Both of your plumbing fixtures will have so-called "open drain" or "indirect drain".
Your Ice Cream machine drain will be draining into your floor sink. Your 3-bowl sinks will combine their drains under the sink in one drain and will, also, drain into your floor sink as well. Your floor sink is the fixtures that is connected into sewer and vented above the roof. In other words, your installation is "surface" installation. If you have a floor sink already installed, you shouldn't have any problems installing them.
But I have seen some old commercial places with 3 compartment sink connected directly into wall. You should call your inspector and ask him/her about this detail. In my location it is not allowed.
If you don't have floor sink, than you will have to concentrate on installing one. (Floor sink is 12"x12" cast iron, enameled, "drain" with cast iron grate over it )
In commercial applications you cannot use plastic. Instead, you use tubular brass or DWV copper. Your drain should slope 1/4" per foot. In your case, his detail should not present any problem...
If this is in your private residence than you could - perhaps - cut corners. But in commercial applications, drains and drain vents are very important for proper functionality of your plumbing fixtures.
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