B727 WAS CLRED TO LAND SLC RWY 17. TWR REQUESTED ACR TO CHANGE TO RWY 16L. AS ACR BEGAN SIDESTEP MANEUVER; PIC OBSERVED CONVERGING CESSNA JET APPEARING TO LINE UP TO RWY 16L. PIC QUERIED TWR ABOUT TFC AND RECEIVED A TCASII TA. PIC MANEUVERED BACK TOWARDS RWY 17 AND RECEIVED A TCASII RA FROM OTHER UNKNOWN TFC OBSERVED ON BASE LEG FOR RWY 17. PIC EXECUTED GAR.

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SLC BEARR ARR; AFTER OGD VOR CLRED HDG 130 DEGS TO INTERCEPT RWY 17 LOC; CLRED VISUAL APCH TO RWY 17. OUTSIDE OF TIFUL INTXN SLOWED TO 170 KTS; CLRED TO TWR FREQ; CLRED TO LAND RWY 17. TWR ASKED IF WE HAD RWY 16L IN SIGHT; AFFIRMATIVE. CLRED TO LAND RWY 16L. AT APPROX 7 NM OUT; TURNED TOWARDS RWY 16L AND MY TCASII WARNING SHOWED AN ACFT AT 3 O'CLOCK POS VERY CLOSE. RECEIVED A CLB RESOLUTION. THE 3 O'CLOCK POS TFC (CESSNA) WAS VISUAL; SLIGHTLY HIGH. I ELECTED TO RETURN TOWARDS RWY 17 AND REQUESTED LNDG CLRNC. I THEN SAW A LIGHT AIRPLANE ON SHORT L BASE TURNING TO FINAL TO RWY 17; BELOW AND AHEAD OF US. I ELECTED TO GAR AND STATED OUR DECISION ON THE FREQ. RECEIVED NO REPLY FROM TWR FOR SOME TIME AFTER RECEIVING CLRNC TO LAND ON RWY 16L. HOWEVER; I DID HEAR TWR TALKING TO OTHER ACFT. AFTER EXECUTING THE GAR; TWR GAVE US DIRECTIONS OF RWY HEADING AND 8000 FT. SUPPLEMENTAL INFO FROM ACN 387115: ACR X WAS CLRED FOR APCH TO RWY 17 AT SLC. CLRED TO SLOW TO 170 KTS AND CONTACT TWR ABOUT 12 MI OUT. TWR CTLR ASKED US IF WE COULD SEE RWY 16L. WE ANSWERED AFFIRMATIVE; HE CLRED US TO LAND ON RWY 16L. AS WE WERE SLIDING OVER TO THE EXTENDED CTRLINE OF RWY 16L; A CESSNA CITATION WAS DIRECTLY ABEAM US; APPARENTLY ALSO LINED UP FOR A RWY 16L APCH. WHILE TRYING TO CLARIFY THE SIT; WE WERE PARALLEL TO THE CESSNA AND RECEIVED A TCASII RA; ADVISING US TO CLB. WITH THE TFC IN SIGHT AND SLIGHTLY ABOVE US IN ALT; WE TURNED SLIGHTLY L TO INCREASE HORIZ SEPARATION. COMMUNICATED WITH TWR THAT IT APPEARED THE CESSNA WAS LINED UP WITH RWY 16L AND THAT WE WERE GOING BACK TOWARD RWY 17. NO RESPONSE WAS RECEIVED IMMEDIATELY FROM TWR; AND WHEN WE SAW A SMALL CIVIL ACFT; TYPE UNKNOWN; ON A TURN TO FINAL FOR RWY 17; WE ELECTED TO CLB AND FLY THE MISSED APCH. APPARENTLY THE CESSNA CITATION WAS ALSO GIVEN A CLB/MISSED APCH INSTRUCTIONS; AS WE HEARD THE CTLR ATTEMPTING TO CONTACT HIM; WITH NO RESPONSE HEARD FROM THE ACFT. PERHAPS CITATION WAS STILL ON APCH FREQ OR OTHER TWR FREQ. REST OF FLT; VECTORED AROUND TO APCH FOR RWY 16R; WAS UNEVENTFUL.