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Deadband is also known as hysteresis and is the amount of a measured variable (pressure, temperature, etc.) between the point where a switch closes and then re-opens. It can be implemented is software if you have an analog input (i.e., open at 1.02 PSI, close at 0.98 PSI) or designed into a switching mechanism. Often the deadband of a switch is fixed and cannot be adjusted. A typical example is in a wall thermostat in your house.
When the battery voltage drops even further, there will be no sound. Even the telltale lights on the instrument panel may dim when you turn the key. Usually, jump-starting the car will get you on the road, but be sure to find the cause of the dead battery or you will repeat this drill soon.