Optimizing the Throughput of Particulate Streams Subject to Blocking

Filtration, flow in narrow channels and traffic flow are examples of processes subject to blocking when the channel conveying the particles becomes too crowded. If the blockage is temporary, which means that after a finite time the channel is flushed and reopened, one expects to observe a maximum throughput for a finite intensity of entering particles. We investigate this phenomenon by introducing a queueing theory inspired, circular Markov model. Particles enter a channel with intensity $\lambda$ and exit at a rate $\mu$. If $N$ particles are present at the same time in the channel, the system becomes blocked and no more particles can enter until the blockage is cleared after an exponentially distributed time with rate $\mu^*$. We obtain an exact expression for the steady state throughput (including the exiting blocked particles) for all values of $N$. For $N=2$ we show that the throughput assumes a maximum value for finite $\lambda$ if $\mu^*/\mu < 1/4$. The time-dependent throughput either monotonically approaches the steady state value, or reaches a maximum value at finite time. We demonstrate that, in the steady state, this model can be mapped to a previously introduced non-Markovian model with fixed transit and blockage times. We also examine an irreversible, non-Markovian blockage process with constant transit time exposed to an entering flux of fixed intensity for a finite time and we show that the first and second moments of the number of exiting particles are maximized for a finite intensity.Comment: 20 pages, 13 figure

Two parallel insurance lines with simultaneous arrivals and risks correlated with inter-arrival times

We investigate an insurance risk model that consists of two reserves which receive income at fixed rates. Claims are being requested at random epochs from each reserve and the interclaim times are generally distributed. The two reserves are coupled in the sense that at a claim arrival epoch, claims are being requested from both reserves and the amounts requested are correlated. In addition, the claim amounts are correlated with the time elapsed since the previous claim arrival. We focus on the probability that this bivariate reserve process survives indefinitely. The infinite- horizon survival problem is shown to be related to the problem of determining the equilibrium distribution of a random walk with vector-valued increments with reflecting boundary. This reflected random walk is actually the waiting time process in a queueing system dual to the bivariate ruin process. Under assumptions on the arrival process and the claim amounts, and using Wiener-Hopf factor- ization with one parameter, we explicitly determine the Laplace-Stieltjes transform of the survival function, c.q., the two-dimensional equilibrium waiting time distribution. Finally, the bivariate transforms are evaluated for some examples, including for proportional reinsurance, and the bivariate ruin functions are numerically calculated using an efficient inversion scheme.Comment: 24 pages, 6 figure

Queues and risk processes with dependencies

We study the generalization of the G/G/1 queue obtained by relaxing the assumption of independence between inter-arrival times and service requirements. The analysis is carried out for the class of multivariate matrix exponential distributions introduced in [12]. In this setting, we obtain the steady state waiting time distribution and we show that the classical relation between the steady state waiting time and the workload distributions re- mains valid when the independence assumption is relaxed. We also prove duality results with the ruin functions in an ordinary and a delayed ruin process. These extend several known dualities between queueing and risk models in the independent case. Finally we show that there exist stochastic order relations between the waiting times under various instances of correlation

Impact of hydrothermalism on the ocean iron cycle

As the iron supplied from hydrothermalism is ultimately ventilated in the iron-limited Southern Ocean, it plays an important role in the ocean biological carbon pump. We deploy a set of focused sensitivity experiments with a state of the art global model of the ocean to examine the processes that regulate the lifetime of hydrothermal iron and the role of different ridge systems in governing the hydrothermal impact on the Southern Ocean biological carbon pump. Using GEOTRACES section data, we find that stabilization of hydrothermal iron is important in some, but not all regions. The impact on the Southern Ocean biological carbon pump is dominated by poorly explored southern ridge systems, highlighting the need for future exploration in this region. We find inter-basin differences in the isopycnal layer onto which hydrothermal Fe is supplied between the Atlantic and Pacific basins, which when combined with the inter-basin contrasts in oxidation kinetics suggests a muted influence of Atlantic ridges on the Southern Ocean biological carbon pump. Ultimately, we present a range of processes, operating at distinct scales, that must be better constrained to improve our understanding of how hydrothermalism affects the ocean cycling of iron and carbon

Queues and risk models with simultaneous arrivals

We focus on a particular connection between queueing and risk models in a multi-dimensional setting. We first consider the joint workload process in a queueing model with parallel queues and simultaneous arrivals at the queues. For the case that the service times are ordered (from largest in the first queue to smallest in the last queue) we obtain the Laplace-Stieltjes transform of the joint stationary workload distribution. Using a multivariate duality argument between queueing and risk models, this also gives the Laplace transform of the survival probability of all books in a multivariate risk model with simultaneous claim arrivals and the same ordering between claim sizes. Other features of the paper include a stochastic decomposition result for the workload vector, and an outline how the two-dimensional risk model with a general two-dimensional claim size distribution (hence without ordering of claim sizes) is related to a known Riemann boundary value problem

Ruin excursions, the G/G/Infinity queue and tax payments in renewal risk models

In this paper we investigate the number and maximum severity of the ruin excursion of the insurance portfolio reserve process in the Cramer-Lundberg model with and without tax payments. We also provide a relation of the Cramer-Lundberg risk model with the G/G/infinity queue and use it to derive some explicit ruin probability formulae. Finally, the renewal risk model with tax is considered, and an asymptotic identity is derived that in some sense extends the tax identity of the Cramer-Lundberg risk model