Center Controller reported due to their workload they did not notice an aircraft descend without a clearance. The aircraft left their frequency and flew below the Minimum IFR Altitude.

Narrative

I was working combined sectors with an SAU (aerial refueling) no radar assistant; radio congestion and pilots not listening. Aircraft X came from ZZZ Approach control headed to ZZZ1 climbing from 6;000 feet to 7;000 feet. As the pilot got over ZZZ2 he requested and approach into ZZZ2 an airport not on the original flight plan. I asked the pilot to if he was going to full stop and advise when he had the weather; NOTAMS and type of approach. As all of that was going on I had two aircraft that departed off ZZZ that were in trail and were gonna be in conflict due to overtaking speed from the rear aircraft that I had to resolve. I was also trying to coordinate the altitudes of the aerial refueling so I could request the airspace and coordinate with adjacent controllers. That turned into a confusing episode due to the aircraft utilizing the airspace had a call sign of Aircraft Y and there was an aircraft on the frequency with the call sign of Aircraft Z [similar call sign].Aircraft X advised me of the approach request then I had to get up and look at the ERID to pull up the approach plates for that approach. The approach started in ZZZ Approaches airspace (180 degrees from his route of flight) airspace that Aircraft X just came from. I issued a clearance to the initial approach fix (ZZZZZ) direct the airport. I received no response from Aircraft X. Tried a couple more times with a radio check and no response. (I learned after watching the tape of the event that Aircraft X started a descent even prior to me issuing the clearance which I did not notice in real time because when I issued it I was looking at the approach plates on the ERIDS). I would never intentionally issue a clearance to an aircraft below the MIA nor intentionally not issue a Safety Alert to one below the MIA. In real time I thought I issued the clearance as the aircraft was still in the climb or level at 7;000 feet (because in my mind there was no reason for Aircraft X to descend especially without authorization). After noticing the descent I tried tried to establish radio communications with no luck. Things were moving fast in real time and I felt as if I was moving from one situation to the next.I got bogged down in the other portions of the sectors and military aerial refueling that I didn't notice a descent below the MIA. At some point throughout all of that I was given a radar assistant (D-side) but apparently that assistant was standing behind me a while trying to get in (tight space because there is a computer; desk; and myself all blocking his entrance). The only reason I knew he was there was because the Supervisor tapped me on my shoulder for me to slide over and allow access for the assistant to get in to his seat.There were some calls from ZZZ Approach to the assistant in reference to Aircraft X but the assistant had little to no idea what was going on and couldn't clearly answer. I told the assistant that I knew but couldn't explain and would call ZZZ Approach back myself; which I did shortly after getting the sectors under control. Apparently Aircraft X switched to ZZZ Approach's frequency and I explained that I issued the clearance to the initial approach fix and airport only with no reply and that I did not issue a descent clearance. Supervisors place radar assistants with radar controller 99% of the time even when there is push back from the controller. I received a letter from my immediate supervisor yesterday after watching and discussing the event that said I should have asked for assistance. I have been a big advocate of the problems with the frequency congestion and pilots stepping over each other with out first monitoring the frequency before keying up but it has fallen on deaf ears and I seem to be seen as the disgruntle employee when I vent and express my dissatisfaction. While listening to the play back of that event I can be heard getting on pilots about listening up and not just keying up and talking without listening first. Justlast week I had situations where I would transmit a clearance on frequency 1 and as I unkey a pilot would key up and transmit on frequency 2 without giving the pilot on frequency 1 to reply. I keyed up and said; 'I've been doing this for XX yrs and Its amazing how I can transmit on one and just as I unkey a pilot on another frequency keys up and just starts talking' a pilot said in a condescending voice; 'we've been stepping on each other for XX yrs'. DEAF EARS and I am left to deal with it and not comment or be dissatisfied. We have to repeat clearances 3 times to pilots and that happens more time not [just] now. Clearances such as frequency change; climb; descent; turn etc. It's not just a problem for me but co workers around me as well.