Section: New Results

Analytic models

Participants : Gerardo Rubino, Bruno Sericola.

Sojourn times in Markovian models. In [74] , we discuss different issues related to the time a Markov chain spends in a part of its state space. This is relevant in many application areas including those interesting Dionysos, namely, performance and dependability analysis of complex systems. For instance, in dependability, the reliability of a system subject to failures and repairs of its components, is, in terms of a discrete-space model of it, the probability that it remains in the subset of operational or up states during the whole time interval $[0,t]$. In performance, the occupancy factor of some server is the probability that, in steady state, the model belongs to the subset of states where the server is busy. This book chapter reviews some past work done by the authors on this topic, and add some new insights on the properties of these sojourn times.

Queuing systems in equilibrium. In the late 70s, Leonard Kleinrock proposed a metric able to capture the tradeoff between the work done by a system and its cost, or, in terms of queueing systems, between throughput and mean response time. The new metric was called power and among its properties, it satisfies a nice one informally called “keep the pipe full”, specifying that the operation point of some queues (mainly the $M/M/1$ one) giving the maximal possible value to the power is when the mean backlog is 1. In [56] , we took back this idea to explore what happens when we consider Jackson queuing networks. After showing that the same property holds for them and exploring other ones, we show that the power metric has some drawbacks when considering multiserver queues and networks of queues. We then propose a new metric that we called effectiveness, identical to power when there is a single queue with a single server, but different otherwise, that avoids these drawbacks. We analyze it and, in particular, we show that the same “keep the pipe full” holds for it.

Transient analysis of queuing systems. In a well-known book [86] , today out of press, a concept of dual of a birth-and-death process is proposed, based on stochastic monotonicity. In past work  [88] we showed that this concept coupled with the classical randomization or uniformization of continuous time Markov chains and lattice path combinatorics, allowed to derive analytical expressions of the transient distribution of several Markovian queueing systems. Recently, we discovered two new things: first, that this dual concept can be generalized to arbitrary systems of ordinary differential equations (ODEs) and still keep its main properties; second, that we can define a similar transformation than uniformization, that can be applied to arbitrary systems of ODEs and again, holding similar properties than the former. We respectively called pseudo-dual and pseudo-randomization the two concepts and associated methods. In [69] , we presented these ideas and first results about them. We illustrated their use, and how they allow to obtain analytical expressions of transient queues' distributions in cases where Anderson's dual doesn't exist (see  [87] .

In [68] , we present results concerning some aspects of the behavior of a queueing system observed during a fixed time period of the form $[0,t]$. The two aspects we looked at in this work are the loss process of a finite capacity model during the considered $[0,t]$, and the maximal backlog reached at a queue over the interval. Following the classical procedure mentioned below, consisting in using uniformization to go to discrete time and then, combinatorial techniques, we develop numerical schemes to analyze both aspects of some basic queueing systems.

Network reliability. In [28] , we consider the classical network design “Capacitated $m$-Ring Star Problem” (CmRSP), where we look for $m$ rings connecting two nodes in a network at minimum cost. We add to this model the fact that links can fail, and propose a new paradigm that we call “Capacitated $m$-Ring Star Problem with Diameter Constrained Reliability ” (in short, CmRSP-DCR), where we look again for a minimal cost spanning graph of the set of nodes in the network that connects the selected source and terminal, while satisfying a Diameter Constrained Reliability (DCR) condition. The DCR is the probability that the two nodes can communicate by means of paths having lengths bounded by some fixed value $d$. We prove that this problem is NP-hard, and we propose a GRASP-based approach to solve it.

Fluid models. In [19] we study congestion periods in a finite fluid buffer when the net input rate depends upon a recurrent Markov process; congestion occurs when the buffer content is equal to the buffer capacity. We consider the duration of congestion periods as well as the associated volume of lost information. We derive their distributions in a typical stationary busy period of the buffer. Our goal is to compute the exact expression of the loss probability in the system, which is usually approximated by the probability that the occupancy of the infinite buffer is greater than the buffer capacity under consideration. Moreover, by using general results of the theory of Markovian arrival processes, we show that the duration of congestion and the volume of lost information have phase-type distributions.