Sofian De Clercq

A multi-class discrete-time queueing system under the FCFS service discipline

The problem with the FCFS server discipline in discrete-time queueing systems is that it doesn’t actually determine what happens if multiple customers enter the system at the same time, which in the discrete-time paradigm translates into ‘during the same time-slot’. In other words, it doesn’t specify in which order such customers are served. When we consider multiple types of customers, each requiring different service time distributions, the precise order of service even starts to affect quantities such as queue content and delays of arbitrary customers, so specifying this order will be prime. In this paper we study a multi-class discrete-time queueing system with a general independent arrival process and generally distributed service times. The service discipline is FCFS and customers entering during the same time-slot are served in random order. It will be our goal to search for the steady-state distribution of queue content and delays of certain types of customers. If one thinks of the time-slot as a continuous but bounded time period, the random order of service is equivalent to FCFS if different customers have different arrival epochs within this time-slot and if the arrival epochs are independent of customer class. For this reason we propose two distinct ways of analysing; one utilizing permutations, the other considering a slot as a bounded continuous time frame.

Delay Analysis of a Discrete-Time Multiclass Slot-Bound Priority System

This paper introduces a new priority mechanism in discrete-time queueing systems that compromises between first-come-first-served (FCFS) and head-of-line priority. In this scheduling discipline—which we dubbed slot-bound priority—customers of different priority classes entering the system during the same time-slot are served in order of their respective priority class. Customers entering during different slots are served on a FCFS basis. In this paper we study the delay in an N-class discrete-time queueing system under slot-bound priority. General independent arrivals and class-specific general service time distributions are assumed. Expressions for the probability generating function of the delay of a random type-j customer are derived, from which the respective moments are easily obtained. The tail behaviour of these distributions is analyzed as well, and some numerical examples show the effect slot-bound priority can have on the performance measures.

Perspective: Essential Study Quality Descriptors for Data from Nutritional Epidemiologic Research

Pooled analysis of secondary data increases the power of research and enables scientific discovery in nutritional epidemiology. Information on study characteristics that determine data quality is needed to enable correct reuse and interpretation of data. This study aims to define essential quality characteristics for data from observational studies in nutrition. First, a literature review was performed to get an insight on existing instruments that assess the quality of cohort, case-control, and cross-sectional studies and dietary measurement. Second, 2 face-to-face workshops were organized to determine the study characteristics that affect data quality. Third, consensus on the data descriptors and controlled vocabulary was obtained. From 4884 papers retrieved, 26 relevant instruments, containing 164 characteristics for study design and 93 characteristics for measurements, were selected. The workshop and consensus process resulted in 10 descriptors allocated to “study design” and 22 to “measurement” domains. Data descriptors were organized as an ordinal scale of items to facilitate the identification, storage, and querying of nutrition data. Further integration of an Ontology for Nutrition Studies will facilitate interoperability of data repositories.

Queue Content Analysis in a 2-Class Discrete-Time Queueing System under the Slot-Bound Priority Service Rule

The paper we present here introduces a new priority mechanism in discrete-time queueing systems. It is a milder form of priority when compared to HoL priority, but it favors customers of one type over the other when compared to regular FCFS. It also provides an answer to the starvation problem that occurs in HoL priority systems. In this new priority mechanism, customers of different priority classes entering the system during the same time slot are served in order of their respective priority class—hence the name slot-bound priority. Customers entering during different slots are served on an FCFS basis. We consider two customer classes (pertaining to two levels of priority) such that type-1 customers are served before type-2 customers that enter the system during the same slot. A general independent arrival process and generally distributed service times are assumed. Expressions for the probability generating function (PGF) of the system content (number of type- customers, ) in regime are obtained using a slot-to-slot analysis. The first moments are calculated, as well as an approximation for the probability mass functions associated with the found PGFs. Lastly, some examples allow us some deeper insight into the inner workings of the slot-bound priority mechanism.

Analysis of a discrete-time queue with time-limited overtake priority

In this paper, we investigate a single-server discrete-time queueing system subject to two independent batch Bernoulli arrival processes, each supplying the queue with different customer classes. The two classes of customers have different priority levels in the queue, and different service-time distributions. The studied priority mechanism is time-limited, i.e., customers of the high-priority class cannot overtake customers of lower priority if the latter arrived at least N slots earlier than the former. The parameter N makes the mechanism versatile, spanning a bridge between absolute (fixed) priority and slot-bound priority (see De Clercq et al. in Math Probl Eng. doi:10.1155/2012/425630, 2012). The time-limited overtake priority mechanism maintains levels of fairness that are unattainable by a pure absolute priority mechanism, and offers more service differentiation than the slot-bound priority alternative studied earlier. By using a censoring argument, we obtain expressions for the steady-state probability generating functions of the delays of both customer classes, as well as the steady-state joint probability generating function of the system content, by using a censoring argument.

Analysis of a Two-Class Priority Queue with Correlated Arrivals from Another Node

Exact analysis of tandem priority queues is a difficult problem. In this paper, we model the output process of the first stage as a three-state Markov chain and analyze the second stage. The arrival process of this second stage is the superposition of this output process and an uncorrelated arrival process. We calculate the joint probability generating function of the number of high- and low-priority packets in the second stage and show that two implicit functions appear in this expression. We demonstrate how to deal with these implicitly defined functions in the calculation of moments.

Transform-Domain solutions of poisson's equation with applications to the asymptotic variance

Recent years have seen a considerable increase of attention devoted to Poissons equation for Markov chains, which now has attained a central place in Markov chain theory, due to the extensive list of areas where Poissons equation pops up: perturbation analysis, Markov decision processes, limit theorems of Markov chains, etc. all find natural expression when viewed from the vantage point of Poissons equation. We describe how the use of generating functions helps solve Poissons equation for different types of structured Markov chains and for driving functions, and point out some applications. In particular, we solve Poissons equation in the transform domain for skip-free Markov chains and Markov chains with linear displacement. Closed-form solutions are obtained for a class of driving functions encompassing polynomial functions and functions with finite support.

A decomposition result for single server discrete-time queues with generalized vacations

For several specific discrete-time queueing models with a vacation policy, the stationary system occupancy at the beginning of a random slot is distributed as the sum of two independent random variables. One of these variables is the stationary number of customers in an equivalent queueing system with no vacations. This paper aims to show that this decomposition can be applied to for a large class of discrete-time queueing systems with vacations. The analysis builds on results obtained by Fuhrmann and Cooper concerning continuous-time queueing systems with Poissonian arrivals. Through some examples we show that the queueing analysis can be considerably simplified using this decomposition property.