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Service oriented architecture is approach to have software resources in an enterprise available and discoverable on network as well defined services. Each service would achieve a predefined business objective and perform discrete units of work. The services are independent and do not depend on the context or state of the other services. They work within distributed systems architecture. Earlier SOA used COM or ORB based on CORBA specifications and recent SOA stress on web services using standard description (WSDL), discovery (UDDI) and messaging (SOAP). Service oriented architecture may or may not use web services but yes web services provide a simple way towards service oriented architecture albeit with the age old security and reliability limitations.
EAI Enterprise application integration is a business need to make diverse applications in an enterprise including partner systems to communicate to each other to achieve a business objective in a seamless reliable fashion irrespective of platform and geographical location of these applications. It is a business need and business never dies it only evolves. I have seen people saying that EAI is a thing of past now SOA is here, it is just like saying “transportation is a thing of past now road is here”. EAI comprises of message acceptance, transformation, translation, routing, message delivery and business process management. Usually messages transportation is asynchronous but for a business need it can be synchronous as well. There are two basic architectures to achieve this, bus and hub/spoke architecture. Both of these can be used to develop services and then it also becomes service orientated architecture.