RPT FROM A FLT INSTRUCTOR IN AN SMA WITH A STUDENT ON ILS APCH; UNDER THE HOOD. WHEN RPTING THE OM; THERE WAS NO TWR RESPONSE. WHEN NEXT MAKING CONTACT THEY WERE CLRED AS #2 ON THE APCH. THE ACFT AHEAD WAS SLOWER AND THERE WAS A RAPID CLOSURE RATE. INSTRUCTOR ELECTED TO MAKE A 360 DEG TURN SINCE HE COULD NOT VERY WELL CLB ABOVE FOR A GAR. ON INFORMING THE CTLR; HE WAS TOLD HE COULD NOT DO THAT.

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AFTER RECEIVING RADAR VECTORS FROM CHICAGO APCH (133.5 MHZ). WE WERE CLRED FOR A PRACTICE ILS 1L APCH AT DPA. WHEN THE TWR WAS CONTACTED; WE WERE TOLD TO RPT THE OM INBOUND. UPON ARRIVING AT THE FIX; MY STUDENT RPTED THE MARKER TO THE TWR WITH NO RESPONSE FROM THE BUSY LCL CTLR. AT THE NEXT POSSIBLE CHANCE (AT LEAST 1 MIN); I REPEATED OUR RPT TO ENSURE THE CTLR KNEW OUR LOCATION. AT THIS TIME THE CTLR ACKNOWLEDGED US AND TOLD US WE WERE #2 BEHIND A CESSNA ON BASE LEG. I IMMEDIATELY LOOKED FOR AND LOCATED THE TFC AT 100 FT ABOVE US AND AT 1500 FT IN FRONT OF US WITH THE GAP DECREASING. (ESTIMATED 30 KTS CLOSURE RATE.) BECAUSE THE ACFT IN QUESTION WAS ABOVE US; I ELECTED TO DEVIATE COURSE RATHER THAN CLB FOR A GAR. I INSTRUCTED MY STUDENT TO BEGIN AN IMMEDIATE R TURN AND RPTED WE WERE EXECUTING A 360 DEG TURN FOR SPACING; AT WHICH TIME THE CTLR TOLD US WE COULD NOT DO THIS WITHOUT HIS PERMISSION (OBVIOUSLY FORGETTING THAT COLLISION AVOIDANCE ULTIMATELY IS MY RESPONSIBILITY). I INFORMED THE CTLR THAT WE NEEDED TO TAKE IMMEDIATE ACTION FOR COLLISION AVOIDANCE; BUT THE CTLR SEEMED UNSYMPATHETIC AND ANGRY TO THE POINT OF DISSUADING US FROM ANY FURTHER APCHS ON THIS DAY. CONTRIBUTING FACTORS WERE FREQ CONGESTION; THE CTLR'S FAILURE TO NOTIFY US OF TFC UNTIL IT WAS TOO LATE; THE TFC COMING FROM THE L (NOT MY SIDE WITH STUDENT UNDER HOOD); AND THE DECISION OF THE CTLR TO SEQUENCE SLOW TFC IN FRONT OF FASTER TFC.