Services that provide information to Boarding Staff about guest's presence in the Airport

Depends on the airline, but in my experience letting the airport staff know that you are there and on your way will do nothing useful.

1. First of all, "airport staff" is too broad. Roles are highly compartmentalized and there isn't a lot of communication. Not only would you have to find a person from the airline (operating carrier, to be precise) but also the right person from that airline (gate agent) and that person won't be where you are
2. The gate agent typically has the information anyway. They have full access to your itinerary and the flight status of your connection

My main experience is with United gate agents and they seem to be utterly ruthless. Some examples

1. I had a tight connection but got out fast, ran fast and made it to the connecting gate with still 2 minutes to spare and want to hop on the plane. My boarding pass was rejected: The gate agent says: "Oh, we didn't think you could make it this fast, so we have already given your seat away"
2. Six of us are in the same delayed plane from Santa Barbara trying to make the last red eye out of LAX to Boston. The gate agent closed the gate about 1 minute before the first of us arrived at the gate. She knew that we were already in the terminal building, she also knew that the delay was United's fault so they would have to put up six people in hotels for the night and re-book all of us, she also knew that it would at max take 5 more minutes to get everyone on board and the flight had plenty of buffer to absorb that, but that didn't stop her.
3. A passenger in LAX arrives literally 30s late at the gate to a flight Melbourne (a really, really long flight). She pleads with the gate agent, but to no avail. The plane is still there, the jetway is still docked, the aircraft doors are still open, but the gate agent didn't budge. I've seen it done correctly and it only takes a minute or so to re-open a gate.

it possible to let the Airport staff known that the passenger is on her/his way to the next connecting flight>

NO.

Does a traveller have some rights or services to let the boarding staff known that s/he is on her/his way and perhaps not miss out the flight?

NO on services, somewhat YES on rights (#4).

Sorry, hate me and downvote all you want, but this is just how it works.

There is no way, unless you have their personal mobile phone #, for YOU to let the gate staff know that you're on your way...BUT!...

Within your specified assumptions:

1. The Gate Agents can easily presume you're on the way because they can see when your aircraft arrived at the Gate and which Gate. From that, they will have an idea of how long it should take a passenger to get there.

2. The passenger services contractor, the one's who provide accessibility services and sometimes the carts, will usually have ways of communicating with gate staff like a radio or handset. The Gate Agent should also be able to see if a passenger required assistance and can take that into account. (Note, I said should be able, I don't know how ever airline in world has their gate terminals laid out.)

3. The number of connecting passengers is considered as they are more likely to wait longer for more passengers. This is heavily dependent on the destination. If there's a slot, crew time or curfew issue, they'll leave anyone behind.

4. The amount of arrival delay on the incoming flight isn't specifically important, it's the time between the delayed arrival and departure time. Here's where contract obligations and any passenger rights come in. If the arrival is delayed but still falls outside of a Minimum Connection Time (MTC) for that airport/combination, then missing the flight is on you. If the delayed arrival time puts you under the MTC, then the airline should reaccommodate you if you late, even if you could have made it. Meaning, an MTC might be 45 minutes, but practical gate-to-gate time for healthy adults is 15 minutes so you can still make the connection.

5. Status, yes it can be a factor as well. I have personally experienced several times "Oh, Mr. 305, we've been waiting for you." I believe them because the Agent closes the door right behind me.