Response Time Approximations in Fork-Join Queues

Fork-join queueing networks model a network of parallel servers in which an arriving job splits into a number of subtasks that are serviced in parallel. Fork-join queues can be used to model disk arrays. A response time approximation of the fork-join queue is presented that attempts to comply with the additional constraints of modelling a disk array. This approximation is compared with existing analytical approximations of the fork-join queueing network

Approximations and Bounds for (n, k) Fork-Join Queues: A Linear Transformation Approach

Compared to basic fork-join queues, a job in (n, k) fork-join queues only needs its k out of all n sub-tasks to be finished. Since (n, k) fork-join queues are prevalent in popular distributed systems, erasure coding based cloud storages, and modern network protocols like multipath routing, estimating the sojourn time of such queues is thus critical for the performance measurement and resource plan of computer clusters. However, the estimating keeps to be a well-known open challenge for years, and only rough bounds for a limited range of load factors have been given. In this paper, we developed a closed-form linear transformation technique for jointly-identical random variables: An order statistic can be represented by a linear combination of maxima. This brand-new technique is then used to transform the sojourn time of non-purging (n, k) fork-join queues into a linear combination of the sojourn times of basic (k, k), (k+1, k+1), ..., (n, n) fork-join queues. Consequently, existing approximations for basic fork-join queues can be bridged to the approximations for non-purging (n, k) fork-join queues. The uncovered approximations are then used to improve the upper bounds for purging (n, k) fork-join queues. Simulation experiments show that this linear transformation approach is practiced well for moderate n and relatively large k.Comment: 10 page

Validation of Large Zoned RAID Systems

Building on our prior work we present an improved model for for large partial stripe following full stripe writes in RAID 5. This was necessary because we observed that our previous model tended to underestimate measured results. To date, we have only validated these models against RAID systems with at most four disks. Here we validate our improved model, and also our existing models for other read and write configurations, against measurements taken from an eight disk RAID array

RAID Organizations for Improved Reliability and Performance: A Not Entirely Unbiased Tutorial (1st revision)

RAID proposal advocated replacing large disks with arrays of PC disks, but as the capacity of small disks increased 100-fold in 1990s the production of large disks was discontinued. Storage dependability is increased via replication or erasure coding. Cloud storage providers store multiple copies of data obviating for need for further redundancy. Varitaions of RAID based on local recovery codes, partial MDS reduce recovery cost. NAND flash Solid State Disks - SSDs have low latency and high bandwidth, are more reliable, consume less power and have a lower TCO than Hard Disk Drives, which are more viable for hyperscalers.Comment: Submitted to ACM Computing Surveys. arXiv admin note: substantial text overlap with arXiv:2306.0876

On Fork-Join Queues and Maximum Ratio Cliques

This dissertation consists of two parts. The first part delves into the problem of response time estimation in fork-join queueing networks. These systems have been seen in literature for more than thirty years. The estimation of the mean response time in these systems has been found to be notoriously hard for most forms of these queueing systems. In this work, simple expressions for the mean response time are proposed as conjectures. Extensive experiments demonstrate the remarkable accuracy of these conjectures. Algorithms for the estimation of response time using these conjectures are proposed. For many of the networks studied in this dissertation, no approximations are known in literature for estimation of their response time. Therefore, the contribution of this dissertation in this direction marks significant progress in the analysis of fork-join queues. The second part of this dissertation introduces a fractional version of the classical maximum weight clique problem, the maximum ratio clique problem, which is to find a maximal clique that has the largest ratio of benefit and cost weights associated with the cliques vertices. This problem is formulated to model networks in which the vertices have a benefit as well as a cost associated with them. The maximum ratio clique problem finds applications in a wide range of areas including social networks, stock market graphs and wind farm location. NP-completeness of the decision version of the problem is established, and three solution methods are proposed. The results of numerical experiments with standard graph instances, as well as with real-life instances arising in finance and energy systems, are reported

Queueing network models of zoned RAID system performance

RAID systems are widely deployed, both as standalone storage solutions and as the building blocks of modern virtualised storage platforms. An accurate model of RAID system performance is therefore critical towards fulfilling quality of service constraints for fast, reliable storage. This thesis presents techniques and tools that model response times in zoned RAID systems. The inputs to this analysis are a specified I/O request arrival rate, an I/O request access profile, a given RAID configuration and physical disk parameters. The primary output of this analysis is an approximation to the cumulative distribution function of I/O request response time. From this, it is straightforward to calculate response time quantiles, as well as the mean, variance and higher moments of I/O request response time. The model supports RAID levels 0, 01, 10 and 5 and a variety of workload types. Our RAID model is developed in a bottom-up hierarchical fashion. We begin by modelling each zoned disk drive in the array as a single M/G/1 queue. The service time is modelled as the sum of the random variables of seek time, rotational latency and data transfer time. In doing so, we take into account the properties of zoned disks. We then abstract a RAID system as a fork-join queueing network. This comprises several queues, each of which represents one disk drive in the array. We tailor our basic fork-join approximation to account for the I/O request patterns associated with particular request types and request sizes under different RAID levels. We extend the RAID and disk models to support bulk arrivals, requests of different sizes and scheduling algorithms that reorder queueing requests to minimise disk head positioning time. Finally, we develop a corresponding simulation to improve and validate the model. To test the accuracy of all our models, we validate them against disk drive and RAID device measurements throughout