Queues with Redundancy: Is Waiting in Multiple Lines Fair?

Problem definition: We study service systems where some (so-called “redundant”) customers join multiple queues simultaneously, enabling them to receive service in any one of the queues, while other customers join a single queue. Academic/practical relevance: The improvement in overall system performance due to redundant customers has been established in prior work. We address the question of fairness—whether the benefit experienced by redundant customers adversely affects others who can only join a single line. This question is particularly relevant to organ transplantation, as critics have contended that multiple listing provides unfair access to organs for patients based on wealth. Methodology: We analyze two queues serving two classes of customers; the redundant class joins both queues, whereas the nonredundant class joins a single queue randomly. We compare this system against a benchmark wherein the redundant class resorts to joining the shortest queue (JSQ) if multiple queue joining were not allowed, capturing the most likely case if multilisting was prohibited: Affluent patients could still afford to list in the region with the shorter wait list. Results: We prove that when the arrival rate of nonredundant customers is balanced across both queues, they actually benefit under redundancy of the other class—that is, redundancy is fair. We also establish that redundancy may be unfair under some circumstances: Nonredundant customers are worse off if their arrival rate is strongly skewed toward one of the queues. We illustrate how these findings apply in the organ-transplantation setting through a numerical study using publicly available data. Managerial implications: Our analysis helps identify when, and by how much, multiple listing may be unfair and, as such, could be a useful tool for policy makers who may be concerned with trying to ensure equitable access to resources, such as organs, across patients with differing wealth levels.

Robotic Sorting Systems: Performance Estimation and Operating Policies Analysis

Many distribution centers use expensive, conveyor-based sorting systems that require large buildings to house them. In areas with tight space, robotic sorting systems offer a new type of solution to sort parcels by destination. Such systems are highly flexible in throughput capacity and are now gradually being introduced, particularly in express companies. This paper studies robotic sorting system with two layouts. The first layout has two tiers: robots drive on the top tier and sort parcels by destination on spiral conveyors connected to roll containers at the lower tier. The second layout has a single tier with input and output points located at the perimeter, connected by robots. For each layout, we consider both the shortest path topology via dual-lane aisles and the detour path topology via single-lane aisles. We build closed queueing networks for performance estimation, design an iterative procedure to investigate robot congestion in the two-tier layout, and use a traffic flow function to estimate robot congestion in the single-tier layout. Random, closest, dedicated, and shortest-queue robot-to-loading-station assignment rules are examined. We validate analytical models by both simulation and a real case of Deppon Express and analyze the optimal system size and operating policies for throughput capacity and operating cost. The results show that the system throughput capacity is significantly affected by robot congestion in the single-tier layout with the detour path topology, but it is only slightly affected in the other systems. A square layout fits the shortest path and a rectangular layout fits the detour path. Both the random assignment rule and the shortest-queue assignment rule are superior for a large number of robots, whereas the dedicated assignment rule is superior for a small number of robots. We apply these insights at Deppon Express for different allocations in peak and off-peak hours. Our analysis shows that a robotic sorting system typically has lower overall annual cost than a traditional cross-belt sorting system when the required throughput capacity is not too large.

Magnitude integration in the Archerfish

We make magnitude-related decisions every day, for example, to choose the shortest queue at the grocery store. When making such decisions, which magnitudes do we consider? The dominant theory suggests that our focus is on numerical quantity, i.e., the number of items in a set. This theory leads to quantity-focused research suggesting that discriminating quantities is automatic, innate, and is the basis for mathematical abilities in humans. Another theory suggests, instead, that non-numerical magnitudes, such as the total area of the compared items, are usually what humans rely on, and numerical quantity is used only when required. Since wild animals must make quick magnitude-related decisions to eat, seek shelter, survive, and procreate, studying which magnitudes animals spontaneously use in magnitude-related decisions is a good way to study the relative primacy of numerical quantity versus non-numerical magnitudes. We asked whether, in an animal model, the influence of non-numerical magnitudes on performance in a spontaneous magnitude comparison task is modulated by the number of non-numerical magnitudes that positively correlate with numerical quantity. Our animal model was the Archerfish, a fish that, in the wild, hunts insects by shooting a jet of water at them. These fish were trained to shoot water at artificial targets presented on a computer screen above the water tank. We tested the Archerfish's performance in spontaneous, untrained two-choice magnitude decisions. We found that the fish tended to select the group containing larger non-numerical magnitudes and smaller quantities of dots. The fish selected the group containing more dots mostly when the quantity of the dots was positively correlated with all five different non-numerical magnitudes. The current study adds to the body of studies providing direct evidence that in some cases animals’ magnitude-related decisions are more affected by non-numerical magnitudes than by numerical quantity, putting doubt on the claims that numerical quantity perception is the most basic building block of mathematical abilities.

Hospital selection in emergency medical service systems: A literature review

Emergency medical service (EMS) systems play a fundamental role in society by providing a vital service in initial emergency care. The purpose of this research is to present the first literature review of hospital selection operational decision within the context of the EMS system. The main findings were the following: The hospital selection problem is integrated with the location, dispatch, routing, and size of the ambulance fleet. The main selection criteria were closeness, hospital care capacities and the shortest queue or greatest number of free beds. The most used performance measures were the shortest transport and waiting time. Solution techniques include discrete event simulation, queuing models, mixed linear integer programming, and CPLEX and Arena software. The application of metaheuristics is scarce; mobile applications and Internet information systems have been implemented for real-time decision making. It is recommended that the design of hospital selection methods be implemented as well as the technological developments, considering the participation of the actors of the EMS system.

Hospital selection in emergency medical service systems: A literature review

Emergency medical service (EMS) systems play a fundamental role in society by providing a vital service in initial emergency care. The purpose of this research is to present the first literature review of hospital selection operational decision within the context of the EMS system. The main findings were the following: The hospital selection problem is integrated with the location, dispatch, routing, and size of the ambulance fleet. The main selection criteria were closeness, hospital care capacities and the shortest queue or greatest number of free beds. The most used performance measures were the shortest transport and waiting time. Solution techniques include discrete event simulation, queuing models, mixed linear integer programming, and CPLEX and Arena software. The application of metaheuristics is scarce; mobile applications and Internet information systems have been implemented for real-time decision making. It is recommended that the design of hospital selection methods be implemented as well as the technological developments, considering the participation of the actors of the EMS system.

Sensitivity of mean-field fluctuations in Erlang loss models with randomized routing

In this paper we study a large system of N servers, each with capacity to process at most C simultaneous jobs; an incoming job is routed to a server if it has the lowest occupancy amongst d (out of N) randomly selected servers. A job that is routed to a server with no vacancy is assumed to be blocked and lost. Such randomized policies are referred to JSQ(d) (Join the Shortest Queue out of d) policies. Under the assumption that jobs arrive according to a Poisson process with rate $N\lambda^{(N)}$ where $\lambda^{(N)}=\sigma-\frac{\beta}{\sqrt{N}\,}$ , $\sigma\in\mathbb{R}_+$ and $\beta\in\mathbb{R}$ , we establish functional central limit theorems for the fluctuation process in both the transient and stationary regimes when service time distributions are exponential. In particular, we show that the limit is an Ornstein–Uhlenbeck process whose mean and variance depend on the mean field of the considered model. Using this, we obtain approximations to the blocking probabilities for large N, where we can precisely estimate the accuracy of first-order approximations.

Asymptotic Optimality of Power-of-d Load Balancing in Large-Scale Systems

We consider a system of N identical server pools and a single dispatcher in which tasks with unit-exponential service requirements arrive at rate [Formula: see text]. In order to optimize the experienced performance, the dispatcher aims to evenly distribute the tasks across the various server pools. Specifically, when a task arrives, the dispatcher assigns it to the server pool with the minimum number of tasks among d(N) randomly selected server pools. We construct a stochastic coupling to bound the difference in the system occupancy processes between the join-the-shortest-queue (JSQ) policy and a scheme with an arbitrary value of d(N). We use the coupling to derive the fluid limit in case [Formula: see text] and [Formula: see text] as [Formula: see text] along with the associated fixed point. The fluid limit turns out to be insensitive to the exact growth rate of d(N) and coincides with that for the JSQ policy. We further establish that the diffusion limit corresponds to that for the JSQ policy as well, as long as [Formula: see text], and characterize the common limiting diffusion process. These results indicate that the JSQ optimality can be preserved at the fluid and diffusion levels while reducing the overhead by nearly a factor O(N) and O([Formula: see text]), respectively.

Steady-State Analysis of the Join-the-Shortest-Queue Model in the Halfin–Whitt Regime

This paper studies the steady-state properties of the join-the-shortest-queue model in the Halfin–Whitt regime. We focus on the process tracking the number of idle servers and the number of servers with nonempty buffers. Recently, Eschenfeldt and Gamarnik proved that a scaled version of this process converges, over finite time intervals, to a two-dimensional diffusion limit as the number of servers goes to infinity. In this paper, we prove that the diffusion limit is exponentially ergodic and that the diffusion scaled sequence of the steady-state number of idle servers and nonempty buffers is tight. Combined with the process-level convergence proved by Eschenfeldt and Gamarnik, our results imply convergence of steady-state distributions. The methodology used is the generator expansion framework based on Stein’s method, also referred to as the drift-based fluid limit Lyapunov function approach in Stolyar. One technical contribution to the framework is to show how it can be used as a general tool to establish exponential ergodicity.