Light Traffic Equivalence in Single-Server Queues

A light traffic limit theorem is proved for random walks in a triangular array setting similar to the heavy traffic situation, the basic assumption being on the moments in the right tail of the increment distribution. When specialized to GI/G/1 queues, this result is shown to contain the known types of light traffic behaviour in this setting (Daley and Rolski) as well as some additional ones. Intuitively, the results state that typically delay in light traffic occurs with just one customer in the system, and then as a result of long service times and/or short interarrival times in a balance which depends on the particular parameters of the model. Particular attention is given to queues with phase-type service times, for example of Coxian type.