arXiv subjects

Isaac Grosof

Publications and source records attributed to Isaac Grosof.

At least 19 recordsLinked to original sources

arXiv2025

Improving Nonpreemptive Multiserver Job Scheduling with Quickswap

Modern data center workloads are composed of multiserver jobs, computational jobs that require multiple servers in order to run. A data center server can run many multiserver jobs in parallel, as long as it has sufficient resources to meet their demands. However, multiserver jobs are generally stateful, meaning that job preemptions incur significant overhead from saving and reloading the state associated with running jobs. Hence, most systems try to avoid these costly job preemptions altogether. Given these constraints, a scheduling policy must determine what set of jobs to run in parallel at each moment in time to minimize the mean response time across a stream of arriving jobs. Unfortunately, simple non-preemptive policies such as FCFS may leave many servers idle, resulting in high mean response times or even system instability. Our goal is to design and analyze non-preemptive scheduling policies for multiserver jobs that maintain high system utilization to achieve low mean response time. One well-known non-preemptive policy, Most Servers First (MSF), prioritizes jobs with higher server requirements and achieves high resource utilization. However, MSF causes extreme variability in job waiting times, and can perform significantly worse than FCFS in practice. To address this, we propose and analyze a class of scheduling policies called MSF-Quick Swap (MSFQ) that performs well. MSFQ reduces the variability of job waiting times by periodically granting priority to other jobs in the system. We provide both stability results and an analysis of mean response time under MSFQ to prove that our policy dramatically outperforms MSF in the case where jobs request one or all the servers. In more complex cases, we evaluate MSFQ in simulation. We show that, with some additional optimization, variants of the MSFQ policy can greatly outperform MSF and FCFS on real-world multiserver job workloads.

cs.PF↗

arXiv2025

Multiserver-job Response Time under Multilevel Scaling

We study the multiserver-job setting in the load-focused multilevel scaling limit, where system load approaches capacity much faster than the growth of the number of servers $n$. We consider the ``1 and $n$'' system, where each job requires either one server or all $n$. Within the multilevel scaling limit, we examine three regimes: load dominated by $n$-server jobs, 1-server jobs, or balanced. In each regime, we characterize the asymptotic growth rate of the boundary of the stability region and the scaled mean queue length. We demonstrate that mean queue length peaks near balanced load via theory, numerics, and simulation.

cs.PF↗

arXiv2025

Improving Multiresource Job Scheduling with Markovian Service Rate Policies

Modern cloud computing workloads are composed of multiresource jobs that require a variety of computational resources in order to run, such as CPU cores, memory, disk space, or hardware accelerators. A single cloud server can typically run many multiresource jobs in parallel, but only if the server has sufficient resources to satisfy the demands of every job. A scheduling policy must therefore select sets of multiresource jobs to run in parallel in order to minimize the mean response time across jobs -- the average time from when a job arrives to the system until it is completed. Unfortunately, achieving low response times by selecting sets of jobs that fully utilize the available server resources has proven to be a difficult problem. In this paper, we develop and analyze a new class of policies for scheduling multiresource jobs, called Markovian Service Rate (MSR) policies. While prior scheduling policies for multiresource jobs are either highly complex to analyze or hard to implement, our MSR policies are simple to implement and are amenable to response time analysis. We show that the class of MSR policies is throughput-optimal in that we can use an MSR policy to stabilize the system whenever it is possible to do so. We also derive bounds on the mean response time under an MSR algorithm that are tight up to an additive constant. These bounds can be applied to systems with different preemption behaviors, such as fully preemptive systems, non-preemptive systems, and systems that allow preemption with setup times. We show how our theoretical results can be used to select a good MSR policy as a function of the system arrival rates, job service requirements, the server's resource capacities, and the resource demands of the jobs.

cs.PF↗

arXiv2025

Graph-Based Product Form

Product-form stationary distributions in Markov chains have been a foundational advance and driving force in our understanding of stochastic systems. In this paper, we introduce a new product-form relationship that we call "graph-based product form". As our first main contribution, we prove that two states of the Markov chain are in graph-based product form if and only if the following two equivalent conditions are satisfied: (i) a cut-based condition, reminiscent of classical results on product-form queueing systems, and (ii) a novel characterization that we call joint-ancestor freeness. The latter characterization allows us in particular to introduce a graph-traversal algorithm that checks product-form relationships for all pairs of states, with time complexity $O(|V|^2 |E|)$, if the Markov chain has a finite transition graph $G = (V, E)$. We then generalize graph-based product form to encompass more complex relationships, which we call "higher-level product form", and we again show these can be identified via a graph-traversal algorithm when the Markov chain has a finite state space. Lastly, we identify several examples from queueing theory that satisfy this product-form relationship.

math.PR↗

arXiv2024

Improving Multiresource Job Scheduling with Markovian Service Rate Policies

cs.PF↗

arXiv2024

Performance of NPG in Countable State-Space Average-Cost RL

We consider policy optimization methods in reinforcement learning settings where the state space is arbitrarily large, or even countably infinite. The motivation arises from control problems in communication networks, matching markets, and other queueing systems. Specifically, we consider the popular Natural Policy Gradient (NPG) algorithm, which has been studied in the past only under the assumption that the cost is bounded and the state space is finite, neither of which holds for the aforementioned control problems. Assuming a Lyapunov drift condition, which is naturally satisfied in some cases and can be satisfied in other cases at a small cost in performance, we design a state-dependent step-size rule which dramatically improves the performance of NPG for our intended applications. In addition to experimentally verifying the performance improvement, we also theoretically show that the iteration complexity of NPG can be made independent of the size of the state space. The key analytical tool we use is the connection between NPG step-sizes and the solution to Poisson's equation. In particular, we provide policy-independent bounds on the solution to Poisson's equation, which are then used to guide the choice of NPG step-sizes.

cs.LG↗

arXiv2024

Analysis of Markovian Arrivals and Service with Applications to Intermittent Overload

In many important real-world queueing settings, arrival and service rates fluctuate over time. We consider the MAMS system, where the arrival and service rates each vary according to an arbitrary finite-state Markov chain, allowing intermittent overload to be modeled. This model has been extensively studied, and we derive results matching those found in the literature via a somewhat novel framework. We derive a characterization of mean queue length in the MAMS system, with explicit bounds for all arrival and service chains at all loads, using our new framework. Our bounds are tight in heavy traffic. We prove even stronger bounds for the important special case of two-level arrivals with intermittent overload. Our framework is based around the concepts of relative arrivals and relative completions, which have previously been used in studying the MAMS system, under different names. These quantities allow us to tractably capture the transient correlational effect of the arrival and service processes on the mean queue length.

cs.PF↗

arXiv2024

Convergence of Natural Policy Gradient for a Family of Infinite-State Queueing MDPs

A wide variety of queueing systems can be naturally modeled as infinite-state Markov Decision Processes (MDPs). In the reinforcement learning (RL) context, a variety of algorithms have been developed to learn and optimize these MDPs. At the heart of many popular policy-gradient based learning algorithms, such as natural actor-critic, TRPO, and PPO, lies the Natural Policy Gradient (NPG) policy optimization algorithm. Convergence results for these RL algorithms rest on convergence results for the NPG algorithm. However, all existing results on the convergence of the NPG algorithm are limited to finite-state settings. We study a general class of queueing MDPs, and prove a $O(1/\sqrt{T})$ convergence rate for the NPG algorithm, if the NPG algorithm is initialized with the MaxWeight policy. This is the first convergence rate bound for the NPG algorithm for a general class of infinite-state average-reward MDPs. Moreover, our result applies to a beyond the queueing setting to any countably-infinite MDP satisfying certain mild structural assumptions, given a sufficiently good initial policy. Key to our result are state-dependent bounds on the relative value function achieved by the iterate policies of the NPG algorithm.

cs.LG↗

arXiv2023

Heavy-Traffic Optimal Size- and State-Aware Dispatching

Dispatching systems, where arriving jobs are immediately assigned to one of multiple queues, are ubiquitous in computer systems and service systems. A natural and practically relevant model is one in which each queue serves jobs in FCFS (First-Come First-Served) order. We consider the case where the dispatcher is size-aware, meaning it learns the size (i.e. service time) of each job as it arrives; and state-aware, meaning it always knows the amount of work (i.e. total remaining service time) at each queue. While size- and state-aware dispatching to FCFS queues has been extensively studied, little is known about optimal dispatching for the objective of minimizing mean delay. A major obstacle is that no nontrivial lower bound on mean delay is known, even in heavy traffic (i.e. the limit as load approaches capacity). This makes it difficult to prove that any given policy is optimal, or even heavy-traffic optimal. In this work, we propose the first size- and state-aware dispatching policy that provably minimizes mean delay in heavy traffic. Our policy, called CARD (Controlled Asymmetry Reduces Delay), keeps all but one of the queues short, then routes as few jobs as possible to the one long queue. We prove an upper bound on CARD's mean delay, and we prove the first nontrivial lower bound on the mean delay of any size- and state-aware dispatching policy. Both results apply to any number of servers. Our bounds match in heavy traffic, implying CARD's heavy-traffic optimality. In particular, CARD's heavy-traffic performance improves upon that of LWL (Least Work Left), SITA (Size Interval Task Assignment), and other policies from the literature whose heavy-traffic performance is known.

cs.PF↗

arXiv2023

The RESET and MARC Techniques, with Application to Multiserver-Job Analysis

Multiserver-job (MSJ) systems, where jobs need to run concurrently across many servers, are increasingly common in practice. The default service ordering in many settings is First-Come First-Served (FCFS) service. Virtually all theoretical work on MSJ FCFS models focuses on characterizing the stability region, with almost nothing known about mean response time. We derive the first explicit characterization of mean response time in the MSJ FCFS system. Our formula characterizes mean response time up to an additive constant, which becomes negligible as arrival rate approaches throughput, and allows for general phase-type job durations. We derive our result by utilizing two key techniques: REduction to Saturated for Expected Time (RESET) and MArkovian Relative Completions (MARC). Using our novel RESET technique, we reduce the problem of characterizing mean response time in the MSJ FCFS system to an M/M/1 with Markovian service rate (MMSR). The Markov chain controlling the service rate is based on the saturated system, a simpler closed system which is far more analytically tractable. Unfortunately, the MMSR has no explicit characterization of mean response time. We therefore use our novel MARC technique to give the first explicit characterization of mean response time in the MMSR, again up to constant additive error. We specifically introduce the concept of "relative completions," which is the cornerstone of our MARC technique.

cs.PF↗

arXiv2022

Optimal Scheduling in the Multiserver-job Model under Heavy Traffic

Multiserver-job systems, where jobs require concurrent service at many servers, occur widely in practice. Essentially all of the theoretical work on multiserver-job systems focuses on maximizing utilization, with almost nothing known about mean response time. In simpler settings, such as various known-size single-server-job settings, minimizing mean response time is merely a matter of prioritizing small jobs. However, for the multiserver-job system, prioritizing small jobs is not enough, because we must also ensure servers are not unnecessarily left idle. Thus, minimizing mean response time requires prioritizing small jobs while simultaneously maximizing throughput. Our question is how to achieve these joint objectives. We devise the ServerFilling-SRPT scheduling policy, which is the first policy to minimize mean response time in the multiserver-job model in the heavy traffic limit. In addition to proving this heavy-traffic result, we present empirical evidence that ServerFilling-SRPT outperforms all existing scheduling policies for all loads, with improvements by orders of magnitude at higher loads. Because ServerFilling-SRPT requires knowing job sizes, we also define the ServerFilling-Gittins policy, which is optimal when sizes are unknown or partially known.

cs.PF↗

arXiv2022

Towards the shortest DRAT proof of the Pigeonhole Principle

The Pigeonhole Principle (PHP) has been heavily studied in automated reasoning, both theoretically and in practice. Most solvers have exponential runtime and proof length, while some specialized techniques achieve polynomial runtime and proof length. Several decades ago, Cook manually constructed $O(n^4)$ extended resolution proofs, where $n$ denotes the number of pigeons.Existing automated techniques only surpass Cook's proofs in similar proof systems for large $n$. We construct the shortest known proofs of PHP in the standard proof format of modern SAT solving, DRAT. Using auxiliary variables and by recursively decomposing the original program into smaller sizes, we manually obtain proofs having length $O(n^3)$ and leading coefficient $5/2$.

cs.LO↗

arXiv2022

Incentive Compatible Queues Without Money

For job scheduling systems, where jobs require some amount of processing and then leave the system, it is natural for each user to provide an estimate of their job's time requirement in order to aid the scheduler. However, if there is no incentive mechanism for truthfulness, each user will be motivated to provide estimates that give their job precedence in the schedule, so that the job completes as early as possible. We examine how to make such scheduling systems incentive compatible, without using monetary charges, under a natural queueing theory framework. In our setup, each user has an estimate of their job's running time, but it is possible for this estimate to be incorrect. We examine scheduling policies where if a job exceeds its estimate, it is with some probability "punished" and re-scheduled after other jobs, to disincentivize underestimates of job times. However, because user estimates may be incorrect (without any malicious intent), excessive punishment may incentivize users to overestimate their job times, which leads to less efficient scheduling. We describe two natural scheduling policies, BlindTrust and MeasuredTrust. We show that, for both of these policies, given the parameters of the system, we can efficiently determine the set of punishment probabilities that are incentive compatible, in that users are incentivized to provide their actual estimate of the job time. Moreover, we prove for MeasuredTrust that in the limit as estimates converge to perfect accuracy, the range of punishment probabilities that are incentive compatible converges to $[0,1]$. Our formalism establishes a framework for studying further queue-based scheduling problems where job time estimates from users are utilized, and the system needs to incentivize truthful reporting of estimates.

cs.GT↗

arXiv2021

Uniform Bounds for Scheduling with Job Size Estimates

We consider the problem of scheduling to minimize mean response time in M/G/1 queues where only estimated job sizes (processing times) are known to the scheduler, where a job of true size $s$ has estimated size in the interval $[\beta s, \alpha s]$ for some $\alpha \geq \beta > 0$. We evaluate each scheduling policy by its approximation ratio, which we define to be the ratio between its mean response time and that of Shortest Remaining Processing Time (SRPT), the optimal policy when true sizes are known. Our question: is there a scheduling policy that (a) has approximation ratio near 1 when $\alpha$ and $\beta$ are near 1, (b) has approximation ratio bounded by some function of $\alpha$ and $\beta$ even when they are far from 1, and (c) can be implemented without knowledge of $\alpha$ and $\beta$? We first show that naively running SRPT using estimated sizes in place of true sizes is not such a policy: its approximation ratio can be arbitrarily large for any fixed $\beta < 1$. We then provide a simple variant of SRPT for estimated sizes that satisfies criteria (a), (b), and (c). In particular, we prove its approximation ratio approaches 1 uniformly as $\alpha$ and $\beta$ approach 1. This is the first result showing this type of convergence for M/G/1 scheduling. We also study the Preemptive Shortest Job First (PSJF) policy, a cousin of SRPT. We show that, unlike SRPT, naively running PSJF using estimated sizes in place of true sizes satisfies criteria (b) and (c), as well as a weaker version of (a).

cs.PF↗

arXiv2021

WCFS: A new framework for analyzing multiserver systems

Multiserver queueing systems are found at the core of a wide variety of practical systems. Many important multiserver models have a previously-unexplained similarity: identical mean response time behavior is empirically observed in the heavy traffic limit. We explain this similarity for the first time. We do so by introducing the work-conserving finite-skip (WCFS) framework, which encompasses a broad class of important models. This class includes the heterogeneous M/G/k, the limited processor sharing policy for the M/G/1, the threshold parallelism model, and the multiserver-job model under a novel scheduling algorithm. We prove that for all WCFS models, scaled mean response time $E[T](1-\rho)$ converges to the same value, $E[S^2]/(2E[S])$, in the heavy-traffic limit, which is also the heavy traffic limit for the M/G/1/FCFS. Moreover, we prove additively tight bounds on mean response time for the WCFS class, which hold for all load $\rho$. For each of the four models mentioned above, our bounds are the first known bounds on mean response time.

cs.PF↗

arXiv2021

Nudge: Stochastically Improving upon FCFS

The First-Come First-Served (FCFS) scheduling policy is the most popular scheduling algorithm used in practice. Furthermore, its usage is theoretically validated: for light-tailed job size distributions, FCFS has weakly optimal asymptotic tail of response time. But what if we don't just care about the asymptotic tail? What if we also care about the 99th percentile of response time, or the fraction of jobs that complete in under one second? Is FCFS still best? Outside of the asymptotic regime, only loose bounds on the tail of FCFS are known, and optimality is completely open. In this paper, we introduce a new policy, Nudge, which is the first policy to provably stochastically improve upon FCFS. We prove that Nudge simultaneously improves upon FCFS at every point along the tail, for light-tailed job size distributions. As a result, Nudge outperforms FCFS for every moment and every percentile of response time. Moreover, Nudge provides a multiplicative improvement over FCFS in the asymptotic tail. This resolves a long-standing open problem by showing that, counter to previous conjecture, FCFS is not strongly asymptotically optimal.

cs.PF↗

arXiv2020

Stability for Two-class Multiserver-job Systems

Multiserver-job systems, where jobs require concurrent service at many servers, occur widely in practice. Much is known in the dropping setting, where jobs are immediately discarded if they require more servers than are currently available. However, very little is known in the more practical setting where jobs queue instead. In this paper, we derive a closed-form analytical expression for the stability region of a two-class (non-dropping) multiserver-job system where each class of jobs requires a distinct number of servers and requires a distinct exponential distribution of service time, and jobs are served in first-come-first-served (FCFS) order. This is the first result of any kind for an FCFS multiserver-job system where the classes have distinct service distributions. Our work is based on a technique that leverages the idea of a "saturated" system, in which an unlimited number of jobs are always available. Our analytical formula provides insight into the behavior of FCFS multiserver-job systems, highlighting the huge wastage (idle servers while jobs are in the queue) that can occur, as well as the nonmonotonic effects of the service rates on wastage.

cs.PF↗

arXiv2020

Optimal Multiserver Scheduling with Unknown Job Sizes in Heavy Traffic

We consider scheduling to minimize mean response time of the M/G/k queue with unknown job sizes. In the single-server case, the optimal policy is the Gittins policy, but it is not known whether Gittins or any other policy is optimal in the multiserver case. Exactly analyzing the M/G/k under any scheduling policy is intractable, and Gittins is a particularly complicated policy that is hard to analyze even in the single-server case. In this work we introduce monotonic Gittins (M-Gittins), a new variation of the Gittins policy, and show that it minimizes mean response time in the heavy-traffic M/G/k for a wide class of finite-variance job size distributions. We also show that the monotonic shortest expected remaining processing time (M-SERPT) policy, which is simpler than M-Gittins, is a 2-approximation for mean response time in the heavy traffic M/G/k under similar conditions. These results constitute the most general optimality results to date for the M/G/k with unknown job sizes. Our techniques build upon work by Grosof et al., who study simple policies, such as SRPT, in the M/G/k; Bansal et al., Kamphorst and Zwart, and Lin et al., who analyze mean response time scaling of simple policies in the heavy-traffic M/G/1; and Aalto et al. and Scully et al., who characterize and analyze the Gittins policy in the M/G/1.

cs.PF↗