Closed Queueing Network Model Research Articles

Fleet sizing and charger allocation in electric vehicle sharing systems

In this paper, we propose a closed queueing network model for performance analysis of electric vehicle sharing systems with a certain number of chargers in each neighborhood. Depending on the demand distribution, we devise algorithms to compute the optimal fleet size and number of chargers required to maximize profit while maintaining a certain quality of service. We show that the profit is concave with respect to the fleet size and the number of chargers at each charging point. If more chargers are installed within the city, we show that it can not only reduce the fleet size, but it also improves the availability of vehicles at all the points within a city. We further show through simulation that two slow chargers may outperform one fast charger when the variance of charging time becomes relatively large in comparison to the mean charging time.

Facilitating Load-Dependent Queueing Analysis Through Factorization (Extended Abstract)

We construct novel exact and approximate solutions for meanvalue analysis and probabilistic evaluation of closed queueing network models with limited load-dependent (LLD) nodes. In this setting, load-dependent functions are assumed to become constant after a finite queue-length threshold. For single-class models, we provide an explicit formula for the normalizing constant that applies to models with arbitrary LLD functions, whilst retaining constant complexity with respect to the total population size. From this result, we then derive corresponding closed-form solutions for the multiclass case and show that these yield a novel mean value analysis approach for LLD models. Significantly, this allows us to determine exactly the correction factor between a load-independent solution and a limited load-dependent one, enabling the reuse of state-of-the-art methods for loadindependent models in the analysis of load-dependent networks.

A Dynamic Model for Managing Volunteer Engagement

Non-profit organizations that provide food, shelter, and other services to people in need, rely on volunteers to deliver their services. Unlike paid labor, non-profit organizations have less control over unpaid volunteers’ schedules, efforts, and reliability. However, these organizations can invest in volunteer engagement activities to ensure a steady and adequate supply of volunteer labor. We study a key operational question of how a non-profit organization can manage its volunteer workforce capacity to ensure consistent provision of services. In particular, we formulate a multiclass closed queueing network model to characterize the optimal engagement activities for the non-profit organization to minimize the costs of enhancing volunteer engagement and reducing abandonments, while maximizing productive work done by volunteers. Because this problem appears intractable, we formulate an approximating Browning control problem in the heavy traffic limit and study the dynamic control of that system. Our solution is a nested threshold policy with explicit congestion thresholds that indicate when the non-profit should optimally pursue various types of volunteer engagement activities. A numerical example calibrated using data from a large food bank shows that our dynamic policy for deploying engagement activities can reduce the food bank's total annual cost of its volunteer operations by 31%, while lowering its throughput by only 0.01%. This improvement in performance does not require any additional resources -- it only requires that the food bank strategically deploy its engagement activities based on the number of volunteers signed up to work volunteer shifts.

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.

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.

State Dependent Control of Closed Queueing Networks

We study the design of state dependent control for a closed queueing network model, inspired by shared transportation systems such as ridesharing. In particular, we focus on the design of assignment policies, wherein the platform can choose which supply unit to dispatch to meet an incoming customer request. The supply unit subsequently becomes available at the destination after dropping the customer. We consider the proportion of dropped demand in steady state as the performance measure. We propose a family of simple and explicit state dependent policies called Scaled MaxWeight (SMW) policies and prove that under the complete resource pooling (CRP) condition (analogous to a strict version of Hall's condition for bipartite matchings), any SMW policy induces an exponential decay of demand-dropping probability as the number of supply units scales to infinity. Furthermore, we show that there is an SMW policy that achieves the optimal exponent among all assignment policies, and analytically specify this policy in terms of the matrix of customer-request arrival rates. The optimal SMW policy protects structurally under-supplied locations.

Fluid-Model-Based Car Routing for Modern Ridesharing Systems

We consider a closed queueing network model of ridesharing systems such as Didi Chuxing, Lyft, and Uber. We focus on empty-car routing, a mechanism by which we control car flow in the network to optimize system-wide utility functions, e.g. the availability of empty cars when a passenger arrives. We establish both process-level and steady-state convergence of the queueing network to a fluid limit in a large market regime where demand for rides and supply of cars tend to infinity, and use this limit to study a fluid-based optimization problem. We prove that the optimal network utility obtained from the fluid-based optimization is an upper bound on the utility in the finite car system for any routing policy, both static and dynamic, under which the closed queueing network has a stationary distribution. This upper bound is achieved asymptotically under the fluid-based optimal routing policy. Simulation results with real-word data released by Didi Chuxing demonstrate that the utility under the fluid-based optimal routing policy converges to the upper bound with a rate of 1/√N, where N is the number of cars in the network.

Blending randomness in closed queueing network models

Random environments are stochastic models used to describe events occurring in the environment a system operates in. The goal is to describe events that affect performance and reliability such as breakdowns, repairs, or temporary degradations of resource capacities due to exogenous factors. Despite having been studied for decades, models that include both random environments and queueing networks remain difficult to analyse. To cope with this problem, we introduce the blending algorithm, a novel approximation for closed queueing network models in random environments. The algorithm seeks to obtain the stationary solution of the model by iteratively evaluating the dynamics of the system in between state changes of the environment. To make the approach scalable, the computation relies on a fluid approximation of the queueing network model. A validation study on 1800 models shows that blending can save a significant amount of time compared to simulation, with an average accuracy that grows with the number of servers in each station. We also give an interpretation of this technique in terms of Laplace transforms and use this approach to determine convergence properties.

Queue decomposition & finite closed queueing network models

Closed, finite queueing networks are applicable to many different manufacturing and service system settings. The incorporation of material handling and transportation networks in finite buffer closed queueing networks is studied. A novel queue decomposition approach using state dependent queues to capture the buffer of finite M/M/1/K queues is shown to be a viable approach for modelling these systems. Each M/M/1/K queue is replaced with a coupled state dependent queue plus an M/M/1 queue. An extended mean value analysis (MVA) algorithm is employed to demonstrate the integration of the state dependent queues for the buffers in the approach. Under certain restrictions concerning the network population, finite queueing networks with the state dependent queues acting as buffers should have a product form distribution. This paper focuses on M/M/1/K queues and their transformation while future papers will treat the multi-server case. Several different closed series (i.e. cyclic), merge, and split topological systems of finite queues are analyzed and presented.

Optimal workload allocation in closed queueing networks with state dependent queues

The problem of optimal workload allocation in closed queueing network models with multi-server exponential infinite capacity workstations and finite capacity state dependent queueing models is examined. The processing rates (i.e. service times) of jobs in the queueing system are the main focus. State dependent queues are appropriate for modeling the transportation and material transfer of the customers and products within the system. By combining these two types of queues, we can model the processing of jobs or customers at service stations and also the material handling transfer of job and customers between these stations. Because the environment is a closed queueing network model, it allows for the dynamic interaction of the jobs and customers in the optimal workload allocation values. This combination of queues and optimal search process provides a broad range of potential applications in manufacturing and service type design operations where the travel time between workstations is important. Conveyors, automated-guided vehicle systems (AGVS), and fork lift trucks are utilized to show the full range of material handling component systems.

An Estimated Delay Based Association Policy for Web Browsing in a Multirate WLAN

Station-Access Point (STA-AP) association is an important function in IEEE 802.11 Wireless LAN (WLAN) management. We obtain an association policy that can be implemented in centralized WLAN management devices, or in STAs, by taking into account explicitly two aspects of practical importance: (a) TCP-controlled short file downloads interspersed with read times (motivated by web browsing), and (b) different STAs associated with an Access Point (AP) at possibly different rates (depending on distance from the AP). Our approach is based on two steps. First, we consider an analytical model to obtain the aggregate AP throughput for long TCP-controlled file downloads when STAs are associated at k different rates r <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> , r <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> , ..., r <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sub> ; this extends earlier work in the literature. Second, we present a 2-node closed queueing network model to approximate the expected average-sized file download time for a user who shares the AP with other users associated at a multiplicity of rates. These analytical results motivate the proposed association policy, called the Estimated Delay based Association (EDA) policy: Associate with the AP at which the expected file download time is the least. Simulations indicate that for a web-browsing type traffic scenario, EDA performs substantially better than other policies that have been proposed earlier. Crucially, the improved performance is sustained even in realistic evaluation scenarios, where the assumptions underpinning the analytical model do not hold. To the best of our knowledge, this is the first work that proposes an association policy tailored specifically for web browsing. Apart from this, our analytical results could be of independent interest.

A Simulation-Based Closed Queueing Network Approximation of Semiconductor Automated Material Handling Systems

Queueing networks have been used to model material handling in flexible manufacturing systems. We explore the use of a closed queueing network model to approximate an intrabay automated material handling system (AMHS) in semiconductor manufacturing. A simulation-based approach is proposed to obtain estimates of system performance measures. Current simulation models in the industry are very complex, require significant development time, and take a long time to run. The proposed simulation-based queueing network approximation can be used as an easy and fast alternative or as a precursor to a full-fledged modeling exercise depending on the type of decision being made and the level of granularity required in the output. Using a simple intrabay configuration, we compare estimates from the approximation approach and a more detailed modeling approach to characterize the performance of the approximation. In addition, a simulation information model is used to compare AMHS models based on the amount and types of information needed to model the system, and to obtain desired output. This classification aids modelers in determining the level of detail to incorporate in a model based on the objectives of the simulation study.

Fleet-sizing and service availability for a vehicle rental system via closed queueing networks

In this paper, we address the problem of determining the optimal fleet size for a vehicle rental company and derive analytical results for its relationship to vehicle availability at each rental station in the company’s network of locations. This work is motivated by the recent surge in interest for bicycle and electric car sharing systems, one example being the French program Vélib (2010). We first formulate a closed queueing network model of the system, obtained by viewing the system from the vehicle’s perspective. Using this framework, we are able to derive the asymptotic behavior of vehicle availability at an arbitrary rental station with respect to fleet size. These results allow us to analyze imbalances in the system and propose some basic principles for the design of system balancing methods. We then develop a profit-maximizing optimization problem for determining optimal fleet size. The large-scale nature of real-world systems results in computational difficulties in obtaining this exact solution, and so we provide an approximate formulation that is easier to solve and which becomes exact as the fleet size becomes large. To illustrate our findings and validate our solution methods, we provide numerical results on some sample networks.

Efficient solutions of a PEPA model of a key distribution centre

In this paper we explore the trade-off between security and performance in considering a model of a key distribution centre. The model is specified using the Markovian process algebra PEPA. The basic model suffers from the commonly encountered state space explosion problem, and so we apply some efficient techniques to solve it. First, we use model reduction techniques and approximation to give a form of the model (in fact, a closed queueing network model) which is more scalable. We then consider the use of a fluid flow approximation based on ordinary differential equations (ODEs) derived from an alternative form of the simplified model. Finally, we evaluate a utility function of this secure key exchange model. Three questions have been proposed; how many clients can a given KDC configure support? how much service capacity must we provide at a KDC to satisfy a given number of clients? and what is the maximum rate at which keys can be refreshed before the KDC performance begins to degrade under a given demand on a given system? These questions are explored through numerical examples.

The design of robotic dairy barns using closed queueing networks

In this paper we present a closed queueing network model for a robotic milking barn. We use an approximative mean-value algorithm to evaluate important performance criteria such as the number of cows waiting, their waiting time and the utilization of the facilities in the barn. It is shown how the results from the queueing network analysis can support the discussion about the optimal design of the robotic barn.

Cache affinity and resequencing in a shared-memory multiprocessing system

Most shared-memory multiprocessor systems have a per-processor memory cache that can improve average memory access time. A task executing on a processor with a cache, accumulates a considerable amount of context at that processor in the form of cache entries. Therefore, a task that restarts after I/O, will probably execute faster on the processor it last ran on, since some of its data may still be present in the cache. A task that is shifted to another processor will have little or no initial cache context, so it must renew its context through cache misses. This not only increases task execution time, but also degrades overall system performance due to an increase in bus traffic. In this paper, we examine processor cache affinity in systems that require resequencing to improve system and program performance. We show that some cache affinity benefits are not exploited due to resequencing delays. Also, simulation is used to study the closed queueing network model of a shared-memory multiprocessing system while it is executing under a variety of workloads.

Dimensioning bandwidth for elastic traffic in high-speed data networks

Simple and robust engineering rules for dimensioning bandwidth for elastic data traffic are derived for a single bottleneck link via normal approximations for a closed-queueing network (CQN) model in heavy traffic. Elastic data applications adapt to available bandwidth via a feedback control such as the transmission control protocol (TCP) or the available bit rate transfer capability in asynchronous transfer mode. The dimensioning rules satisfy a performance objective based on the mean or tail probability of the per-flow bandwidth. For the mean objective, we obtain a simple expression for the effective bandwidth of an elastic source. We provide a new derivation of the normal approximation in CQNs using more accurate asymptotic expansions and give an explicit estimate of the error in the normal approximation. A CQN model was chosen to obtain the desirable property that the results depend on the distribution of the file sizes only via the mean, and not the heavy-tail characteristics. We view the exogenous "load" in terms of the file sizes and consider the resulting flow of packets as dependent on the presence of other flows and the closed-loop controls. We compare the model with simulations, examine the accuracy of the asymptotic approximations, quantify the increase in bandwidth needed to satisfy the tail-probability performance objective as compared with the mean objective, and show regimes where statistical gain can and cannot be realized.

Towards the accurate modelling of high-speed communication networks with product-form discrete-time networks of queues

High-speed communication networks and computer systems can be modelled by using discrete-time networks of queues that have a product-form equilibrium queue-length distribution. Although these models can be solved exactly to give a steady-state performance analysis, the models themselves are only approximations to reality in that they do not take into account restrictions imposed by such things as switch capacities, finite buffer sizes, channel capacities or data rates. These problems are addressed in the paper, where methods of refining a closed queueing network model to allow for such restrictions are considered. Conditions are derived for the refined network models to have a product-form equilibrium distribution, and examples of applying the results to obtain product-form networks are given.

Task routing and resequencing in a multiprocessor system

The effects of task routing and resequencing on system and program performance of a multiprocessor system are studied using simulation techniques. A closed queueing network model is considered. It consists of the CPU and the I/O channel. The CPU consists of four homogeneous and independent processors each serving its own queue. Resequencing of jobs after CPU service ensures that jobs leave the CPU on a first-in-first-out basis. We examine the performance of four task routing strategies in conjunction with the subsequent resequencing of jobs for various coefficients of variation of processor service times (exponential and Branching Erlang distributions) and for different degrees of multiprogramming.

A decision support model for the initial design of FMS

This study proposes a decision support model for the design and evaluation of FMS. The model optimizes the profit by analyzing the interaction between candidate part types and FMS configuration under the constraints of production requirement and cost effectiveness. The model is solved by using the closed queueing network model and optimization model iteratively. The throughput of current FMS configuration is evaluated by the closed queueing network model, and then its results are fed to the optimization model for the selection of best combination of part types and production resources. The resulting solution gives answers to a series of decision problems at the initial design stage of FMS. An example problem is presented to show the superiority of the proposed approach.