This work is devoted to the study of an integro-differential system of equations modelling the genetic adaptation of a pathogen by taking into account both mutation and selection processes. First, we study the asymptotic behaviour of the system and prove that it eventually converges to a stationary state. Next, we more closely investigate the behaviour of the system in the presence of multiple EAs. Under suitable assumptions and based on a small mutation variance asymptotic, we describe the existence of a long transient regime during which the pathogen population remains far from its asymptotic behaviour and highly concentrated around some phenotypic value that is different from the one described by its asymptotic behaviour. In that setting, the time needed for the system to reach its large time configuration is very long and multiple evolutionary attractors may act as a barrier of evolution that can be very long to bypass.

A tracking problem is consideredin the context of a class $\mathcal{S}$ of multi-input,multi-output, nonlinear systems modelled by controlled functionaldifferential equations. The class contains, as a prototype, allfinite-dimensional, linear, m-input, m-output, minimum-phasesystems with sign-definite “high-frequency gain". The first controlobjective is tracking of reference signals r by the output y ofany system in $\mathcal{S}$ : given $\lambda \geq 0$ , construct afeedback strategy which ensures that, for every r (assumed boundedwith essentially bounded derivative) and every system of class $\mathcal{S}$ , the tracking error $e = y-r$ is such that, in the case $\lambda >0$ , $\limsup_{t\rightarrow\infty}\|e(t)\|<\lambda$ or, inthe case $\lambda=0$ , $\lim_{t\rightarrow\infty}\|e(t)\| = 0$ . Thesecond objective is guaranteed output transient performance: theerror is required to evolve within a prescribed performance funnel $\mathcal{F}_\varphi$ (determined by a function φ). Forsuitably chosen functions α, ν and θ, bothobjectives are achieved via a control structure of the form $u(t)=-\nu (k(t))\theta (e(t))$ with $k(t)=\alpha (\varphi(t)\|e(t)\|)$ , whilst maintaining boundedness of the control andgain functions u and k. In the case $\lambda=0$ , the feedbackstrategy may be discontinuous: to accommodate this feature, aunifying framework of differential inclusions is adopted in theanalysis of the general case $\lambda \geq 0$ .

We consider birth-death processes taking values in but allow the death rate in state 0 to be positive, so that escape from is possible. Two such processes with transition functions are said to be similar if, for all there are constants cij such that for all t ≥ 0. We determine conditions on the birth and death rates of a birth-death process for the process to be a member of a family of similar processes, and we identify the members of such a family. These issues are also resolved in the more general setting in which the two processes are called similar if there are constants cij and ν such that for all t ≥ 0.

In this paper, transient characteristics related to excursions of the occupation process of M/M/∞ queues are studied, when the excursion level is large and close to the mean offered load. We show that the classical diffusion approximation by an Ornstein–Uhlenbeck (OU) process captures well the average values of the transient variables considered, while the asymptotic distributions of these variables depart from those corresponding to the OU process. They exhibit, however, equivalent tail behaviour at infinity and numerical evidence shows that they are amazingly close to each other over the whole half-line.

We define a new family of special functions that we call lattice Bessel functions. They are indexed by the N-dimensional integer lattice such that they reduce to modified Bessel functions when N = 1, and the exponential function when N = 0. The transition probabilities for an M/M/1 queue going from one state to another before becoming idle (exiting at 0) can be solved in terms of modified Bessel functions. In this paper, we use lattice Bessel functions to solve the analogous problem involving the exit time from the interior of the positive orthant of the N-dimensional lattice for a series Jackson network with N nodes. These special functions allow us to derive asymptotic expansions for the taboo transition probabilities, as well as for the tail of the exit-time distribution.

Earlier work by Hadidi and Conolly and contemporary work by the author point to the great operational advantages of state-dependent queueing models. Let pin (t) be the state probabilities and p∗in the corresponding L.T.'s relative to the single server birth-and-death queueing process with parameters λn = λ/(n + 1), n ≥ 0, μn = μ, n ≥ 1. We have obtained an exact formulation of p∗i0, p∗in (n ≥ 1) being determined recursively. An exact expression for p10(t) is given in the case of low traffic intensities, and this has been approximated efficiently. Numerical evaluations show that the steady-state is reached very rapidly.

The transient behaviour of the waiting time process in GI/M/S is studied algebraically by means of a two-dimensional Markovian process {(vn, ln} , where the variables vn denote the times of full occupation immediately after the arrival instants Tn and where ln = max {0,– 1 + number of idle servers at (Tn + 0)}.

This paper is concerned with an infinite discrete dam fed by inputs which form a moving average sequence. Generating functions are derived for the joint time dependent distribution of the content, accumulated input and total dry time, the distribution of first emptiness, and the stationary content distribution. Also investigated is the problem of first emptiness before overflow for the finite dam.