In this paper, we consider the update bandwidth in distributed storage systems~(DSSs). The update bandwidth, which measures the transmission efficiency of the update process in DSSs, is defined as the total amount of data symbols transferred in the network when the data symbols stored in a node are updated. This paper contains the following contributions. First, we establish the closed-form expression of the minimum update bandwidth attainable by irregular array codes. Second, after defining a class of irregular array codes, called Minimum Update Bandwidth~(MUB) codes, which achieve the minimum update bandwidth of irregular array codes, we determine the smallest code redundancy attainable by MUB codes. Third, the code parameters, with which the minimum code redundancy of irregular array codes and the smallest code redundancy of MUB codes can be equal, are identified, which allows us to define MR-MUB codes as a class of irregular array codes that simultaneously achieve the minimum code redundancy and the minimum update bandwidth. Fourth, we introduce explicit code constructions of MR-MUB codes and MUB codes with the smallest code redundancy. Fifth, we establish a lower bound of the update complexity of MR-MUB codes, which can be used to prove that the minimum update complexity of irregular array codes may not be achieved by MR-MUB codes. Last, we construct a class of $(n = k + 2, k)$ vertical maximum-distance separable (MDS) array codes that can achieve all of the minimum code redundancy, the minimum update bandwidth and the optimal repair bandwidth of irregular array codes.