Delay Tail Research Articles

When Does the Gittins Policy Have Asymptotically Optimal Response Time Tail in the M/G/1?

Looking Beyond Mean Delay in Queue Scheduling A central aim of queueing theory is to design scheduling policies that reduce delays. Much of literature focuses on minimizing mean delay, but, in practice, it can be more important to reduce the delay tail—that is, the probability of especially large delays. There is a subtle interplay between these objectives. For instance, the SRPT policy (Shortest Remaining Processing Time) prioritizes short jobs over long jobs, which optimizes mean delay. Sometimes, SRPT also has optimal delay tail, but other times, its delay tail is pessimal. Thus, good mean delay need not imply good delay tail. The article When Does the Gittins Policy Have Asymptotically Optimal Response Time Tail in the M/G/1? explores the interplay between mean delay and delay tail for systems with unknown job durations. The article studies the Gittins policy, which optimizes mean delay in this setting. Curiously, its delay tail can be optimal, pessimal, or in between, depending on the job duration distribution. Fortunately, the worst case can be avoided: the article shows how to modify Gittins to avoid pessimal delay tail while maintaining near-optimal mean delay.

Admission control for cellular networks with direct QoS monitoring

This paper proposes an admission control algorithm for cellular networks based on the direct and dynamic monitoring of quality of services (QoS) performance metrics--both system delay tail and resi...

Delay tails in MapReduce scheduling

MapReduce/Hadoop production clusters exhibit heavy-tailed characteristics for job processing times. These phenomena are resultant of the workload features and the adopted scheduling algorithms. Analytically understanding the delays under different schedulers for MapReduce can facilitate the design and deployment of large Hadoop clusters. The map and reduce tasks of a MapReduce job have fundamental difference and tight dependence between them, complicating the analysis. This also leads to an interesting starvation problem with the widely used Fair Scheduler due to its greedy approach to launching reduce tasks. To address this issue, we design and implement Coupling Scheduler, which gradually launches reduce tasks depending on map task progresses. Real experiments demonstrate improvements to job response times by up to an order of magnitude. Based on extensive measurements and source code investigations, we propose analytical models for the default FIFO and Fair Scheduler as well as our implemented Coupling Scheduler. For a class of heavy-tailed map service time distributions, i.e., regularly varying of index -a, we derive the distribution tail of the job processing delay under the three schedulers, respectively. The default FIFO Scheduler causes the delay to be regularly varying of index -a+1. Interestingly, we discover a criticality phenomenon for Fair Scheduler, the delay under which can change from regularly varying of index -a to -a+1, depending on the maximum number of reduce tasks of a job. Other more subtle behaviors also exist. In contrast, the delay distribution tail under Coupling Scheduler can be one order lower than Fair Scheduler under some conditions, implying a better performance.

Delay Distributions in Appointment Systems with Generally and Non-Identically Distributed Service Times and No-Shows

We derive the probability distribution of each patient's wait time and doctor's idle time and overtime in single-server systems where healthcare providers respond to appointment-driven arrivals with generally and non-identically distributed service times. Patient no-show probabilities and appointment intervals are also non-identical. The formulation is due to a convolution-based busy period analysis that can be implemented via numerical integration methods. A special case yields the distribution of customer wait times and server idle times in transient-phase D/M/1 queues for appointments at fixed-intervals, where analytical distribution functions facilitate a mathematical proof that delays are unfair to patients. We demonstrate more generally that appointment system assessment of delay tail probabilities (rather than expectations) considers the non-linear disutility for delays, and indeed reverses conclusions made in prior analysis.

Athermal Martensites, Temperature-Time-Transformation Diagrams and Thermal Hysteresis: Monte Carlo Simulations of Strain Pseudospins

We explore the kinetics of a three-state strain pseudospin model for a square/rectangle ferroelastic transition, described by a temperature dependent hamiltonian without quenched disorder, using temperature quench Monte Carlo simulations. The model hamiltonian includes power law anisotropic long range interactions, which lock the domain walls in a symmetry breaking diagonal direction. In athermal parameter regime, there are fast conversions at the athermal transition temperature, but with delay tails above it, as in experiment. The conversion delay tails have a Vogel-Fulcher divergence at transition to austenite. The incubation delays and their insensitivity to elastic energy scales are attributed to entropy barriers. Temperature cycling shows hysteretic behavior in physical quantities.

On the Origins of Heavy-Tailed Delay in Dynamic Spectrum Access Networks

This paper provides an asymptotic analysis of the transmission delay experienced by SUs for dynamic spectrum access (DSA) networks. It is shown that DSA induces only light-tailed delay if both the busy time of PU channels and the message size of SUs are light tailed. On the contrary, if either the busy time or the message size is heavy tailed, then the SUs' transmission delay is heavy tailed. For this latter case, it is proven that if one of either the busy time or the message size is light tailed and the other is regularly varying with index \alpha, the transmission delay is regularly varying with the same index. As a consequence, the delay has an infinite mean provided \alpha < 1 and an infinite variance provided \alpha < 2. Furthermore, if both the busy time and the message size are regularly varying with different indices, then the delay tail distribution is as heavy as the one with the smaller index. Moreover, the impact of spectrum mobility and multiradio diversity on the delay performance of SUs is studied. It is shown that both spectrum mobility and multiradio diversity can greatly mitigate the heavy-tailed delay by increasing the orders of its finite moments.

Control of end-to-end delay tails in a multiclass network: LWDF discipline optimality

We consider a multiclass queueing network with $N$ customer classes, each having an arbitrary fixed route through the network. (Thus, the network is not necessarily feedforward.) We show that the largest weighted delay first (LWDF) discipline is an optimal scheduling discipline in the network in the following sense. Let $w_i$ be the (random) instantaneous largest end-to-end delay of a class $i$ customer in the network in stationary regime. For any set of positive constants $\alpha_1, \ldots, \alpha_N$, the LWDF discipline associated with this set maximizes (among all disciplines) the quantity \[ (1)\hspace*{15pt}\min_{i=1,\ldots,N} \left[ \alpha_i \lim_{n\to\infty} \frac{-1}{n} \log P \lb w_i > n \rb \right] = \lim_{n\to\infty} \frac{-1}{n} \log P \lb r > n \rb,\hspace*{15pt} \] where $r \doteq \max_{i} w_i/\alpha_i$ is the maximal weighted delay in the network. [This result is a generalization of the single-server result proved by A. L. Stolyar and K. Ramanan in Ann. Appl. Probab. 11 (2001) 1--48.] As the key element of the proof, we establish the following critical node property: In a LWDF network, there exists a most likely path to build large $r$, which is a most likely path to do so in one of the network nodes in isolation. Such a most likely path has a very simple structure: its parameters [and the optimal value of (1)] can be computed by solving a finite-dimensional optimization problem for each network node.

Analytic and Simulation Modeling for AAL2 Connection Admission Control

Fine time-scale simulation and analysis are combined with coarse time-scale simulation in the design and performance evaluation of connection admission control (CAC) techniques applied to connections that utilize the asynchronous transfer mode (ATM) Adaptation Layer Type 2 (AAL2). Fine time-scale (packet-level) simulation and then analytical modeling are used to determine the AAL2-level connection load that can be supported by a given ATM-level connection capacity, under a delay tail quality of service (QoS) specification. Self-configuring CAC for AAL2 with load estimation (SCALE) is introduced, which uses ATM signaling to establish and release ATM-level connections as necessary to support a set of AAL2-level connections. Using the results of the earlier analytical modeling, coarse time-scale (connection-level) simulation experiments demonstrate SCALE parameter trade-offs and confirm that SCALE can be an effective solution to the large time-of-day load variations expected from a set of AAL2 connections.

Many-Sources Delay Asymptotics with Applications to Priority Queues

In this paper, we study discrete-time priority queueing systems fed by a large number of arrival streams. We first provide bounds on the actual delay asymptote in terms of the virtual delay asymptote. Then, under suitable assumptions on the arrival process to the queue, we show that these asymptotes are the same. As an application of this result, we then consider a priority queueing system with two queues. Using the earlier result, we derive an upper bound on the tail probability of the delay. Under certain assumptions on the rate function of the arrival process, we show that the upper bound is tight. We then consider a system with Markovian arrivals and numerically evaluate the delay tail probability and validate these results with simulations.

Delay asymptotics for a priority queueing system

In this paper, we study discrete-time priority queueing systems fed by a large number of arrival streams. We first provide bounds on the actual delay asymptote in terms of the virtual delay asymptote. Then, under suitable assumptions on the arrival process to the queue, we show that these asymptotes are the same. We then consider a priority queueing system with two queues. Using the earlier result, we derive an upper bound on the tail probability of the delay. Under certain assumptions on the rate function of the arrival process, we show that the upper bound is tight. We then consider a system with Markovian arrivals and numerically evaluate the delay tail probability and validate these results with simulations.