This paper presents an analysis of several simple on-line algorithms for processing requests for connections in distributed networks. These algorithms are called call admission algorithms. Each request comes with a source, a destination, and a bandwidth requirement. The call admission algorithm decides whether to accept a request, and if so, when to schedule it and which path the connection should use through the network. The duration of the request is unknown to the algorithm when the request is made. We analyze the performance of the algorithms on simple networks such as linear arrays, trees, and networks with small separators. We use three measures to quantify their performance: makespan, maximum response time, and data-admission ratio. Our results include a proof that greedy algorithms are log-competitive with respect to makespan on n-node trees for arbitrary durations and bandwidth, a proof that on an n-node tree no algorithm can be better than Omega (log log n/log log log n)- competitive with respect to makespan, and a proof that no algorithm can be better than Omega(log n)-competitive with respect to call-admission and data-admission ratio on a linear array, if each request can be delayed for at most some constant times its (known) duration. (AN).