This is a standard exercise in Shodokan Aikido (required for every test). We call it either "Randori" or "Jiyu Waza". We don't standardize it the way you're asking.
(In Chinese martial arts, the term may be Sanshou, but I'm not sure that is standardized. Closer to Kumite)
My school used to do 4 attackers, 90 seconds, with each attacker starting their attack run at the moment the preceding attacker made contact. The defender is expected to evade and execute techniques. Attackers attack in order. someone is the referee and will call "STOP" if there is anything dangerous (people get too enthusiastic, someone gets injured, etc.)
We don't rest between attackers.
That is probably still the archetype, but:
Timekeeping is a pain - we generally now do a set number of attacks. Each attacker attacks four or five or X times.
For lower belts the attack is restricted to a "loping stiffarm" - slightly faster than a walk, slower than a run, hand held out in front to simulate a wide variety of attacks. Lower belts are expected to learn to evade, do hand passes, and control the floor (don't get trapped against walls, be aware of where the next attacker is, etc.) The worst that happens is that you get a palmtouch where you weren't expecting it. This lowers the threat threshold for juniors, and helps them feel comfortable.
We go back and forth on blocks. One of the defender's best strategies is to move so that the next attacker blocks the following attacker. (if attackers are A, B, C, move so that A is between the defender and B). Defender also wants to alter the technique so that A is thrown into the path of B. For lower and mid-belts, B will call "block" and the attack proceeds to C. Blocking is a very realistic strategy, and there are times when this is the focus of the exercise. On the other hand at some point we become so good at blocking that we cease to do techniques, so some days we don't permit blocking. You'll want to play with both.
Sometimes we have too few healthy attackers (I work with a bunch of seniors, who have correspondingly high injury rates). In which case we use 3 attackers (and up the number of attacks).
for higher belts, attackers are authorized to use a variety of attacks (shomen uchi, yokomen uchi, reverse punch, reverse yokomen uchi, haymaker, jab, etc.) (Uppercut and kicks have a tendency to develop into dangerous and artificial patterns, so we avoid those). Basically any attack which is interesting and which doesn't tend towards dangerous. We slow the pace so that the next attacker begins the run only after the defender has finished a technique. Yes, this is artificial, but (a) we've mastered the earlier drills and (b) we want to focus on finishing techniques.
Some groups (I've only seen this in Chinese martial arts) use a circle, and if if the attacker is forced out of the circle, the attack is deemed to have been countered.
Every belt test also includes knife randori - we generally limit this to one attacker. We also do tournaments with knife randori
In my Taji Chuan (Chinese Martial arts) school, we do a variation of this. One student performs the Taiji form; other students are free to attack whenever they feel like it. The student performing the form must adapt the form to counter the attack. (and return to the form, but we're not strict about that).
Because my Taiji school has more junior than senior students, we tend to restrict the attacks to slow, abstracted thrusts and grabs. This reduces the threat level and allows the student to concentrate on the purpose of the exercise. As the students become more comfortable, shift upwards.
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