the PowerCenter Repository Service is responsible for managing the metadata inside the PowerCenter Repository.
The Model Repository Service is responsible for managing the metadata inside the Model Repository. This service is part of the Data Quality (Mercury) product and has i't own design utility, the so called Developer.
Basically, they have the same function but for different product lines, so which service you need depends on your product you want to use.
The "repository" is the basis for "classic" PowerCenter. The first versions have been published in 1995, if I'm not mistaken. One of the heavily used features of the "classic" PowerCenter repository are the so-called Metadata Exchange views (MX views) which can be used to extract most of the information from the repository in a way that doesn't interfere with "normal" repository operations; in particular that means that if the repository DB is DB2 you won't incur deadlocks, table locks, or similar trouble when accessing the repository contents via the MX views (which happens quite regularly if you source the tables). Furthermore the MX views usually don't change with new versions
The model repository is the basis for the product lines IDQ (Informatica Data Quality), IDS (Informatica Data Services), and PCX (PowerCenter Express, not to be mixed up with PowerCenter). Its structure has no similarity with the PowerCenter repository, the views on this database are (at least that's what I've heard) not as detailed as the MX views for PowerCenter repositories.
A repository (or model repository, for that matter) stores information about data structures and how they are to be processed. For example, if you load data from Oracle tables and write them to flat files, then the respective repository holds information about the table and file structures, how data are transferred, where the Oracle table lies and with which user ID it's accessed, what's the file name and location, and many more things.
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