There are sports where warmups can be identified to be useful, for example leg stretching for running/jogging.
In my opinion, BJJ is more akin to swimming. Its a whole body activity. You don't jog before going swimming. You do laps as warmups before a swimming race. Maybe at most you stretch muscles that normally ache when you swim before swimming.
That is why currently for cardio, I make my students do takedowns for 1/2 an hour rather than skipping or running. Why? Being a BJJ class I think if you are going to do a workout, you might as well pick up some skills doing it. So while skipping gives you good cardio and make you good at skipping, I would rather my students get good at takedowns and improve their cardio. Same cardio workout, different skills practiced.
Instead of situps, replace it with armbars/triangle/omaplata's from guard. You'll get a good stomach workout in addition to getting better at armbars etc.
Also, the best way to improve grappling stamina is simple. Grapple! There are guys who can run marathons, can skip for hours, or can swim 300 laps in the pool. But they can't last 15 mins on the mat with a good blue belt. Why? Because those exercises, while it does keep you feeling fit, does not give you grappling endurance. They do however give you the mental toughness to tough it out when you think you have run out of steam. Ultimately however, I believe if you only have a limited amount of time, 2 hours of grappling will benefit you more than 2 hours of running in terms of grappling endurance.
Being that I only offer BJJ classes twice a week, and open mats on Saturday, I personally feel that this is the best use of the time in classes.
If you are training BJJ nearly every day of the week, then running, skipping and all kinds of conditioning training is useful to add on to your BJJ training. This is because you may suffer from burnout or suffer repetitive movement injuries from grappling too often, using your same movements all the time.
However, as we only offer grappling training 3 times a week, then I believe it is best to concentrate on techniques so that your body will memorize those techniques, and hopefully provide sufficient workout doing these techniques.
I have heard of students who boast on how tough their warmups are. However, can you or anyone do this several times a week, every week for years with no goal? If your goal is tournaments, yes you can do this as you build your fitness to peak at the time of your tournaments. But if you don't it will be impossible to mentally do this indefinitely.
The reason I say this is, unless you are a fitness trainer or a professional athlete, there will come a point in your life where you may not be able to train your fitness anymore. It may be because you got married, have a child, or even job or financial constraints.
This happens to even top athletes who retire, they grow fat and out of shape (have you seen Mark Kerr lately or any ex Lion's Den fighters?). It is a reality of life. Thus, to me it is best to give my students something they can keep, good technique.
The gym that I teach at KDT, has an excellent fitness training class already for those who want the extra training. But in my class, I prefer to concentrate on what I believe is my main responsibility to my students who pay me.... teach BJJ.
Ultimately I believe BJJ should give you the skills that last, even if/when you grow old, get fat and lazy, or for whatever reason you no longer are able to do intensive fitness training.
While it is true, that the fit and strong grappler with 5 moves who train like mad for competitions may beat the average joe grappler, who has a more complete game, but does not train in fitness, in the long term the average joe grappler will be able to have a longer lasting game, as he does not rely on his fitness and strength, which is temporary, but technique which lasts his lifetime.
However, I do indeed run my classes differently from other instructors, and the primary reason is to instill the skills in as little time as possible to my students. My recent blue belts on average have gone from white to blue belt in roughly a year, and they're good blue belts too!
Alternatively, I can take a page from my student who trained in Abu Dhabi and Dubai. Their classes go 4 hours in 48 degree celsius heat, with 1 and a half hours of that a grueling workout!
Sam Wee is the head instructor for Brazilian Jiu Jitsu (BJJ) at the KDT Academy (www.kdta.com), Malaysia and has been teaching BJJ since 2003.