Virginia Giorno

On the M / M /1 Queue with Catastrophes and Its Continuous Approximation

For the M/M/1 queue in the presence of catastrophes the transition probabilities, densities of the busy period and of the catastrophe waiting time are determined. A heavy-traffic approximation to this discrete model is then derived. This is seen to be equivalent to a Wiener process subject to randomly occurring jumps for which some analytical results are obtained. The goodness of the approximation is discussed by comparing the closed-form solutions obtained for the continuous process with those obtained for the M/M/1 catastrophized queue.

On the evaluation of first-passage-time probability densities via non-singular integral equations

The algorithm given by Buonocore et al. (1) to evaluate first-passage-time p.d.f.s for Wiener and Ornstein-Uhlenbeck processes through a time-dependent boundary is extended to a wide class of time-homogeneous one-dimensional diffusion proc- esses. Several examples are thoroughly discussed along with some computational results. equation was proposed to determine first-passage-time p.d.f.s through time- dependent boundaries for time-homogeneous one-dimensional diffusion processes with natural boundaries. This equation was seen to be particularly manageable for computational purposes in the cases of Wiener and Ornstein-Uhlenbeck processes. Indeed, for these the kernel of the integral equation can be made continuous, thus overcoming some crucial difficulties arising from the singular nature of the integral equations for the first-passage-time p.d.f. Such continuity was made possible by a suitable choice of two arbitrary time-dependent functions that appear in the kernel of the integral equations. Here, in view of the relevance of first-passage-time problems in numerous applied fields (see, for instance, Buonocore et al. (1) and references therein), we intend to provide an extension of the results of Buonocore et al. (1) in two directions: on the one hand, we shall prove that the method for regularizing the kernel of the integral equation is valid not only for the Wiener and the Ornstein-Uhlenbeck processes but can also be used for a more general class of diffusion processes whose free transition

ON THE ASYMPTOTIC BEHAVIOUR OF FIRST- PASSAGE-TIME DENSITIES FOR ONE-DIMENSIONAL DIFFUSION PROCESSES AND VARYING BOUNDARIES

Making use of the integral equations given in [1], [2] and [3], the asymptotic behaviour of the first-passage time (FPT) p.d.f.s through certain time-varying boundaries, including periodic boundaries, is determined for a class of onedimensional diffusion processes with steady-state density. Sufficient conditions are given for the cases both of single and of pairs of asymptotically constant and asymptotically periodic boundaries, under which the FPT densities asymptotically exhibit an exponential behaviour. Explicit expressions are then worked out for the processes that can be obtained from the Omstein-Uhlenbeck process by spatial transformations. Some new asymptotic results for the FPT density of the Wiener process are finally proved, together with a few miscellaneous results.

ON THE TWO-BOUNDARY FIRST-CROSSING-TIME PROBLEM FOR DIFFUSION PROCESSES

The first-crossing-time problem through two time-dependent boundaries for one-dimensional diffusion processes is considered. The first-crossing p.d.f.s from below and from above are proved to satisfy a new system of Volterra integral equations of the second kind involving two arbitrary continuous functions. By a suitable choice of such functions a system of continuous-kernel integral equations is obtained and an efficient algorithm for its solution is provided. Finally, conditions on the drift and infinitesimal variance of the diffusion process are given such that the system of integral equations reduces to a non-singular single integral equation for the first-crossing-time p.d.f.

A Double-ended Queue with Catastrophes and Repairs, and a Jump-diffusion Approximation

Consider a system performing a continuous-time random walk on the integers, subject to catastrophes occurring at constant rate, and followed by exponentially-distributed repair times. After any repair the system starts anew from state zero. We study both the transient and steady-state probability laws of the stochastic process that describes the state of the system. We then derive a heavy-traffic approximation to the model that yields a jump-diffusion process. The latter is equivalent to a Wiener process subject to randomly occurring jumps, whose probability law is obtained. The goodness of the approximation is finally discussed.

On a symmetry-based constructive approach to probability densities for two-dimensional diffusion processes

The method earlier introduced for one-dimensional diffusion processes [6] is extended to obtain closed form expressions for the transition p.d.f.s of two-dimensional diffusion processes in the presence of absorbing boundaries and for the first-crossing time p.d.f.s through such boundaries. Use of such a method is finally made to analyse a two-dimensional linear process.

Inferring the effect of therapy on tumors showing stochastic Gompertzian growth

The present work deals with a Gompertz-type diffusion process, which includes in the drift term a time-dependent function C(t) representing the effect of a therapy able to modify the dynamics of the underlying process. However, in experimental studies is not immediate to deduce the functional form of C(t) from a treatment protocol. So a statistical approach is proposed in order to estimate this function when a control group and one or more treated groups are observed. In order to validate the proposed strategy a simulation study for several interesting functional forms of C(t) has been carried out. Finally, an application to infer the net effect of cisplatin and doxorubicin+cyclophosphamide in actual murine models is presented.

On some time non homogeneous diffusion approximations to queueing systems

Time-non-homogeneous diffusion approximations to single server-single queue-FCFs discipline systems are considered. Under various assumptions on the nature of the time-dependent functions appearing in the infinitesimal moments the transient and the regime behaviour of the approximating diffusions are analysed in some detail. Special attention is then given to the study of a diffusion approximation characterized by a linear drift and by a periodically time-varying infinitesimal variance. Unlike the behaviour of transition functions and moments, the p.d.f. of the busy period is seen to be unaffected by the presence of such periodicity.

On some diffusion approximations to queueing systems

For a class of models of adaptive queueing systems an exact diffusion approximation is derived with the aim of obtaining information on the evolution of the systems. Our approximating diffusion process includes the Wiener and the Ornstein-Uhlenbeck processes with reflecting boundaries at 0. The goodness of the approximations is thoroughly discussed and the closed-form solutions obtained for the diffusion processes are compared with those holding for the queueing system in order to investigate the conditions under which reliable information can be obtained from the approximating continuous models. For the latter the transient behaviour is quantitatively analysed and the distribution of the busy period is determined in closed form. ADAPTIVE QUEUEING SYSTEMS; M/M/1 MODEL; WIENER PROCESS; ORNSTEINUHLENBECK PROCESS; FIRST-PASSAGE TIME; BUSY PERIOD

On some time non-homogeneous queueing systems with catastrophes

Computations for non-homogeneous birth-and-death processes with catastrophes.Applications to queueing and population dynamics problems.Effect of the periodicity of the rates on the performance of systems with catastrophes. Non-stationary queueing systems subject to catastrophes occurring with time varying intensity are considered. The effect of a catastrophe is to make the queue instantly empty. The transition probabilities, the related moments and the first visit time density to zero state are analyzed. Particular attention is dedicated to queueing systems in the presence of catastrophes with periodic intensity function. Various applications are provided, including the non-stationary birth-death process with immigration, the queueing systems M ( t ) / M ( t ) / 1 and M ( t ) / M ( t ) / ∞ .