The desktop app is not always alerting when a call is coming in.  We have this situation with both Engage and Communicate users.  For an Engage user, they don't get alerted, so they have to answer and transfer using the deskphone, which is sometimes problematic.  They have to immediately press F5 to refresh so that they can answer the next call that comes in.  For a Communicate user that uses WebRTC exclusively, they have no indication of an incoming call and only know they've missed a call if the caller leaves a voicemail which is delivered in Outlook.  The inbox does not show a voicemail until after pressing F5 to refresh the desktop app.  For those Communicate users with a deskphone, the phone rings, so they can get their call and may not realize the app didn't alert them.

Do you have PureCloud Edges located on your premises, or are you using PureCloud Voice? 

Based on what you describe, it sounds like the calls are coming into an Edge on-premises and being routed to the physical phone quickly, but the Internet connection is not keeping up so the alert sent to the UI doesn't get there when the call gets to the phone.

The web-based phone (WebRTC) is completely dependent on the Internet connection to know that a call is coming in, which is why it doesn't alert and calls end up going to voicemail.

I'd start by getting some packet captures on the Internet connection (whatever device you can...port mirroring on the switch, for example, to a laptop) and see if the messages to the Client are coming in at the same time the calls are alerting the hard phones.

We do have PureCloud Edges.  Thanks for the suggestion.