Характеристики производительности системы массового обслуживания с расщеплением запросов

Objectives. The problem of investigating a fork-join queuing system is considered. It is required to build the process of the system functioning, to find the condition for the existence of a stationary distribution, and propose algorithms for calculating the stationary distribution and the main stationary performance characteristics. The special interest of the study is to obtain the lower and upper bounds of the mean sojourn time of a customer in the system.Methods. Methods of probability theory, queuing theory and matrix theory are used.Results. The functioning of the system is described in terms of a multidimensional Markov chain. A constructive condition for the existence of a stationary distribution is found, and algorithms for calculating the stationary distribution and stationary performance characteristics of the system are proposed. Analytical expressions are obtained for the lower and upper bounds of the mean sojourn time of customers in the system.Conclusion. The functioning of the fork-join queuing system with a stationary Poisson flow has been studied. Each of the arriving customers forks into two tasks that go to two subsystems, each of which consists of a server and a buffer. We assume that the buffer to one of the servers is unlimited, and to the second server has a finite volume. Service times have, generally speaking, different phase distributions (PH-Phase type distributions). For this system, a condition for the existence of a stationary distribution is obtained, algorithms for calculating the stationary distribution and a number of stationary performance measures of the system are proposed. Analytical expressions for the lower and upper bounds of the mean sojourn time of a customer in the system from the moment it enters the system to the moment of synchronization, which is a critical performance indicator of the fork-join queues, are obtained. The results of the study can be used for modeling various computer and communication systems, in particular, systems that perform parallel computing, customer processing in distributed databases, and parallel disk access.Цели. Рассматривается задача построения и исследования математической модели стохастической системы с расщеплением и сборкой запросов. Требуется построить процесс функционирования системы, найти условие существования стационарного распределения, предложить алгоритмы его вычисления и основных стационарных характеристик производительности системы. Особый интерес вызывает задача получения нижней и верхней границ математического ожидания времени пребывания запроса в системе.Методы. Используются методы теории вероятностей, теории массового обслуживания и теории матриц.Результаты. Функционирование системы описано в терминах многомерной цепи Маркова. Найдено конструктивное условие существования стационарного распределения, предложены алгоритмы его вычисления и стационарных характеристик производительности системы. Получены аналитические выражения для нижней и верхней границ математического ожидания времени пребывания запросов в системе.Заключение. Исследован стационарный режим функционирования системы массового обслуживания с расщеплением и сборкой запросов, поступающих в систему в стационарном пуассоновском потоке. Каждый из поступающих запросов расщепляется на два задания, которые идут в две подсистемы, состоящие из обслуживающего прибора и буфера. Времена обслуживания заданий имеют разные фазовые распределения (PH-Phase type distributions). Для данной системы найдено условие существования стационарного распределения, предложены алгоритмы вычисления стационарного распределения и ряда стационарных характеристик производительности системы. Получены аналитические выражения для нижней и верхней границ математического ожидания времени пребывания запроса в системе от момента его поступления в систему до момента синхронизации, которое является критическим показателем производительности системы с расщеплением и сборкой запросов

Topics in Modeling and Analysis of Low-Latency Systems

Cloud-based architectures have become integral elements of modern networking infrastructure and are characterized by a large number of servers operating in parallel. Optimizing performance in these systems, with a particular focus on specific metrics such as system response time and the probability of loss, is critical to ensure user satisfaction. To address this challenge, this thesis analyzes load balancing policies that are designed to efficiently assign incoming user requests to the servers such that the system performance is optimized. In particular, the thesis focuses on a specialized category known as "randomized dynamic load balancing policies". These policies optimize system performance by dynamically adapting assignment decisions based on the current state of the system while interacting with a randomly selected subset of servers. Given the complex interdependencies among servers and the large size of these systems, an exact analysis of these systems is intractable. Consequently, the thesis studies these systems in the system size limit. It employs relevant limit theorems, including mean-field techniques and Stein's approach, as crucial mathematical tools. Furthermore, the thesis evaluates the accuracy of these limits when applied to systems of finite size, providing valuable insights into the practical applicability of the proposed load balancing policies. Motivated by different types of user requests or jobs, the thesis focuses on two main job categories: single-server jobs which can only run on a single server to represent non-parallelizable requests, and multiserver jobs, which can run on multiple servers simultaneously modeling parallelizable requests. The first part of the thesis studies single-server jobs in a system comprising a large number of processor sharing servers operating in parallel, where servers have different processing speeds and unlimited queueing buffers. The objective is to design randomized load balancing policies that minimize the average response time of jobs. A novel policy is introduced that allocates incoming jobs to servers based on predefined thresholds, state information from a randomly sampled subset of servers, and their processing speeds. The policy subsumes a broad class of other load balancing policies by adjusting the threshold levels, offering a unified framework for concurrent analysis of multiple load balancing policies. It is shown that under this policy, the system achieves the maximal stability region. Moreover, it is shown that as the system size approaches infinity, the transient and stationary stochastic occupancy measure of the system converges to a deterministic mean-field limit and the unique fixed point of this mean-field limit, respectively. As a result, the study of the asymptotic average response time of jobs becomes feasible through the fixed point of the mean-field limit. The analysis continues by studying error estimation related to asymptotic values in finite-sized systems. It is shown that when the mean delay of the finite-size system is approximated by its asymptotic value, the error is proportional to the inverse square root of the system size. Subsequently, the thesis analyzes adaptive multiserver jobs in loss systems, where they can be parallelized across a variable number of servers, up to a maximum degree of parallelization. In loss systems, each server can process only a finite number of jobs simultaneously and blocks any additional jobs beyond this capacity. Therefore, the goal is to devise randomized job assignment schemes that optimize the average response time of accepted jobs and the blocking probability while interacting with a sampled subset of servers. A load balancing policy is proposed, where the number of allocated servers for processing each job depends on the state information of a randomly sampled subset of servers and the maximum degree of parallelization. Employing Stein's method, it is shown that, provided that the sampling size grows at an appropriate rate, the difference between the steady-state system and a suitable deterministic system that exhibits optimality, decreases to zero as the system size increases. Thus, as the system size approaches infinity, the steady-state system achieves a zero blocking probability and optimal average response time for accepted jobs. Additionally, the thesis analyzes error estimation for these asymptotic values in finite-sized systems and establishes the error bounds as a function of the number of servers in the system

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum