In Industrial Internet of Things (IIoT), a number of mutually untrusted devices with diverse computing resources and application requirements often need to work together to serve industrial production. When applying permissionless blockchain to process transactions among these devices, we are concerned with two types of fairness: mining fairness (i.e., miners should obtain commensurate rewards according to their respective invested computing power) and transaction-processing fairness (i.e., transaction packing and confirming is in a desired fair order). To address the two types of fairness simultaneously, this paper proposes GT-Chain, in which miners select transactions according to Geometric-distribution and propagate transactions with a Time-to-live limit for achieving transaction-processing fairness as well as balancing the processing efficiency and fairness. Besides, GT-Chain inherits properties of a famous blockchain called FruitChain for achieving mining fairness. We then conduct a probability analysis to quantify the efficiency of transaction processing and define a Euclidean-distance-based fair degree to measure the fairness of transaction processing. Extensive simulations verify that GT-Chain can well achieve our design objectives, and our theoretical model is accurate. This paper is helpful in better designing blockchain protocols for IIoT.