arXiv subjects

David Koops

Publications and source records attributed to David Koops.

4 recordsLinked to original sources

arXiv2018

Predicting the confirmation time of Bitcoin transactions

We study the probabilistic distribution of the confirmation time of Bitcoin transactions, conditional on the current memory pool (i.e., the queue of transactions awaiting confirmation). The results of this paper are particularly interesting for users that want to make a Bitcoin transaction during `heavy-traffic situations', when the transaction demand exceeds the block capacity. In such situations, Bitcoin users tend to bid up the transaction fees, in order to gain priority over other users that pay a lower fee. We argue that the time until a Bitcoin transaction is confirmed can be modelled as a particular stochastic fluid queueing process (to be precise: a Cram\'er-Lundberg process). We approximate the queueing process in two different ways. The first approach leads to a lower bound on the confirmation probability, which becomes increasingly tight as traffic decreases. The second approach relies on a diffusion approximation with a continuity correction, which becomes increasingly accurate as traffic intensifies. The accuracy of the approximations under different traffic loads are evaluated in a simulation study.

cs.PF↗

arXiv2017

Infinite-server queues with Hawkes input

In this paper we study the number of customers in infinite-server queues with a self-exciting (Hawkes) arrival process. Initially we assume that service requirements are exponentially distributed and that the Hawkes arrival process is of a Markovian nature. We obtain a system of differential equations that characterizes the joint distribution of the arrival intensity and the number of customers. Moreover, we provide a recursive procedure that explicitly identifies (transient and stationary) moments. Subsequently, we allow for non-Markovian Hawkes arrival processes and non-exponential service times. By viewing the Hawkes process as a branching process, we find that the probability generating function of the number of customers in the system can be expressed in terms of the solution of a fixed-point equation. We also include various asymptotic results: we derive the tail of the distribution of the number of customers for the case that the intensity jumps of the Hawkes process are heavy-tailed, and we consider a heavy-traffic regime. We conclude the paper by discussing how our results can be used computationally and by verifying the numerical results via simulations.

math.PR↗

arXiv2017

Linear Stochastic Fluid Networks: Rare-Event Simulation and Markov Modulation

We consider a linear stochastic fluid network under Markov modulation, with a focus on the probability that the joint storage level attains a value in a rare set at a given point in time. The main objective is to develop efficient importance sampling algorithms with provable performance guarantees. For linear stochastic fluid networks without modulation, we prove that the number of runs needed (so as to obtain an estimate with a given precision) increases polynomially (whereas the probability under consideration decays essentially exponentially); for networks operating in the slow modulation regime, our algorithm is asymptotically efficient. Our techniques are in the tradition of the rare-event simulation procedures that were developed for the sample-mean of i.i.d. one-dimensional light-tailed random variables, and intensively use the idea of exponential twisting. In passing, we also point out how to set up a recursion to evaluate the (transient and stationary) moments of the joint storage level in Markov-modulated linear stochastic fluid networks.

math.PR↗

arXiv2016

Networks of $\cdot/G/\infty$ Server Queues with Shot-Noise-Driven Arrival Intensities

We study infinite-server queues in which the arrival process is a Cox process (or doubly stochastic Poisson process), of which the arrival rate is given by shot noise. A shot-noise rate emerges as a natural model, if the arrival rate tends to display sudden increases (or: shots) at random epochs, after which the rate is inclined to revert to lower values. Exponential decay of the shot noise is assumed, so that the queueing systems are amenable for analysis. In particular, we perform transient analysis on the number of customers in the queue jointly with the value of the driving shot-noise process. Additionally, we derive heavy-traffic asymptotics for the number of customers in the system by using a linear scaling of the shot intensity. First we focus on a one dimensional setting in which there is a single infinite-server queue, which we then extend to a network setting.

math.PR↗