Large distributed client-server systems often contain subsystems which are either identical to each other, or very nearly so, and this simplifies the system description for planning purposes. These replicated components and subsystems all have the same workload and performance parameters. It is known how to exploit this symmetry to simplify the solution of some kinds of performance models, using state aggregation in Markov Chains. This work considers the same problem for layered queueing models, using mean value analysis. The mean values are found for each group of replicas just once, and then are inserted appropriately into the solution of the system as a whole. An algorithm has been implemented in the Layered Queueing Network Solver (LQNS), including approximations to deal with interactions among the replicas, and is evaluated for accuracy and for efficiency. The resulting solver is insensitive (in time of solution) to the number of replicas in a group, and can efficiently calculate waiting times and throughputs for systems with tens of thousands of nodes and processes.
Replication is a technique used in distributed systems to improve performance, availability, and reliability. In replication schemes, often a J out of N voting pattern (also called quorum) is used in which the quorum waits for J replies to arrive. Integrating a quorum scheme into the Layered Queueing Network (LQN) performance modeling language necessitates the computation of the quorum response time as the J th order statistic. To do so, we need the exact (or an accurate estimation of the) time distribution of individual replies. This distribution was estimated in previous work but only for the special case of (J = N ) and yields large errors for J N . This paper presents a new analytic approach for the derivation of the distributions. Under a number of assumptions, we derive closed form expressions for the probability distribution functions of the replies. The application of our new approach on a number of LQN models shows that, even for models that violate those assumptions, it is far more accurate than previous approaches and it yields an error less than 10% for most example models.
We present an implementation of model checking for the probabilistic π-calculus, a process algebra which supports modelling of concurrency, mobility and discrete probabilistic behaviour. Formal verification techniques for this calculus have clear applications in several domains, including mobile ad-hoc network protocols and random security protocols. Despite this, no implementation of automated verification exists. Building upon the (non-probabilistic) π-calculus model checker MMC, we first show an automated procedure for constructing the Markov decision process representing a probabilistic π-calculus process. This can then be verified using existing probabilistic model checkers such as PRISM. Secondly, we demonstrate how for a large class of systems a more efficient, compositional approach can be applied, which uses our extension of MMC on each parallel component of the system and then translates the results into a highlevel model description for the PRISM tool. The feasibility of our techniques is demonstrated through three case studies from the π-calculus literature.
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