Performance Analysis of Modified SRPT in Multiple-Processor Multitask Scheduling

In this paper we study the multiple-processor multitask scheduling problem in both deterministic and stochastic models. We consider and analyze Modified Shortest Remaining Processing Time (M-SRPT) scheduling algorithm, a simple modification of SRPT, which always schedules jobs according to SRPT whenever possible, while processes tasks in an arbitrary order. The M-SRPT algorithm is proved to achieve a competitive ratio of $\Theta(\log \alpha +\beta)$ for minimizing response time, where $\alpha$ denotes the ratio between maximum job workload and minimum job workload, $\beta$ represents the ratio between maximum non-preemptive task workload and minimum job workload. In addition, the competitive ratio achieved is shown to be optimal (up to a constant factor), when there are constant number of machines. We further consider the problem under Poisson arrival and general workload distribution (\ie, $M/GI/N$ system), and show that M-SRPT achieves asymptotic optimal mean response time when the traffic intensity $\rho$ approaches $1$, if job size distribution has finite support. Beyond finite job workload, the asymptotic optimality of M-SRPT also holds for infinite job size distributions with certain probabilistic assumptions, for example, $M/M/N$ system with finite task workload

Large deviations analysis of scheduling policies for a web server

With increasing demand and availability of bandwidth resources, there has been tremendous growth in the scale and speed of web servers. In web servers, scheduling plays an important role in resource allocation (for instance, bandwidth allocation, processor allocation, etc). However, as the scale of a system increases so does the number of activities/events in the system (e.g., job arrivals), as a consequence of which the analysis of scheduling becomes increasingly harder. In particular, the possible ways in which scheduling failure (e.g., queue overflow, excessively large delay, instability of a system) can occur becomes increasingly greater, thus making it more difficult to understand the behavior and develop design rules for scheduling algorithms. However, a well-known observation from large devi viations theory that large scale systems fails in a “most likely way” can potentially be used to simplify the design and analysis of scheduling. In this thesis, we study the implications and applications of this effect on scheduling in a web server accessed by a large number of sources. We analyze the delay distribution of scheduling policies for web servers under a many sources large deviation regime which models web servers in a large scale system well. Due to the difficulties brought on considering a large number of sources, only a small number of scheduling policies, such as First-Come-First-Serve (FCFS), General-ProcessorSharing (GPS), and Priority Queueing policies have been analyzed under the many sources regime. In particular, in a single queue single server setup the delay characteristics of only FCFS, Shortest-Job-First (SJF), and Longest-Job-First (LJF) has been analyzed. In this thesis, we study the Two-Dimensional-Queueing (2DQ) framework, a unifying queueing framework that allows the identification of the “most likely way” in which delay occurs, to analyze the delay of various unexplored scheduling policies. In conjunction with the 2DQ framework, we develop a new “cycle based” technique for understanding the large deviations tail probability of more complex policies. Using the combination of the 2DQ framework and the cycle based analysis, we first analyze two interesting scheduling policies, i.e., Shortest-Remaining-Processing-Time (SRPT) policy (which is mean delay optimal) and Processer-Sharing (PS) policy (which is a “fair” policy). We derive the asymptotic delay distributions (rate functions) of both policies and study their behavior across job sizes. Next, we address three problems in implementing the aforementioned scheduling policies: (i) end receivers may have bandwidth constraints that are not taken account in SRPT, (ii) the remaining processing time information might not be available to the web-server, and (iii) most actual implementations are variants of SRPT to reflect other implementation constraints and/or to jointly optimize other metrics in addition to delay, i.e., jitter, fairness, etc. To address these, we first develop finite-SRPT that takes into account the bandwidth constraint at the end receiver, and show that the policy shifts between SRPT and a PS-like policy depending on the bandwidth constraint. Second, we study the Least-Attained-Service (LAS) policy which is viewed as a good substitute for SRPT when the remaining job size is not available and we analyze the penalty associated with not using the remaining size information directly. Lastly, we analyze a class of scheduling policies known as SMART that contains many variants of SRPT with different fairness properties and show that all policies in the class have the same tail probability of delay across job sizes for a many sources regime. The results of this thesis facilitate the understanding of various scheduling policies under the many sources regime and provides an analytical queueing framework that can be used to understand other scheduling policies.Electrical and Computer Engineerin

Scheduling for the tail: Robustness versus optimality

When scheduling to minimize the sojourn time tail, the goals of optimality and robustness are seemingly at odds. Over the last decade, results have emerged which show that scheduling disciplines that are near-optimal under light (exponential) tailed workload distributions do not perform well under heavy (power) tailed workload distributions, and vice-versa. Very recently, it has been shown that this conflict between optimality and robustness is fundamental, i.e., no policy that does not learn information about the workload can be optimal across both light-tailed and heavy-tailed workloads. In this paper we show that one can exploit very limited workload information (the system load) in order to design a scheduler that provides robust performance across heavy-tailed and light-tailed workloads

Heavy-Tailed Limits for Medium Size Jobs and Comparison Scheduling

We study the conditional sojourn time distributions of processor sharing (PS), foreground background processor sharing (FBPS) and shortest remaining processing time first (SRPT) scheduling disciplines on an event where the job size of a customer arriving in stationarity is smaller than exactly k>=0 out of the preceding m>=k arrivals. Then, conditioning on the preceding event, the sojourn time distribution of this newly arriving customer behaves asymptotically the same as if the customer were served in isolation with a server of rate (1-\rho)/(k+1) for PS/FBPS, and (1-\rho) for SRPT, respectively, where \rho is the traffic intensity. Hence, the introduced notion of conditional limits allows us to distinguish the asymptotic performance of the studied schedulers by showing that SRPT exhibits considerably better asymptotic behavior for relatively smaller jobs than PS/FBPS. Inspired by the preceding results, we propose an approximation to the SRPT discipline based on a novel adaptive job grouping mechanism that uses relative size comparison of a newly arriving job to the preceding m arrivals. Specifically, if the newly arriving job is smaller than k and larger than m-k of the previous m jobs, it is routed into class k. Then, the classes of smaller jobs are served with higher priorities using the static priority scheduling. The good performance of this mechanism, even for a small number of classes m+1, is demonstrated using the asymptotic queueing analysis under the heavy-tailed job requirements. We also discuss refinements of the comparison grouping mechanism that improve the accuracy of job classification at the expense of a small additional complexity.Comment: 26 pages, 2 figure

Is Tail-Optimal Scheduling Possible?

This paper focuses on the competitive analysis of scheduling disciplines in a large deviations setting. Although there are policies that are known to optimize the sojourn time tail under a large class of heavy-tailed job sizes (e.g., processor sharing and shortest remaining processing time) and there are policies known to optimize the sojourn time tail in the case of light-tailed job sizes (e.g., first come first served), no policies are known that can optimize the sojourn time tail across both light- and heavy-tailed job size distributions. We prove that no such work-conserving, nonanticipatory, nonlearning policy exists, and thus that a policy must learn (or know) the job size distribution in order to optimize the sojourn time tail

Delay Analysis And Optimality Of scheduling In Multi-hop Wireless Network

The delay is one of the important metric considered in the wireless network and wire-line network.In single hop wire-line network only one hop(router) is present from source to destination .In single hop network the interference problems occurred and the trac control is dicult,the high amount of delay and the low amount of packet delivery ratio, because of routes changes dynamically and finally leads to low performance of the network.The delay analysis of a packets plays a vital role in the network.In real time applications the fixed time is given, so that the given amount of time all the packets should be delivered from source to destination.In multi-hop wireless network decomposition of packets into multiple paths,if any two nodes meet at same point bottleneck is occurred.In order to overcome from bottleneck used new queuing technique.For knowing the behavior of the each path in the network lower bound analysis is used.Dierent policies are used for scheduling the packets, which gives better optimality

Size-based scheduling vs fairness for datacenter flows: a queuing perspective

Contrary to the conclusions of a recent body of work where approximate shortest remaining processing time first (SRPT) flow scheduling is advocated for datacenter networks, this paper aims to demonstrate that per-flow fairness remains a preferable objective. We evaluate abstract queuing models by analysis and simulation to illustrate the non-optimality of SRPT under the reasonable assumptions that datacenter flows occur in batches and bursts and not, as usually assumed, individually at the instants of a Poisson process. Results for these models have significant implications for the design of bandwidth sharing strategies for datacenter networks. In particular, we propose a novel "virtual fair scheduling" algorithm that enforces fairness between batches and is arguably simple enough to be implemented in high speed devices.Comment: 16 pages, 5 figure

Many-Sources Large Deviations for Max-Weight Scheduling

In this paper, a many-sources large deviations principle (LDP) for the transient workload of a multi-queue single-server system is established where the service rates are chosen from a compact, convex and coordinate-convex rate region and where the service discipline is the max-weight policy. Under the assumption that the arrival processes satisfy a many-sources LDP, this is accomplished by employing Garcia's extended contraction principle that is applicable to quasi-continuous mappings. For the simplex rate-region, an LDP for the stationary workload is also established under the additional requirements that the scheduling policy be work-conserving and that the arrival processes satisfy certain mixing conditions. The LDP results can be used to calculate asymptotic buffer overflow probabilities accounting for the multiplexing gain, when the arrival process is an average of \emph{i.i.d.} processes. The rate function for the stationary workload is expressed in term of the rate functions of the finite-horizon workloads when the arrival processes have \emph{i.i.d.} increments.Comment: 44 page

Filter Scheduling Function Model In Internet Server: Resource Configuration, Performance Evaluation And Optimal Scheduling

ABSTRACT FILTER SCHEDULING FUNCTION MODEL IN INTERNET SERVER: RESOURCE CONFIGURATION, PERFORMANCE EVALUATION AND OPTIMAL SCHEDULING by MINGHUA XU August 2010 Advisor: Dr. Cheng-Zhong Xu Major: Computer Engineering Degree: Doctor of Philosophy Internet traffic often exhibits a structure with rich high-order statistical properties like selfsimilarity and long-range dependency (LRD). This greatly complicates the problem of server performance modeling and optimization. On the other hand, popularity of Internet has created numerous client-server or peer-to-peer applications, with most of them, such as online payment, purchasing, trading, searching, publishing and media streaming, being timing sensitive and/or financially critical. The scheduling policy in Internet servers is playing central role in satisfying service level agreement (SLA) and achieving savings and efficiency in operations. The increasing popularity of high-volume performance critical Internet applications is a challenge for servers to provide individual response-time guarantees. Existing tools like queuing models in most cases only hold in mean value analysis under the assumption of simplified traffic structures. Considering the fact that most Internet applications can tolerate a small percentage of deadline misses, we define a decay function model characterizes the relationship between the request delay constraint, deadline misses, and server capacity in a transfer function based filter system. The model is general for any time-series based or measurement based processes. Within the model framework, a relationship between server capacity, scheduling policy, and service deadline is established in formalism. Time-invariant (non-adaptive) resource allocation policies are design and analyzed in the time domain. For an important class of fixed-time allocation policies, optimality conditions with respect to the correlation of input traffic are established. The upper bound for server capacity and service level are derived with general Chebshev\u27s inequality, and extended to tighter boundaries for unimodal distributions by using VysochanskiPetunin\u27s inequality. For traffic with strong LRD, a design and analysis of the decay function model is done in the frequency domain. Most Internet traffic has monotonically decreasing strength of variation functions over frequency. For this type of input traffic, it is proved that optimal schedulers must have a convex structure. Uniform resource allocation is an extreme case of the convexity and is proved to be optimal for Poisson traffic. With an integration of the convex-structural principle, an enhance GPS policy improves the service quality significantly. Furthermore, it is shown that the presence of LRD in the input traffic results in shift of variation strength from high frequency to lower frequency bands, leading to a degradation of the service quality. The model is also extended to support server with different deadlines, and to derive an optimal time-variant (adaptive) resource allocation policy that minimizes server load variances and server resource demands. Simulation results show time-variant scheduling algorithm indeed outperforms time-invariant optimal decay function scheduler. Internet traffic has two major dynamic factors, the distribution of request size and the correlation of request arrival process. When applying decay function model as scheduler to random point process, corresponding two influences for server workload process is revealed as, first, sizing factor--interaction between request size distribution and scheduling functions, second, correlation factor--interaction between power spectrum of arrival process and scheduling function. For the second factor, it is known from this thesis that convex scheduling function will minimize its impact over server workload. Under the assumption of homogeneous scheduling function for all requests, it shows that uniform scheduling is optimal for the sizing factor. Further more, by analyzing the impact from queueing delay to scheduling function, it shows that queueing larger tasks vs. smaller ones leads to less reduction in sizing factor, but at the benefit of more decreasing in correlation factor in the server workload process. This shows the origin of optimality of shortest remain processing time (SRPT) scheduler

Opportunistic scheduling of flows with general size distribution in wireless time-varying channels

In this paper we study how to design an opportunistic scheduler when flow sizes have a general service time distribution with the objective of minimizing the expected holding cost. We allow the channel condition to have two states which in particular covers the important special case of ON/OFF channels. We formulate the problem as a multi-armed restless bandit problem, a particular class of Markov decision processes. Since an exact solution is out of reach, we characterize in closed-form the Whittle index, which allows us to define a heuristic scheduling rule for the problem. We then particularize the index to the important subclass of distributions with a decreasing hazard rate. We finally evaluate the performance of the proposed Whittle-index based scheduler by simulation of a wireless network. The numerical results show that the performance of the proposed scheduler is very satisfactory