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We consider a queueing network in which there are constraints on which queues may be served simultaneously; such networks may be used to model input-queued switches and wireless networks. The scheduling policy for such a network specifies which queues to serve at any point in time. We consider a family of scheduling policies, related to the maximum-weight policy of Tassiulas and Ephremides [IEEE Trans. Automat. Control 37 (1992) 1936-1948], for single-hop and multihop networks. We specify a fluid model and show that fluid-scaled performance processes can be approximated by fluid model solutions. We study the behavior of fluid model solutions under critical load, and characterize invariant states as those states which solve a certain network-wide optimization problem. We use fluid model results to prove multiplicative state space collapse. A notable feature of our results is that they do not assume complete resource pooling.

Motivated by queues with many servers, we study Brownian steady-state approximations for continuous time Markov chains (CTMCs). Our approximations are based on diffusion models (rather than a diffusion limit) whose steady-state, we prove, approximates that of the Markov chain with notable precision. Strong approximations provide such \"limitless\" approximations for process dynamics. Our focus here is on steady-state distributions, and the diffusion model that we propose is tractable relative to strong approximations. Within an asymptotic framework, in which a scale parameter n is taken large, a uniform (in the scale parameter) Lyapunov condition imposed on the sequence of diffusion models guarantees that the gap between the steady-state moments of the diffusion and those of the properly centered and scaled CTMCs shrinks at a rate of √n. Our proofs build on gradient estimates for solutions of the Poisson equations associated with the (sequence of) diffusion models and on elementary martingale arguments. As a by-product of our analysis, we explore connections between Lyapunov functions for the fluid model, the diffusion model and the CTMC.

This paper develops a Gaussian model for an open network of queues having a path-dependent net-input process, whose evolution depends on its early history, and satisfies a non-ergodic law of large numbers. We show that the Gaussian model arises as the heavy-traffic limit for a sequence of open queueing networks, each with a multivariate generalization of a Polya arrival process. We show that the net-input and queue-length processes for the Gaussian model satisfy non-ergodic laws of large numbers with tractable distributions.

In this paper, we develop a diffusion approximation for the transient distribution of the workload process in a standard single-server queue with a non-stationary Polya arrival process, which is a path-dependent Markov point process. The path-dependent arrival process model is useful because it has the arrival rate depending on the history of the arrival process, thus capturing a self-reinforcing property that one might expect in some applications. The workload approximation is based on heavy-traffic limits for (i) a sequence of Polya processes, in which the limit is a Gaussian–Markov process, and (ii) a sequence of P/GI/1 queues in which the arrival rate function approaches a constant service rate uniformly over compact intervals.

In this study, we explore optimal service allocation within the Chinese hierarchical healthcare system with green channels, providing valuable insights for practitioners to understand how optimal service allocation is affected by various realistic factors. These green channels are designed to streamline referrals from community healthcare centers to comprehensive hospitals. We aim to determine the optimal capacity allocation for these green channels within comprehensive hospitals. Our research employs techniques from queuing theory and stochastic processes, e.g., diffusion analysis, to develop a mathematical model that approximates the optimal allocation of resources. We uncover both closed-form and numerical solutions for this optimal capacity allocation. By analyzing the impact of various cost factors, we find that an increase in costs within the green channel results in a lower optimal service rate. Additionally, patient preferences for specific treatments influence allocation, reducing the optimal share of services provided by general hospitals. The optimal solution is also affected by the proportions of different patient types. Through extensive simulations, we validate the accuracy of our model approximations under heavy traffic conditions, further examining sources of error to ensure robustness. Our findings provide valuable insights into optimizing resource allocation in hierarchical healthcare systems, ensuring efficient and cost-effective patient care.

A system manager makes dynamic pricing and dispatch control decisions in a queueing network model motivated by ride hailing applications. A novel feature of the model is that it incorporates travel times. Unfortunately, this renders the exact analysis of the problem intractable. Therefore, we study this problem in the heavy traffic regime. Under the assumptions of complete resource pooling and common travel time and routing distributions, we solve the problem in closed form by analyzing the corresponding Bellman equation. Using this solution, we propose a policy for the queueing system and illustrate its effectiveness in a simulation study.

In Braverman et al. [3], the authors justify the steady-state diffusion approximation of a multiclass queueing network under static buffer priority policy in heavy traffic. A major assumption in [3] is the moment state space collapse (moment-SSC) property of the steady-state queue length. In this paper, we prove that moment-SSC holds under a corresponding state space collapse condition on the fluid model. Our approach is inspired by Dai and Meyn [8], which was later adopted by Budhiraja and Lee [4] to justify the diffusion approximation for generalized Jackson networks. We will verify that the fluid state space collapse holds for various networks.

We study a double-ended queue consisting of two classes of customers. Whenever there is a pair of customers from both classes, they are matched and leave the system. The matching is instantaneous following the first-come–first-match principle. If a customer cannot be matched immediately, he/she will stay in a queue. We also assume customers are impatient with generally distributed patience times. Under suitable heavy traffic conditions, we establish simple linear asymptotic relationships between the diffusion-scaled queue length process and the diffusion-scaled offered waiting time processes and show that the diffusion-scaled queue length process converges weakly to a diffusion process that admits a unique stationary distribution.

Upon arrival to a ticket queue, a customer is offered a slip of paper with a number on it—indicating the order of arrival to the system—and is told the number of the customer currently in service. The arriving customer then chooses whether to take the slip or balk, a decision based on the perceived queue length and associated waiting time. Even after taking a ticket, a customer may abandon the queue, an event that will be unobservable until the abandoning customer would have begun service. In contrast, a standard queue has a physical waiting area so that abandonment is apparent immediately when it takes place and balking is based on the actual queue length at the time of arrival. We prove heavy traffic limit theorems for the generalized ticket and standard queueing processes, discovering that the processes converge together to the same limit, a regulated Ornstein–Uhlenbeck process. One conclusion is that for a highly utilized service system with a relatively patient customer population, the ticket and standard queue performances are asymptotically indistinguishable on the scale typically uncovered under heavy traffic approaches. Next, we heuristically estimate several performance metrics of the ticket queue, some of which are of a sensitivity typically undetectable under diffusion scaling. The estimates are tested using simulation and are shown to be quite accurate under a general collection of parameter settings.

We study the transient and limiting behavior of a queue with a Pólya arrival process. The Pólya process is interesting because it exhibits path-dependent behavior, e.g. it satisfies a non-ergodic law of large numbers: the average number of arrivals over time [0, t] converges almost surely to a nondegenerate limit as $t \\rightarrow \\infty$. We establish a heavy-traffic diffusion limit for the $\\sum_{i=1}^{n} P_i/GI/1$ queue, with arrivals occurring exogenously according to the superposition of n independent and identically distributed Pólya point processes. That limit yields a tractable approximation for the transient queue-length distribution, because the limiting net input process is a Gaussian Markov process with stationary increments. We also provide insight into the long-run performance of queues with path-dependent arrival processes. We show how Little’s law can be stated in this context, and we provide conditions under which there is stability for a queue with a Pólya arrival process.