Section: New Results

Performance Evaluation

Participants : Gerardo Rubino, Bruno Sericola.

Fluid Queues. Stochastic fluid flow models and, in particular, those driven by Markov chains, have been intensively studied in the last two decades. Not only they have been proven to be efficient tools to mimic Internet traffic flows at a macroscopic level but they are useful tools in many areas of applications such as manufacturing systems or in actuarial sciences, to cite but a few. We propose in [61] a chapter which focus on such a model in the context of performance analysis of a potentially congested system. The latter is modeled by means of a finite-capacity system whose content is described by a Markov driven stable fluid flow. We describe step-by-step a methodology to compute exactly the loss probability of the system. Our approach is based on the computation of hitting probabilities jointly with the peak level reached during a busy period, both in the infinite and finite buffer case. Accordingly we end up with differential Riccati equations that can be solved numerically. Moreover, we are able to characterize the complete distribution of both the duration of congestion and of the total information lost during such a busy period.

Connecting irreducible and absorbing Markov chains. Irreducible Markov chains in continuous time are the basic tool for instance in performance evaluation (typically, a queuing model), where in a large majority of cases, we are interested in the behavior of the modeled system in steady-state. Most metrics used are based on the stationary distribution of the model, under unicity natural conditions. Absorbing Markov chains, also in continuous time, play the equivalent role in dependability evaluation, because realistic models must have a finite lifetime, which corresponds here to the absorption time of the chain. In this case, the object of interest is this lifetime, steady-state gives no useful information about the system, and most of the used metrics are defined based on that object. In [30] with describe different connections between the two worlds together with some consequences of those relations in both areas, that is, both in performance and in dependability.

Transient analysis of Markov queueing models. Analyzing the transient behavior of a queueing system is much harder than studying its steady state, the difference being basically that of moving from a linear system to a linear differential system. However, a huge amount of efforts has been put on the former problem, from all kinds of points of view: trials to find closed-forms of the main state distributions, algorithms for numerical evaluations, approximations of different types, exploration of other transient metrics than the basic state distributions, etc. In [62] we focus on the first two elements, the derivation of closed-forms for the main transient state distributions, and the development of numerical techniques. The chapter is organized as a survey, and the main goal is to position and to underline the role of the uniformization technique, for both finding closed-forms and for developing efficient numerical evaluation procedures. In some cases, we extend the discussion to other related transient metrics that are relevant for applications.