In this paper we consider the stationary PH/M/c queue with deterministic impatience times (PH/M/c+D). We show that the probability density function of the virtual waiting time takes the form of a matrix exponential whose exponent is given explicitly by system parameters.

We give an exponential tail approximation for the extinction time of a subcritical multitype branching process arising from the SIR epidemic model on a random graph with given degrees, where the type corresponds to the vertex degree. As a corollary we obtain a Gumbel limit law for the extinction time, when beginning with a large population. Our contribution is to allow countably many types (this corresponds to unbounded degrees in the random graph epidemic model, as the number of vertices tends to∞). We only require a second moment for the offspring-type distribution featuring in our model.

We investigate protected nodes in random recursive trees. The exact mean of the number of such nodes is obtained by recurrence, and a linear asymptotic equivalent follows. A nonlinear recurrence for the variance shows that the variance grows linearly, too. It follows that the number of protected nodes in a random recursive tree, upon proper scaling, converges in probability to a constant.

In this paper we propose a strategy that gives an optimal lower bound of the average gain for the two-envelope problem within the McDonnell and Abbott (2009) and McDonnell et al. (2011) framework. We obtain this result with partial information about the probability distribution of the envelope's contents.