arXiv subjects

Jonatha Anselmi

Publications and source records attributed to Jonatha Anselmi.

11 recordsLinked to original sources

arXiv2025

Non-Stationary Gradient Descent for Optimal Auto-Scaling in Serverless Platforms

To efficiently manage serverless computing platforms, a key aspect is the auto-scaling of services, i.e., the set of computational resources allocated to a service adapts over time as a function of the traffic demand. The objective is to find a compromise between user-perceived performance and energy consumption. In this paper, we consider the \emph{scale-per-request} auto-scaling pattern and investigate how many function instances (or servers) should be spawned each time an \emph{unfortunate} job arrives, i.e., a job that finds all servers busy upon its arrival. We address this problem by following a stochastic optimization approach: we develop a stochastic gradient descent scheme of the Kiefer--Wolfowitz type that applies \emph{over a single run of the state evolution}. At each iteration, the proposed scheme computes an estimate of the number of servers to spawn each time an unfortunate job arrives to minimize some cost function. Under natural assumptions, we show that the sequence of estimates produced by our scheme is asymptotically optimal almost surely. In addition, we prove that its convergence rate is $O(n^{-2/3})$ where $n$ is the number of iterations. From a mathematical point of view, the stochastic optimization framework induced by auto-scaling exhibits non-standard aspects that we approach from a general point of view. We consider the setting where a controller can only get samples of the \emph{transient} -- rather than stationary -- behavior of the underlying stochastic system. To handle this difficulty, we develop arguments that exploit properties of the mixing time of the underlying Markov chain. By means of numerical simulations, we validate the proposed approach and quantify its gain with respect to common existing scale-up rules.

math.OC↗

arXiv2024

Balanced Splitting: A Framework for Achieving Zero-wait in the Multiserver-job Model

We present a new framework for designing nonpreemptive and job-size oblivious scheduling policies in the multiserver-job queueing model. The main requirement is to identify a static and balanced sub-partition of the server set and ensure that the servers in each set of that sub-partition can only handle jobs of a given class and in a first-come first-served order. A job class is determined by the number of servers to which it has exclusive access during its entire execution and the probability distribution of its service time. This approach aims to reduce delays by preventing small jobs from being blocked by larger ones that arrived first, and it is particularly beneficial when the job size variability intra resp. inter classes is small resp. large. In this setting, we propose a new scheduling policy, Balanced-Splitting. We provide a sufficient condition for the stability of Balanced-Splitting and show that the resulting queueing probability, i.e., the probability that an arriving job needs to wait for processing upon arrival, vanishes in both the subcritical (the load is kept fixed to a constant less than one) and critical (the load approaches one from below) many-server limiting regimes. Crucial to our analysis is a connection with the M/GI/s/s queue and Erlang's loss formula, which allows our analysis to rely on fundamental results from queueing theory. Numerical simulations show that the proposed policy performs better than several preemptive/nonpreemptive size-aware/oblivious policies in various practical scenarios. This is also confirmed by simulations running on real traces from High Performance Computing (HPC) workloads. The delays induced by Balanced-Splitting are also competitive with those induced by state-of-the-art policies such as First-Fit-SRPT and ServerFilling-SRPT, though our approach has the advantage of not requiring preemption, nor the knowledge of job sizes.

cs.PF↗

arXiv2024

Load Balancing with Job-Size Testing: Performance Improvement or Degradation?

In the context of decision making under explorable uncertainty, scheduling with testing is a powerful technique used in the management of computer systems to improve performance via better job-dispatching decisions. Upon job arrival, a scheduler may run some \emph{testing algorithm} against the job to extract some information about its structure, e.g., its size, and properly classify it. The acquisition of such knowledge comes with a cost because the testing algorithm delays the dispatching decisions, though this is under control. In this paper, we analyze the impact of such extra cost in a load balancing setting by investigating the following questions: does it really pay off to test jobs? If so, under which conditions? Under mild assumptions connecting the information extracted by the testing algorithm in relationship with its running time, we show that whether scheduling with testing brings a performance degradation or improvement strongly depends on the traffic conditions, system size and the coefficient of variation of job sizes. Thus, the general answer to the above questions is non-trivial and some care should be considered when deploying a testing policy. Our results are achieved by proposing a load balancing model for scheduling with testing that we analyze in two limiting regimes. When the number of servers grows to infinity in proportion to the network demand, we show that job-size testing actually degrades performance unless short jobs can be predicted reliably almost instantaneously and the network load is sufficiently high. When the coefficient of variation of job sizes grows to infinity, we construct testing policies inducing an arbitrarily large performance gain with respect to running jobs untested.

cs.PF↗

arXiv2024

Asynchronous Load Balancing and Auto-scaling: Mean-Field Limit and Optimal Design

We develop a Markovian framework for load balancing that combines classical algorithms such as Power-of-$d$ with auto-scaling mechanisms that allow the net service capacity to scale up or down in response to the current load on the same timescale as job dynamics. Our framework is inspired by serverless platforms, such as Knative, where servers are software functions that can be flexibly instantiated in milliseconds according to scaling rules defined by the users of the serverless platform. The main question is how to design such scaling rules to minimize user-perceived delay performance while ensuring low energy consumption. For the first time, we investigate this problem when the auto-scaling and load balancing processes operate asynchronously (or proactively), as in Knative. In contrast to the synchronous (or reactive) paradigm, asynchronism brings the advantage that jobs do not necessarily need to wait any time a scale-up decision is taken. In our main result, we find a general condition on the structure of scaling rules able to drive mean-field dynamics to delay and relative energy optimality, i.e., a situation where both the user-perceived delay and the relative energy waste induced by idle servers vanish in the limit where the network demand grows to infinity in proportion to the nominal service capacity. The identified condition suggests to scale up the current net capacity if and only if the mean demand exceeds the rate at which servers become idle and active. Finally, we propose a family of scaling rules that satisfy our optimality condition. Numerical simulations demonstrate that these rules provide better delay performance than existing synchronous auto-scaling schemes while inducing almost the same power consumption.

math.OC↗

arXiv2023

Learning Optimal Admission Control in Partially Observable Queueing Networks

We present an efficient reinforcement learning algorithm that learns the optimal admission control policy in a partially observable queueing network. Specifically, only the arrival and departure times from the network are observable, and optimality refers to the average holding/rejection cost in infinite horizon. While reinforcement learning in Partially Observable Markov Decision Processes (POMDP) is prohibitively expensive in general, we show that our algorithm has a regret that only depends sub-linearly on the maximal number of jobs in the network, $S$. In particular, in contrast with existing regret analyses, our regret bound does not depend on the diameter of the underlying Markov Decision Process (MDP), which in most queueing systems is at least exponential in $S$. The novelty of our approach is to leverage Norton's equivalent theorem for closed product-form queueing networks and an efficient reinforcement learning algorithm for MDPs with the structure of birth-and-death processes.

cs.LG↗

arXiv2023

Reinforcement Learning in a Birth and Death Process: Breaking the Dependence on the State Space

In this paper, we revisit the regret of undiscounted reinforcement learning in MDPs with a birth and death structure. Specifically, we consider a controlled queue with impatient jobs and the main objective is to optimize a trade-off between energy consumption and user-perceived performance. Within this setting, the \emph{diameter} $D$ of the MDP is $Ω(S^S)$, where $S$ is the number of states. Therefore, the existing lower and upper bounds on the regret at time$T$, of order $O(\sqrt{DSAT})$ for MDPs with $S$ states and $A$ actions, may suggest that reinforcement learning is inefficient here. In our main result however, we exploit the structure of our MDPs to show that the regret of a slightly-tweaked version of the classical learning algorithm {\sc Ucrl2} is in fact upper bounded by $\tilde{\mathcal{O}}(\sqrt{E_2AT})$ where $E_2$ is related to the weighted second moment of the stationary measure of a reference policy. Importantly, $E_2$ is bounded independently of $S$. Thus, our bound is asymptotically independent of the number of states and of the diameter. This result is based on a careful study of the number of visits performed by the learning algorithm to the states of the MDP, which is highly non-uniform.

cs.LG↗

arXiv2021

Stability and Optimization of Speculative Queueing Networks

We provide a queueing-theoretic framework for job replication schemes based on the principle "\emph{replicate a job as soon as the system detects it as a \emph{straggler}}". This is called job \emph{speculation}. Recent works have analyzed {replication} on arrival, which we refer to as \emph{replication}. Replication is motivated by its implementation in Google's BigTable. However, systems such as Apache Spark and Hadoop MapReduce implement speculative job execution. The performance and optimization of speculative job execution is not well understood. To this end, we propose a queueing network model for load balancing where each server can speculate on the execution time of a job. Specifically, each job is initially assigned to a single server by a frontend dispatcher. Then, when its execution begins, the server sets a timeout. If the job completes before the timeout, it leaves the network, otherwise the job is terminated and relaunched or resumed at another server where it will complete. We provide a necessary and sufficient condition for the stability of speculative queueing networks with heterogeneous servers, general job sizes and scheduling disciplines. We find that speculation can increase the stability region of the network when compared with standard load balancing models and replication schemes. We provide general conditions under which timeouts increase the size of the stability region and derive a formula for the optimal speculation time, i.e., the timeout that minimizes the load induced through speculation. We compare speculation with redundant-$d$ and redundant-to-idle-queue-$d$ rules under an $S\& X$ model. For light loaded systems, redundancy schemes provide better response times. However, for moderate to heavy loadings, redundancy schemes can lose capacity and have markedly worse response times when compared with a speculative scheme.

cs.PF↗

arXiv2019

Power-of-$d$-Choices with Memory: Fluid Limit and Optimality

In multi-server distributed queueing systems, the access of stochastically arriving jobs to resources is often regulated by a dispatcher, also known as load balancer. A fundamental problem consists in designing a load balancing algorithm that minimizes the delays experienced by jobs. During the last two decades, the power-of-$d$-choice algorithm, based on the idea of dispatching each job to the least loaded server out of $d$ servers randomly sampled at the arrival of the job itself, has emerged as a breakthrough in the foundations of this area due to its versatility and appealing asymptotic properties. In this paper, we consider the power-of-$d$-choice algorithm with the addition of a local memory that keeps track of the latest observations collected over time on the sampled servers. Then, each job is sent to a server with the lowest observation. We show that this algorithm is asymptotically optimal in the sense that the load balancer can always assign each job to an idle server in the large-system limit. This holds true if and only if the system load $λ$ is less than $1-\frac{1}{d}$. If this condition is not satisfied, we show that queue lengths are tightly bounded by $\left\lceil - \frac{ \log (1-λ)}{\log (λd +1)} \right\rceil$. This is in contrast with the classic version of the power-of-$d$-choice algorithm, where at the fluid scale a strictly positive proportion of servers containing $i$ jobs exists for all $i\ge 0$, in equilibrium. Our results quantify and highlight the importance of using memory as a means to enhance performance in randomized load balancing.

math.PR↗

arXiv2015

Control of parallel non-observable queues: asymptotic equivalence and optimality of periodic policies

We consider a queueing system composed of a dispatcher that routes deterministically jobs to a set of non-observable queues working in parallel. In this setting, the fundamental problem is which policy should the dispatcher implement to minimize the stationary mean waiting time of the incoming jobs. We present a structural property that holds in the classic scaling of the system where the network demand (arrival rate of jobs) grows proportionally with the number of queues. Assuming that each queue of type $r$ is replicated $k$ times, we consider a set of policies that are periodic with period $k \sum_r p_r$ and such that exactly $p_r$ jobs are sent in a period to each queue of type $r$. When $k\to\infty$, our main result shows that all the policies in this set are equivalent, in the sense that they yield the same mean stationary waiting time, and optimal, in the sense that no other policy having the same aggregate arrival rate to \emph{all} queues of a given type can do better in minimizing the stationary mean waiting time. This property holds in a strong probabilistic sense. Furthermore, the limiting mean waiting time achieved by our policies is a convex function of the arrival rate in each queue, which facilitates the development of a further optimization aimed at solving the fundamental problem above for large systems.

cs.PF↗

arXiv2014

Closed queueing networks under congestion: non-bottleneck independence and bottleneck convergence

We analyze the behavior of closed product-form queueing networks when the number of customers grows to infinity and remains proportionate on each route (or class). First, we focus on the stationary behavior and prove the conjecture that the stationary distribution at non-bottleneck queues converges weakly to the stationary distribution of an ergodic, open product-form queueing network. This open network is obtained by replacing bottleneck queues with per-route Poissonian sources whose rates are determined by the solution of a strictly concave optimization problem. Then, we focus on the transient behavior of the network and use fluid limits to prove that the amount of fluid, or customers, on each route eventually concentrates on the bottleneck queues only, and that the long-term proportions of fluid in each route and in each queue solve the dual of the concave optimization problem that determines the throughputs of the previous open network.

math.PR↗

arXiv2013

Ergodic transition in a simple model of the continuous double auction

We study a phenomenological model for the continuous double auction, equivalent to two independent $M/M/1$ queues. The continuous double auction defines a continuous-time random walk for trade prices. The conditions for ergodicity of the auction are derived and, as a consequence, three possible regimes in the behavior of prices and logarithmic returns are observed. In the ergodic regime, prices are unstable and one can observe an intermittent behavior in the logarithmic returns. On the contrary, non-ergodicity triggers stability of prices, even if two different regimes can be seen.

q-fin.TR↗