A Tandem Fluid Network with L\'evy Input in Heavy Traffic

In this paper we study the stationary workload distribution of a fluid tandem queue in heavy traffic. We consider different types of L\'evy input, covering compound Poisson, $\alpha$-stable L\'evy motion (with $1<\alpha<2$), and Brownian motion. In our analysis we separately deal with L\'evy input processes with increments that have finite and infinite variance. A distinguishing feature of this paper is that we do not only consider the usual heavy-traffic regime, in which the load at one of the nodes goes to unity, but also a regime in which we simultaneously let the load of both servers tend to one, which, as it turns out, leads to entirely different heavy-traffic asymptotics. Numerical experiments indicate that under specific conditions the resulting simultaneous heavy-traffic approximation significantly outperforms the usual heavy-traffic approximation

Evaluation of phosphate in fertilizers by means of the alkaline ammonium citrate extraction according to Petermann.

The phosphate extraction with alkaline ammonium citrate solution for the control of phosphate in fertilizers was subjected to a critical examination. The contribution of inferior phosphate forms such as tricalcium phosphates and apatites to the available P2O5 percentage can be excluded to a great extent by using a narrow sample solvent ratio. However, the dissolution rate of dicalcium phosphate in alkaline ammonium citrate solution limits the narrowing of the ratio. The dissolution rate of dicalcium phosphate can be increased, if instead of intermittent manual shaking, mechanical shaking is applied. Extraction of 1.0 g sample with 200 ml of alkaline ammonium citrate solution at 65 deg C under mechanical shaking for 1.5 hours is preferred to other methods for quality control of phosphate in fertilizers. The extraction method with neutral ammonium citrate solution is not reliable, because this method makes little or no distinction between high quality and low quality fertilizers. (Abstract retrieved from CAB Abstracts by CABI’s permission

Computable bounds in fork-join queueing systems

In a Fork-Join (FJ) queueing system an upstream fork station splits incoming jobs into N tasks to be further processed by N parallel servers, each with its own queue; the response time of one job is determined, at a downstream join station, by the maximum of the corresponding tasks' response times. This queueing system is useful to the modelling of multi-service systems subject to synchronization constraints, such as MapReduce clusters or multipath routing. Despite their apparent simplicity, FJ systems are hard to analyze. This paper provides the first computable stochastic bounds on the waiting and response time distributions in FJ systems. We consider four practical scenarios by combining 1a) renewal and 1b) non-renewal arrivals, and 2a) non-blocking and 2b) blocking servers. In the case of non blocking servers we prove that delays scale as O(logN), a law which is known for first moments under renewal input only. In the case of blocking servers, we prove that the same factor of log N dictates the stability region of the system. Simulation results indicate that our bounds are tight, especially at high utilizations, in all four scenarios. A remarkable insight gained from our results is that, at moderate to high utilizations, multipath routing 'makes sense' from a queueing perspective for two paths only, i.e., response times drop the most when N = 2; the technical explanation is that the resequencing (delay) price starts to quickly dominate the tempting gain due to multipath transmissions

System occupancy of a two-class batch-service queue with class-dependent variable server capacity

Due to their wide area of applications, queueing models with batch service, where the server can process several customers simultaneously, have been studied frequently. An important characteristic of such batch-service systems is the size of a batch, that is the number of customers that are processed simultaneously. In this paper, we analyse a two-class batch-service queueing model with variable server capacity, where all customers are accommodated in a common first-come-first served single-server queue. The server can only process customers that belong to the same class, so that the size of a batch is determined by the number of consecutive same-class customers. After establishing the system equations that govern the system behaviour, we deduce an expression for the steady-state probability generating function of the system occupancy at random slot boundaries. Also, some numerical examples are given that provide further insight in the impact of the different parameters on the system performance

Analysis of a multi-server queueing model of ABR

In this paper we present a queueing model for the performance analysis of Available Bit Rate (ABR) traffic in Asynchronous Transfer Mode (ATM) networks. We consider a multi-channel service station with two types of customers, denoted by high priority and low priority customers. In principle, high priority customers have preemptive priority over low priority customers, except on a fixed number of channels that are reserved for low priority traffic. The arrivals occur according to two independent Poisson processes, and service times are assumed to be exponentially distributed. Each high priority customer requires a single server, whereas low priority customers are served in processor sharing fashion. We derive the joint distribution of the numbers of customers (of both types) in the system in steady state. Numerical results illustrate the effect of high priority traffic on the service performance of low priority traffic

Repair systems with exchangeable items and the longest queue mechanism

We consider a repair facility consisting of one repairman and two arrival streams of failed items, from bases 1 and 2. The arrival processes are independent Poisson processes, and the repair times are independent and identically exponentially distributed. The item types are exchangeable, and a failed item from base 1 could just as well be returned to base 2, and vice versa. The rule according to which backorders are satisfied by repaired items is the longest queue rule: at the completion of a service (repair), the repaired item is delivered to the base that has the largest number of failed items. We point out a direct relation between our model and the classical longer queue model. We obtain simple expressions for several probabilities of interest, and show how all two-dimensional queue length probabilities may be obtained. Finally, we derive the sojourn time distributions

Queues with delays in two-state strategies and Lévy input

We consider a reflected Lévy process without negative jumps, starting at the origin. When the reflected process first upcrosses level K, a timer is activated. After D time units, the timer expires and the Lévy exponent of the Lévy process is changed. As soon as the process hits zero again, the Lévy exponent reverses to the original function. If the process has reached the origin before the timer expires then the Lévy exponent does not change. Using martingale techniques, we analyze the steady-state distribution of the resulting process, reflected at the origin. We pay special attention to the cases of deterministic and exponential timers, and to the following three special Lévy processes: (i) a compound Poisson process plus negative drift (corresponding to an M/G/1 queue), (ii) Brownian motion, and (iii) a Lévy process that is a subordinator until the timer expires. © Applied Probability Trust 2008

Large deviations for polling systems

Related INRIA Research report available at : http://hal.inria.fr/docs/00/07/27/62/PDF/RR-3892.pdfInternational audienceWe aim at presenting in short the technical report, which states a sample path large deviation principle for a resealed process n-1 Qnt, where Qt represents the joint number of clients at time t in a single server 1-limited polling system with Markovian routing. The main goal is to identify the rate function. A so-called empirical generator is introduced, which consists of Q t and of two empirical measures associated with S t the position of the server at time t. The analysis relies on a suitable change of measure and on a representation of fluid limits for polling systems. Finally, the rate function is solution of a meaningful convex program

A queueing model with randomized depletion of inventory

In this paper, we study an M/M/1 queue, where the server continues to work during idle periods and builds up inventory. This inventory is used for new arriving service requirements, but it is completely emptied at random epochs of a non-homogeneous Poisson process, whose rate depends on the current level of the acquired inventory. For several shapes of depletion rates, we derive differential equations for the stationary density of the workload and the inventory level and solve them explicitly. Finally, numerical illustrations are given for some particular examples, and the effects of this depletion mechanism are discussed