My furnace heats for about six and a half minutes (blower running) then goes into the exhaust cycle for about 14 minutes. The furnace is a Bryant Plus 90 furnace that is around 20 years old.
The control fire switch. Easy fix, about $6.00 for the part
If your furnace is short-cycling (when it is very cold outdoors) the house will gradually cool down and not reach the temperature setting on the thermostat. If this is what is happening you need to call a heating company and schedule a service appointment. Failure to have it professionally checked could result in the destruction of your furnace or, at the least, a shortening of the life of the equipment. By the way, there is no cycle time for the burners to heat. The burners should be aflame, without interruption, until the house reaches the temperature set on the thermostat.
Sounds like the high temperature limit switch is tripping. Check your air-flow. If you aren't moving enough air through the heat exchanger the high limit will cycle the burners off until it cools down.