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.     

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.     

Shot-noise queueing models

Queueing Systems - We provide a survey of so-called shot-noise queues: queueing models with the special feature that the server speed is proportional to the amount of work it faces. Several results...     

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.     

Redundancy scheduling with scaled Bernoulli service requirements

Queueing Systems - Redundancy scheduling has emerged as a powerful strategy for improving response times in parallel-server systems. The key feature in redundancy scheduling is replication of a job...     

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...     

Queues with random back-offs

We consider a broad class of queueing models with random state-dependent vacation periods, which arise in the analysis of queue-based back-off algorithms in wireless random-access networks. In contrast to conventional models, the vacation periods may be initiated after each service completion, and can be randomly terminated with certain probabilities that depend on the queue length. We first present exact queue-length and delay results for some specific cases and we derive stochastic bounds for a much richer set of scenarios. Using these, together with stochastic relations between systems with different vacation disciplines, we examine the scaled queue length and delay in a heavy-traffic regime, and demonstrate a sharp trichotomy, depending on how the activation rate and vacation probability behave as function of the queue length. In particular, the effect of the vacation periods may either (i) completely vanish in heavy-traffic conditions, (ii) contribute an additional term to the queue lengths and delays of similar magnitude, or even (iii) give rise to an order-of-magnitude increase. The heavy-traffic trichotomy provides valuable insight into the impact of the back-off algorithms on the delay performance in wireless random-access networks.