Hi, how old are you? Are you obese? It is unlikely to be asthma as its episodes happen early morning and is associated with cough. Most probably it is sleep apnea.
“Obstructive sleep apnea (OSA) is the most common category of sleep-disordered breathing. The muscle tone of the body ordinarily relaxes during sleep and at the level of the throat the human airway is composed of collapsible walls of soft tissue which can obstruct breathing during sleep. Mild, occasional sleep apnea, such as many people experience during an upper respiratory infection may not be important, but chronic, severe obstructive sleep apnea requires treatment to prevent sleep deprivation and other complications”. Taken from website http://en.wikipedia.org/wiki/Sleep_apnea
Could be sleep apnea...Unless you really cant get air into your lungs. With athsma it kinda feels like you're breathing through a straw.