An absorbing Markov Chain model is formulated to describe the queue formation and dissipation process at a service facility. The model yields the average time for a queue to dissipate and the probability for the queue to reach a certain length, using the properties of the fundamental matrix derived from the canonical form of the transition probability matrix of the Markov chain. The model is useful in evaluating the time for a queue to dissipate at a facility which provides service intermittently, for example, at a loading point of a transportation facility. At these locations, a vehicle cannot depart until all the waiting passengers (or cargo) are aboard the vehicle. The delay to a user is thus affected not only by the number of persons ahead in the queue but also by the ones behind him in the queue and the ones who join the queue during the boarding process. The total waiting time of the first person in the queue is approximately equal to the vehicle standing time and he experiences the longest delay since he also had waited the longest before the vehicle arrived. The last person in the queue experiences the shortest delay. This paper formulates the general purpose model for calculating delay, queue dissipation time, and queue length fluctuation under such conditions. The model may be applied to a number of queuing situations in which dissipation of the entire queue is the main concern, including problems of the dissipation of traffic back—up at a traffic accident site or road construction site.