  We revisit a single-server retrial queue with two independent Poisson streams
(corresponding to two types of customers) and two orbits. The size of each
orbit is infinite. The exponential server (with a rate independent of the type
of customers) can hold at most one customer at a time and there is no waiting
room. Upon arrival, if a type $i$ customer $(i=1,2)$ finds a busy server, it
will join the type $i$ orbit. After an exponential time with a constant
(retrial) rate $\mu_i$, an type $i$ customer attempts to get service. This
model has been recently studied by Avrachenkov, Nain and
Yechiali~\cite{ANY2014} by solving a Riemann-Hilbert boundary value problem.
One may notice that, this model is not a random walk in the quarter plane.
Instead, it can be viewed as a random walk in the quarter plane modulated by a
two-state Markov chain, or a two-dimensional quasi-birth-and-death (QBD)
process. The special structure of this chain allows us to deal with the
fundamental form corresponding to one state of the chain at a time, and
therefore it can be studied through a boundary value problem. Inspired by this
fact, in this paper, we focus on the tail asymptotic behaviour of the
stationary joint probability distribution of the two orbits with either an idle
or busy server by using the kernel method, a different one that does not
require a full determination of the unknown generating function. To take
advantage of existing literature results on the kernel method, we identify a
censored random walk, which is an usual walk in the quarter plane. This
technique can also be used for other random walks modulated by a finite-state
Markov chain with a similar structure property.

Probability