Before revision:  Then, we study several structural properties of the minimum cost path of the maximum queue length . Given these properties, we prove that the maximum queue length follows a sample path with linear increment.
After revision:  Then, we study several structural properties of the minimum-cost-path to overflow of the queue with the largest length, which is basically equivalent to the decay rate of the largest queue-overflow probability . Given these properties, we prove that the maximum queue length follows a sample path with linear increment.