We propose a tractable expression for the outage probability in single-cell uplink non-orthogonal multiple access (NOMA) systems serving an arbitrary number of users. The expression is obtained by approximating the inter-user interference using a shifted-gamma distributed random variable. We then formulate and propose an efficient iterative algorithm for the outage-constrained min-max power allocation problem for the NOMA system. To give a rigorous comparison, we solve the outage-constrained min-max power allocation problem for the orthogonal multiple access (OMA) counterpart where both the user power allocation and the radio resource division pattern are optimized. Simulations confirm the accuracy of the derived outage probability expression for the NOMA system. Also, we demonstrate that fairness among users in terms of transmission power can be achieved by NOMA. Moreover, NOMA can bring significant power savings to the users as compared with OMA.