Saab 340 Undifferentiated
Air Traffic Control Fully Certified
Air Traffic Control Time Certified In Pos 1 (yrs) 29
Communication Breakdown; Distraction; Time Pressure; Workload; Situational Awareness
ATC Issue All Types; Conflict Airborne Conflict; Deviation / Discrepancy - Procedural Clearance; Deviation / Discrepancy - Procedural Published Material / Policy; Inflight Event / Encounter Weather / Turbulence
Air Traffic Control Separated Traffic
Human Factors; Procedure
Working the south departure position while providing OJT (On Job Training) with frontal passage weather impacting all departures routed to the east of MIA. Traffic volume was at maximum capacity.  During the session I had to stop departures twice to minimize my volume. The supervisor stopped departures one more additional time in order to minimize traffic. During a period of high traffic my developmental turned Aircraft X to a 90 degree heading in order to avoid a conflict simultaneously the tower departed Aircraft Z that did not tag up because of a wrong squawk code while attempting to correct the squawk code issue Aircraft X continued to travel easterly and came in conflict with Aircraft Y on downwind to MIA.  Both aircraft were at 6000 feet.  No CA/MSAW alert sounded.  I continued to work the position for a period of time because of a staffing shortage.Traffic was extreme and ZMA had in trail restrictions.After stopping departures and restarting them I asked the tower for additional in trail and the tower supervisor instructed the local controller to depart aircraft faster. The tower controller told the supervisor that ZMA had flow restrictions but the tower supervisor told him to disregard the restrictions and launch harder because OJT was being performed on the departure procedure.  All of this information was brought to management's attention and both the tower supervisor and tower controller were interviewed.In my opinion all the parts of this puzzle; weather reroutes; heightened arrivals to TMB due to NASCAR activity; skydive operations; etc were not taken into consideration with regards to the position I was working causing an overload of traffic.
MIA TRACON Departure Controller reports of being inundated with traffic and stopping departures at least three times to get sector under control. Controller has an operational error with an aircraft they turned away from another departure; but this aircraft then became a conflict with an arrival on the downwind. Controller reported the Front Line Manager in Tower instructed the Local Controller to disregard the flow and launch harder because OJT was being performed in the departure sector.