routing cannot guarantee the delivery delay when the sink path and speed are all fixed. For example, a vehicle may take 2 h for a round trip, incurring high data collection delay, which is not acceptable for many applications. On the contrary, the immediate multi-hop routing cannot utilize the sink mobility for energy conservation. In real life, there are many types of sensing data that have different delay requirements varying in different conditions. For instance, the smoke concentration in an industrial park might be reported every 2 min in normal circumstances but 1 s in emergency situations. How can various delay constraints be satisfied as well as achieving high energy efficiency in this new context? To address this problem, in this work, we propose a Delay-aware Energy-efficient Routing algorithm for WSNs with a path-fixed and uncontrollable Mobile sink (DERM).