Sabina Rossi

In queueing networks, product-form solutions are of fundamental importance to efficiently compute performance metrics in complex models of computer systems. The product-form property entails that the steady-state probabilities of the joint stochastic process underlying the network can be expressed as the normalized product of functions that only depend on the local state of the components. In many relevant cases, product-forms are the only way to perform exact quantitative analyses of large systems. In this work, we introduce a novel class of product-form queueing networks where servers are always busy. Applications include model of systems where successive refinements on jobs improve the processes quality but are not strictly required to obtain a result. To this aim, we define a job movement policy that admits instantaneous migrations of jobs from non-empty waiting buffers to empty ones. Thus, the resulting routing scheme is state-dependent. This class of networks maximizes the system throughput. This model can be implemented with arbitrary topology, including feedback, and both in an open and closed setting. As far as closed systems are concerned, we give a convolution algorithm and the corresponding mean value analysis to compute expected performance indices for closed models.

This paper explores the concept of proportional lumpability as an extension of the original definition of lumpability, addressing the challenges posed by the state space explosion problem in computing performance indices for large stochastic models. Lumpability traditionally relies on state aggregation techniques and is applicable to Markov chains demonstrating structural regularity. Proportional lumpability extends this idea, proposing that the transition rates of a Markov chain can be modified by certain factors, resulting in a lumpable new Markov chain. This concept facilitates the derivation of precise performance indices for the original process. This paper establishes the well-defined nature of the problem of computing the coarsest proportional lumpability that refines a given initial partition, ensuring a unique solution exists. Additionally, a polynomial time algorithm is introduced to solve this problem, offering valuable insights into both the concept of proportional lumpability and the broader realm of partition refinement techniques. The effectiveness of proportional lumpability is demonstrated through a case study that consists of designing a model to investigate selfish mining behaviors on public blockchains. This research contributes to a better understanding of efficient approaches for handling large stochastic models and highlights the practical applicability of proportional lumpability in deriving exact performance indices.