We're not using Avaya CTI, but a purpose-built CTI that's unique to us. I'm assuming that much of what we do will apply to you, because the required tasks are the same -- CTI all does more or less the same things, whoever you are.
The way we do it is this:
Our T-Servers are behind the switch. We're using TDM, not IP telephony. Our routing is done by the PBX, not by Genesys.
When a human agent starts her shift, she logs into a CTI application on her desktop with two things: her i.d. (associated with her skills) and her telephone extension. The CTI application reports these two things and the PC name to the T-Server so the T-Server will know which PC and which phone extension are associated.
The active T-Server monitors the extensions of the GVPs and the human agents.
When a caller zeroes out, the GVP returns the call to the PBX, with a VDN that tells where to route it (e.g., to an agent with a skill for MO start-service). The PBX notifies the T-Server what agent extension the call will be going to.
At the time the GVP does this, it also send a message to the T-Server with data it collected during the call: the caller's account number and phone number.
When the PBX decides what agent's extension to route the call to, the PBX notifies the T-Server that a call is going there.
The T-Server looks up the PC that's associated with that extension (it knows this because of the CTI login the agent did when she reported for work). The T-Server then sends the information about the call (phone number and account number, as collected by the GVP) to an application on the agent's PC.
The PC (agent desktop) application takes the account number from the message and plugs this into the terminal emulator the agents use to interact with the customer database. (If necessary, the application will back up to the initial screen so it can enter the data at a place where it knows the data can be retrieved.) It hits "ENTER" on the screen, and the data populates. In other words, the agent gets a screen pop.
The stuff that's required in order to do this is:
T-Server configurations to collect and associate the data
GVP setup so it will send the call data to the T-Server
PBX setup -- agent i.d.s, skills, VDNs, and so on
CTI login application on the agent desktop
CTI screen pop application on the agent PC
Appropriate configurations on the agent PC: .NET server names, etc.
I hope I haven't left anything out.
As I say, I don't know Avaya CTI for Genesys (though we used to do it on our Conversant VRUs, and it was similar, though less complex). I hope this is similar enough to help.