Closed Queueing Network Research Articles

Using queuing theory to describe adaptive mathematical models of computing systems with resource virtualization and its verification using a virtual server with a configuration similar to the configuration of a given model

The article describes the issues of preparation and verification of mathematical models of computing systems with resource virtualization. The object of this study is to verify of mathematical models of computer systems with virtualization experimentally by creating a virtual server on the host platform and monitoring its characteristics under load. Known models cannot be applied to the aircraft with virtualization, because they do not allow a comprehensive analysis to determine the most effective option for the implementation of the initial allocation of resources and its optimization for a specific sphere and task of use. The article for the study used a closed queueing network. Simple models for the analysis of various structures of computer systems are experimentally obtained. To implement the properties of adaptability in the models, triggers are used that monitor and adjust the power of the processing channel in individual Queuing systems, depending on the specified conditions. Experiments prove the obtained results reliable and usable as a flexible tool for studying the virtualization properties when structuring computing systems. This knowledge could be of use for businesses interested in optimizing the server configuration for their IT infrastructure.

Operating policies in multi-warehouse drone delivery systems

Drones are increasingly used to deliver packages with high efficiency in several areas. We study a multi-warehouse drone delivery system, considering the allocation rule that all warehouses share the drones and the allocation rule that each warehouse owns its drones. Both plug-in charge and battery swap strategies are investigated for battery management. We examine the random and closest drone to warehouse assignment rules, and design a heuristic to improve the drone to warehouse assignment rule. A closed queueing network is built to estimate the maximum throughput capacity and a cost minimisation model is developed for cost analysis. We validate the analytical model by simulation and conduct numerical experiments to analyse the operating polices. The results show that the closest drone to warehouse assignment rule outperforms the random drone to warehouse assignment rule when the number of drones is not large, and our heuristic can improve the throughput capacity by about 13.31%. The battery swap strategy provides a better throughput capacity than the plug-in charge strategy in most cases, while it needs more investment. Moreover, the shared allocation rule gives a larger throughput capacity than the dedicated allocation rule, and it reduces the operating cost by about 30.70%.

Optimization Strategies for Dockless Bike Sharing Systems via two Algorithms of Closed Queuing Networks

The dockless bike sharing system (DBSS) has been globally adopted as a sustainable transportation system. Due to the robustness and tractability of the closed queuing network (CQN), it is a well-behaved method to model DBSSs. In this paper, we view DBSSs as CQNs and use the mean value analysis (MVA) algorithm to calculate a small size DBSS and the flow equivalent server (FES) algorithm to calculate the larger size DBSS. This is the first time that the FES algorithm is used to study the DBSS, by which the CQN can be divided into different subnetworks. A parking region and its downlink roads are viewed as a subnetwork, so the computation of CQN is reduced greatly. Based on the computation results of the two algorithms, we propose two optimization functions for determining the optimal fleet size and repositioning flow, respectively. At last, we provide numerical experiments to verify the two algorithms and illustrate the optimal fleet size and repositioning flow. This computation framework can also be used to analyze other on-demand transportation networks.

Dimensioning On-Demand Vehicle Sharing Systems

We consider the problem of optimal fleet sizing in a vehicle sharing system. Vehicles are available for short-term rental and are accessible from multiple locations. A vehicle rented at one location can be returned to any other location. The size of the fleet must account not only for the nominal load and for the randomness in demand and rental duration but also for the randomness in the number of vehicles that are available at each location due to vehicle roaming (vehicles not returning to the same location from which they were picked up). We model the dynamics of the system using a closed queueing network and obtain explicit and closed form lower and upper bounds on the optimal number of vehicles (the minimum number of vehicles needed to meet a target service level). Specifically, we show that starting with any pair of lower and upper bounds, we can always obtain another pair of lower and upper bounds with gaps between the lower and upper bounds that are independent of demand and bounded by 1/(1-alpha), where alpha is the prescribed service level. We show that the generated bounds are asymptotically exact under several regimes. We use features of the bounds to construct a simple and closed form approximation that we show to be always within the generated lower and upper bounds and is exact under the asymptotic regimes considered. Extensive numerical experiments show that the approximate and exact values are nearly indistinguishable for a wide range of parameter values. The approximation is highly interpretable with buffer capacity expressed in terms of three explicit terms that can be interpreted as follows: (1) standard buffer capacity that is protection against randomness in demand and rental times, (2) buffer capacity that is protection against vehicle roaming, and (3) a correction term. Our analysis reveals important differences between the optimal sizing of standard queueing systems (where servers always return to the same queue upon service completion) and that of systems where servers, upon service completion, randomly join any one of the queues in the system. We show that the additional capacity needed to buffer against vehicle roaming can be substantial even in systems with vanishingly small demand.

Analysis and Optimization of a Controlled Model for a Closed Queueing Network

We consider a closed network consisting of two queuing systems: the main system simulates a packet transmission queue over an unreliable communication channel, and the auxiliary multiserver system contains lost packets for resending. The service rate in the main system is controllable and is supposed to be optimized with the aim of minimizing the time of successful transmission, taking into account the cost of using network resources. We obtain optimality conditions in two cases: 1) in the model based on fluid approximation in the presence of heavy load; 2) in the steady state using stationary strategies.

Fleet Sizing for Electric Car Sharing Systems in Discrete Event System Frameworks

This paper proposes a two-level strategy to determine the optimal fleet size of electric car (EC) sharing systems (ECSSs) in a networks of a set of stations. At the first level, the system is modeled as a discrete event system in a closed queueing network framework that allows describing the asymptotic system behavior and determining the optimal fleet size that maximizes the network profit. At the second level the ECSS dynamics is modeled by timed Petri net, in order to take into account some particular aspects, such as the user flows in different time periods of the day or the exit of the customers from the stations when they do not find available ECs. A simulation campaign analysis enlightens the effectiveness of the presented complementary design strategy.

Fluid approximation of closed queueing networks with discriminatory processor sharing

As a multi-class variant of the classical egalitarian processor-sharing (EPS) discipline, discriminatory processor sharing (DPS) provides a suitable paradigm to model systems where share exists to control the service access of heterogeneous jobs. Although DPS is a more fine-grained scheduling discipline than EPS, the behavior of closed queueing networks (QNs) with DPS remains poorly understood. We propose in this paper a fluid approach to transient and steady-state analysis of closed QNs comprising delay and DPS stations. Our reference model features an arbitrary routing topology, phase-type service time distributions and class switching. The proposed approach has been validated against simulation for both transient and steady-state analysis. Moreover, we introduce a refined method for approximating response time distributions at either station or system level through transient analysis, showing its improvement upon prior art in handling non-exponentially distributed service times.

A Diffusion Approximation Framework for Ride-hailing with Travel Delays

This paper develops a closed queueing network model of a ride-hailing system by viewing cars as the jobs moving through various nodes of the network perpetually. The queueing network model crucially incorporates the travel times between different nodes of the network. We prove a novel heavy-traffic limit theorem for this queueing network, providing an approximation for the original ride-hailing system.

Design and implementation of an elastic processor with hyperthreading technology and virtualization for elastic server models

The server models functioning in the industry are required to be more elastic in nature. They are constantly scaling-up and scaling-down on required computation power depending on different conditions. These elastic cloud platforms use accelerators like DSP’s, TPU’s, GPU’s, FPGA’s, and multi-core processors to provide exponential computing power and outsource their services. This process is not only costly and non-efficient but is also responsible for damaging the server’s hardware architecture. Furthermore, these additions degrade the level of threading and symmetric parallel processing capability of the architecture. Intel uses hyperthreading technology (HTT) to split the workload between hardware and operating system to avoid additions, but that too is only possible up until a certain limit. This paper presents design methodology and implementation of an elastic-natured 32-bit RISC-pipelined processor inspired from Intel Xeon and MIPS to function as a standard integrated platform for server models. It implements concepts of hyperthreading technology (HTT) and virtualization on hardware basis. It will allow to derive multiple outputs from units on hardware basis to enhance security and performance without compromising compatibility. The designed elastic core uses a probabilistic node-based closed-queuing network model for server analysis and implementation. Hence, elastic behavior from individual core microarchitecture to server model architecture enables a generic automated scaling self-aware optimization architecture.

Analysis of the Shuttle-Based Storage and Retrieval System

The shuttle-based storage and retrieval system (SBS/RS) is composed of a shuttle sub-system that is responsible for horizontal movements and a lift sub-system that is responsible for vertical movements. As the combination of the two sub-systems yields high flexibility, low operating cost and large storage capacity, SBS/RS is becoming more and more popular, but also raises managerial issue of how to coordinate the shuttle and the lift. Based on its operational processes, this study models the tier-to-tier SBS/RS system as a semi-open queuing network (SOQN). By removing or adding the synchronization nodes, the SOQN is further transformed into two different closed queuing networks (CQNs), and applies the approximate mean value analysis (AMVA) algorithm to solve the model and estimate its performance. The system performance is measured by the utilization of shuttles, the utilization of lift, and the task cycle time, under various design configurations. Simulation is carried out to validate the effectiveness of the analytical model and algorithm. Compared with the simulation results, the established semi-open queuing network can accurately estimate task cycle time for different rack configurations. The proposed solution method can help to identify the optimal number of shuttles and guide the design of the SBS/RS system.

Operational strategies for managing container terminals. An approach based on closed queuing networks

Operational strategies for managing container terminals. An approach based on closed queuing networks

Анализ замкнутых сетей массового обслуживания с групповым обслуживанием

We consider a closed queuing network with batch service and movements of customers in continuous time. Each node in the queueing network is an infinite capacity single server queueing system under a RANDOM discipline. Customers move among the nodes following a routing matrix. Customers are served in batches of a fixed size. If a number of customers in a node is less than the size, the server of the system is idle until the required number of customers arrive at the node. An arriving at a node customer is placed in the queue if the server is busy. The batсh service time is exponentially distributed. After a batсh finishes its execution at a node, each customer of the batch, regardless of other customers of the batch, immediately moves to another node in accordance with the routing probability. This article presents an analysis of the queueing network using a Markov chain with continuous time. The qenerator matrix is constructed for the underlying Markov chain. We obtain expressions for the performance measures. Some numerical examples are provided. The results can be used for the performance analysis manufacturing systems, passenger and freight transport systems, as well as information and computing systems with parallel processing and transmission of information.

Operational strategies for managing container terminals. An approach based on closed queueing networks.

The port performance can be improved by a well-defined operations plan. We present a closed Jackson network (CJN) approach to analyse port operations from vessel to yard. The tool is not only focused on the berthing line, but also on the service level at the storage yard. It aims to determine the best way to operate a containership using two crucial parameters: container stowage and truck traffic. Thus, the CJN model determines the impact of cargo stowage on port performance (μ1) and the impact of traffic on port performance (μ2). Both are crucial to set the best strategy and ensure the efficient use of handling equipments. The CJN model is flexible in the sense that managers may plan different scenarios. The study suggests that terminal operators can achieve substantial benefits by means of the best resource allocation in each scenario.

Queueing Network Realization of an Epidemiological Model for Efficient Evaluation of Computer Transmitted Infections

Abstract This paper reexamines an epidemiological model with 4 population groups (vigilant/non-vigilant susceptible/infectious) built to study the effect of user vigilance on computer transmitted infections (CTIs) in computer networks. The model serves as an example through which a model conversion process is delineated, which aims at enhancing computational efficiency in the evaluation of the global prevalence of CTIs. More specifically, the conventional node-centric networked Markov chain (NCMC) is remodeled as a population-centric Markov chain (PCMC) to reduce the state-space size from an exponential to a polynomial function of the number of computing nodes N in a strongly connected network, where external attack and internal spread processes are aggregated. The PCMC is then realized as a closed queueing network of 4 M/M/N/N queueing nodes, corresponding to the 4 population groups. The results of evaluating the evolution of mean populations for the 4-population network of up to 150,000 computing nodes show that the queueing network realization slows the growth of computational complexity from exponential to linear with respect to the network size without resorting to mean field approximations. The paper briefly discusses on how the queueing network framework can accommodate node-centric Markov chains (NCMCs) of arbitrary directed networks of heterogeneous nodes, and its potential to significantly reduce the complexity in the evaluation of mean population dynamics for the more general class of large networks.

Novel Solutions for Closed Queueing Networks with Load-Dependent Stations

Load-dependent closed queueing networks are difficult to approximate since their analysis requires to consider statedependent service demands. Standard evaluation techniques, such as mean-value analysis, are not equally efficient in the load-dependent setting, where mean queue-lengths are insufficient alone to recursively determine the model equilibrium performance. As such, novel exact techniques to address this class of models can benefit performance engineering practice by offering alternative trade-offs between accuracy and computational cost. In this paper, we derive novel exact solutions for the normalizing constant of state probabilities in the load-dependent setting. For single-class load-dependent models, we provide an explicit exact formula for the normalizing constant that is valid for models with arbitrary load-dependent rates. From this result, we derive two novel integral forms for the normalizing constant in multiclass load-dependent models, which involve integration in the real and complex domains, leading to novel numerical approximations. The paper also illustrates through experiments the computational gains and accuracy of the obtained expressions.

Optimizing Large On-Demand Transportation Networks via Approximate Analysis

On-demand transportation systems (OTS) are increasingly popular worldwide. Modeling these systems as closed queueing networks (CQN) or semi-open queueing networks (SOQN) makes optimizations tractable, and controls robust to input data. Prior literature has obtained mean performance measures (e.g., vehicle availability) by decomposing OTS into networks for which product-form equilibrium distributions exist. However, such an approach is unsatisfactory in terms of computational complexity for large networks such as micromobility networks. This paper presents an alternative approach to obtaining mean performance measures with mild computational complexity and high fidelity. Leveraging this approach, we explore day-to-day vehicle redistribution and network pricing problems for OTS operations in New York City. These results support the potential for rebuilding a more accessible and sustainable OTS, which is of particular significance as multimodal transport continues to emerge globally.

Modeling the productivity and stability of a terminal operation system with quay crane double cycling

This paper focuses on the overall handling efficiency and the system’s stability of container terminals with double cycling. A closed queuing network is used to model terminal operations, and theoretical upper and lower bounds of the system performance is obtained to show the maximum service availability of the terminals. The generally distributed service time indicates the impact of variations in service time on the system’s stability. Finally, real-world data is used to demonstrate the validity and applicability of the proposed model, and the analysis results provide policies to improve the terminal productivity.

Performance Analysis and Improvement of the Bike Sharing System Using Closed Queuing Networks with Blocking Mechanism

The Bike Sharing System is a sustainable urban transport solution that consists of a fleet of bikes placed in various stations. Users will be satisfied if they find available bikes at their departure station and free docks at the destination. Despite the regulation operations of the system provider (i.e., redistribution of bikes by truck) deeper modifications (bike fleet size or station capacity) are often necessary to ensure a satisfactory service rate. In this paper, we model a sub-graph of a Bike Sharing System using the closed queuing network with a Repetitive-Service-Random-Destination blocking mechanism. This model is solved using the Maximum Entropy Method. This model faithfully reproduces the system dynamics considering the limited capacity of stations. We analyze the performance, particularly, via an overall performance indicator of the system. The various control and monitoring decisions (fleet-size, capacity of stations, incoming and outgoing flow of bikes) are applied to find out the best performance levels. The results demonstrate that the overall performance is robust enough regarding the fleet size changes but it degrades with the increase of the stations’ capacity. Finally, the arrival and departure flows control is an efficient and powerful operational leverage.

Robot-storage zone assignment strategies in mobile fulfillment systems

The robotic mobile fulfillment system (MFS) is widely used for automating storage pick and pack activities in e-commerce distribution centers. In this system, the items are stored on movable storage shelves, also known as inventory pods, and brought to the order pick stations by robotic drive units. We develop stylized performance evaluation models to analyze both order picking and replenishment processes in a mobile fulfillment system storage zone, based on multi-class closed queueing network models. To analyze robot assignment strategies for multiple storage zones, we develop a two-stage stochastic model. For a single storage zone, we compare dedicated and pooled robot systems for pod retrieval and replenishment. For multiple storage zones, we also analyze the effect of assigning robots to least congested zones on system throughput in comparison to random zone assignment. The models are validated using detailed simulations. For single zones, the expected throughput time for order picking reduces to one-third of its initial value by using pooled robots instead of dedicated robots; however, the expected replenishment time estimate increases up to three times. For multiple zones, we find that robots that are assigned to storage zones with dedicated and shortest queues provide a greater throughput than robots assigned at random to the zones.

Analysis of a fork/join station with inputs from a finite population subnetwork with multi-server stations

Forks/join stations are commonly used to model synchronization constraints in queuing networks. This paper presents an exact analysis of a fork/join station with inputs from stations composed of multiple exponential servers. The queue length process at the input buffers is analyzed exactly using the underlying Markov process. The semi-Markov kernel characterizing the departure process is analyzed to derive expressions for the marginal and joint distributions of inter-departure times from the fork/join station. These analyses are used to study the effect of inputs from multiple servers on key performance measures at a fork/join station. Comparison studies show that inputs from multiple servers have a significant effect on performance measures such as throughput, synchronization delays and queue lengths at the individual buffers. These insights are important to obtain better representation of synchronization constraints in closed queuing network models of computer networks, fabrication/assembly systems and material control strategies for manufacturing systems.