Analysis of Buffer Starvation with Application to Objective QoE Optimization of Streaming Services

Our purpose in this paper is to characterize buffer starvations for streaming services. The buffer is modeled as an M/M/1 queue, plus the consideration of bursty arrivals. When the buffer is empty, the service restarts after a certain amount of packets are \emph{prefetched}. With this goal, we propose two approaches to obtain the \emph{exact distribution} of the number of buffer starvations, one of which is based on \emph{Ballot theorem}, and the other uses recursive equations. The Ballot theorem approach gives an explicit result. We extend this approach to the scenario with a constant playback rate using T\`{a}kacs Ballot theorem. The recursive approach, though not offering an explicit result, can obtain the distribution of starvations with non-independent and identically distributed (i.i.d.) arrival process in which an ON/OFF bursty arrival process is considered in this work. We further compute the starvation probability as a function of the amount of prefetched packets for a large number of files via a fluid analysis. Among many potential applications of starvation analysis, we show how to apply it to optimize the objective quality of experience (QoE) of media streaming, by exploiting the tradeoff between startup/rebuffering delay and starvations.Comment: 9 pages, 7 figures; IEEE Infocom 201

Perfect and Nearly Perfect Sampling of Work-conserving Queues

We present sampling-based methods to treat work-conserving queueing systems. A variety of models are studied. Besides the First Come First Served (FCFS) queues, many efforts are putted on the accumulating priority queue (APQ), where a customer accumulates priority linearly while waiting. APQs have Poisson arrivals, multi-class customers with corresponding service durations, and single or multiple servers. Perfect sampling is an approach to draw a sample directly from the steady-state distribution of a Markov chain without explicitly solving for it. Statistical inference can be conducted without initialization bias. If an error can be tolerated within some limit, i.e. the total variation distance between the simulated draw and the stationary distribution can be bounded by a specified number, then we get a so called nearly perfect sampling. Coupling from the past (CFTP) is one approach to perfect sampling, but it usually requires a bounded state space. One strategy for perfect sampling on unbounded state spaces relies on construction of a reversible dominating process. If only the dominating property is guaranteed, then regenerative method (RM) becomes an alternative choice. In the case where neither the reversibility nor dominance hold, a nearly perfect sampling method will be proposed. It is a variant of dominated CFTP that we call the CFTP Block Absorption (CFTP-BA) method. Time-varying queues with periodic Poisson arrivals are being considered in this thesis. It has been shown that a particular limiting distribution can be obtained for each point in time in the periodic cycle. Because there are no analytical solutions in closed forms, we explore perfect (or nearly perfect) sampling of these systems

Bloody fast blood collection

This thesis consists of four parts: The first part contains an introduction, the second presents approaches for the evaluation of waiting times at blood collection sites, the third uses these to present approaches that improve waiting times at blood collection sites. The final part shows the application of two of the approaches to data from real blood collection sites, followed by the conclusions that can be drawn from this thesis. Part I: Introduction, contains two chapters. Chapter 1 introduces the context for this thesis: blood banks in general, the Dutch blood bank Sanquin and blood collection sites. The chapter sketches some of the challenges faced with respect to blood collection sites. As blood donors are voluntary and non-remunerated, delays and waiting times within blood collection sites should be kept at acceptable levels. However, waiting times are currently not incorporated in staff planning or in other decisions with respect to blood collection sites. These blood collection sites will be the primary focus of this thesis. This thesis provides methods that do take waiting times into account, aiming to decrease waiting times at blood collection sites and leveling work pressure for staff members, without the need for additional staff. Chapter 2 then presents a technical methods that will be used most of the chapters in this thesis: uniformization. Uniformization can be used to transform Continuous Time Markov Chains (CTMCs) — that are very hard to analyze — into Discrete Time Markov Chains (DTMCs) — that are much easier to analyze. The chapter shows how the method works, provides an extensive overview of the literature related to the method, the (technical) intuition behind the method as well as several extensions and applications. Although not all of the extensions and applications are necessary for this thesis, it does provide an overview of one of the most valuable methods for this thesis. Part II: Evaluation, contains two chapters that propose and adapt several methods to compute waiting times and queues at blood collection sites. A blood collection site is best modeled as a time-dependent queueing network, requiring non-standard approaches. Chapter 3 considers a stationary, i.e. not time-dependent model of blood collection sites as a first step. A blood collection site consists of three main stations: Registration, Interview and Donation. All three of the stations can have their own queue. This means that even the stationary model is non-trivial for some computations. However, for the stationary model, an analytic so-called product form expression is derived. Based on this product form, two more results are shown. The first result is that the standard waiting time distributions from M|M|s queues are applicable, as if the queue is in isolation. It is then concluded that no closed form expression exist for the total waiting or delay time distribution, as the distributions of the three stations in tandem are not independent. Therefore a numerical approach is presented to compute the total delay time distribution of a collection site. All of the results are supported by numerical examples based on a Dutch blood collection site. The approach for the computation of the total delay time distribution can also be combined with the approach from Chapter 4 for an extension to a time-dependent setting. Chapter 4 shows an approach to deal with these time-dependent aspects in queueing systems, as often experienced by blood collection sites and other service systems, typically due to time-dependent arrivals and capacities. Easy and quick to use queueing expressions generally do not apply to time-dependent situations. A large number of computational papers has been written about queue length distributions for time-dependent queues, but these are mostly theoretical and based on single queues. This chapter aims to combine computational methods with more realistic time-dependent queueing networks, with an approach based on uniformization. Although uniformization is generally perceived to be too computationally prohibitive, we show that our method is very effective for practical instances, as shown with an example of a Dutch blood collection site. The objective of the results is twofold: to show that a time-dependent queueing network approach can be beneficial and to evaluate possible improvements for Dutch blood collection sites that can only be properly assessed with a time-dependent queueing method. Part III: Optimization, contains four chapters that aim to improve service levels at Sanquin. The first three chapters focus on three different methods to decrease queues at blood collection sites. Chapters 5 and 6 focus on improving the service by optimizing staff allocation to shifts and stations. Chapter 7 focuses on improving the arrival process with the same goal. Chapter 8 is focused at improving inventory management of red blood cells. Donors do not arrive to blood collection sites uniformly throughout the day, but show clear preferences for certain times of the day. However, the arrival patterns that are shown by historical data, are not used for scheduling staff members at blood collection sites. As a first significant step to shorten waiting times we can align staff capacity and shifts with walk-in arrivals. Chapter 5 aims to optimize shift scheduling for blood collection sites. The chapter proposes a two-step procedure. First, the arrival patterns and methods from queueing theory are used to determine the required number of staff members for every half hour. Second, an integer linear program is used to compute optimal shift lengths and starting times, based on the required number of staff members. The chapter is concluded with numerical experiments that show, depending on the scenario, a reduction of waiting times, a reduction of staff members or a combination of both. At a blood collection site three stations (Registration, Interview and Donation) can roughly be distinguished. Staff members at Dutch blood collection sites are often trained to work at any of these stations, but are usually allocated to one of the stations for large fractions of a shift. If staff members change their allocation this is based on an ad hoc decision. Chapter 6 aims to take advantage of this mostly unused allocation flexibility to reduce queues at blood collection sites. As a collection site is a highly stochastic process, both in arrivals and services, an optimal allocation of staff members to the three stations is unknown, constantly changing and a challenge to determine. Chapter 6 provides and applies a so-called Markov Decision Process (MDP) to compute optimal staff assignments. Extensive numerical and simulation experiments show the potential reductions of queues when the reallocation algorithm would be implemented. Based on Dutch blood collection sites, reductions of 40 to 80% on the number of waiting donors seem attainable, depending on the scenario. Chapter 7 also aims to align the arrival of donors with scheduled staff, similarly to Chapter 5. Chapter 7 tries to achieve this by changing the arrivals of donors. By introducing appointments for an additional part of donors, arrivals can be redirected from the busiest times of the day to quiet times. An extended numerical queueing model with priorities is introduced for blood collection sites, as Sanquin wants to incentive donors to make appointments by prioritizing donors with appointments over donors without appointments. Appointment slots are added if the average queue drops below certain limits. The correct values for these limits, i.e. the values that plan the correct number of appointments, are then determined by binary search. Numerical results show that the method succeeds in decreasing excessive queues. However, the proposed priorities might result in unacceptably high waiting times for donors without appointments, and caution is therefore required before implementation. Although this thesis mainly focuses on blood collection sites, many more logistical challenges are present at a blood bank. One of these challenges arises from the expectation that Sanquin can supply hospitals with extensively typed red blood cell units directly from stock. Chapter 8 deals with this challenge. Currently, all units are issued according to the first-in-first-out principle, irrespective of their specific typing. These kind of issuing policies lead to shortages for rare blood units. Shortages for rare units could be avoided by keeping them in stock for longer, but this could also lead to unnecessary wastage. Therefore, to avoid both wastage and shortages, a trade-off between the age and rarity of a specific unit in stock should be made. For this purpose, we modeled the allocation of the inventory as a circulation flow problem, in which decisions about which units to issue are based on both the age and rarity of the units in stock. We evaluated the model for several settings of the input parameters. It turns out that, especially if only a few donors are typed for some combinations of antigens, shortages can be avoided by saving rare blood products. Moreover, the average issuing age remains unchanged. Part IV: Practice and Outlook concludes this thesis. The first of two chapters in this part shows the combined application of two approaches from this thesis to data from three collection sites in the Netherlands. The final chapter of this thesis presents the conclusions that can be drawn from this thesis and discusses an outlook for further research. Chapter 9 shows the combined application of the methods in Chapters 5 and 6 to three real collection sites in Dutch cities: Nijmegen, Leiden and Almelo. The collection sites in Nijmegen and Leiden are both large fixed collection sites. The collection site in Almelo is a mobile collection site. The application of each one of the methods individually reduce waiting times significantly, and the combined application of the methods reduces waiting times even further. Simultaneously, small reductions in the number of staff hours are attainable. The results from Chapter 9 summarize the main message of this thesis: waiting time for blood donors at blood collection sites can be reduced without the need for more staff members when the working times of staff members are used more effectively and efficiently, and controlling the arrival process of donors. The approaches presented in this thesis can be used for this purpose. This is not only beneficial for blood donors, but will also result in more balanced workload for staff members, as fluctuations in this workload are reduced significantly

Stochastic Processes with Applications

Stochastic processes have wide relevance in mathematics both for theoretical aspects and for their numerous real-world applications in various domains. They represent a very active research field which is attracting the growing interest of scientists from a range of disciplines.This Special Issue aims to present a collection of current contributions concerning various topics related to stochastic processes and their applications. In particular, the focus here is on applications of stochastic processes as models of dynamic phenomena in research areas certain to be of interest, such as economics, statistical physics, queuing theory, biology, theoretical neurobiology, and reliability theory. Various contributions dealing with theoretical issues on stochastic processes are also included

Efficient rare-event simulation for the maximum of heavy-tailed random walks

Let $(X_n:n\geq 0)$ be a sequence of i.i.d. r.v.'s with negative mean. Set $S_0=0$ and define $S_n=X_1+... +X_n$. We propose an importance sampling algorithm to estimate the tail of $M=\max \{S_n:n\geq 0\}$ that is strongly efficient for both light and heavy-tailed increment distributions. Moreover, in the case of heavy-tailed increments and under additional technical assumptions, our estimator can be shown to have asymptotically vanishing relative variance in the sense that its coefficient of variation vanishes as the tail parameter increases. A key feature of our algorithm is that it is state-dependent. In the presence of light tails, our procedure leads to Siegmund's (1979) algorithm. The rigorous analysis of efficiency requires new Lyapunov-type inequalities that can be useful in the study of more general importance sampling algorithms.Comment: Published in at http://dx.doi.org/10.1214/07-AAP485 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Queueing theory approach to rain fade analysis at microwave and millimeter bands in tropical Africa.

Ph. D. University of KwaZulu-Natal, Durban 2014.With an overwhelming demand of larger bandwidth required for high capacity data with content-rich services ranging from high-speed video streaming to multimedia content, there is a continuous need to migrate to higher microwave bands, particularly beyond the regular Ku and Ka bands (between 11 - 40 GHz). The presence of precipitation at these microwave and millimeter bands (3-300 GHz) generally induce rain fade, which is a constraint to network providers intending to achieve optimal service delivery, at acceptable signal to noise ratios (SNRs). In practice, fade countermeasures – static or dynamic – are necessary to combat the consequences of chronic fluctuations of rainfall resulting in signal deterioration and impairment over communication links. However, the implementation of dynamic fade countermeasures is systematically tied upon the available Channel State Information (CSI), which is often timevariant relative to the occurrence of precipitation events. Time-variation of rainfall events are perceptible in measurable rainfall microstructural parameters which vary intensely in space and time. These spatio-temporal variations yield the generation of observable random patterns of signal attenuation during rain events, often in a stochastic manner. To this end, researchers have emphasized on understanding the underlying behaviour of generic rainfall microstructural parameters such as rainfall rate, rainfall Drop Size Distribution (DSD) and radar reflectivity. Therefore, the investigation of these stochastic properties of rainfall processes is primary in the determination of recognisable patterns of rainfall rate and other microstructures. This thesis introduces the queueing theory approach via the Markov Chain technique to investigate the time-varying characteristics of the rainfall process from distrometer data in subtropical and equatorial Africa. Rainfall data obtained from these two climatic locations, at one minute integration time, were processed from sites in Durban, South Africa and Butare, Rwanda, over a specified measurement period. Initial investigation and comparison of rainfall microstructures undertaken at both sites clearly show key differences in their probability distribution profiles at Stratiform-Convective (SC) bounds. The underlying queue discipline of rainfall spikes and their queue metrics are determined and appraised for system performance using rainfall time series database. The results show rain spike generation processes vividly exhibit a First-Come, First- Served (FCFS) semi-Markovian distributed traffic of M/Ek/s discipline, with a varying degree of servers, for different rainfall regimes. Comparison of queue statistics results over different rainfall regimes at the two locations reveal significant differences in their queue metrics and performances. The knowledge obtained from the queue statistics and SC probability analysis are further employed in the determination and classification of rainfall cells, rainfall growth models and path attenuation prediction. The results are compared and validated with data collected from a 6.73km, 19.5 GHz terrestrial link in Durban