In this paper we study the instability of certain nth order differential equations by means of Liapunov's functions and obtain some sufficient conditions for the equations being unstable. A method of constructing Liapunov's function for instability is being perfected. In particular the problem Ezeio posed in 1982 has been resolved to some extent.

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].

Starting with some mild assumptions on the parametrization of the service process, perturbation analysis (PA) estimates are obtained for stationary and ergodic single-server queues. Besides relaxing the stochastic assumptions, our approach solves some problems associated with the traditional regenerative approach taken in most of the previous work in this area. First, it avoids problems caused by perturbations interfering with the regenerative structure of the system. Second, given that the major interest is in steady-state performance measures, it examines directly the stationary version of the system, instead of considering performance measures expressed as Cesaro limits. Finally, it provides new estimators for general (possibly discontinuous) functions of the workload and other steady-state quantities.

The object of the paper is to establish an instability theorem for sixth order differential equations of the form (1.1). The proof is based on the use of Krasovskii criteria.