Inspired by P systems initiated by Gheorghe Pãun, we study a computation model over a multiset of communicating objects. The objects in our model are instances of finite automata. They interact with each other by firing external transitions between two objects. Our model, called service automata, is intended to specify, at a high level, a service provided on top of network devices abstracted as communicating objects. We formalize the concept of processes, running over a multiset of objects, of a service automaton and study the computing power of both single-process and multiprocess service automata. In particular, in the multiprocess case, regular maximal parallelism is defined for inter-process synchronization. It turns out that single-process service automata are equivalent to vector addition systems and hence can define nonregular processes. Among other results, we also show that Presburger reachability problem for single-process service automata is decidable, while it becomes undecidable in the multiprocess case. Hence, multiprocess service automata are strictly more powerful than single-process service automata.