We show that planar graphs have bounded queue-number, thus proving a conjecture of Heath et al. 66 from 1992. The key to the proof is a new structural tool called
layered partitions
, and the result ...that every planar graph has a vertex-partition and a layering, such that each part has a bounded number of vertices in each layer, and the quotient graph has bounded treewidth. This result generalises for graphs of bounded Euler genus. Moreover, we prove that every graph in a minor-closed class has such a layered partition if and only if the class excludes some apex graph. Building on this work and using the graph minor structure theorem, we prove that every proper minor-closed class of graphs has bounded queue-number.
Layered partitions have strong connections to other topics, including the following two examples. First, they can be interpreted in terms of strong products. We show that every planar graph is a subgraph of the strong product of a path with some graph of bounded treewidth. Similar statements hold for all proper minor-closed classes. Second, we give a simple proof of the result by DeVos et al. 31 that graphs in a proper minor-closed class have low treewidth colourings.
We study matching queues with abandonment. The simplest of these is the two-sided queue with servers on one side and customers on the other, both arriving dynamically over time and abandoning if not ...matched by the time their patience elapses. We identify nonasymptotic and universal scaling laws for the matching loss due to abandonment, which we refer to as the “cost of impatience.” The scaling laws characterize the way in which this cost depends on the arrival rates and the (possibly different) mean patience of servers and customers. Our characterization reveals four operating regimes identified by an operational measure of patience that brings together mean patience and utilization. The four regimes subsume the regimes that arise in asymptotic (heavy-traffic) approximations. The scaling laws, specialized to each regime, reveal the fundamental structure of the cost of impatience and show that its order of magnitude is fully determined by (i) a “winner-take-all” competition between customer impatience and utilization, and (ii) the ability to accumulate inventory on the server side. Practically important is that when servers are impatient, the cost of impatience is, up to an order of magnitude, given by an insightful expression where only the minimum of the two patience rates appears. Considering the trade-off between abandonment and capacity costs, we characterize the scaling of the optimal safety capacity as a function of costs, arrival rates, and patience parameters. We prove that the ability to hold inventory of servers means that the optimal safety capacity grows logarithmically in abandonment cost and, in turn, slower than the square-root growth in the single-sided queue. This paper was accepted by Baris Ata, stochastic models and simulation. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2023.01513 .
•We derive point queue models as limits of link-based queueing models.•We provide two definitions of demand and supply of a point queue.•We present four point queue models, four approximate models, ...and discrete versions.•We analytically solve Vickrey’s point queue model and stationary states in all models.•All existing point queue models are shown to be special cases of proposed models.
In transportation and other types of facilities, various queues arise when the demands of service are higher than the supplies, and many point and fluid queue models have been proposed to study such queueing systems. However, there has been no unified approach to deriving such models, analyzing their relationships and properties, and extending them for networks. In this paper, we derive point queue models as limits of two link-based queueing model: the link transmission model and a link queue model. With two definitions for demand and supply of a point queue, we present four point queue models, four approximate models, and their discrete versions. We discuss the properties of these models, including equivalence, well-definedness, smoothness, and queue spillback, both analytically and with numerical examples. We then analytically solve Vickrey’s point queue model and stationary states in various models. We demonstrate that all existing point and fluid queue models in the literature are special cases of those derived from the link-based queueing models. Such a unified approach leads to systematic methods for studying the queueing process at a point facility and will also be helpful for studies on stochastic queues as well as networks of queues.
Edge computing applications typically require generated data to be preprocessed at the source and then transmitted to an edge server. In such cases, transmission time and preprocessing time are ...coupled, yielding a tradeoff between them to achieve the targeted objective. This paper presents analysis of such a system with the objective of optimizing freshness of received data at the edge server. We model this system as two queues in tandem whose service times are independent but the transmission service time is monotonically dependent on the computation service time in mean value. This dependence captures the natural decrease in transmission time due to lower offloaded computation. We analyze various queue management schemes in this tandem queue where the compute queue has a single server, Poisson packet arrivals, general independent service and no extra buffer to save incoming packets. The transmit queue has a single server receiving packets from the compute queue with memoryless service time. We consider the transmit queue in two forms: (i) No data buffer and (ii) One unit data buffer and last come first serve with discarding. We analyze various non-preemptive as well as preemptive cases. We perform stationary distribution analysis and obtain closed form expressions for average age of information (AoI) and average peak AoI. Our numerical results illustrate analytical findings on how computation and transmission times could be traded off to optimize AoI and reveal a consequent tradeoff between average AoI and average peak AoI.
We study three non-equivalent queueing models in continuous time that each generalise the classical M/M/1 queue in a different way. Inter-event times in all models are Mittag-Leffler distributed, ...which is a heavy tail distribution with no moments. For each of the models we answer the question of the queue being at zero infinitely often (the ‘recurrence’ regime) or not (the transient regime). Aside from this question, the different analytical properties of each models allow us to answer a number of questions such as existence and description of equilibrium distributions, mixing times, asymptotic behaviour of return probabilities and moments and functional limit theorems.
In large-scale service systems, it is a common practice to organize customers with similar service requirements into a single queue served by a group of servers. This pooled queue structure is deemed ...highly efficient because the servers’ idleness will be minimized. In “To Pool or Not to Pool: Queueing Design for Large-Scale Service Systems,” Cao, He, Huang, and Liu demonstrate that the dedicated queue structure, under which each server has her own queue, could be more advantageous for improving the system’s service level. Moreover, the servers’ additional idleness induced by the dedicated queue structure will be negligible when the system scale is large. By solving a staffing problem, this study also intends to help service system designers answer the following question: To achieve a specified service-level objective in a more efficient manner, should the servers have a common queue or separate queues?
There are two basic queue structures commonly adopted in service systems: the pooled structure, where waiting customers are organized into a single queue served by a group of servers, and the dedicated structure, where each server has her own queue. Although the pooled structure, known to minimize the servers’ idle time, is widely used in large-scale service systems, this study reveals that the dedicated structure, along with the join-the-shortest-queue routing policy, could be more advantageous for improving certain performance measures, such as the probability of a customer’s waiting time being within a delay target. The servers’ additional idleness resulting from the dedicated structure will be negligible when the system scale is large. Using a fluid model substantiated by asymptotic analysis, we provide a performance comparison between the two structures for a moderately overloaded queueing system with customer abandonment. We intend to help service system designers answer the following question: To reach a specified service-level target, which queue structure will be more cost effective? Aside from structure design, our results are of practical value for performance analysis and staffing deployment.
•Short vehicle trajectories are collected and used from mobile sensors.•Missing acceleration/deceleration process is estimated from short vehicle trajectories.•Models are trained using sample short ...vehicle trajectories from multiple cycles.•The queue profile and maximum queue length of a cycle can be estimated.
Queue length is one of the key measures in assessing arterial performances. Under heavy congestion, queues are difficult to estimate from either fixed-location sensors (such as loop detectors) or mobile sensors since they may exceed the region of detection, which is defined as long queue in the literature. While the long queue problem has been successfully addressed in the past using fixed-location sensors, whether this can be done using mobile traffic sensors remains unclear. In this paper, a queue length estimation method is proposed to solve this long queue problem using short vehicle trajectories obtained from mobile sensors. The method contains vehicle trajectory reconstruction models to estimate the missing deceleration or acceleration process of a vehicle. Long queue estimation models are then developed using the reconstructed vehicle trajectories. The proposed method can provide estimates of the queue profile and the maximum queue length of a cycle. The method is tested in a field experiment with reasonable results.
Dynamic resource control and routing are important for realising the intelligent control of data transmission in wireless multi-hop networks. It is well known that back-pressure routing based on a ...max-weight policy maximises network throughput and optimises resource allocation in multi-hop wireless networks with time-varying channels. Due to the slow routing convergence and complex control of data queues, however, back-pressure routing also results in large end-to-end delays and a waste of network resources, particularly when the network loads are light or moderate. In this study, a differential back-pressure routing scheme with single-queue management is proposed to improve the packet delay performance and simplify the management of data queues. Unlike in traditional back-pressure routing, the authors use the differences in the rates of change in data queue length to calculate data pressure. Compared with the method of data backlog calculation based on queue length differences, this method achieves faster routing convergence. They also consider the ceiling problem for a single queue and enhance the effect of the queue cap on the routing metric by means of dynamic weighting. Simulation results show that the proposed routing algorithm achieves a 20% decrease in end-to-end delay in grid and random networks.
•Discriminant models and Kalman filters are used to estimate lane-based queue lengths.•Once calibrated, commonly available detector data are used to estimate queue lengths.•Simulations and case ...studies demonstrate the effectiveness of the proposed method.
In this study, we develop a real-time estimation approach for lane-based queue lengths. Our aim is to determine the numbers of queued vehicles in each lane, based on detector information at isolated signalized junctions. The challenges involved in this task are to identify whether there is a residual queue at the start time of each cycle and to determine the proportions of lane-to-lane traffic volumes in each lane. Discriminant models are developed based on time occupancy rates and impulse memories, as calculated by the detector and signal information from a set of upstream and downstream detectors. To determine the proportions of total traffic volume in each lane, the downstream arrivals for each cycle are estimated by using the Kalman filter, which is based on upstream arrivals and downstream discharges collected during the previous cycle. Both the computer simulations and the case study of real-world traffic show that the proposed method is robust and accurate for the estimation of lane-based queue lengths in real time under a wide range of traffic conditions. Calibrated discriminant models play a significant role in determining whether there are residual queued vehicles in each lane at the start time of each cycle. In addition, downstream arrivals estimated by the Kalman filter enhance the accuracy of the estimates by minimizing any error terms caused by lane-changing behavior.
It is generally accepted that operating with a combined (i.e., pooled) queue rather than separate (i.e., dedicated) queues is beneficial because pooling queues reduces long-run average sojourn time. ...In fact, this is a well-established result in the literature when jobs cannot make decisions and servers and jobs are identical. An important corollary of this finding is that pooling queues improves social welfare in the aforementioned setting. We consider an observable multiserver queueing system that can be operated with either dedicated queues or a pooled one. Customers are delay-sensitive, and they decide to join or balk based on queue length information upon arrival; they are not subject to an external admission control. In this setting, we prove that, contrary to the common understanding, pooling queues can increase the long-run average sojourn time so much that the pooled system results in strictly smaller social welfare (and strictly smaller consumer surplus) than the dedicated system under certain conditions. Specifically, pooling queues hurts performance when the arrival-rate-to-service-rate ratio is large (e.g., greater than one) and the normalized service benefit is also large. We prove that the performance loss due to pooling queues can be significant. Our numerical studies demonstrate that pooling queues can decrease the social welfare (and consumer surplus) by more than 95%. The benefit of pooling is commonly believed to increase with system size. In contrast, we show that when delay-sensitive customers make rational joining decisions, the magnitude of the performance loss due to pooling can strictly increase with the system size.
This paper was accepted by Terry Taylor, operations management.
PDF
