On the relaxation time for single server queues

This paper gives a natural definition of the relaxation time for general single server queues. First, it describes a GI/GI/1 queue as a limit of GI/GPH/1 queues. Each GI/GPH/1 queue is transferred to some equivalent bulk arrival GI^x/M/1 queue, which is formulated by a spatially homogeneous Markov chain with reflecting barrier at zero. An upper bound, which is easy to calculate, of the relaxation time of the Markov chain is derived. It is shown that the relaxation time of the GI/GI/1 queue, defined as a limit of the relaxation times of GI/GPH/1 queues, has a particularly simple upper bound. Some particular cases are finally treated, where the upper bound obtained is shown to be tight for M/M/1 and M/D/1 queues.