Queue Length Asymptotics for Generalized Max-Weight Scheduling in the presence of Heavy-Tailed Traffic

We investigate the asymptotic behavior of the steady-state queue length distribution under generalized max-weight scheduling in the presence of heavy-tailed traffic. We consider a system consisting of two parallel queues, served by a single server. One of the queues receives heavy-tailed traffic, and the other receives light-tailed traffic. We study the class of throughput optimal max-weight-alpha scheduling policies, and derive an exact asymptotic characterization of the steady-state queue length distributions. In particular, we show that the tail of the light queue distribution is heavier than a power-law curve, whose tail coefficient we obtain explicitly. Our asymptotic characterization also contains an intuitively surprising result - the celebrated max-weight scheduling policy leads to the worst possible tail of the light queue distribution, among all non-idling policies. Motivated by the above negative result regarding the max-weight-alpha policy, we analyze a log-max-weight (LMW) scheduling policy. We show that the LMW policy guarantees an exponentially decaying light queue tail, while still being throughput optimal

Fast symbolic computation of the worst-case delay in tandem networks and applications

International audienceComputing deterministic performance guarantees is a defining issue for systems with hard real-time constraints , like reactive embedded systems. In this paper, we use burst-rate constrained arrivals and rate-latency servers to deduce tight worst-case delay bounds in tandem networks under arbitrary multiplexing. We present a constructive method for computing the exact worst-case delay, which we prove to be a linear function of the burstiness and latencies; our bounds are hence symbolic in these parameters. Our algorithm runs in quadratic time in the number of servers. We also present an application of our algorithm to the case of stochastic arrivals and server capacities. For a generalization of the exponentially bounded burstiness (EBB) model, we deduce a polynomial-time algorithm for stochastic delay bounds that strictly improve the state-of-the-art separated flow analysis (SFA) type bounds

Sharp Bounds in Stochastic Network Calculus

The practicality of the stochastic network calculus (SNC) is often questioned on grounds of potential looseness of its performance bounds. In this paper it is uncovered that for bursty arrival processes (specifically Markov-Modulated On-Off (MMOO)), whose amenability to \textit{per-flow} analysis is typically proclaimed as a highlight of SNC, the bounds can unfortunately indeed be very loose (e.g., by several orders of magnitude off). In response to this uncovered weakness of SNC, the (Standard) per-flow bounds are herein improved by deriving a general sample-path bound, using martingale based techniques, which accommodates FIFO, SP, EDF, and GPS scheduling. The obtained (Martingale) bounds gain an exponential decay factor of ${\mathcal{O}}(e^{-\alpha n})$ in the number of flows $n$. Moreover, numerical comparisons against simulations show that the Martingale bounds are remarkably accurate for FIFO, SP, and EDF scheduling; for GPS scheduling, although the Martingale bounds substantially improve the Standard bounds, they are numerically loose, demanding for improvements in the core SNC analysis of GPS

Delay analysis for wireless applications using a multiservice multiqueue processor sharing model

The ongoing development of wireless networks supporting multimedia applications requires service providers to efficiently deliver complex Quality of Service (QoS) requirements. The wide range of new applications in these networks significantly increases the difficulty of network design and dimensioning to meet QoS requirements. Medium Access Control (MAC) protocols affect QoS achieved by wireless networks. Research on analysis and performance evaluation is important for the efficient protocol design. As wireless networks feature scarce resources that are simultaneously shared by all users, processor sharing (PS) models were proposed for modelling resource sharing mechanisms in such systems. In this thesis, multi-priority MAC protocols are proposed for handling the various service traffic types. Then, an investigation of multiservice multiqueue PS models is undertaken to analyse the delay for some recently proposed wireless applications. We start with an introduction to MAC protocols for wireless networks which are specified in IEEE standards and then review scheduling algorithms which were proposed to work with the underlying MAC protocols to cooperatively achieve QoS goals. An overview of the relevant literature is given on PS models for performance analysis and evaluation of scheduling algorithms. We propose a multiservice multiqueue PS model using a scheduling scheme in multimedia wireless networks with a comprehensive description of the analytical solution. Firstly, we describe the existing multiqueue processor sharing (MPS) model, which uses a fixed service quantum at each queue, and correct a subtle incongruity in previous solutions presented in the literature. Secondly, a new scheduling framework is proposed to extend the previous MPS model to a general case. This newly proposed analytical approach is based on the idea that the service quantum arranged by a MAC scheduling controller to service data units can be priority-based. We obtain a closed-form expression for the mean delay of each service class in this model. In summary, our new approach simplifies MAC protocols for multimedia applications into an analytical model that includes more complex and realistic traffic models without compromising details of the protocol and significantly reduces the number of MAC headers, thus the overall average delay will be decreased. In response to using the studied multiservice multiqueue PS models, we apply the MPS model to two wireless applications: Push to Talk (PTT) service over GPRS/GSM networks and the Worldwide Interoperability for Microwave Access (WiMAX) networks. We investigate the uplink delay of PTT over traditional GPRS/GSM networks and the uplink delay for WiMAX Subscriber Station scheduler under a priority-based fair scheduling. MAC structures capable of supporting dynamically varying traffic are studied for the networks, especially, with the consideration of implementation issues. The model provides useful insights into the dynamic performance behaviours of GPRS/GSM and WiMAX networks with respect to various system parameters and comprehensive traffic conditions. We then evaluate the model under some different practical traffic scenarios. Through modelling of the operation of wireless access systems, under a variety of multimedia traffic, our analytical approaches provide practical analysis guidelines for wireless network dimensioning

Scheduling analysis with martingales

This paper proposes a new characterization of queueing systems by bounding a suitable exponential transform with a martingale. The constructed martingale is quite versatile in the sense that it captures queueing systems with Markovian and autoregressive arrivals in a unified manner; the second class is particularly relevant due to Wold’s decomposition of stationary processes. Moreover, using the framework of stochastic network calculus, the martingales allow for a simple handling of typical queueing operations: (1) flows’ multiplexing translates into multiplying the corresponding martingales, and (2) scheduling translates into time-shifting the martingales. The emerging calculus is applied to estimate the per-flow delay for FIFO, SP, and EDF scheduling. Unlike state-of-the-art results, our bounds capture a fundamental exponential leading constant in the number of multiplexed flows, and additionally are numerically tight

Large Deviations and the Generalized Processor Sharing Scheduling: Upper and Lower Bounds Part I: Two-Queue Systems

We prove asymptotic upper and lower bounds on the asymptotic decay rate of per-session queue length tail distributions for a single constant service rate server queue shared by multiple sessions with the generalized processor sharing (GPS) scheduling discipline. The simpler case of a GPS system with only two queues needs special attention, as under this case, it is shown that the upper bounds and lower boundsmatch, thus yielding exact bounds. This result is established in this part (Part I) of the paper. The general case is much more complicated, and is treated separately in Part II of the paper [42], where tight upper and lower bound results are proved by examining the dynamics of bandwidth sharing nature of GPS scheduling. The proofs use sample-path large deviation principle and are based on some recent large deviation results for a single queue with a constant service rate server. These results have implications in call admission control for high-speed communication networks

Performance modelling with adaptive hidden Markov models and discriminatory processor sharing queues

In modern computer systems, workload varies at different times and locations. It is important to model the performance of such systems via workload models that are both representative and efficient. For example, model-generated workloads represent realistic system behaviour, especially during peak times, when it is crucial to predict and address performance bottlenecks. In this thesis, we model performance, namely throughput and delay, using adaptive models and discrete queues. Hidden Markov models (HMMs) parsimoniously capture the correlation and burstiness of workloads with spatiotemporal characteristics. By adapting the batch training of standard HMMs to incremental learning, online HMMs act as benchmarks on workloads obtained from live systems (i.e. storage systems and financial markets) and reduce time complexity of the Baum-Welch algorithm. Similarly, by extending HMM capabilities to train on multiple traces simultaneously it follows that workloads of different types are modelled in parallel by a multi-input HMM. Typically, the HMM-generated traces verify the throughput and burstiness of the real data. Applications of adaptive HMMs include predicting user behaviour in social networks and performance-energy measurements in smartphone applications. Equally important is measuring system delay through response times. For example, workloads such as Internet traffic arriving at routers are affected by queueing delays. To meet quality of service needs, queueing delays must be minimised and, hence, it is important to model and predict such queueing delays in an efficient and cost-effective manner. Therefore, we propose a class of discrete, processor-sharing queues for approximating queueing delay as response time distributions, which represent service level agreements at specific spatiotemporal levels. We adapt discrete queues to model job arrivals with distributions given by a Markov-modulated Poisson process (MMPP) and served under discriminatory processor-sharing scheduling. Further, we propose a dynamic strategy of service allocation to minimise delays in UDP traffic flows whilst maximising a utility function.Open Acces