We consider a system consisting of two not necessarily identical
exponential servers having a common Poisson arrival process. Upon
arrival, customers inspect the first queue and join it if it is
shorter than some threshold n. Otherwise, they join the second
queue. This model was dealt with, among others, by Altman et al. [Stochastic Models 20 (2004) 149–172].
We first derive an explicit
expression for the Laplace-Stieltjes transform of the distribution
underlying the arrival (renewal) process to the second queue. Second,
we observe that given that the second server
is busy, the two queue lengths are independent.
Third, we develop two computational schemes for the
stationary distribution of the two-dimensional Markov process underlying this
model, one with a complexity of
, the other with a complexity of
, where δ is the tolerance criterion.