This paper introduces a framework for modeling and specifying the global behavior of e-service compositions. Under this framework, peers (individual e-services) communicate through asynchronous messages and each peer maintains a queue for incoming messages. A global "watcher" keeps track of messages as they occur. We propose and study a central notion of a "conversation", which is a sequence of (classes of) messages observed by the watcher. We consider the case where the peers are represented by Mealy machines (finite state machines with input and output). The sets of conversations exhibit unexpected behaviors. For example, there exists a composite e-service based on Mealy peers whose set of conversations is not context free (and not regular). (The set of conversations is always context sensitive.) One cause for this is the queuing of messages; we introduce an operator "prepone" that simulates queue delays from a global perspective and show that the set o...