A Boeing 737-800 pilot reported a compressor stall; in the #1 engine; at cruise altitude; resulting in an inflight engine shutdown; and diversion to a suitable airport.

Narrative

Number 1 Eng Fail (Compressor Stall) @FL360. Acft was at M.74 (MCP (Mode Control Panel) SPEED) due ZZZ ATC Flow Control. When ATC cleared us NORMAL SPEED; speed control was given back to VNAV (FMC Speed) ECON. Acft power advanced to accelerate to M.78. Upon application of power; Eng 1 compressor stalled (growled; shuddered; shaked; followed by a loud bang). RPM rolled back to ~33-30% N1. Eng 1 attempted to recover RPM but upon addl (Additional) Compressor Stall rolled back to ~30% N1. Eng 1 shutdown by crew in descent …estimate windmill time 15-25 mins. No indications of fire; overheat or fluid loss…however; Pax (Passengers) seated behind Eng 1 saw large white smoke pour out of Eng 1 at compressor stall. WX conditions were CAVOK SKC and Smooth Air; No Turbulence. ZZZ ATC initially did not want us to descend (due traffic deconfliction) and we complied for about 1 min before we told them we are descending (airspeed was decreasing)…they cleared us diversion direct ZZZ and multiple altitudes in the descent. [Priority handling was requested] upon initial call to ATC to inform them of our situation. ATC gave us radar vectors to ILS Rwy XX….Eng Out Flaps 15 landing uneventful…Taxi cleared with ARFF (Airport Rescue and Firefighting) inspection of Acft (nothing found). Taxied to Gate.