Stability in queues with impatient customers

This paper deals with GI/G/1 queueing systems with impatient customers that have individual deadlines until their beginning of service. The impatience law depends on the number of waiting customers and on the elapsed service time of the customer in service. An exhaustive analysis of the asymptotic behavior of the model, combining ideas of stochastic dominance of well-known processes and some properties of models with finite capacity, is provided. We prove that the model is ergodic, null recurrent or transient if the corresponding traffic parameter in a simple associated model is respectively lower than, equal to, or greater than one.