Queueing Network Research Articles

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 because of 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 a function that depends only on 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. This paper was accepted by Baris Ata, stochastic models and simulation.

Modeling of Railway Stations Based on Queuing Networks

Among the micro-logistic transport systems, railway stations should be highlighted, such as one of the most important transport infrastructure elements. The efficiency of the transport industry as a whole depends on the quality of their operation. Such systems have a complex multi-level structure, and the incoming traffic flow often has a stochastic character. It is known that the most effective approach to study the operation of such systems is mathematical modeling. Earlier, we proposed an approach to transport hub modeling using multiphase queuing systems with a batch Markovian arrival process (BMAP) as an incoming flow. In this paper, we develop the method by applying more complex models based on queuing networks that allow us to describe in detail the route of requests within an object with a non-linear hierarchical structure. This allows us to increase the adequacy of modeling and explore a new class of objects—freight railway stations and marshalling yards. Here we present mathematical models of two railway stations, one of which is a freight railway station located in Russia, and the other is a marshalling yard in the USA. The models have the form of queuing networks with BMAP flow. They are implemented as simulation software, and a numerical experiment is carried out. Based on the numerical results, some “bottlenecks” in the structure of the studied stations are determined. Moreover, the risk of switching to an irregular mode of operation is assessed. The proposed method is suitable for describing a wide range of cargo and passenger transport systems, including river ports, seaports, airports, and multimodal transport hubs. It allows a primary analysis of the hub operation and does not need large statistical information for parametric identification.

Autonomic Resource Management for Fog Computing

Fog computing is a distributed computing paradigm that extends cloud computing capabilities to the edge of the network and aims at reducing high latency and network congestion, which are characteristics of cloud computing. This recent paradigm enables portions of a transaction to be executed at a fog server and other portions at the cloud. Fog servers are generally not as robust as cloud servers; at peak loads, the data that cannot be processed by fog servers is processed by cloud servers. The data that need to be processed by the cloud is sent over a Wide Area Network (WAN). Therefore, only a fraction of the total data needs to travel through the WAN, as compared with a pure cloud computing paradigm. Additionally, the fog/cloud computing paradigm reduces the cloud processing load when compared with the pure cloud computing model. This article presents a multiclass closed-form analytic queuing network model that is used by an autonomic controller to dynamically change the fraction of processing between edge and cloud servers in order to maximize a utility function of response time and cost. The model was validated using both synthetic and real IoT traces. A detailed design of the autonomic controller is presented and a series of experiments compare the efficacy and efficiency of the controller versus a brute force optimal controller and versus an uncontrolled system using synthetic and real traces. The results show that the controller is able to maintain a high utility in the presence of wide variations of request arrival rates.

A System Optimal Speed Advisory Framework for a Network of Connected and Autonomous Vehicles

The technological advancements involving information and communication technologies (ICT), such as Connected and Automated Vehicles (CAVs) and the Intelligent Transport Systems (ITS), have enabled new efficient traffic control and management strategies to mitigate traffic congestion. Specifically, the combined traffic flow-speed advisory systems based on CAVs and ITS technologies could provide the individual vehicle with the optimal speed to reduce the fuel consumption, the number of stops, simultaneously reduce the network-wide traffic congestion and improve road safety. This article develops a novel bi-level control framework underpinned by the mutual interaction between a system optimal traffic flow control strategy at a network level and a speed control policy for an individual vehicle at a link level within a connected traffic environment. Our framework proposes the novel group-based method to guarantee the consistency and interaction between the macroscopic and microscopic models. To this end, it efficiently optimizes vehicular trajectories while meeting the network-wide objectives which have not been investigated previously in the literature. We propose an efficient algorithm for this problem that iteratively solves mixed-integer linear programming (MILP) models for each upper and lower level. Numerical results indicate the effectiveness of the proposed speed advisory method in vehicular emission reduction, favorable network queue formation, and its positive influence on traffic flow patterns over the network.

On the Throughput Optimization in Large-Scale Batch-Processing Systems

We analyze a data-processing system with n clients producing jobs which are processed in batches by m parallel servers; the system throughput critically depends on the batch size and a corresponding sub-additive speedup function that arises due to overhead amortization. In practice, throughput optimization relies on numerical searches for the optimal batch size which is computationally cumbersome. In this paper, we model this system in terms of a closed queueing network assuming certain forms of service speedup; a standard Markovian analysis yields the optimal throughput in w n4 time. Our main contribution is a mean-field model that has a unique, globally attractive stationary point, derivable in closed form. This point characterizes the asymptotic throughput as a function of the batch size that can be calculated in O(1) time. Numerical settings from a large commercial system reveal that this asymptotic optimum is accurate in practical finite regimes.

On asymptotic merging of nodes set for multichannel stochastic networks

In this paper, a multichannel queueing network and the problem of asymptotic merging for its set of nodes are considered. Rate of input flow arriving to the network depends on time. Service times in the network nodes are generally distributed. We introduce a multivariate service process as the number of calls being processed in the nodes. Under heavy traffic conditions, functional limit theorems of Gaussian approximation type are proved for the normalized service process of both original and merged networks. Limit Gaussian processes are constructed, and their correlation characteristics are written via the network parameters. In the case of the merging of the network nodes set, the limit process is a multidimensional diffusion Gaussian process. Moreover, due to the approach with merging, dimension of the limit process is reduced.

Long-Term Behavior of Dynamic Equilibria in Fluid Queuing Networks

Steady State in Equilibrium for Flows over Time

Stochastic modeling of multiline orders in integrated storage‐order picking system

AbstractDue to demanding service levels in e‐commerce order fulfillment, modeling and analysis of integrated storage and order picking processes in warehouses deserve special attention. The upstream storage system can have a significant impact on the performance of the downstream order picking process. With a particular focus on multiline e‐commerce orders, we develop an analytical modeling framework for integrated analysis of upstream (shuttle‐based storage and retrieval system) and downstream (pick system) networks. To capture the consolidation delays in fulfilling multiline orders, the downstream pick system is modeled with a closed queuing network that includes synchronization nodes. The configuration of the synchronization station is adapted to model the variety of order profiles handled at the pick station. For the downstream closed queuing network, we propose a decomposition‐based solution methodology that results in good solution accuracy. The resulting semi‐open queuing network (SOQN) of the integrated system is analyzed using the matrix‐geometric method (MGM). To improve the accuracy of analytical estimates of the measures, we propose a hybrid simulation/analytical framework, where the performance measures of complex subnetworks are obtained from simulation. We also develop a detailed simulation model of the physical system for validating the analytical and hybrid estimates of the performance measures. The results from experiments indicate that the hybrid simulation/analytical approach reduces the error in the throughput time estimates to 3% from 18% obtained from the analytical model. Then, we investigate the effect of the upstream network configuration (such as the number of storage aisles) and the downstream network configuration (such as the mixed vs. dedicated picking, CONWIP control for orders, order batching) on the order throughput times. Our analysis provides a threshold on the maximum numbers of allowable orders (CONWIP control) and number of aisles beyond which the improvement in average throughput time of the integrated system is marginal. Numerical experiments with high‐order arrivals also highlight that mixed picking in the downstream network can result in significant throughput time reduction in comparison to dedicated picking.

Supervised learning-based approximation method for single-server open queueing networks with correlated interarrival and service times

Efficient performance evaluation methods are needed to design and control production systems. We propose a method to analyse single-server open queueing network models of manufacturing systems composed of delay, batching, merge and split blocks with correlated interarrival and service times. Our method (SLQNA) is based on using a supervised learning approach to determine the mean, the coefficient of variation, and the first-lag autocorrelation of the inter-departure time process as functions of the mean, coefficient of variation and first-lag autocorrelations of the interarrival and service times for each block, and then using the predicted inter-departure time process as the input to the next block in the network. The training data for the supervised learning algorithm is obtained by simulating the systems for a wide range of parameters. Gaussian Process Regression is used as a supervised learning algorithm. The algorithm is trained once for each block. SLQNA does not require generating additional training data for each unique network. The results are compared with simulation and also with the approximations that are based on Markov Arrival Process modelling, robust queueing, and G/G/1 approximations. Our results show that SLQNA is flexible, computationally efficient, and significantly more accurate and faster compared to the other methods.

Large deviations for acyclic networks of queues with correlated Gaussian inputs

We consider an acyclic network of single-server queues with heterogeneous processing rates. It is assumed that each queue is fed by the superposition of a large number of i.i.d. Gaussian processes with stationary increments and positive drifts, which can be correlated across different queues. The flow of work departing from each server is split deterministically and routed to its neighbors according to a fixed routing matrix, with a fraction of it leaving the network altogether. We study the exponential decay rate of the probability that the steady-state queue length at any given node in the network is above any fixed threshold, also referred to as the ‘overflow probability’. In particular, we first leverage Schilder’s sample-path large deviations theorem to obtain a general lower bound for the limit of this exponential decay rate, as the number of Gaussian processes goes to infinity. Then, we show that this lower bound is tight under additional technical conditions. Finally, we show that if the input processes to the different queues are nonnegatively correlated, non-short-range dependent fractional Brownian motions, and if the processing rates are large enough, then the asymptotic exponential decay rates of the queues coincide with the ones of isolated queues with appropriate Gaussian inputs.

Little’s Laws for Extreme Values in Multi-Server Multi-Core Open Queueing Networks

The paper is devoted to the analysis of queueing systems in the context of the network and communication theory (called a multi-server multi-core open queueing network). The ob ject of this research on the queueing theory is theorems about the Functional Strong Laws of Large Numbers (FSLLN) in multi-server multi-core open queueing networks, working under overload heavy traffic conditions. FSLLN is known as a fluid limit or fluid approximation. In this paper, FSLLN are proved for extreme values of important probabilistic characteristics of the multi-server multicore open queueing network, investigated as well as the virtual waiting time of a job and the queue length of jobs. As applications of the proved theorems Little’s laws in a multi-server multi-core open queueing network are presented.

Multi server queuing system with crashes and alternative repair strategies

In the modern-day world, a large number of queuing systems have been used in a variety of applications. Their usages involve huge amounts of complex computation coupled with inter and intra communication. However, in the vast majority of these systems, the available capacity is limited. Owing to these limitations, servers in general are prone to failures from time to time. Open network queuing system problems with breakdowns and its alternative repair strategies vastly increasing in computer and communication networks, needs proper modeling to solve day to day applications. The performance modeling of a multi-node system has been determined in this work taking into consideration arrivals at different periods. Proper analysis has been carried out thereby producing amazing results.

G-network models to support planning for disaster relief distribution

One of the key activities during disaster response is distributing relief items to victims. This is a challenging task due to dynamically changing victim needs and disaster aftermath conditions. We model the distribution operations where items like tarpaulins and blankets are distributed by volunteers, to victims at temporary distribution areas called relief centers (RC). We investigate the impact victim movements have on the distribution performance. We model each RC as a queue, and the distribution operation as a generalised queuing network (G-network). We investigate product form solutions for the proposed G-network model, and prove a new product form result for G-networks with signals and batch transfer under certain conditions. We leverage this result to develop product form approximations that apply across a broad range of settings. We apply the G-network model to a case study using the Nepal earthquake relief distribution data, and quantify the impact of victim movement on network performance.

Joint optimization of service chain caching and task offloading in mobile edge computing

Caching and offloading in Mobile Edge Computing (MEC) are hot topics recently. Existing caching strategies at the edge ignore the programming ability of edge network and design strategies independently thus network resource is under utilization and the quality of experience (QOE) for end users is far from satisfactory. In this paper, we design intelligently joint caching and offloading strategies under the assumption that applications can be in the form of divisible service chain. Different from common approaches that target on reducing response latency only for users, our system take the leasing cost into consideration thus is more efficient for Application Service Providers (ASP). To fulfill our design, we novelly utilize open Jackson queuing network to formulate this joint optimization problem under long term cost restrictions and design a pipeline of algorithm to search for the optimal solution. More specifically, we design a cost adaptive algorithm derived from Lyapunov drift-plus-penalty function so that the long-term problem can be optimized in the slot-by-slot basis. Moreover, we propose to exploit resource-based utility function and device-number-based relative distance to jointly find optimal caching and offloading scheme. Extensive simulation results demonstrate that our approach can effectively reduce the average service latency of the MEC system and keep a low average leasing cost.

Modeling and Optimization in Resource Sharing Systems: Application to Bike-Sharing with Unequal Demands

The imbalanced distribution of shared bikes in the dockless bike-sharing system (a typical example of the resource-sharing system), which may lead to potential customer churn and lost profit, gradually becomes a vital problem for bike-sharing firms and their users. To resolve the problem, we first formulate the bike-sharing system as a Markovian queueing network with higher-demand nodes and lower-demand nodes, which can provide steady-state probabilities of having a certain number of bikes at one node. A model reduction method is then designed to reduce the complexity of the proposed model. Subsequently, we adopt an operator-based relocation strategy to optimize the reduced network. The objective of the optimization model is to maximize the total profit and act as a decision-making tool for operators to determine the optimal relocation frequency. The results reveal that it is possible for most of the shared bikes to gather at one low-demand node eventually in the long run under the influence of the various arrival rates at different nodes. However, the decrease of the number of bikes at the high-demand nodes is more sensitive to the unequal demands, especially when the size of the network and the number of bikes in the system are large. It may cause a significant loss for operators, to which they should pay attention. Meanwhile, different estimated values of parameters related with revenue and cost affect the optimization results differently.

Non-cooperative queueing games on a network of single server queues

This paper introduces non-cooperative games on a network of single server queues with fixed routes. A player has a set of routes available and has to decide which route(s) to use for its customers. Each player’s goal is to minimize the expected sojourn time of its customers. We consider two cases: a continuous strategy space, where each player is allowed to divide its customers over multiple routes, and a discrete strategy space, where each player selects a single route for all its customers. For the continuous strategy space, we show that a unique pure-strategy Nash equilibrium exists that can be found using a best-response algorithm. For the discrete strategy space, we show that the game has a Nash equilibrium in mixed strategies, but need not have a pure-strategy Nash equilibrium. We show the existence of pure-strategy Nash equilibria for four subclasses: (i) N-player games with equal arrival rates for the players, (ii) 2-player games with identical service rates for all nodes, (iii) 2-player games on a 2times 2-grid, and (iv) 2-player games on an Atimes B-grid with small differences in the service rates.

Dynamic Large-Scale Server Scheduling for IVF Queuing Network in Cloud Healthcare System

As one of the most effective medical technologies for the infertile patients, in vitro fertilization (IVF) has been more and more widely developed in recent years. However, prolonged waiting for IVF procedures has become a problem of great concern, since this technology is only mastered by the large general hospitals. To deal with the insufficiency of IVF service capacity, this paper studies an IVF queuing network in an integrated cloud healthcare system, where the two key medical services, that is, egg retrieval and transplantation, are assigned to accomplish in the general hospital, while the routine medical tests are assigned into the community hospital. Based on continuous-time Markov procedure, a dynamic large-scale server scheduling problem in this complicated service network is modeled with consideration of different arrival rates of multiple type of patients and different service capacities of multiple servers that can be defined as doctors of the general hospital. To solve this model, a reinforcement learning (RL) algorithm is proposed, where the reward functions are designed for four conflicting subcosts: setup cost, patient waiting cost, penalty cost for unsatisfied patient personal preferences, and medical cost of patient. The experimental results show that the optimal service rule of each server’s queue obtained by the RL method is significantly superior to the traditional service rule.

Performance Analysis of Multi-processor Two-Stage Tandem Call Center Retrial Queues with Non-Reliable Processors

We analyze a multi-processor two-stage tandem call center retrial queueing network in which the processors are subject to active breakdowns and repairs at stage-I. A level-dependent quasi-birth-and-death (LDQBD) process is formulated and a sufficient condition for ergodicity of the system is discussed. Under the stability condition, the stationary distribution of the number of calls in the system, the mean number of calls in the orbit, the mean waiting time of calls in the orbit and the mean busy period of the system along with other descriptors of the system are determined by using the matrix-analytic techniques. Besides, the availability analysis of the processors is also studied. Further, we have also discussed the characteristics of the first-passage time to reach the orbit critical level, the number of calls served in the call center during this period and their corresponding moments. Finally, extensive numerical results are presented to highlight the impact of the system parameters on the performance measures of the two-stage tandem call center retrial queueing system under investigation.

Robust Control of Arrivals Into a Queuing Network

Queuing networks have been widely-used to model congestion in transportation systems. Due to their interconnected nature, delays in a queuing network can propagate as customers traverse through the network; similarly, downstream resources can be underutilized due to poor control policies. This paper considers the regulation of arrivals into a queuing network in order to maintain a desired level of occupancy (queue length) in the system. The dynamics of the queuing network is represented by a fluid-flow model, which is then used to develop a robust controller for tracking the desired queue length. The controller is based on a sliding mode control approach, with predictor-based feedback to account for propagation delays. For a single queue, we determine sufficient conditions for tracking the queue length, and bounds on the tracking error. We also present an analysis of the tracking performance for queues in tandem. We demonstrate our approach for the example of airport surface congestion control. The proposed robust control framework is based on a queuing network model of the airport, and is used to tactically manage aircraft departures in order to reduce congestion on the airport tarmac.

A queuing network simulation optimization method for coordination control of passenger flow in urban rail transit stations

The imbalances between supply and demand in an urban rail transit system have received extensive attention. Addressing these imbalances by controlling the movements of passengers in the system is a real challenge. To complete the peak passenger flow control scheme, a novel discrete event simulation (DES) optimization model based on queuing theory is proposed to minimize the urban rail transit company losses and the passenger time delays considering effects of congestion propagation among facilities at busy stations. The proposed approach can generate a control scheme that consists of the controlling number of entering passengers and the controlling parameters of facilities inside the station. The first stage of the method models the subway network using queuing theory and then builds a DES model that is based on an urban rail transit queuing network. In the second stage, a service-security-economic-oriented optimization model is established and combined with a queuing network simulation model to implement the simulation optimization. We present three numerical experiments to compare the optimization results in different passenger flow scenarios. The results demonstrate that the DES optimization method in this paper can provide a reasonable and reliable control scheme in daily operations.