Existence of an invariant measure for stochastic evolutions driven by an eventually compact semigroup

It is shown that for an SDE in a Hilbert space, eventual compactness of the driving semigroup together with compact perturbations can be used to establish the existence of an invariant measure. The result is applied to stochastic functional differential equations and the heat equation perturbed by delay and noise, which are both shown to be driven by an eventually compact semigroup.     

Two-stage queueing network models for quality control and testing

We study sojourn times in a two-node open queueing network with a processor sharing node and a delay node, with Poisson arrivals at the PS node. Motivated by quality control and blood testing applications, we consider a feedback mechanism in which customers may either leave the system after service at the PS node or move to the delay node; from the delay node, they always return to the PS node for new quality controls or blood tests. We propose various approximations for the distribution of the total sojourn time in the network; each of these approximations yields the exact mean sojourn time, and very accurate results for the variance. The best of the three approximations is used to tackle an optimization problem that is mainly inspired by a blood testing application.     

Mechanism of immunoglobulin G4 Fab-arm exchange

Immunoglobulin G (IgG) antibodies are symmetrical molecules that may be regarded as covalent dimers of 2 half-molecules, each consisting of a light chain and a heavy chain. Human IgG4 is an unusually dynamic antibody, with half-molecule exchange ("Fab-arm exchange") resulting in asymmetrical, bispecific antibodies with two different antigen binding sites, which contributes to its anti-inflammatory activity. The mechanism of this process is unknown. To elucidate the elementary steps of this intermolecular antibody rearrangement, we developed a quantitative real-time FRET assay to monitor the kinetics of this process. We found that an intrinsic barrier is the relatively slow dissociation of the CH3 domains that noncovalently connect the heavy chains, which becomes rate determining in case disulfide bonds between the heavy chains are reduced or absent. Under redox conditions that mimic the previously estimated in vivo reaction rate, i.e., 1 mM of reduced glutathione, the overall rate is ca. 20 times lower because only a fraction of noncovalent isomers is present (with intra- rather than interheavy chain disulfide bonds), formed in a relatively fast pre-equilibrium from covalent isomers. Interestingly, Fab arms stabilize the covalent isomer: the amount of noncovalent isomers is ca. 3 times higher for Fc fragments of IgG4 (lacking Fab domains) compared to intact IgG4, and the observed rate of exchange is 3 times higher accordingly. Thus, kinetic data obtained from a sensitive and quantitative real-time FRET assay as described here yield accurate data about interdomain interactions such as those between Fab and/or Fc domains. The results imply that in vivo, the reaction is under control of local redox conditions.     

Invariant Measures for Monotone SPDEs with Multiplicative Noise Term

We study diffusion processes corresponding to infinite dimensional semilinear stochastic differential equations with local Lipschitz drift term and an arbitrary Lipschitz diffusion coefficient. We prove tightness and the Feller property of the solution to show existence of an invariant measure. As an application we discuss stochastic reaction diffusion equations.     

Marginal queue length approximations for a two-layered network with correlated queues

We consider an extension of the classical machine-repair model, where we assume that the machines, apart from receiving service from the repairman, also serve queues of products. The extended model can be viewed as a layered queueing network, where the first layer consists of the queues of products and the second layer is the ordinary machine-repair model. As the repair time of one machine may affect the time the other machine is not able to process products, the downtimes of the machines are correlated. This correlation leads to dependence between the queues of products in the first layer. Analysis of these queue length distributions is hard, as the exact dependence structure for the downtimes, or the queue lengths, is not known. Therefore, we obtain an approximation for the complete marginal queue length distribution of any queue in the first layer, by viewing such a queue as a single server queue with correlated server downtimes. Under an explicit assumption on the form of the downtime dependence, we obtain exact results for the queue length distribution for that single server queue. We use these exact results to approximate the machine-repair model. We do so by computing the downtime correlation for the latter model and by subsequently using this information to fine-tune the parameters we introduced to the single server queue. As a result, we immediately obtain an approximation for the queue length distributions of products in the machine-repair model, which we show to be highly accurate by extensive numerical experiments.     

A continuous-flow crushing device for on-line delta2H analysis of fluid inclusion water in speleothems

A method for the isotope analysis of fluid inclusion water in speleothem calcite is presented. The technique is based on a commercially available continuous-flow pyrolysis furnace (ThermoFinnigan TC-EA). The main adaptation made to the standard TC-EA configuration is the addition of a crusher and cold trap unit, which is connected to the carrier gas inlet at the top of the TC-EA reactor tube. A series of tests conducted with this device shows that: (1) standard waters, injected in the crusher, and passed through a cryogenic trapping routine, yield accurate delta(2)H values; (2) crushed cubes of speleothem calcite from two Peruvian caves with rather dissimilar seepage water delta(2)H values yield fluid inclusion delta(2)H values in good accordance with these drip waters. The clear advantage of this continuous-flow technique for fluid inclusion isotope analysis is that it is relatively quick compared with other techniques. Since the conditions of water sample introduction into the TC-EA are identical for delta(2)H and delta(18)O analysis, we expect that only limited adaptations to the extraction procedure are required to provide delta(18)O analysis of fluid inclusion samples with the same device.     

Sojourn times in a processor sharing queue with multiple vacations

We study an M/G/1 processor sharing queue with multiple vacations. The server only takes a vacation when the system has become empty. If he finds the system still empty upon return, he takes another vacation, and so on. Successive vacations are identically distributed, with a general distribution. When the service requirements are exponentially distributed we determine the sojourn time distribution of an arbitrary customer. We also show how the same approach can be used to determine the sojourn time distribution in an M/M/1-PS queue of a polling model, under the following constraints: the service discipline at that queue is exhaustive service, the service discipline at each of the other queues satisfies a so-called branching property, and the arrival processes at the various queues are independent Poisson processes. For a general service requirement distribution we investigate both the vacation queue and the polling model, restricting ourselves to the mean sojourn time.     

Stability results for stochastic delayed recurrent neural networks with discrete and distributed delays

We present new conditions for asymptotic stability and exponential stability of a class of stochastic recurrent neural networks with discrete and distributed time varying delays. Our approach is based on the method using fixed point theory, which do not resort to any Liapunov function or Liapunov functional. Our results neither require the boundedness, monotonicity and differentiability of the activation functions nor differentiability of the time varying delays. In particular, a class of neural networks without stochastic perturbations is also considered. Examples are given to illustrate our main results.     

Busy period analysis of the state dependent M/M/1/K queue

In this paper, we study the transient behavior of a state dependent M/M/1/K queue during the busy period. We derive in closed-form the joint transform of the length of the busy period, the number of customers served during the busy period, and the number of losses during the busy period. For two special cases called the threshold policy and the static policy we determine simple expressions for their joint transform.     

Polling: past,present, and perspective

This is a survey on polling systems, focussing on the basic single-server multi-queue polling system in which the server visits the queues in cyclic order. The main goals of the paper are: (i) to discuss a number of the key methodologies in analyzing polling models; (ii) to give an overview of recent polling developments; and (iii) to present a number of challenging open problems.