Performance analysis of polling systems with retrials and glue periods

We consider gated polling systems with two special features: (i) retrials and (ii) glue or reservation periods. When a type-i customer arrives, or retries, during a glue period of the station i, it will be served in the following service period of that station. Customers arriving at station i in any other period join the orbit of that station and will retry after an exponentially distributed time. Such polling systems can be used to study the performance of certain switches in optical communication systems. When the glue periods are exponentially distributed, we obtain equations for the joint generating functions of the number of customers in each station. We also present an algorithm to obtain the moments of the number of customers in each station. When the glue periods are generally distributed, we consider the distribution of the total workload in the system, using it to derive a pseudo-conservation law which in turn is used to obtain accurate approximations of the individual mean waiting times. We also investigate the problem of choosing the lengths of the glue periods, under a constraint on the total glue period per cycle, so as to minimize a weighted sum of the mean waiting times.     

Milling of Organic Solids in a Jet Mill. Part 1: Determination of the Selection Function and Related Mechanical Material Properties

The particle size distribution of pharmaceutically active materials and other fine chemicals determines the performance of the final product to a large extent. Often milling of these particles is necessary. It is not possible to determine the milling conditions solely on the basis of the particle size distribution of the starting material, because the (mechanical) properties of the material also determine the desired milling conditions. It is often not possible to optimize milling conditions experimentally because the amount of material available is frequently highly limited. A theoretical approach towards predicting the best milling conditions is needed. The purpose of this study was to develop a method to predict the desired milling conditions given a specific (organic) solid material. The selection function and the breakage distribution function are usually the starting points in modeling the milling process. The selection function is the parameter that includes the material and mill properties. Dimensional analysis made it possible to correlate the selection function with material properties. A set of theories available in the literature enable prediction of the material properties. For different compounds (lactose, paracetamol, a steroidal compound, and two heterocyclic compounds) the selection functions were calculated. The calculations predict differences: lactose reduces slowly in size, while one of the heterocyclic compounds shows the most intense fracture.     

The Asymptotic Workload Behavior of Two Coupled Queues

We consider a system of two coupled queues Q ₁ and Q ₂. When both queues are backlogged, they are each served at unit rate. However, when one queue empties, the service rate at the other queue increases. Thus, the two queues are coupled through the mechanism for dynamically sharing surplus service capacity. We derive the asymptotic workload behavior at Q ₁ for various scenarios where at least one of the two queues has a heavy-tailed service time distribution. First of all, we consider a situation where the traffic load at Q ₁ is below the nominal unit service rate. We show that if the service time distribution at Q ₁ is heavy-tailed, then the workload behaves exactly as if Q ₁ is served in isolation at a constant rate, which only depends on the service time distribution at Q ₂ through its mean. In addition, we establish that if the service time distribution at Q ₁ is exponential, then the workload distribution is either exponential or semi-exponential, depending on whether the traffic load at Q ₂ exceeds the nominal service rate or not. Next, we focus on a regime where the traffic load at Q ₁ exceeds the nominal service rate, so that Q ₁ relies on the surplus capacity from Q ₂ to maintain stability. In that case, the workload distribution at Q ₁ is determined by the heaviest of the two service time distributions, so that Q ₁ may inherit potentially heavier-tailed characteristics from Q ₂.     

Waiting Time Asymptotics in the Single Server Queue with Service in Random Order

We consider the single server queue with service in random order. For a large class of heavy-tailed service time distributions, we determine the asymptotic behavior of the waiting time distribution. For the special case of Poisson arrivals and regularly varying service time distribution with index ?ν, it is shown that the waiting time distribution is also regularly varying, with index 1?ν, and the pre-factor is determined explicitly. Another contribution of the paper is the heavy-traffic analysis of the waiting time distribution in the M/G/1 case. We consider not only the case of finite service time variance, but also the case of regularly varying service time distribution with infinite variance.     

Optimization of headspace solid-phase microextraction gas chromatography-atomic emission detection analysis of monomethylmercury

Optimum conditions for headspace solid-phase microextraction (HS-SPME) in the analysis of monomethylmercury (MeHg) have been determined. Sodium tetra(n-)propylborate (NaBPr(4)) is used as derivatization reagent to promote volatility. A simple aluminium bar was used to cool the SPME fiber to about 2 degrees C during the equilibration phase just before extraction. HS-SPME was performed using different fibers. The 100 microm polydimethylsiloxane (PDMS) and 65 microm polydimethylsiloxane-divinylbenzene (PDMS-DVB) fibers showed the best results. Although the extraction efficiency for MeHg derivative of the polydimethylsiloxane-Carboxen (PDMS-CAR) fiber is similar to the other fibers, desorption of MeHg derivative from a PDMS-CAR fiber is poor. Factors affecting the HS-SPME process such as adsorption and desorption times, ionic strength (salting-out) and extraction temperature have been evaluated and optimized thoroughly. The highest extraction efficiency for the PDMS fiber was obtained by extraction at a low temperature (2 degrees C) immediately after equilibration at 30 degrees C. With the PDMS-DVB and PDMS-CAR fiber improvement of extraction efficiency at lower temperatures is negligible. Repeated extraction out of the same vial revealed that about 30% of MeHg derivative is extracted from the headspace with a PDMS fiber at 2 degrees C and about 70% with a PDMS-DVB fiber. Repeated extraction with two different fiber coatings showed that the PDMS-CAR fiber also extracts about 70% but that the desorption is incomplete. Attempts to improve the desorption failed due to degradation of the MeHg derivate at high injection temperatures. The limit of detection (3sigma) was 16 pg/L MeHg. The relative standard deviation (n = 8) for 100 pg/L of MeHg was found to be 5%. Linearity of the HS-SPME-GC-atomic emission detection method was established over at least two orders of magnitude in the range 0-2000 pg/L. Recovery of a surface water sample spiked at 2 ng/L was 85%. The suitability of the procedure was demonstrated by analysis of a surface water sample that showed a concentration of 100 pg/L MeHg. The optimized method can be used with standard commercial equipment without further adaptations.     

Asymptotic behavior and stability of second order neutral delay differential equations

We study the asymptotic behavior of a class of second order neutral delay differential equations by both a spectral projection method and an ordinary differential equation method approach. We discuss the relation of these two methods and illustrate some features using examples. Furthermore, a fixed point method is introduced as a third approach to study the asymptotic behavior. We conclude the paper with an application to a mechanical model of turning processes.     

The equivalence between processor sharing and service in random order

In this note we explore a useful equivalence relation for the delay distribution in the G/M/1 queue under two different service disciplines: (i) processor sharing (PS); and (ii) random order of service (ROS). We provide a direct probabilistic argument to show that the sojourn time under PS is equal (in distribution) to the waiting time under ROS of a customer arriving to a non-empty system. We thus conclude that the sojourn time distribution for PS is related to the waiting-time distribution for ROS through a simple multiplicative factor, which corresponds to the probability of a non-empty system at an arrival instant. We verify that previously derived expressions for the sojourn time distribution in the M/M/1 PS queue and the waiting-time distribution in the M/M/1 ROS queue are indeed identical, up to a multiplicative constant. The probabilistic nature of the argument enables us to extend the equivalence result to more general models, such as the M/M/1/K queue and ·/M/1 nodes in product-form networks.     

Convergence of the all-time supremum of a Lévy process in the heavy-traffic regime

In this paper we derive a technique for obtaining limit theorems for suprema of Lévy processes from their random walk counterparts. For each a>0, let $\{Y^{(a)}_{n}:n\ge1\}In this paper we derive a technique for obtaining limit theorems for suprema of Lévy processes from their random walk counterparts. For each a>0, let {Y^(a)_n:n 3 1}\{Y^{(a)}_{n}:n\ge1\} be a sequence of independent and identically distributed random variables and {X^(a)_t:t 3 0}\{X^{(a)}_{t}:t\ge0\} be a Lévy process such that X₁^(a)=^dY₁^(a)X_{1}^{(a)}\stackrel{d}{=}Y_{1}^{(a)}, \mathbbEX₁^(a) < 0\mathbb{E}X_{1}^{(a)}<0 and \mathbbEX₁^(a)-0\mathbb{E}X_{1}^{(a)}\uparrow0 as a↓0. Let S^(a)_n=?_k=1ⁿ Y^(a)_kS^{(a)}_{n}=\sum _{k=1}^{n} Y^{(a)}_{k}. Then, under some mild assumptions, , for some random variable and some function Δ(⋅). We utilize this result to present a number of limit theorems for suprema of Lévy processes in the heavy-traffic regime.     

Workload distributions in ASIP queueing networks

Queueing Systems - The workload of a generalized n-site asymmetric simple inclusion process (ASIP) is investigated. Three models are analyzed. The first model is a serial network for which the...