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Title: Load Balancing Under Strict Compatibility Constraints

Consider a system with N identical single-server queues and a number of task types, where each server is able to process only a small subset of possible task types. Arriving tasks select [Formula: see text] random compatible servers and join the shortest queue among them. The compatibility constraints are captured by a fixed bipartite graph between the servers and the task types. When the graph is complete bipartite, the mean-field approximation is accurate. However, such dense compatibility graphs are infeasible for large-scale implementation. We characterize a class of sparse compatibility graphs for which the mean-field approximation remains valid. For this, we introduce a novel notion, called proportional sparsity, and establish that systems with proportionally sparse compatibility graphs asymptotically match the performance of a fully flexible system. Furthermore, we show that proportionally sparse random compatibility graphs can be constructed, which reduce the server degree almost by a factor [Formula: see text] compared with the complete bipartite compatibility graph. more »« less

Weng, Wentao; Zhou, Xingyu; Srikant, R.(
, Proceedings of the ACM on Measurement and Analysis of Computing Systems)

null
(Ed.)

Applications in cloud platforms motivate the study of efficient load balancing under job-server constraints and server heterogeneity. In this paper, we study load balancing on a bipartite graph where left nodes correspond to job types and right nodes correspond to servers, with each edge indicating that a job type can be served by a server. Thus edges represent locality constraints, i.e., an arbitrary job can only be served at servers which contain certain data and/or machine learning (ML) models. Servers in this system can have heterogeneous service rates. In this setting, we investigate the performance of two policies named Join-the-Fastest-of-the-Shortest-Queue (JFSQ) and Join-the-Fastest-of-the-Idle-Queue (JFIQ), which are simple variants of Join-the-Shortest-Queue and Join-the-Idle-Queue, where ties are broken in favor of the fastest servers. Under a "well-connected'' graph condition, we show that JFSQ and JFIQ are asymptotically optimal in the mean response time when the number of servers goes to infinity. In addition to asymptotic optimality, we also obtain upper bounds on the mean response time for finite-size systems. We further show that the well-connectedness condition can be satisfied by a random bipartite graph construction with relatively sparse connectivity.

Dumitriu, Ioana; Zhu, Yizhe(
, Random Matrices: Theory and Applications)

We compute the eigenvalue fluctuations of uniformly distributed random biregular bipartite graphs with fixed and growing degrees for a large class of analytic functions. As a key step in the proof, we obtain a total variation distance bound for the Poisson approximation of the number of cycles and cyclically non-backtracking walks in random biregular bipartite graphs, which might be of independent interest. We also prove a semicircle law for random [Formula: see text]-biregular bipartite graphs when [Formula: see text]. As an application, we translate the results to adjacency matrices of uniformly distributed random regular hypergraphs.

The fundamental problem in the study of parallel-server systems is that of finding and analyzing routing policies of arriving jobs to the servers that efficiently balance the load on the servers. The most well-studied policies are (in decreasing order of efficiency) join the shortest workload (JSW), which assigns arrivals to the server with the least workload; join the shortest queue (JSQ), which assigns arrivals to the smallest queue; the power-of-[Formula: see text] (PW([Formula: see text])), which assigns arrivals to the shortest among [Formula: see text] queues that are sampled from the total of [Formula: see text] queues uniformly at random; and uniform routing, under which arrivals are routed to one of the [Formula: see text] queues uniformly at random. In this paper we study the stability problem of parallel-server systems, assuming that routing errors may occur, so that arrivals may be routed to the wrong queue (not the smallest among the relevant queues) with a positive probability. We treat this routing mechanism as a probabilistic routing policy, named a [Formula: see text]-allocation policy, that generalizes the PW([Formula: see text]) policy, and thus also the JSQ and uniform routing, where [Formula: see text] is an [Formula: see text]-dimensional vector whose components are the routing probabilities. Our goal is to study the (in)stability problem of the system under this routing mechanism, and under its “nonidling” version, which assigns new arrivals to an idle server, if such a server is available, and otherwise routes according to the [Formula: see text]-allocation rule. We characterize a sufficient condition for stability, and prove that the stability region, as a function of the system’s primitives and [Formula: see text], is in general smaller than the set [Formula: see text]. Our analyses build on representing the queue process as a continuous-time Markov chain in an ordered space of [Formula: see text]-dimensional real-valued vectors, and using a generalized form of the Schur-convex order.

Lacker, Daniel; Ramanan, Kavita; Wu, Ruoyu(
, Probability Theory and Related Fields)

Consider a system of homogeneous interacting diffusive particles labeled by the nodes of a unimodular Galton–Watson tree, where the state of each node evolves infinitesi- mally like a d-dimensional diffusion whose drift coefficient depends on (the histories of) its own state and the states of neighboring nodes, and whose diffusion coefficient depends only on (the history of) its own state. Under suitable regularity assumptions on the coefficients, an autonomous characterization is obtained for the marginal dis- tribution of the dynamics of the neighborhood of a typical node in terms of a certain local equation, which is a new kind of stochastic differential equation that is nonlinear in the sense of McKean. This equation describes a finite-dimensional non-Markovian stochastic process whose infinitesimal evolution at any time depends not only on the structure and current state of the neighborhood, but also on the conditional law of the current state given the past of the states of neighborhing nodes until that time. Such marginal distributions are of interest because they arise as weak limits of both marginal distributions and empirical measures of interacting diffusions on many sequences of sparse random graphs, including the configuration model and Erdös–Rényi graphs whose average degrees converge to a finite non-zero limit. The results obtained complement classical results in the mean-field regime, which characterize the limiting dynamics of homogeneous interacting diffusions on complete graphs, as the num- ber of nodes goes to infinity, in terms of a corresponding nonlinear Markov process. However, in the sparse graph setting, the topology of the graph strongly influences the dynamics, and the analysis requires a completely different approach. The proofs of existence and uniqueness of the local equation rely on delicate new conditional independence and symmetry properties of particle trajectories on unimodular Galton– Watson trees, as well as judicious use of changes of measure.

Hunt, I.; Husain, S.; Simon, J.; Obeid, I.; Picone, J.(
, IEEE Signal Processing in Medicine and Biology Symposium (SPMB))

Obeid, Iyad; Picone, Joseph; Selesnick, Ivan
(Ed.)

The Neural Engineering Data Consortium (NEDC) is developing a large open source database of high-resolution digital pathology images known as the Temple University Digital Pathology Corpus (TUDP) [1]. Our long-term goal is to release one million images. We expect to release the first 100,000 image corpus by December 2020. The data is being acquired at the Department of Pathology at Temple University Hospital (TUH) using a Leica Biosystems Aperio AT2 scanner [2] and consists entirely of clinical pathology images. More information about the data and the project can be found in Shawki et al. [3]. We currently have a National Science Foundation (NSF) planning grant [4] to explore how best the community can leverage this resource. One goal of this poster presentation is to stimulate community-wide discussions about this project and determine how this valuable resource can best meet the needs of the public.
The computing infrastructure required to support this database is extensive [5] and includes two HIPAA-secure computer networks, dual petabyte file servers, and Aperio’s eSlide Manager (eSM) software [6]. We currently have digitized over 50,000 slides from 2,846 patients and 2,942 clinical cases. There is an average of 12.4 slides per patient and 10.5 slides per case with one report per case. The data is organized by tissue type as shown below:
Filenames:
tudp/v1.0.0/svs/gastro/000001/00123456/2015_03_05/0s15_12345/0s15_12345_0a001_00123456_lvl0001_s000.svs
tudp/v1.0.0/svs/gastro/000001/00123456/2015_03_05/0s15_12345/0s15_12345_00123456.docx
Explanation:
tudp: root directory of the corpus
v1.0.0: version number of the release
svs: the image data type
gastro: the type of tissue
000001: six-digit sequence number used to control directory complexity
00123456: 8-digit patient MRN
2015_03_05: the date the specimen was captured
0s15_12345: the clinical case name
0s15_12345_0a001_00123456_lvl0001_s000.svs: the actual image filename consisting of a repeat of the case name, a site code (e.g., 0a001), the type and depth of the cut (e.g., lvl0001) and a token number (e.g., s000)
0s15_12345_00123456.docx: the filename for the corresponding case report
We currently recognize fifteen tissue types in the first installment of the corpus. The raw image data is stored in Aperio’s “.svs” format, which is a multi-layered compressed JPEG format [3,7]. Pathology reports containing a summary of how a pathologist interpreted the slide are also provided in a flat text file format. A more complete summary of the demographics of this pilot corpus will be presented at the conference.
Another goal of this poster presentation is to share our experiences with the larger community since many of these details have not been adequately documented in scientific publications. There are quite a few obstacles in collecting this data that have slowed down the process and need to be discussed publicly. Our backlog of slides dates back to 1997, meaning there are a lot that need to be sifted through and discarded for peeling or cracking. Additionally, during scanning a slide can get stuck, stalling a scan session for hours, resulting in a significant loss of productivity. Over the past two years, we have accumulated significant experience with how to scan a diverse inventory of slides using the Aperio AT2 high-volume scanner. We have been working closely with the vendor to resolve many problems associated with the use of this scanner for research purposes. This scanning project began in January of 2018 when the scanner was first installed. The scanning process was slow at first since there was a learning curve with how the scanner worked and how to obtain samples from the hospital. From its start date until May of 2019 ~20,000 slides we scanned. In the past 6 months from May to November we have tripled that number and how hold ~60,000 slides in our database. This dramatic increase in productivity was due to additional undergraduate staff members and an emphasis on efficient workflow.
The Aperio AT2 scans 400 slides a day, requiring at least eight hours of scan time. The efficiency of these scans can vary greatly. When our team first started, approximately 5% of slides failed the scanning process due to focal point errors. We have been able to reduce that to 1% through a variety of means: (1) best practices regarding daily and monthly recalibrations, (2) tweaking the software such as the tissue finder parameter settings, and (3) experience with how to clean and prep slides so they scan properly. Nevertheless, this is not a completely automated process, making it very difficult to reach our production targets. With a staff of three undergraduate workers spending a total of 30 hours per week, we find it difficult to scan more than 2,000 slides per week using a single scanner (400 slides per night x 5 nights per week). The main limitation in achieving this level of production is the lack of a completely automated scanning process, it takes a couple of hours to sort, clean and load slides. We have streamlined all other aspects of the workflow required to database the scanned slides so that there are no additional bottlenecks.
To bridge the gap between hospital operations and research, we are using Aperio’s eSM software. Our goal is to provide pathologists access to high quality digital images of their patients’ slides. eSM is a secure website that holds the images with their metadata labels, patient report, and path to where the image is located on our file server. Although eSM includes significant infrastructure to import slides into the database using barcodes, TUH does not currently support barcode use. Therefore, we manage the data using a mixture of Python scripts and manual import functions available in eSM. The database and associated tools are based on proprietary formats developed by Aperio, making this another important point of community-wide discussion on how best to disseminate such information.
Our near-term goal for the TUDP Corpus is to release 100,000 slides by December 2020. We hope to continue data collection over the next decade until we reach one million slides. We are creating two pilot corpora using the first 50,000 slides we have collected. The first corpus consists of 500 slides with a marker stain and another 500 without it. This set was designed to let people debug their basic deep learning processing flow on these high-resolution images. We discuss our preliminary experiments on this corpus and the challenges in processing these high-resolution images using deep learning in [3]. We are able to achieve a mean sensitivity of 99.0% for slides with pen marks, and 98.9% for slides without marks, using a multistage deep learning algorithm. While this dataset was very useful in initial debugging, we are in the midst of creating a new, more challenging pilot corpus using actual tissue samples annotated by experts. The task will be to detect ductal carcinoma (DCIS) or invasive breast cancer tissue. There will be approximately 1,000 images per class in this corpus. Based on the number of features annotated, we can train on a two class problem of DCIS or benign, or increase the difficulty by increasing the classes to include DCIS, benign, stroma, pink tissue, non-neoplastic etc.
Those interested in the corpus or in participating in community-wide discussions should join our listserv, nedc_tuh_dpath@googlegroups.com, to be kept informed of the latest developments in this project. You can learn more from our project website: https://www.isip.piconepress.com/projects/nsf_dpath.

Rutten, Daan, and Mukherjee, Debankur. Load Balancing Under Strict Compatibility Constraints. Retrieved from https://par.nsf.gov/biblio/10323989. Mathematics of Operations Research . Web. doi:10.1287/moor.2022.1258.

@article{osti_10323989,
place = {Country unknown/Code not available},
title = {Load Balancing Under Strict Compatibility Constraints},
url = {https://par.nsf.gov/biblio/10323989},
DOI = {10.1287/moor.2022.1258},
abstractNote = {Consider a system with N identical single-server queues and a number of task types, where each server is able to process only a small subset of possible task types. Arriving tasks select [Formula: see text] random compatible servers and join the shortest queue among them. The compatibility constraints are captured by a fixed bipartite graph between the servers and the task types. When the graph is complete bipartite, the mean-field approximation is accurate. However, such dense compatibility graphs are infeasible for large-scale implementation. We characterize a class of sparse compatibility graphs for which the mean-field approximation remains valid. For this, we introduce a novel notion, called proportional sparsity, and establish that systems with proportionally sparse compatibility graphs asymptotically match the performance of a fully flexible system. Furthermore, we show that proportionally sparse random compatibility graphs can be constructed, which reduce the server degree almost by a factor [Formula: see text] compared with the complete bipartite compatibility graph.},
journal = {Mathematics of Operations Research},
author = {Rutten, Daan and Mukherjee, Debankur},
}

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