1. A computer program product for processing quiesce requests of a computer system, the computer program product comprising: a computer readable storage medium readable by a processing circuit and storing instructions for executing by the processing circuit for performing a method comprising: determining whether a quiesce request received at a processor of the computer system is of a fast-quiesce type, and responsive to the quiesce request being of the fast-quiesce type, handling a fast-quiesce type interruption at the processor; responsive to the quiesce request being other than of the fast-quiesce type, setting an I-AM-QUIESCED indicator and a HOLD-QUSYS indicator at the processor to, in part, inform a system controller of the computer system that the processor has entered a quiesced state, and determining whether other processors of the computer system have entered the quiesced state; and responsive to the other processors of the computer system having entered the quiesced state, determining by the processor whether any total system quiesce requests are pending, and responsive to no total system quiesce request being pending, blocking system quiesce interrupts at the processor for a currently pending quiesce request sequence, and thereafter resetting the I-AM-QUIESCED indicator, and the HOLD-QUSYS indicator at the processor to enable processing at the processor.