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

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

A dissolved cobalt plume in the oxygen minimum zone of the eastern tropical South Pacific

© The Author(s), 2016. This article is distributed under the terms of the Creative Commons Attribution License. The definitive version was published in Biogeosciences 13 (2016): 5697-5717, doi:10.5194/bg-13-5697-2016.Cobalt is a nutrient to phytoplankton, but knowledge about its biogeochemical cycling is limited, especially in the Pacific Ocean. Here, we report sections of dissolved cobalt and labile dissolved cobalt from the US GEOTRACES GP16 transect in the South Pacific. The cobalt distribution is closely tied to the extent and intensity of the oxygen minimum zone in the eastern South Pacific with highest concentrations measured at the oxycline near the Peru margin. Below 200 m, remineralization and circulation produce an inverse relationship between cobalt and dissolved oxygen that extends throughout the basin. Within the oxygen minimum zone, elevated concentrations of labile cobalt are generated by input from coastal sources and reduced scavenging at low O2. As these high cobalt waters are upwelled and advected offshore, phytoplankton export returns cobalt to low-oxygen water masses underneath. West of the Peru upwelling region, dissolved cobalt is less than 10 pM in the euphotic zone and strongly bound by organic ligands. Because the cobalt nutricline within the South Pacific gyre is deeper than in oligotrophic regions in the North and South Atlantic, cobalt involved in sustaining phytoplankton productivity in the gyre is heavily recycled and ultimately arrives from lateral transport of upwelled waters from the eastern margin. In contrast to large coastal inputs, atmospheric deposition and hydrothermal vents along the East Pacific Rise appear to be minor sources of cobalt. Overall, these results demonstrate that oxygen biogeochemistry exerts a strong influence on cobalt cycling.This work was funded by NSF awards OCE-1233733 to MAS, OCE-1232814 to BST, and OCE-1237011 to JAR

Capacity Analysis of Sequential Zone Picking Systems

This paper develops a capacity model for sequential zone picking systems. These systems are popular internal transport and order-picking systems because of their scalability, flexibility, high-throughput ability, and fit for use for a wide range of products and order profiles. The major disadvantage of such systems is congestion and blocking under heavy use, leading to long order throughput times. To reduce blocking and congestion, most systems use the block-and-recirculate protocol to dynamically manage workload. In this paper, the various elements of the system, such as conveyor lanes and pick zones, are modeled as a multiclass block-and-recirculate queueing network with capacity constraints on subnetworks. Because of this blocking protocol, the stationary distribution of the queueing network is highly intractable. We propose an approximation method based on jumpover blocking. Multiclass jump-over queueing networks admit a product-form stationary distribution and can be efficiently evaluated by mean value analysis and Norton’s theorem. This method can be applied during the design phase of sequential zone picking systems to determine the number of segments, number and length of zones, buffer capacities, and storage allocation of products to zones to meet performance targets. For a wide range of parameters, the results show that the relative error in the system throughput is typically less than 1% compared with simulation

Tracing differences in iron supply to the Mid-Atlantic Ridge valley between hydrothermal vent sites: implications for the addition of iron to the deep ocean

Supply of iron (Fe) to the surface ocean supports primary productivity, and while hydrothermal input of Fe to the deep ocean is known to be extensive it remains poorly constrained. Global estimates of hydrothermal Fe supply rely on using dissolved Fe (dFe) to excess He (xs3He) ratios to upscale fluxes, but observational constraints on dFe/xs3He may be sensitive to assumptions linked to sampling and interpolation. We examined the variability in dFe/xs3He using two methods of estimation, for four vent sites with different geochemistry along the Mid-Atlantic Ridge. At both Rainbow and TAG, the plume was sampled repeatedly and the range of dFe/xs3He was 4 to 63 and 4 to 87 nmol:fmol, respectively, primarily due to differences in plume age. To account for background xs3He and shifting plume position, we calibrated He values using contemporaneous dissolved Mn (dMn). Applying this approach more widely, we found dFe/xs3He ratios of 12, 4–8, 4–44, and 4–86 nmol fmol−1 for the Menez Gwen, Lucky Strike, Rainbow, and TAG hydrothermal vent sites, respectively. Differences in plume dFe/xs3He across sites were not simply related to the vent endmember Fe and He fluxes. Within 40 km of the vents, the dFe/xs3He ratios decreased to 3–38 nmol fmol−1, due to the precipitation and subsequent settling of particulates. The ratio of colloidal Fe to dFe was consistently higher (0.67–0.97) than the deep N. Atlantic (0.5) throughout both the TAG and Rainbow plumes, indicative of Fe exchange between dissolved and particulate phases. Our comparison of TAG and Rainbow shows there is a limit to the amount of hydrothermal Fe released from vents that can form colloids in the rising plume. Higher particle loading will enhance the longevity of the Rainbow hydrothermal plume within the deep ocean assuming particles undergo continual dissolution/disaggregation. Future studies examining the length of plume pathways required to escape the ridge valley will be important in determining Fe supply from slow spreading mid-ocean ridges to the deep ocean, along with the frequency of ultramafic sites such as Rainbow. Resolving the ridge valley bathymetry and accounting for variability in vent sources in global biogeochemical models will be key to further constraining the hydrothermal Fe flux.</p

Tracing differences in iron supply to the Mid-Atlantic Ridge valley between hydrothermal vent sites: implications for the addition of iron to the deep ocean

Supply of iron (Fe) to the surface ocean supports primary productivity, and while hydrothermal input of Fe to the deep ocean is known to be extensive it remains poorly constrained. Global estimates of hydrothermal Fe supply rely on using dissolved Fe (dFe) to excess He (xs³He) ratios to upscale fluxes, but observational constraints on dFe/xs³He may be sensitive to assumptions linked to sampling and interpolation. We examined the variability in dFe/xs³He using two methods of estimation, for four vent sites with different geochemistry along the Mid-Atlantic Ridge. At both Rainbow and TAG, the plume was sampled repeatedly and the range of dFe/xs³He was 4 to 63 and 4 to 87 nmol:fmol, respectively, primarily due to differences in plume age. To account for background xs³He and shifting plume position, we calibrated He values using contemporaneous dissolved Mn (dMn). Applying this approach more widely, we found dFe/xs³He ratios of 12, 4–8, 4–44, and 4–86 nmol fmol−1 for the Menez Gwen, Lucky Strike, Rainbow, and TAG hydrothermal vent sites, respectively. Differences in plume dFe/xs³He across sites were not simply related to the vent endmember Fe and He fluxes. Within 40 km of the vents, the dFe/xs³He ratios decreased to 3–38 nmol fmol−1, due to the precipitation and subsequent settling of particulates. The ratio of colloidal Fe to dFe was consistently higher (0.67–0.97) than the deep N. Atlantic (0.5) throughout both the TAG and Rainbow plumes, indicative of Fe exchange between dissolved and particulate phases. Our comparison of TAG and Rainbow shows there is a limit to the amount of hydrothermal Fe released from vents that can form colloids in the rising plume. Higher particle loading will enhance the longevity of the Rainbow hydrothermal plume within the deep ocean assuming particles undergo continual dissolution/disaggregation. Future studies examining the length of plume pathways required to escape the ridge valley will be important in determining Fe supply from slow spreading mid-ocean ridges to the deep ocean, along with the frequency of ultramafic sites such as Rainbow. Resolving the ridge valley bathymetry and accounting for variability in vent sources in global biogeochemical models will be key to further constraining the hydrothermal Fe flux