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The problem of designing a minimum cost network with multipoint linkages which connects several remote terminals to a data processing center is studied. The important aspects of a teleprocessing network are queue behavior at the terminals and the cost and reliability of the entire system. In this paper it is assumed that the rate and manner in which information is requested at the terminals is known and that acceptable line loadings are given. An algorithm that determines (in principle) the optimum minimum cost network subject to reliability constraints is developed. A heuristic based on Vogel's approximation method (VAM) and two other heuristics presented by Martin and Esau-Williams were compared with each other and with the optimal algorithm. The Esau-Williams heuristic seems to be the one that gives the best solution and Martin's requires the least processing time. It is shown experimentally that Martin's and Esau-Williams heuristics are, in fact, near-optimal heurstics in the sense that the solutions provided by these heuristics are generally very near the optimal solution. In this paper we make the assumption that all lines of the network have the same capacity.