Server replication is an approach often used to improve the ability of a service to handle a large number of clients. One of the important factors in the efficient utilization of replicated servers is the ability to direct client requests to the best server, according to some optimality criteria. In this paper we target an environment in which servers are distributed across the Internet, and clients identify servers using our application-layer any-casting service. Our goal is to allocate servers to clients in a way that minimizes a client's response time. To that end, we develop an approach for estimating the performance that a client would experience when accessing particular servers. Such information is maintained in a resolver that clients can query to obtain the identity of the server with the best response time. Our performance collection technique combines server push with client probes to estimate the expected response time. A set of experiments is used to demonstrate the properties of our performance determination approach and to show its advantages when used within the application-layer anycasting architecture.