Section: New Results

Analytic models

Participants : Gerardo Rubino, Bruno Sericola.

Sojourn times in Markovian models. In [98], 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, in the 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 (see our book  [111] for a synthesis of these works), 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 many queues that maximizes their power also leads to a mean backlog equal to exactly one customer. Last year  [110] we explored what happens with this metric when we consider Jackson queuing networks. After showing that the same property holds for them, we showed that the power metric has some drawbacks, mainly when considering multiserver queues and networks of queues. We then proposed 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 showed that the same “keep the pipe full” holds for it. In the keynote [34] we presented these ideas together with some new results (for example, the analysis of G-queues from this point of view).

For other analytical-oriented work, see [72] for new applications of queueing theory used at the Markovian level, and [72] for applications of stochastic analysis to general problems where performance and dependability are simultaneously taken into account in the same model.