Allocating limited resources such as bandwidth and power in a multi-hop wireless network can be formulated as a Network Utility Maximization (NUM) problem. In this approach, both source transmitting and link relaying nodes exchange information allowing for the NUM problem to be solved in an iterative distributed manner. Previous NUM formulations of wireless networks have considered the parameters of data rate and reliability in the utility function which measures an application's performance. However, it is well known that delay is an important factor in the performance of many applications. In this paper, we consider an additional constraint based on the delay requirements of the sources. This augmented NUM formulation allows an application to tradeoff rate, power and queuing delay according to its needs, thereby providing greater flexibility. Power allocation among different transmitters is a subtle issue to deal with in this problem, since the capacity of the wireless links are interference limited. A distributed iterative algorithm solving the NUM is presented along with its convergence. The performance of the algorithm is examined via simulations which confirm the expectations from the theory.