arXiv subjects

Alan Scheller-Wolf

Publications and source records attributed to Alan Scheller-Wolf.

18 recordsLinked to original sources

arXiv2026

SPLIT: SymPathy for Large jobs Improves Tail latency

We study the asymptotic response time tail in the M/G/n multi-server queue with heavy-tailed (regularly varying) job sizes, a setting representative of modern computing workloads. For single-server systems, tail optimization is well understood: under heavy-tailed job sizes, policies such as SRPT that strictly prioritize short jobs are strongly tail optimal, and giving any priority to large jobs is harmful. For multi-server systems, the question has been almost entirely open. This paper gives the first strongly tail-optimal scheduling policies for the M/G/n queue with heavy-tailed job sizes. Our central finding is that the multi-server case is intrinsically different from the single-server case: giving a small amount of ``sympathy'' to large jobs is essential for strong tail optimality. We establish strong (or arbitrarily close to strong) tail optimality across the full stability region, both with and without knowledge of job sizes.

cs.PF↗

arXiv2026

Sequential vs. Simultaneous Entanglement Swapping under Optimal Link-Layer Control

Connection-less, packet-switched quantum network architectures distribute entanglement across multi-hop paths through sequential entanglement swapping, in which each node acts on purely local state information. The architectural advantages over the connection-oriented alternative -- simultaneous SWAP-ASAP -- are compelling, but sequential swapping holds partial chains in intermediate buffers between successive swaps, exposing them to memory decoherence in a way simultaneous SWAP-ASAP avoids by design. We present a proof-of-principle study at fixed chain length $n = 4$ in which each elementary link is governed by a fixed reinforcement-learning policy optimizing the secret-key rate of the six-state protocol, leaving the network-layer protocol as the sole independent variable. Sweeping the network-layer memory coherence time $T_c^{\mathrm{ext}}$ over four orders of magnitude reveals a clear regime structure governed by the dimensionless ratio $T_c^{\mathrm{ext}}/\tau$, where $\tau$ is the per-link entanglement heralding latency. Simultaneous SWAP-ASAP delivers a constant rate across the full sweep. Sequential swapping, by contrast, collapses to zero end-to-end deliveries below $T_c^{\mathrm{ext}}/\tau = 25$, and begins recovering at $T_c^{\mathrm{ext}}/\tau = 50$. It remains limited by the simultaneous rate, which it saturates only at the relaxed end of the sweep. These results suggest that the connection-less penalty is a near-term phenomenon tied to present-day memory coherence rather than a fundamental property of sequential swapping.

quant-ph↗

arXiv2026

LookAhead: The Optimal Non-decreasing Index Policy for a Time-Varying Holding Cost problem

In practice, the cost of delaying a job can grow as the job waits. Such behavior is modeled by the Time-Varying Holding Cost (TVHC) problem, where each job's instantaneous holding cost increases with its current age (a job's age is the time since it arrived). The goal of the TVHC problem is to find a scheduling policy that minimizes the time-average total holding cost across all jobs. However, no optimality results are known for the TVHC problem outside of the asymptotic regime. In this paper, we study a simple yet still challenging special case: A two-class M/M/1 queue in which class 1 jobs incur a non-decreasing, time-varying holding cost and class 2 jobs incur a constant holding cost. Our main contribution is deriving the first optimal (non-decreasing) index policy for this special case of the TVHC problem. Our optimal policy, called LookAhead, stems from the following idea: Rather than considering each job's current holding cost when making scheduling decisions, we should look at their cost some $X$ time into the future, where this $X$ is intuitively called the ``lookahead amount." This paper derives that optimal lookahead amount.

cs.PF↗

arXiv2025

Improving Upon the generalized c-mu rule: a Whittle approach

Scheduling a stream of jobs whose holding cost changes over time is a classic and practical problem. Specifically, each job is associated with a holding cost (penalty), where a job's instantaneous holding cost is some increasing function of its class and current age (the time it has spent in the system since its arrival). The goal is to schedule the jobs to minimize the time-average total holding cost across all jobs. The seminal paper on this problem, by Van Mieghem in 1995, introduced the generalized c-mu rule for scheduling jobs. Since then, this problem has attracted significant interest but remains challenging due to the absence of a finite-dimensional state space formulation. Consequently, subsequent works focus on more tractable versions of this problem. This paper returns to the original problem, deriving a heuristic that empirically improves upon the generalized c-mu rule and all existing heuristics. Our approach is to first translate the holding cost minimization problem to a novel Restless Multi-Armed Bandit (R-MAB) problem with a finite number of arms. Based on our R-MAB, we derive a novel Whittle Index policy, which is both elegant and intuitive.

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↗

arXiv2023

Optimal Entanglement Distillation Policies for Quantum Switches

In an entanglement distribution network, the function of a quantum switch is to generate elementary entanglement with its clients followed by entanglement swapping to distribute end-to-end entanglement of sufficiently high fidelity between clients. The threshold on entanglement fidelity is any quality-of-service requirement specified by the clients as dictated by the application they run on the network. We consider a discrete-time model for a quantum switch that attempts generation of fresh elementary entanglement with clients in each time step in the form of maximally entangled qubit pairs, or Bell pairs, which succeed probabilistically; the successfully generated Bell pairs are stored in noisy quantum memories until they can be swapped. We focus on establishing the value of entanglement distillation of the stored Bell pairs prior to entanglement swapping in presence of their inevitable aging, i.e., decoherence: For a simple instance of a switch with two clients, exponential decay of entanglement fidelity, and a well-known probabilistic but heralded two-to-one distillation protocol, given a threshold end-to-end entanglement fidelity, we use the Markov Decision Processes framework to identify the optimal action policy - to wait, to distill, or to swap that maximizes throughput. We compare the switch's performance under the optimal distillation-enabled policy with that excluding distillation. Simulations of the two policies demonstrate the improvements that are possible in principle via optimal use of distillation with respect to average throughput, average fidelity, and jitter of end-to-end entanglement, as functions of fidelity threshold. Our model thus helps capture the role of entanglement distillation in mitigating the effects of decoherence in a quantum switch in an entanglement distribution network, adding to the growing literature on quantum switches.

quant-ph↗

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

Capacity Management in a Pandemic with Endogenous Patient Choices and Flows

Motivated by the experiences of a healthcare service provider during the Covid-19 pandemic, we aim to study the decisions of a provider that operates both an Emergency Department (ED) and a medical Clinic. Patients contact the provider through a phone call or may present directly at the ED: patients can be COVID (suspected/confirmed) or non-COVID, and have different severities. Depending on the severity, patients who contact the provider may be directed to the ED (to be seen in a few hours), be offered an appointment at the Clinic (to be seen in a few days), or be treated via phone or telemedicine, avoiding a visit to a facility. All patients make joining decisions based on comparing their own risk perceptions versus their anticipated benefits: They then choose to enter a facility only if it is beneficial enough. Also, after initial contact, their severities may evolve, which may change their decision. The hospital system's objective is to allocate service capacity across facilities so as to minimize costs from patient deaths or defections. We model the system using a fluid approximation over multiple periods, possibly with different demand profiles. While the feasible space for this problem can be extremely complex, it is amenable to decomposition into different sub-regions that can be analyzed individually, the global optimal solution can be reached via provably parsimonious computational methods over a single period and over multiple periods with different demand rates. Our analytical and computational results indicate that endogeneity results in non-trivial and non-intuitive capacity allocations that do not always prioritize high severity patients, for both single and multi-period settings.

cs.GT↗

arXiv2021

Controlling arrival and service rates to reduce sensitivity of queueing systems with customer abandonment

The Erlang A model--an M/M/s queue with exponential abandonment--is often used to represent a service system with impatient customers. For this system, the popular square-root staffing rule determines the necessary staffing level to achieve the desirable QED (quality-and-efficiency-driven) service regime; however, the rule also implies that properties of large systems are highly sensitive to parameters. We reveal that the origin of this high sensitivity is due to the operation of large systems at a point of singularity in a phase diagram of service regimes. We can avoid this singularity by implementing a congestion-based control (CBC) scheme--a scheme that allows the system to change its arrival and service rates under congestion. We analyze a modified Erlang A model under the CBC scheme using a Markov chain decomposition method, derive non-asymptotic and asymptotic normal representations of performance indicators, and confirm that the CBC scheme makes large systems less sensitive than the original Erlang A model.

math.PR↗

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↗

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

A Marginal Analysis Framework to Incorporate the Externality Effect of Ordering Perishables

Finding the optimal policy for multi-period perishable inventory systems requires solving computationally-expensive stochastic dynamic programs (DP). To avoid the difficulty of solving DP models, we propose a framework that uses an externality term to capture the long-term impact of ordering decisions on the average cost over an infinite horizon. By approximating the externality term, we yield a tractable approximate optimality condition, which is solved through standard marginal analysis. The resulted policy is near-optimal in long-run average cost and ordering decisions.

math.OC↗

arXiv2020

Approximate Performance Measures for a Single Station Two-Stage Reneging Queue

We study a single station two-stage reneging queue with Poisson arrivals, exponential services, and two levels of exponential reneging behaviors, extending the popular Erlang A model that assumes a constant reneging rate. We derive approximate analytical formulas representing performance measures for our two-stage queue following the Markov chain decomposition approach. Our formulas not only give accurate results spanning the heavy-traffic to the light-traffic regimes, but also provide insight into capacity decisions.

math.PR↗

arXiv2019

Simple Near-Optimal Scheduling for the M/G/1

We consider the problem of preemptively scheduling jobs to minimize mean response time of an M/G/1 queue. When we know each job's size, the shortest remaining processing time (SRPT) policy is optimal. Unfortunately, in many settings we do not have access to each job's size. Instead, we know only the job size distribution. In this setting the Gittins policy is known to minimize mean response time, but its complex priority structure can be computationally intractable. A much simpler alternative to Gittins is the shortest expected remaining processing time (SERPT) policy. While SERPT is a natural extension of SRPT to unknown job sizes, it is unknown whether or not SERPT is close to optimal for mean response time. We present a new variant of SERPT called monotonic SERPT (M-SERPT) which is as simple as SERPT but has provably near-optimal mean response time at all loads for any job size distribution. Specifically, we prove the mean response time ratio between M-SERPT and Gittins is at most 3 for load $\rho \leq 8/9$ and at most 5 for any load. This makes M-SERPT the only non-Gittins scheduling policy known to have a constant-factor approximation ratio for mean response time.

cs.PF↗

arXiv2019

Markov Chain Decomposition Based On Total Expectation Theorem

A divide-and-conquer approach to analyzing Markov chains (MCs) is not utilized as widely as it could be, despite its potential benefits. One primary reason for this is the fact that most MC decomposition approaches involve a complex and inflexible methodology: decomposed subchains must be disjoint, transition rates of these decomposed subchains must be altered in a way tailored to the particular MC model, and the procedure to aggregate suchains needs to incorporate a nonlinear normalization constraint, complicating the analytical expression of performance measures. In contrast, we propose a versatile yet simple decomposition method for continuous time MCs based on the total expectation theorem. Leveraging the properties of this theorem, our method has great flexibility in the choice of subchains, and the procedure to obtain expected values of interest is simply a linear summation of subchains' properties, which is not affected by the normalization constraint. We prove that to maintain the correct distribution of decomposed subchains one may use our novel termination scheme, a modification of transition rates, that ensures partial flow conservation at boundary states. This termination scheme is applicable to MCs with any structure, since the scheme depends only on the boundary-state distribution, not on the structure of the MCs. To demonstrate the generality and capability of our method, we analytically solve various models, such as a congestion-based staffing queue and a Markov-modulated Mt/Mt/1 queue. As not all systems admit an analytical solution, we complement this analysis with numerical studies of MCs with various sizes using the algorithm based on our method.

math.PR↗

arXiv2018

Optimal Scheduling and Exact Response Time Analysis for Multistage Jobs

Scheduling to minimize mean response time in an M/G/1 queue is a classic problem. The problem is usually addressed in one of two scenarios. In the perfect-information scenario, the scheduler knows each job's exact size, or service requirement. In the zero-information scenario, the scheduler knows only each job's size distribution. The well-known shortest remaining processing time (SRPT) policy is optimal in the perfect-information scenario, and the more complex Gittins policy is optimal in the zero-information scenario. In real systems the scheduler often has partial but incomplete information about each job's size. We introduce a new job model, that of multistage jobs, to capture this partial-information scenario. A multistage job consists of a sequence of stages, where both the sequence of stages and stage sizes are unknown, but the scheduler always knows which stage of a job is in progress. We give an optimal algorithm for scheduling multistage jobs in an M/G/1 queue and an exact response time analysis of our algorithm.

cs.PF↗

arXiv2017

SOAP: One Clean Analysis of All Age-Based Scheduling Policies

We consider an extremely broad class of M/G/1 scheduling policies called SOAP: Schedule Ordered by Age-based Priority. The SOAP policies include almost all scheduling policies in the literature as well as an infinite number of variants which have never been analyzed, or maybe not even conceived. SOAP policies range from classic policies, like first-come, first-serve (FCFS), foreground-background (FB), class-based priority, and shortest remaining processing time (SRPT); to much more complicated scheduling rules, such as the famously complex Gittins index policy and other policies in which a job's priority changes arbitrarily with its age. While the response time of policies in the former category is well understood, policies in the latter category have resisted response time analysis. We present a universal analysis of all SOAP policies, deriving the mean and Laplace-Stieltjes transform of response time.

cs.PF↗

arXiv2017

Delay Asymptotics and Bounds for Multi-Task Parallel Jobs

We study delay of jobs that consist of multiple parallel tasks, which is a critical performance metric in a wide range of applications such as data file retrieval in coded storage systems and parallel computing. In this problem, each job is completed only when all of its tasks are completed, so the delay of a job is the maximum of the delays of its tasks. Despite the wide attention this problem has received, tight analysis is still largely unknown since analyzing job delay requires characterizing the complicated correlation among task delays, which is hard to do. We first consider an asymptotic regime where the number of servers, $n$, goes to infinity, and the number of tasks in a job, $k^{(n)}$, is allowed to increase with $n$. We establish the asymptotic independence of any $k^{(n)}$ queues under the condition $k^{(n)} = o(n^{1/4})$. This greatly generalizes the asymptotic-independence type of results in the literature where asymptotic independence is shown only for a fixed constant number of queues. As a consequence of our independence result, the job delay converges to the maximum of independent task delays. We next consider the non-asymptotic regime. Here we prove that independence yields a stochastic upper bound on job delay for any $n$ and any $k^{(n)}$ with $k^{(n)}\le n$. The key component of our proof is a new technique we develop, called "Poisson oversampling". Our approach converts the job delay problem into a corresponding balls-and-bins problem. However, in contrast with typical balls-and-bins problems where there is a negative correlation among bins, we prove that our variant exhibits positive correlation.

cs.PF↗