Kubernetes design pattern for an existing application which spawns a new process for every single client session?

I'm currently working on a legacy application which uses a ServiceFactory process to spawn new processes which handle current and all subsequent requests from a client. This one process per client session is designed to retain large volume of data queried and analyzed in-memory to serve the follow-up requests from the same client. Once the client terminates the connection, the process exits.

What would be an equivalent implementation using Kubernetes?