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In the Internet of Things (IoT) networks, the data traffic would be very bursty and unpredictable. It is therefore very difficult to analyze and guarantee the delay performance for delay-sensitive IoT applications in fog networks, such as emergency monitoring, intelligent manufacturing, and autonomous driving. To address this challenging problem, a Bursty Elastic Task Scheduling (BETS) algorithm is developed to best accommodate bursty task arrivals and various requirements in IoT networks, thus optimizing service experience for delay-sensitive applications with only limited communication resources in time-varying and competing environments. To better describe the stability and consistence of Quality of Service (QoS) in realistic scenarios, a new performance metric "Bursty Service Experience Index (BSEI)" is defined and quantified as delay jitter normalized by the average delay. Finally, the numeral results shows that the performance of BETS is fully evaluated, which can achieve 5-10 times lower BSEI than traditional task scheduling algorithms, e.g. Proportional Fair (PF) and the Max Carrier-to-Interference ratio (MCI), under bursty traffic conditions. These results demonstrate that BETS can effectively smooth down the bursty characteristics in IoT networks, and provide much predictable and acceptable QoS for delay-sensitive applications.