In this paper we study the problem of on-line call control in a communication Ž network, namely, the problem of accepting or rejecting an incoming call a request . for a connection between two points in a network without having the knowledge of future calls. The problem is a part of the more general problem of bandwidth allocation and management. Intuition suggests that knowledge of future call arrivals can be crucial to the performance of the system. In this paper, however, we present preemptive deterministic on-line call control algorithms. We use competitive analysis to measure their performanceᎏi.e., we compare our algorithms to their off-line, clairvoyant counterpartsᎏand prove optimality for some of them. We consider two specific networks, a line of nodes and a single edge, and investigate a variety of cases concerning the ¨alue of the calls. The value is accrued only if the call terminates successfully; otherwiseᎏif the call is rejected, or prematurely terminatedᎏno value is gained. The performance of the algorithm is then measured by the cumulative value achieved, when given a sequence of calls. The variety of call value criteria that we studyᎏconstant; proportional to the length of the call's route; proportional to its holding timeᎏcaptures many of the natural cost assignments to network services.