Jackson Networks Research Articles

Minimal specialization: Coevolution of network structure and dynamics

The changing topology of a network is driven by the need to maintain or optimize network function. As this function is often related to moving quantities such as traffic, information, etc., efficiently through the network, the structure of the network and the dynamics on the network directly depend on the other. To model this interplay of network structure and dynamics we use the dynamics on the network, or the dynamical processes the network models, to influence the dynamics of the network structure, i.e., to determine where and when to modify the network structure. We model the dynamics on the network using Jackson network dynamics and the dynamics of the network structure using minimal specialization, a variant of the more general network growth model known as specialization. The resulting model, which we refer to as the integrated specialization model, coevolves both the structure and the dynamics of the network. We show that by reducing dynamic bottlenecks, i.e. reducing the network’s maximal asymptotic load, this model simultaneously produces networks with real-world like properties. This includes right-skewed degree distributions, sparsity, the small-world property, and non-trivial equitable partitions. Additionally, when compared to other growth models, the integrated specialization model creates networks with small diameter, minimizing distances across the network. The result are networks that have both dynamic and structural features that allow quantities to more efficiently move through the network.

The BAR Approach for Multiclass Queueing Networks with SBP Service Policies

The basic adjoint relationship (BAR) approach is an analysis technique based on the stationary equation of a Markov process. This approach was introduced to study heavy-traffic, steady-state convergence of generalized Jackson networks in which each service station has a single job class. We extend it to multiclass queueing networks operating under static-buffer-priority (SBP) service disciplines. Our extension makes a connection with Palm distributions that allows one to attack a difficulty arising from queue-length truncation, which appears to be unavoidable in the multiclass setting. For multiclass queueing networks operating under SBP service disciplines, our BAR approach provides an alternative to the “interchange of limits” approach that has dominated the literature in the last twenty years. The BAR approach can produce sharp results and allows one to establish steady-state convergence under three additional conditions: stability, state space collapse (SSC) and a certain matrix being “tight.” These three conditions do not appear to depend on the interarrival and service-time distributions beyond their means, and their verification can be studied as three separate modules. In particular, they can be studied in a simpler, continuous-time Markov chain setting when all distributions are exponential. As an example, these three conditions are shown to hold in reentrant lines operating under last-buffer-first-serve discipline. In a two-station, five-class reentrant line, under the heavy-traffic condition, the tight-matrix condition implies both the stability condition and the SSC condition. Whether such a relationship holds generally is an open problem.

Managing port disruption through sailing speed optimization for sustainable maritime transportation

Ports, as nodes in maritime transportation, frequently face disruptions leading to congestion, adversely affecting the efficiency and sustainability of the global supply chain. This study considers the speeding-up strategy to alleviate port congestion. We model the transportation network as a closed Jackson network and adopt a tailored mean-value analysis algorithm to solve the model. To deliver the increased sailing speed, we further put forward a bi-section search algorithm. Our numerical analysis results demonstrate the feasibility of increasing speed to alleviate port congestion. However, when port congestion exceeds a certain threshold, the system faces collapse, rendering the increase in speed ineffective. Additionally, we recommend shipping companies consider using clean energy when employing the speeding-up strategy to mitigate congestion, thus safeguarding the environment. Our study combines theoretical methodologies and analytical models, providing insights regarding speeding up the vessels. The findings of this study offer guidelines for real-world sustainable maritime practice.

Performance Modeling and Optimization for a Fog-Based IoT Platform

A fog-based IoT platform model involving three layers, i.e., IoT devices, fog nodes, and the cloud, was proposed using an open Jackson network with feedback. The system performance was analyzed for individual subsystems, and the overall system was based on different input parameters. Interesting performance metrics were derived from analytical results. A resource optimization problem was developed and solved to determine the optimal service rates at individual fog nodes under some constraint conditions. Numerical evaluations for the performance and the optimization problem are provided for further understanding of the analysis. The modeling and analysis, as well as the optimization design method, are expected to provide a useful reference for the design and evaluation of fog computing systems.

Stochastic optimization of level-dependent perishable inventory system by Jackson network

In this research work, we developed a model to investigate the long-run behavior of the solutions of continuous review ( s , S ) inventory policy for level-dependent perishable products with positive service time on Jackson queueing network. The service facility is assumed that the waiting capacities of each queue are different. In this model, we assume two independent single server nodes in a M / M / 1 queueing system with an associated inventory, where each node represents a queue. In a Poisson process, customers enter the system. The service times ( μ i > 0 ) and lead times ( θ i > 0 ) are independent and also spread exponentially at the node i for 1 ⩽ i ⩽ 2 . If the inventory levels of goods from the warehouse i are below s i at any moment, an order for Q i = ( ( S i - s i ) > 0 ; ∀ i ) units is made, where S i is the highest capacity of storage for each warehouse in the system. In the steady state case, the matrix analytic technique is used to derive the combined probability distribution for the number of customers and inventory level. The overall anticipated cost rate is determined using some essential system performance measures. Numerical and graphical examples are presented to analyze the effect of variation of parameters on the system.

Accuracy of Estimation of the Vector of Queue Lengths for Open Jackson Networks

The accuracy of approximation of the vector of queue lengths for an open Jackson network with regenerative input flow and unreliable servers is estimated. A theorem on the accuracy of approximation of the vector of queue lengths in open Jackson networks is put forward, i.e., an estimate for the probability of deviations of the norm of the difference between the process of queue lengths and the constructed reflected Brownian motion is obtained. As a corollary, an estimate of the Wasserstein distance is given.

Модели и методы вычисления задержек при предоставлении услуг пользователем на сервисных платформах сеансовых инфокоммуникационных услуг

The purpose of this article is to develop methods for calculating the qualitative characteristics of infocommunication services whose workloads consist of user sessions. The scope of application of these methods affects the development and maintenance of infocommunication multi-user services, in terms of ensuring a high quality of user service. The quality of service (QoS) parameters under consideration determine the characteristics of both the system as a whole and individual elements of its architecture, and for this type of service, the qualitative characteristics are the waiting time for the start of user session maintenance, the load factor of server equipment, the number of user sessions simultaneously supported by the system during its error-free operation. Multi-user services are considered from the point of view of queuing theory as a network of queuing systems, with the duration of the user session and the time of receipt of requests for services by gaming services distributed exponentially. To evaluate the parameters of the models of infocommunication multi-user services, Jackson networks were chosen as a mathematical apparatus, which allow us to consider the necessary systems of the type M/M/v. As an approbation of the results, a mathematical analysis of multi-user gaming services was proposed, since their popularity, and hence the flow of users to game servers, is increasing at a tremendous rate. As an additional tool for assessing the qualitative characteristics of the session models under consideration, simulation modeling was proposed, which allows us to evaluate the accuracy of the proposed analytical method and analyze the characteristics of complex systems with time distribution laws other than exponential, for example, long-tailed distributions (Weibull, Pareto, etc.)

Modeling and Analysis of Stochastic Perishable Inventory System with Impatient Customers at Service Facility

In this paper, we consider a two‐commodity stochastic (s, S) perishable inventory system of the multiqueue Jackson network at a service facility with reneging and jockeying of the customers. The waiting room capacity of the first queue is M/M/1/∞ and the second queue is M/M/1/C2, where C2 < ∞. In this network, different queues have different service rates. Service times are exponentially distributed. Customers join the system at a rate of λi; (i = 1,2) using a Poisson process. We assumed that some customers maybe impatient due to the long waiting time in the queue and can leave the system without service or may switch from the longest queue to the shortest queue to reduce the waiting time. If the inventory levels of the items from the warehouse i at any time reaches below or reorder level si, an order Qi = ((Si − si) > 0; i = 1,2) units is placed to bring the inventory level up to Si. The joint probability distribution of the number of customers in the system and also the stock level of the warehouse is determined in the equilibrium case. Several important measures of steady‐state system performance are derived. Several examples are presented to demonstrate the reneging and jockeying behavior of the customers for this system. A sensitivity analysis is performed to investigate the effect of changing the parameters on the total expected cost of the system.

Fluid limits for earliest-deadline-first networks

This paper analyzes fluid scale asymptotics of two models of generalized Jackson networks employing the earliest deadline first (EDF) policy. One applies the ‘soft’ EDF policy, where deadlines are used to determine priority but jobs do not renege, and the other implements ‘hard’ EDF, where jobs renege when deadlines expire, and deadlines are postponed with each migration to a new station. The arrival rates, deadline distribution and service capacity are allowed to fluctuate over time at the fluid scale. Earlier work on EDF network fluid limits, used as a tool to obtain stability of these networks, addressed only the soft version of the policy, and moreover did not contain a full fluid limit result. In this paper, tools that extend the notion of the measure-valued Skorokhod map are developed and used to establish for the first time fluid limits for both the soft and hard EDF network models.

Method of QoS evaluation of FPGA as a service

The subject of study in this article is the evaluation of the performance issues of cloud services implemented using FPGA technology. The goal is to improve the performance of cloud services built on top of multiple FPGA platforms known as FPGA-as-a-Service (FaaS). Task: to analyze the delays in communications between host computer and FPGA; propose the steps of development to reduce the delay and perform the evaluation of the response time for the FPGA-based accelerator depending on number of involved cards; consider the reliability aspect of such systems implemented using programmable logic. According to the tasks, the following results were obtained. The FPGA-as-a-Service where FPGA resources are provided through a set of hardware/software toolset is considered. The usage of queueing theory for cloud-based services is analyzed. The contribution of the parts of FPGA-as-a-Service to the final delay of the service is discussed. The process of modeling of work the services based on FPGA accelerator cards with use of Jackson's network is analyzed in detail. The model of the delays of FaaS that considers the parameters of accelerator FPGA cards is offered. The formula of the total response time of the service combined based on the response of the components of is obtained. The proposed steps of reduce data processing delays include increase the size of data blocks for processing in FPGA by each kernel, change the communication model with kernel from sequential to pipelined, following timing closure technique and use more FPGA accelerator cards in parallel to divide the enquiring delay. Based on the proposed model the evaluation of response time of FaaS was done. The advantage of the use of many FPGAs in parallel for same data processing task instead of implementation of requests thread for each accelerator card is shown. Conclusions. The main contribution of this study is a step forward to the modeling of FPGA-based services that can be used for FPGA-based artificial intelligence (AI) applications. It helps to improve the performance of the system by means of reducing the delays at different stages of requests processing. Another side of this result is the reliability aspect that is based on modified manner of service operation in case of use the proposed steps of system optimization. It helps to improve the processing of requests to FaaS. The proposed method is the next step after prototyping of such systems because it helps to turn the FaaS from the object for development to the tool for deployment of new technologies like AI applications.

Maximizing hospital capacity to serve pandemic patient surge in hot spots via queueing theory and microsimulation

This paper presents a mixed-integer mathematical program with embedded equations developed from concepts of queueing theory and Jackson networks for estimating the steady-state maximum potential hospital capacity for COVID-19 patient care in extreme surge conditions, where a hospital must turn nearly all of its existing resources toward the care of pandemic patients. Estimating the potential maximum hospital capacity for pandemic patient care can aid in assessing regional healthcare capacity during surges in pandemic patient demand, predicting shortfalls, and designing preparedness and response actions. To obtain such estimates and inform action, the program determines a best assignment of a heterogeneous staff of nurses and doctors to key units appropriate for their skills to create the optimal allocation of staffed beds. An alternative trial-and-error approach is offered that decision-makers without optimization or software expertise can use to obtain similar estimates. Under a given assignment of resources, a variety of key performance indicators can be obtained through the direct use of the queueing equations. Results of comparisons to outcomes from a detailed discrete event simulation model of an identical hospital design show the accuracy of the equations to be high despite the added simplifications needed for the use of a closed-form equation-based methodology.

A queueing Network approach for capacity planning and patient Scheduling: A case study for the COVID-19 vaccination process in Colombia

This paper considers the problem of patient scheduling and capacity planning for the vaccination process during the COVID-19 pandemic. The proposed solution is based on a non-linear mathematical modeling approach representing the dynamics of an open Jackson Network and a Generalized Network. To test these models, we proposed three objective functions and analyzed different configurations of the process corresponding to various levels of the models’ parameters as well as the conditions present in the case study. To assess the computational performance of the models, we also experimented with larger instances in terms of number of steps or stations used and number of patients scheduled. The computational results show how parameters such as the minimum percentage of patients served, the maximum occupation allowed per station and the objective functions used have an impact on the configuration of the process. The proposed approach can support the decision-making process in vaccination centers to efficiently assign human and material resources to maximize the number of patients vaccinated while ensuring reasonable waiting times, number of patients in queue and servers’ utilization rates, which in turn are key to avoid overcrowding and other negative conditions in the system that could increase the risk of infections.

Low-latency service function chain migration in edge-core networks based on open Jackson networks

Multi-access Edge Computing (MEC) offers cloud computing capabilities at the edge of the network. Growing demand for low-latency services requires Service Function Chains (SFCs) to be scaled up beyond MEC network to core network. To adapt to network dynamics and provide low-latency services, being able to migrate SFCs when needed is of paramount importance. However, migration of SFCs among edge and core networks such that average latency is optimized as well as considering resource consumption is an intractable challenge because improper migration of Virtual Network Functions (VNFs) results in failure of meeting the requirements of network policies. In this paper, we investigate SFCs in edge-core networks and model the Latency-aware Edge-Core SFCs Migration problem based on open Jackson networks. Two SFC migration algorithms, i.e., Profit-driven Heuristic Search (PHS) and Average Utilization Based (AUB), are proposed to efficiently optimize average latency of all SFCs in edge-core networks. Extensive evaluation results show that PHS optimizes average latency by 19.5%, while AUB can further reduce average latency by up to 36.9% by allowing a marginally higher number of VNF migrations.

(<i>s</i>, <i>S</i>) stochastic inventory system in Jackson network

In this work, we develop and analyse an (s, S) stochastic perishable inventory system at each node into Jackson network with a service facility in which the waiting hall of first queue has infinite capacity and second queue has finite capacity for the customers. Service times are exponentially distributed. We assume that demands arrive in the system according to a Poisson process with rate λi(> 0, ∀i) and demands only single unit item at a time. The maximum storage capacity of the ith warehouse is fixed as Si; i = 1, 2. Whenever the inventory level reaches the reorder level si(0 ≤ si < Si), an order Qi(= Si – si) units is placed to bring the level to Si. The lead-time is exponentially distributed. The items of inventory have exponential life times. The joint probability distribution of the number of customers in the system and the inventory level is obtained in the steady state case. Matrix analytical method is applied to solve for the steady state occupancy probabilities. Various system performance measures in the steady state are derived. A suitable cost function is defined and the long-run total expected cost rate is calculated. Sensitivity analysis has been carried out to study the effect of variation of parameters. Numerical examples and graphical illustrations are provided to illustrate the proposed model.

Accuracy of estimation of the vector of queue lengths for open Jackson networks

Работа посвящена оценке точности аппроксимации вектора длин очередей для открытой сети Джексона с регенерирующим входящим потоком и ненадежными приборами. Получена теорема о точности аппроксимации вектора длин очередей в открытых сетях Джексона, т.е. оценка вероятности отклонения нормы разности процесса длин очередей и построенного отраженного броуновского движения. В качестве следствия получена оценка расстояния Вассерштейна.

Solving a new bi-objective multi-echelon supply chain problem with a Jackson open-network issue under uncertainty

In today’s world, changes in the economic and industrial sectors are taking place more quickly than in the past. Based on today’s competitive market, considering the needs and desires of the customer is very important. Thus, designing the supply chain network to minimize the costs and improve the system efficiency, the service facility, and attention to the timely service delivery to the customer should be considered. Also, deployment costs to determine the number and position of facilities are regarded as another strategic issue. Therefore, to become closer to the real-world application, by considering the Jackson network model (queuing theory), the present study aims to design a closed-loop supply chain network to reduce the waiting time in the queue. Two objectives, namely minimizing the cost and waiting time incurred in queues formed in producers and distributors, are investigated. Queuing mathematical expressions are used both in dealing with objectives and constraints. To develop a model, which is close to real by considering uncertainty, the parameters, including demand rate and shipping cost of products, were considered as a trapezoidal fuzzy number. Also, we crisp this fuzzy uncertainty using both Jimenez and chance-constrained programming approaches. Due to the suggested model’s complexity and nonlinearity, we linearized the mathematical model as much as possible, then two meta-heuristic algorithms, namely multi-objective particle swarm optimization and non-dominated sorted genetic algorithm (NSGA-II), are used to solve the model. First, the structural parameters of these algorithms are tuned by the Taguchi method. Then, numerous numerical test problems ranging from small to large scales are used to compare these algorithms considering Pareto front measuring indexes and various hypothesis tests.

Bi-objective optimization for customers’ satisfaction improvement in a Public Bicycle Sharing System

In recent years, the popularity of the Public Bicycle Sharing System (PBSS) for urban transportation is increasing. The fleet size of the system and the capacity of its stations are some key factors in establishing a successful sharing system. These factors affect the number of rejected demands and the lack of free docks for returning bicycles because of demand variation at stations. This paper presents a bi-objective optimization solution for PBSS in which the first objective function minimizes the mean number of rejected requests for renting bikes and free docks for returning bikes per each satisfied demand. In other words, this function tries to minimize the mean number of dissatisfied users per each replied demand for renting. The second objective function minimizes the total number of bicycles and docks of the system. The goal is to minimize users’ dissatisfaction with the least possible fleet size considering capacity constraints. The model is discussed under Jackson Network and the Mean Value Analysis (MVA). Non-dominated Sorting Genetic Algorithm (NSGA-II) is used to solve and examine the efficiency of the proposed model via different numerical examples.

Kelly and Jackson networks with interchangeable, cooperative servers

AbstractIn open Kelly and Jackson networks, servers are assigned to individual stations, serving customers only where they are assigned. We investigate the performance of modified networks where servers cooperate. A server who would be idle at the assigned station will serve customers at another station, speeding up service there. We assume interchangeable servers: the service rate of a server at a station depends only on the station, not the server. This gives work conservation, which is used in various ways. We investigate three levels of server cooperation, from full cooperation, where all servers are busy when there is work to do anywhere in the network, to one-way cooperation, where a server assigned to one station may assist a server at another, but not the converse. We obtain the same stability conditions for each level and, in a series of examples, obtain substantial performance improvement with server cooperation, even when stations before modification are moderately loaded.

Age of Information of Jackson Networks With Finite Buffer Size

We investigate the Age of Information (AoI) of Jackson Networks with finite buffer size. We provide a closed form expression of an upper bound of the average AoI by using Stochastic Hybrid Systems (SHS) along with linear algebra tools. Since the computation of the bound requires inverting a N × N matrix, where N is the number of queues in the network, we provide a simple iterative algorithm and prove its convergence with geometric convergence rate. Numerical results allow us to assess the accuracy of our model in various settings.

Flexible Functional Split and Fronthaul Delay: A Queuing-Based Model

We study the delay over virtual RAN (vRAN) topologies, entailing base stations that are divided into centralized and distributed units, as well as the packet-switched fronthaul network that connects them. We consider the use of flexible functional split, where the functions that are executed at each of these two entities can be dynamically shifted. We propose a queuing-based model, which is able to precisely mimic the behavior of such nodes, and we validate it by means of extensive simulations. We also exploit Jackson Networks theory to establish the end-to-end delay over the fronthaul network, allowing us to assess the impact of having different networking policies and conditions (for instance, background traffic or heterogeneous technologies). Thanks to the simulator we can also broaden the analysis, by studying the delay variability. In addition, we conduct an in-depth analysis of the performance exhibited by a realistic network setup, whose particular characteristics might hinder the services performance, due to the longer dwell times at each split configuration. The results evince the validity of the proposed model, even under realistic conditions. We show that it might not be enough to guarantee an average stable operation of the centralized/distributed units, but the traffic load should remain below the slowest service rate, to avoid reaching unacceptable delays. An increase of $> 100 \times $ is observed in the delay, using the realistic network setup, when these conditions do not hold.