Date: 202107
Local time of day: 1801-2400
Location: ZZZ.ARTCC, US
Light: Night
Aircraft: B737-700
Operator: Air Carrier
Operating under: Part 121
Flight plan: IFR
Mission: Passenger
Flight phase: Cruise
Airspace: Class A ZZZ
ATC / advisory: Center ZZZ
Reporter organization: Air Carrier
Reporter function: Pilot Flying; Captain
Reporter qualification: Flight Crew Multiengine; Flight Crew Air Transport Pilot (ATP); Flight Crew Instrument
Human factors: Confusion; Time Pressure; Communication Breakdown
Second reporter function: Pilot Not Flying; First Officer
Anomaly: ATC Issue All Types; Airspace Violation All Types; Deviation / Discrepancy - Procedural FAR; Deviation / Discrepancy - Procedural Published Material / Policy
Detector: Person Air Traffic Control; Person Flight Crew
When detected: In-flight
Result: Air Traffic Control Issued Advisory / Alert; Air Traffic Control Issued New Clearance; Flight Crew Exited Penetrated Airspace; Flight Crew Requested ATC Assistance / Clarification
Contributing factors: Environment - Non Weather Related; Human Factors; Airspace Structure
Primary problem: Environment - Non Weather Related

Narrative: Primarily; en route ATC was task saturated the entire day working multiple frequencies while providing speed and vector control; versus sequencing traffic on the arrivals. On the leg from ZZZ to ZZZ1; we were handed off to ZZZ near the waypoint ZZZZZ.  After multiple failed attempts to reach ATC; we returned to the previous frequency to try an alternate frequency. This frequency was very busy as ATC was working multiple frequencies and also providing multiple aircraft with vectors and speed control. We were given 10 degrees left.  ATC told us to expect direct ZZZZZ1.  As we approached ZZZZZ2; we tried multiple times to reach ATC on the frequency assigned. Unable to reach them we tried the alternate frequency multiple times with no success. Shortly after those attempts; ATC transmitted on Guard frequency for us to turn south immediately. We replied that we are turning south and had attempted to reach them multiple times on two separate frequencies.  Their reply was; 'Yeah; right!'


Second narrative: On (date) at approximately (time); we lost communication with Center; while on a vector and ended up in Special Use Airspace. About 10 minutes prior; we were switched to a frequency from previous Controller. We were unable to get contact on this frequency; so we went back to our previous assigned frequency and were given a new frequency. We were able to contact Center on this frequency. While with Center on this frequency; we were given a vector for spacing with an expectation to rejoin our arrival. After a few minutes; without hearing anything we began to query the Controller. We had no contact. We tried a few times; when the Captain said we should try the other frequency we were given earlier. As we switched over; we hear Center simulcasting on Guard; to come up. As we contacted Center; the Controller asked us where we had been and we explain that we were up on the frequency where they were talking to us earlier. The Controller then proceeds to give us a vector and tells us that we are in Special Use Airspace. Once we cleared the Airspace we continued with the arrival uneventfully.


Synopsis: A flight crew reported they flew into military Restricted Airspace when were assigned a vector off course and issued a frequency change. They could not establish communications on the new frequency in a timely manner due to frequency congestion and saturation on the new frequency.