I agree with Ray that using heart rate has problems.
The 220-age guideline doesn't estimate max HR correctly for most people, it's just a starting point (apparently it was developed back in WW2). If you want to find your max HR, you'll need to do a max heart rate testing protocol
, there are several out there but they generally involve increasing effort every X minutes for as long as you can. If you have a variable resistance (fluid) trainer or a sufficiently long climb just warm up, start out at a moderately hard pace for the first minute, then increase effort every minute after that until you pop.
Of course if you have any heart or health issues you should check with a doctor before doing any max heart rate testing protocol.
Max HR and measured HR during a workout on any particular day can vary though, depending on how much sleep you've had, stress levels, caffeine levels, sickness or overtraining, even your position on the bike and hydration levels. You can't compare HR values between people either - physiological, fitness, and genetic differences makes it impossible.
I guess what I'd ask is why are you concerned that you're over training?
How do you generally feel with your current training load on rest days or before the next workout? What you can handle as far as training load is going to depend on your current fitness level and what load your body is used to. If you're concerned about over training based on 220-age max heart rate alone, I don't think that's a cause for concern.
If you really want to be precise about your training load and ramping up your load in a manageable way to hit some sort of fitness target by a particular date (eg for a race), power meter is the way to go. Of course at a gym this is problematic unless you can find one that has a Wattbike or you bring power pedals like the PowerTap P1 and fit them yourself before a workout with permission. Weekly TSS (training stress) based on power is pretty much the gold standard for training metrics these days.
The only thing I use HR for now is to keep myself in check when I'm supposed to be on a Zone 1/2 recovery ride, and when I'm training hard, I often wear a Fitbit overnight to measure my min heart rate. A min heart rate 5 bpm or more higher than normal can indicate overtraining, but again it's really hard to get consistent HR measurements.