Let S be a countable set and let Q = (qij, i, j ∈ S) be a conservative q-matrix over S with a single instantaneous state b. Suppose that we are given a real number μ ≥ 0 and a strictly positive probability measure m = (mj, j ∈ S) such that ∑i∈Smiqij = −μmj, j ≠ b. We prove that there exists a Q-process P(t) = (pij(t), i, j ∈ S) for which m is a μ-invariant measure, that is ∑i∈Smipij(t) = e−μtmj, j∈S. We illustrate our results with reference to the Kolmogorov ‘K1’ chain and a birth-death process with catastrophes and instantaneous resurrection.

In this paper, we give an alternative derivation of Kendall's representation for symmetric transition functions which relies on the backward and/or forward integral recursions. The proof uses a lemma concerning approximation by finite sections (which is useful in its own right) and is similar to the original proof for birth and death processes by Lederman and Reuter. Finally, we obtain a general result guaranteeing the existence of representations of transition functions such as those obtained by Pruitt and Iglehart.

In this paper we study the M/M/2/2 queue with repeated attempts. It is shown that the part generating functions of the steady state probabilities can be expressed in of generalized hypergeometric unctions.

In this paper the limit behaviour of α-recurrent Markov renewal processes and semi-Markov processes is studied by using the recent results on the concept of α-recurrence for Markov renewal processes. Section 1 contains the preliminary results, which are needed later in the paper. In Section 2 we consider the limit behaviour of the transition probabilities Pij(t) of an α-recurrent semi-Markov process. Section 4 deals with quasi-stationarity. Our results extend the results of Cheong (1968), (1970) and of Flaspohler and Holmes (1972) to the case in which the functions to be considered are directly Riemann integrable. We also try to correct the errors we have found in these papers. As a special case from our results we consider continuous-time Markov processes in Sections 3 and 5.