Stavros Lopatatzidis

Robust queueing theory: an initial study using imprecise probabilities

We study the robustness of performance predictions of discrete-time finite-capacity queues by applying the framework of imprecise probabilities. More concretely, 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 independence in this framework, namely repetition independence and epistemic irrelevance. In the first approach, we assume the existence of time-homogeneous probabilities for arrival and departure, which leads us to consider a collection of stationary queues. In the second, the stationarity assumption is dropped and we allow the arrival and departure probabilities to vary 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 for these two approaches. One of our observations is 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 a steady-state queue.

Imprecise stochastic processes in discrete time

We justify and discuss expressions for joint lower and upper expectations in imprecise probability trees, in terms of the sub- and supermartingales that can be associated with such trees. These imprecise probability trees can be seen as discrete-time stochastic processes with finite state sets and transition probabilities that are imprecise, in the sense that they are only known to belong to some convex closed set of probability measures. We derive various properties for their joint lower and upper expectations, and in particular a law of iterated expectations. We then focus on the special case of imprecise Markov chains, investigate their Markov and stationarity properties, and use these, by way of an example, to derive a system of non-linear equations for lower and upper expected transition and return times. Most importantly, we prove a game-theoretic version of the strong law of large numbers for submartingale differences in imprecise probability trees, and use this to derive point-wise ergodic theorems for imprecise Markov chains. We develop global models for imprecise stochastic processes in discrete time.We define joint lower and upper expectations and study their properties.We do not impose the usual restriction that random variables should be bounded.We apply our results to study the special case of imprecise Markov chains.We prove point-wise ergodic theorems for imprecise Markov chains.

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. Necessary conditions for excluding given independencies from basis computation.Properties of an independency relation that help reduce the size of a representative basis.An algorithm for basis computation that improves on earlier ones in terms of result and efficiency.

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.

Exploiting Stability for Compact Representation of Independency Models

The notion of stability in semi-graphoid independency models was introduced to describe the dynamics of (probabilistic) independency upon inference. We revisit the notion in view of establishing compact representations of semi-graphoid models in general. Algorithms for this purpose typically build upon dedicated operators for constructing new independency statements from a starting set of statements. In this paper, we formulate a generalised strong-contraction operator to supplement existing operators, and prove its soundness. We then embed the operator in a state-of-the-art algorithm and illustrate that the thus enhanced algorithm may establish more compact model representations.

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. We define imprecise birth-death chains in discrete time.We develop global models for them using sub- and supermartingales.We present simple methods for computing bounds on expected first passage times.We investigate the special case where the local models are linear-vacuous.We illustrate our methods with numerical examples.