Distributed storage systems are rapidly emerging as a desirable paradigm for cloud infrastructures. Through redundancy, such systems can improve the reliability of data backends. Interestingly, distributed storage can also enhance file access times for incoming requests, as it takes advantage of statistical averaging. Still, conducting performance characterization for ensuing delay profiles remains a challenge. Several recent contributions in this area construct bounds on the performance of redundant systems. These are then used to explore the latency-redundancy tradeoff, and assess the relative values of candidate implementations. Along these lines, this work establishes novel upper and lower bounds on the mean sojourn time of a request entering a distributed storage system. These bounds are based on stationary distributions of dominating quasi-birth-death processes and, in many settings, they can be made progressively tighter at the expense of additional computations.