###### Abstract  
The links of low power wireless sensor networks are error prone and the transmission on a wireless link is determined probabilistically by the packet reception rate (PRR) of the link. On the other hand, there is a very strict requirement in the end-to-end reliability and delay of sensor data in industrial wireless sensor networks (IWSNs). The existing approaches to provide the end-to-end reliability in IWSNs is retransmitting the packet when failure occurs. These approaches transmit a packet multiple times in successive time slots to provide the required reliability. These approaches, however, can increase the average delay of packets and the number of packets buffered in a queue. This paper proposes a new scheme to estimate the probabilistic amount of packets, called queue level (QL), in the buffer of each node based on the PRRs of the wireless links. This paper also proposes a QL-based centralized scheduling algorithm to assign time slots efficiently in TDMA-based IWSNs. The proposed scheduling algorithm gives higher priority to the nodes with higher QL. By assigning time slots first to the node with the highest QL, we can reduce the average end-to-end delay of packets and reduce the amount of buffered packets in the queue while satisfying the required end-to-end reliability. The performance of the proposed scheduling algorithm have been evaluated through a simulation using the Cooja simulator and compared with the existing approach. In the simulation on an sample network with the target end-to-end reliability of 99%, all of the flows were shown to guarantee the target reliability in both algorithms: on average, 99.76% in the proposed algorithm and 99.85% in the existing approach. On the other hand, the proposed algorithm showed much better performance than the existing approach in terms of the average end-to-end delay of packets (about 47% less) and the number of maximally buffered packets in the queue of each node (maximally, more than 90% less). IEEE 802.15.4e; industrial wireless sensor networks (IWSNs); link scheduling; queue estimation; reliability in IWSNs +
Footnote †: journal: Accepted: 28 September 2021  
There have been many studies in the scheduling algorithms on IWSNs, which are classified into centralized and decentralized scheduling schemes [5]. Decentralized schemes rely on nodes with partial knowledge of their neighborhood and traffic information. Such knowledge is utilized for distributed decision making processes. Decentralized schemes can better react to the dynamics of the network due to link status changes or the nodes' mobility [6]. However, centralized schemes are more suitable for IWSNs that have strict reliability and real-time requirements and utilize a centralized architecture and network management [7; 8]. In industrial environments, sensor nodes are located in a harsh condition with many obstacles, high humidity levels, vibrations, dirt and dust, or other conditions that challenge performance. These harsh environmental conditions lower the performance of the wireless links and makes reliable message delivery even more difficult. Mostly, sensor nodes have a scarce resource in terms of processing power, memory, and energy. Constrained by the limited physical size, sensor nodes have limited battery energy supply and their memories are limited and have restricted computational capabilities [7]. To transmit sensor data reliably in wireless sensor networks with error prone links, retransmission after packet loss is the most widely used method [9]. [PERSON], et al. [10] proposed two scheduling methods using a hypergraph-based approach to maximize the end-to-end reliability of sensor data with a delay constraint in IWSN: dedicated and shared scheduling methods. Their algorithm consists of two phases: the schedule construction phase and schedule improvement phase. In the construction phase, one time slot is allocated to a link group that consists of links that can transmit simultaneously on the path from the source to destination.