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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 of conversations of each Mealy e-service is closed under prepone. We illustrate that the global prepone fails to completely capture the queue delay effects and refine prepone to a "local" version on conversations seen by individual peers. On the other hand, Mealy implementations of a composite e-service will always generate conversations whose "projections" are consistent with individual e-services. We use projection-join to reflect such situations. However, there are still Mealy peers whose set of conversations is not the local prepone and projection-join closure of any regular language. Therefore, we propose conversation specifications as a formalism to define the conversations allowed by an e-service composition. We give two technical results concerning the interplay between the local behaviors of Mealy peers and the global behaviors of their compositions. One result shows that for each regular language, its local prepone and projection-join closure corresponds to the set of conversations by some Mealy peers effectively constructed from . The second result gives a condition on the shape of a composition which guarantees that the set of conversations that can be realized is the local prepone and projection-join closure of a regular language.