We make some comments on recent papers involving the integrated Cauchy functional equation or specialized versions of it, and reveal in particular that these papers give an inaccurate picture of the current state of the literature on the topic.

A number of jobs on two interconnected queues are to be processed by m identical servers. The servers operate in parallel, so that every server can process any job. Jobs in queue i, i = 1, 2, incur an instantaneous holding cost Ci during the time they remain in the system. The service time for jobs in queue i, denoted by Xi, is a random variable with a general distribution. The interconnection process is independent of the service process. We establish sufficient conditions on the service times, the holding costs and the interconnection process under which the non-preemptive scheduling strategy that gives priority to queue 1 minimizes the total expected α -discounted cost. We call this strategy P1. We present counterexamples showing that if any of the sufficient conditions is not satisfied P1 may not be optimal, and that the optimal policy for the single-server problem is not necessarily optimal for the multiserver problem.

Perturbation analysis estimators for expectations of possibly discontinuous functions of the time-stationary workload were derived in [2]. The expressions obtained may, however, not be valid if the customer-stationary distribution of the workload has atoms (at points other than zero). This was pointed out by Brémaud and Lasgouttes in [1]. In this note we clearly state the additional condition required for the validity of the expressions in [2]. We furthermore show how our approximation scheme can also be used to obtain the correct expressions for the right and left derivatives given in [1].

We consider a simple random walk starting at 0 and leading to 0 after 2n steps. By a generating functions approach we achieve closed formulae for the moments of the random variables ‘number of visits to the origin'.