Robustness in queueing systems

We study the robustness of performance predictions of discrete- time finite-capacity queues by applying the framework of imprecise probabilities. More precisely, we consider the Geo/Geo/1/L model with probabilities of arrival and departure that are no longer fixed, but are allowed to vary within given intervals. We distinguish between two concepts of stochastic independence in this framework, namely repetition independence and epistemic irrelevance. In the first approach, we assume the existence of a time-homogeneous probability for an arrival and a respective one for a departure, which leads us to consider a collection of stationary queues. In the second, the assumption of stationarity is dropped and we allow the arrival and departure probabilities to differ from time point to time point; they may even depend on the complete history of queue lengths. We calculate bounds on the expected queue length, the probability of a particular queue length and the probability of turning on the server. For the expected queue length both approaches coincide. For the other performance measures, we observe and discuss various differences between the bounds obtained by these two approaches. Among our observations is the fact that ergodicity may break down due to imprecision. Bounds on expected time averages of certain functions on the state space are not necessarily equal to the bounds on the expectation of that function at random instants in steady state

Computing concise representations of semi-graphoid independency models

The conditional independencies from a joint probability distribution constitute a model which is closed under the semi-graphoid properties of independency. These models typically are exponentially large in size and cannot be feasibly enumerated. For describing a semi-graphoid model therefore, a more concise representation is used, which is composed of a representative subset of the independencies involved, called a basis, and letting all other independencies be implicitly defined by the semi-graphoid properties; for computing such a basis, an appropriate algorithm is available. Based upon new properties of semi-graphoid models in general, we introduce an improved algorithm that constructs a smaller basis for a given independency model than currently existing algorithms

Concise representations and construction algorithms for semi-graphoid independency models

The conditional independencies from a joint probability distribution constitute a model which is closed under the semi-graphoid properties of independency. These models typically are exponentially large in size and cannot be feasibly enumerated. For describing a semi-graphoid model therefore, researchers have proposed a more concise representation. This representation is composed of a representative subset of the independencies involved, called a basis, and lets all other independencies be implicitly defined by the semi-graphoid properties. An algorithm is available for computing such a basis for a semi-graphoid independency model. In this paper, we identify some new properties of a basis in general which can be exploited for arriving at an even more concise representation of a semi- graphoid model. Based upon these properties, we present an enhanced algorithm for basis construction which never returns a larger basis for a given independency model than currently existing algorithms

Computing lower and upper expected first passage and return times in imprecise birth-death chains

We provide simple methods for computing exact bounds on expected first-passage and return times in finite-state birth–death chains, when the transition probabilities are imprecise, in the sense that they are only known to belong to convex closed sets of probability mass functions. In order to do that, we model these so-called imprecise birth–death chains as a special type of time-homogeneous imprecise Markov chain, and use the theory of sub- and supermartingales to define global lower and upper expectation operators for them. By exploiting the properties of these operators, we construct a simple system of non-linear equations that can be used to efficiently compute exact lower and upper bounds for any expected first-passage or return time. We also discuss two special cases: a precise birth–death chain, and an imprecise birth–death chain for which the transition probabilities belong to linear-vacuous mixtures. In both cases, our methods simplify even more. We end the paper with some numerical examples

A pointwise ergodic theorem for imprecise Markov Chains

We prove a game-theoretic version of the strong law of large numbers for submartingale differences, and use this to derive a pointwise ergodic theorem for discrete-time Markov chains with finite state sets, when the transition probabilities are imprecise, in the sense that they are only known to belong to some convex closed set of probability measures