Dieter Fiems

A note on the discretization of Little's result

By considering discrete-time queueing systems as special cases of continuous-time queueing systems, we derive a discrete-time equivalent of Littles result. Our result is general in the sense that no assumptions are made regarding the exact details of arrival and departure processes within a discrete-time unit.

Discrete-time queues with generally distributed service times and renewal-type server interruptions

In this contribution, we investigate a discrete-time single-server queue subjected to server interruptions. Server interruptions are modeled as an on/off process with geometrically distributed on-periods and generally distributed off-periods. As message lengths can exceed one time-slot, different operation modes are considered, depending on whether service of an interrupted message continues, partially restarts or completely restarts after an interruption. For all alternatives, we establish expressions for the steady-state probability generating functions (pgf) of the buffer contents at message departure times and random slot boundaries, of the unfinished work at random slot boundaries, the message delay, and the lengths of the idle and busy periods. From these results, closed-form expressions for various performance measures, such as mean and variance of the buffer occupancy and message delay, can be established. As an application, we show that this model is able to assess performance of a multi-class priority scheduling system. We then illustrate our approach with some numerical examples.

Queueing systems with different types of server interruptions

We consider a queueing system with disruptive and non-disruptive server interruptions. Both disruptive and non-disruptive interruptions may start when there is a customer in service. The customer repeats its service after a disruptive interruption, and continues its service after a non-disruptive interruption. Using a transform approach, we obtain various performance measures such as the moments of the queue content and waiting times. We illustrate our approach by means of some numerical examples.

A performance model for an asynchronous optical buffer

We investigate the behaviour of an asynchronous optical buffer by means of a continuous-time queuing model. Through a limit procedure, previously obtained results for a discrete-time queuing model are translated to a continuous-time setting. We also show that the same results can be obtained by a direct analysis using Laplace transforms. Closed-form expressions are obtained for the cases of exponentially distributed burst sizes, deterministic burst sizes and mixtures of deterministic burst sizes. The performance of asynchronous optical buffers shows the same characteristics as that of synchronous optical buffers: a reduction in throughput due to the creation of voids on the outgoing channel and a burst loss probability that is strongly influenced by the choice of fiber delay line granularity. The optimal value of the latter depends on the burst size distribution and the offered load.

A unified model for synchronous and asynchronous FDL buffers allowing closed-form solution

Novel switching approaches like Optical Burst/Packet Switching have buffering implemented with Fiber Delay Lines (FDLs). Previous performance models of the resulting buffer only allowed for solution by numerical means, and only for one time setting: continuous, or discrete. With a Markov chain approach, we constructed a generic framework that encompasses both time settings. The output includes closed-form expressions of loss probabilities and waiting times for a rather realistic setting. This allows for exact performance comparison of the classic M/D/1 buffer and FDL M/D/1 buffer, revealing that waiting times are (more than) doubled in the case of FDL buffering.

Modeling the performance of FDL buffers with wavelength conversion

In optical burst switching and optical packet switching, contention of bursts (or packets) can be dealt with most effectively through a combination of wavelength conversion and optical buffering. While this is generally accepted in the optical networking community, and validated through simulation, analytic performance results for optical buffers were limited to the single-wavelength case, and the performance gain from wavelength conversion was never traced analytically for general assumptions. Quantifying this gain analytically is the scope of the current contribution. Relying on generating functions, we developed a fiber delay line (FDL) buffer model with wavelength conversion, which assumes the buffer located at the output of an optical switch, having access to multiple wavelengths. This document presents our model, validates its accuracy, and compares its output for different burst sizes (fixed or varying), scheduling policies and buffer sizes. Several numerical examples assess the applicability of our approximation, and show that our approach yields accurate results.

Randomly Interrupted GI-G-1 Queues Service Strategies and Stability Issues

We consider a discrete-time queueing system subjected to random server interruptions. As customers arriving in the queue require generally distributed service times, the server can be interrupted during a customers service. Therefore, nine different service strategies are proposed and analyzed using a probability generating functions approach. Performance measures under investigation include moments of steady-state buffer contents at random slot boundaries in equilibrium and moments of the customer delay. In particular we focus on the stability requirements for the strategies under consideration.

Performance analysis of the IEEE 802.16e sleep mode for correlated downlink traffic

In this paper, we evaluate the performance of the IEEE 802.16e sleep mode mechanism in wireless access networks. This mechanism reduces the energy consumption of a mobile station (MS) by allowing it to turn off its radio interface (sleep mode) when there is no traffic present at its serving base station (BS). After a sleep period expires, the MS briefly checks the BS for data packets and switches off for the duration of another sleep period if none are available. Specifically for IEEE 802.16e, each additional sleep period doubles in length, up to a certain maximum. Clearly, the sleep mode mechanism can extend the battery life of the MS considerably, but also increases the delay at the BS buffer. For the performance analysis, we use a discrete-time queueing model with general service times and multiple server vacations. The vacations represent the sleep periods and have a length depending on the number of preceding vacations. Unlike previous studies, we take the (short-range) traffic correlation into account by assuming a D-BMAP arrival process, i.e. the distribution of the number of packet arrivals per slot is modulated by the transitions in a Markov chain with N background states. As results, we obtain the distribution of the number of packets in the queue at various sets of time epochs, the distribution of the packet delay and the antenna activity rate. We apply these results to the IEEE 802.16e sleep mode mechanism with correlated downlink traffic. By means of some examples, we show the influence of both the configuration parameters and the traffic correlation on the delay and the energy consumption.

Computationally efficient evaluation of appointment schedules in health care

We consider the problem of evaluating and constructing appointment schedules for patients in a health care facility where a single physician treats patients in a consecutive manner, as is common for general practitioners, clinics and for outpatients in hospitals. Specifically, given a fixed-length session during which a physician sees K patients, each patient has to be given an appointment time during this session in advance. Optimising a schedule with respect to patient waiting times, physician idle times, session overtime, etc. usually requires a heuristic search method involving a huge number of repeated schedule evaluations. Hence, our aim is to obtain accurate predictions at very low computational cost. This is achieved by (1) using Lindley’s recursion to allow for explicit expressions and (2) choosing a discrete-time (slotted) setting to make those expressions easy to compute. We assume general, possibly distinct, distributions for the patients’ consultation times, which allows to account for multiple treatment types, emergencies and patient no-shows. The moments of waiting and idle times are obtained and the computational complexity of the algorithm is discussed. Additionally, we calculate the schedule’s performance in between appointments in order to assist a sequential scheduling strategy.

Delay versus energy consumption of the IEEE 802.16e sleep-mode mechanism

We propose a discrete-time queueing model for the evaluation of the IEEE 802.16e sleep-mode mechanism of power saving class (PSC) I in wireless access networks. Contrary to previous studies, we model the downlink traffic by means of a discrete batch Markov arrival process (D-BMAP) with N phases, which allows to take traffic correlation into account. The tradeoff between energy saving and increased packet delay is discussed. In many situations, the sleep-mode performance improves for heavily correlated traffic. Also, when compared to other strategies, the exponential sleep-period update strategy of PSC I may not always be the best.