Heavy Traffic Regimes Research Articles

N-qubit operations on sphere and queueing scaling limits for programmable quantum computer

We study n-qubit operation rules on (n+1)-sphere with the target to help developing a (photon or other technique) based programmable quantum computer. In the meanwhile, we derive the scaling limits (called reflecting Gaussian random fields on a (n+1)-sphere) for n-qubit quantum computer based queueing systems under two different heavy traffic regimes. The queueing systems are with multiple classes of users and batch quantum random walks over the $(n+1)$-sphere as arrival inputs. In the first regime, the qubit number $n$ is fixed and the scaling is in terms of both time and space. Under this regime, performance modeling during deriving the scaling limit in terms of balancing the arrival and service rates under first-in first-out and work conserving service policy is conducted. In the second regime, besides the time and space scaling parameters, the qubit number $n$ itself is also considered as a varying scaling parameter with the additional aim to find a suitable number of qubits for the design of a quantum computer. This regime is in contrast to the well-known Halfin-Whitt regime.

Analysis of Discrete-Time Queues with Branching Arrivals

We consider a discrete-time single server queueing system, where arrivals stem from a multi-type Galton–Watson branching process with migration. This branching-type arrival process exhibits intricate correlation, and the performance of the corresponding queueing process can be assessed analytically. We find closed-form expressions for various moments of both the queue content and packet delay. Close inspection of the arrival process at hand, however, reveals that sample paths consist of large independent bursts of arrivals followed by geometrically distributed periods without arrivals. Allowing for non-geometric periods without arrivals, and correlated bursts, we apply π-thinning on the arrival process. As no closed-form expressions can be obtained for the performance of the corresponding queueing system, we focus on approximations of the main performance measures in the light and heavy traffic regimes.

Skorokhod Reflection Problem for Delayed Brownian Motion with Applications to Fractional Queues

Several queueing systems in heavy traffic regimes are shown to admit a diffusive approximation in terms of the Reflected Brownian Motion. The latter is defined by solving the Skorokhod reflection problem on the trajectories of a standard Brownian motion. In recent years, fractional queueing systems have been introduced to model a class of queueing systems with heavy-tailed interarrival and service times. In this paper, we consider a subdiffusive approximation for such processes in the heavy traffic regime. To do this, we introduce the Delayed Reflected Brownian Motion by either solving the Skorohod reflection problem on the trajectories of the delayed Brownian motion or by composing the Reflected Brownian Motion with an inverse stable subordinator. The heavy traffic limit is achieved via the continuous mapping theorem. As a further interesting consequence, we obtain a simulation algorithm for the Delayed Reflected Brownian Motion via a continuous-time random walk approximation.

Customer-Server Population Dynamics in Heavy Traffic

We study a many-server queueing model with server vacations, where the population size dynamics of servers and customers are coupled: a server may leave for vacation only when no customers await, and the capacity available to customers is directly affected by the number of servers on vacation. We focus on scaling regimes in which server dynamics and queue dynamics fluctuate at matching time scales so that their limiting dynamics are coupled. Specifically, we argue that interesting coupled dynamics occur in (a) the Halfin–Whitt regime, (b) the nondegenerate slowdown regime, and (c) the intermediate near Halfin–Whitt regime, whereas the dynamics asymptotically decouple in the other heavy-traffic regimes. We characterize the limiting dynamics, which are different for each scaling regime. We consider relevant respective performance measures for regimes (a) and (b)—namely, the probability of wait and the slowdown. Although closed-form formulas for these performance measures have been derived for models that do not accommodate server vacations, it is difficult to obtain closed-form formulas for these performance measures in the setting with server vacations. Instead, we propose formulas that approximate these performance measures and depend on the steady-state mean number of available servers and previously derived formulas for models without server vacations. We test the accuracy of these formulas numerically.

Queue and Loss Distributions in Finite-Buffer Queues

We derive simple bounds on the queue distribution in finite-buffer queues with Markovian arrivals. The bounds capture a truncated exponential behavior, involving joint horizontal and vertical shifts of an exponential function; this is fundamentally different than existing results capturing horizontal shifts only.We also obtain similar bounds on the loss distribution, which is a key metric to understand the impact of finite-buffer queues on real-time applications. Simulations show that the bounds are accurate in heavy-traffic regimes, and improve existing ones by orders of magnitude.

Time-Varying Robust Queueing

The natural queueing models for many operations research applications have time-varying arrival rates. In addition, the natural models often are not Markov stochastic processes, so that they are not amenable to exact mathematical analysis. In “Time-Varying Robust Queueing,” Ward Whitt and Wei You propose a time-varying robust queueing algorithm to approximate the time-varying distribution of the workload (virtual waiting time) in a non-Markovian single-server queue with a time-varying arrival-rate function. They apply simulation and asymptotic methods to examine the performance of periodic robust queueing. They show that periodic robust queueing converges to a proper limit in appropriate long-cycle and heavy-traffic regimes and coincides with long-cycle fluid limits and heavy-traffic diffusion limits for long cycles. Simulation examples show that the mean and the full distribution (specified by the quantiles) of the periodic steady-state workload are remarkably well approximated.

Queue and Loss Distributions in Finite-Buffer Queues

We derive simple bounds on the queue distribution in finite-buffer queues with Markovian arrivals. Our technique relies on a subtle equivalence between tail events and stopping times orderings. The bounds capture a truncated exponential behavior, involving joint horizontal and vertical shifts of an exponential function; this is fundamentally different than existing results capturing horizontal shifts only. Using the same technique, we obtain similar bounds on the loss distribution, which is a key metric to understand the impact of finite-buffer queues on real-time applications. Simulations show that the bounds are accurate in heavy-traffic regimes, and improve existing ones by orders of magnitude. In the limiting regime with utilization ρ=1 and iid arrivals, the bounds on the queue size distribution are insensitive to the arrivals distribution.

Beyond Heavy-Traffic Regimes: Universal Bounds and Controls for the Single-Server Queue

Central-limit (Brownian) approximations are widely used for the performance analysis and optimization of queueing networks because of their tractability relative to the original queueing models. The stationary distributions of the approximations are used as proxies for those of the queues. The convergence of suitably scaled and centered processes provides mathematical support for the use of these Brownian models. As with the central limit theorem, to establish convergence, one must impose assumptions directly on the primitives or indirectly on the parameters of a related optimization problem. These assumptions reflect an interpretation of the underlying parameters—a classification into so-called heavy-traffic regimes that specify a scaling relationship between the utilization and the arrival rate. Here, it matters whether a utilization of 90% in a queue with an arrival rate of λ = 100 is read as [Formula: see text] or as ρ(λ) ≡ 0.9, because different interpretations lead to different limits and, in turn, to different approximations. However, from a heuristic point of view, there is an immediate Brownian (i.e., normal) analogue of the queueing model that is derived directly from the primitives and requires no scaling interpretation of the parameters. In this model, the drift is that of the original queue, and the noise term is replaced by a Brownian motion with the same variance. This is intuitive and appealing as a tool, but it lacks mathematical justification. In this paper, we prove that for the fundamental M/GI/1 + GI queue, this direct intuitive approach works: the Brownian model is accurate uniformly over a family of patience distributions and universally in the heavy-traffic regime. The validity of this approach extends to dynamic control in that the solution of the directly derived diffusion control problem is universally accurate. To build mathematical support for the accuracy of this model, we introduce a framework built around “queue families” that allows us to treat various patience distributions simultaneously, and it uncovers the role of a concentration property of the queue. The e-companion is available at https://doi.org/10.1287/opre.2017.1715 .

An ɛ-Nash Equilibrium with High Probability for Strategic Customers in Heavy Traffic

A multiclass queue with many servers is considered, where customers make a join-or-leave decision upon arrival based on queue length information, without knowing the state of other queues. A game theoretic formulation is proposed and analyzed, that takes advantage of a phenomenon unique to heavy traffic regimes, namely, Reiman’s snaphshot principle, by which waiting times are predicted with high precision by the information available upon arrival. The payoff considered is given as a random variable, which depends on the customer’s decision, accounting for waiting time in the queue and penalty for leaving. The notion of an equilibrium is only meaningful in an asymptotic framework, which is taken here to be the Halfin-Whitt heavy traffic regime. The main result is the identification of an ɛ-Nash equilibrium with probability approaching 1. On the way to proving this result, new diffusion limit results for systems with finite buffers are obtained.

Stein's Method for Mean Field Approximations in Light and Heavy Traffic Regimes

Mean-field analysis is an analytical method for understanding large-scale stochastic systems such as large-scale data centers and communication networks. The idea is to approximate the stationary distribution of a large-scale stochastic system using the equilibrium point (called the mean-field limit) of a dynamical system (called the mean-field model). This approximation is often justified by proving the weak convergence of stationary distributions to its mean-field limit. Most existing mean-field models concerned the light-traffic regime where the load of the system, denote by ρ, is strictly less than one and is independent of the size of the system. This is because a traditional mean-field model represents the limit of the corresponding stochastic system. Therefore, the load of the mean-field model is ρ= limN→ ∞ρ(N), where ρ(N)is the load of the stochastic system of sizeN.Now if ρ(N)→ 1asN→ ∞ (i.e., in the heavy-traffic regime), then ρ=1. For most systems, the mean-field limits when ρ=1 are trivial and meaningless. To overcome this difficulty of traditional mean-field models, this paper takes a different point of view on mean-field models. Instead of regarding a mean-field model as the limiting system of large-scale stochastic system, it views the equilibrium point of the mean-field model, called a mean-field solution, simply as an approximation of the stationary distribution of the finite-size system. Therefore both mean-field models and solutions can be functions ofN.This paper first outlines an analytical method to bound the approximation error based on Stein's method and the perturbation theory. We further present two examples: theM/M/Nqueueing system and the supermarket model under the power-of-two-choices algorithm. For both applications, the method enables us to characterize the system performance under a broad range of traffic loads. For the supermarket model, this is the first paper that rigorously quantifies the steady-state performance of the-power-of-two-choices in the heavy-traffic regime. These results in the heavy-traffic regime cannot be obtained using the traditional mean-field analysis and the interchange of the limits.

Stein's Method for Mean-Field Approximations in Light and Heavy Traffic Regimes

Mean-field analysis is an analytical method for understanding large-scale stochastic systems such as large-scale data centers and communication networks. The idea is to approximate the stationary distribution of a large-scale stochastic system using the equilibrium point (called the mean-field limit) of a dynamical system (called the mean-field model). This approximation is often justified by proving the weak convergence of stationary distributions to its mean-field limit. Most existing mean-field models concerned the light-traffic regime where the load of the system, denote by ρ, is strictly less than one and is independent of the size of the system. This is because a traditional mean-field model represents the limit of the corresponding stochastic system. Therefore, the load of the mean-field model is ρ=lim N-> ∞ ρ (N) , where ρ (N) is the load of the stochastic system of size N. Now if ρ (N) -> 1 as N -> ∞ (i.e., in the heavy-traffic regime), then ρ=1. For most systems, the mean-field limits when ρ=1 are trivial and meaningless. To overcome this difficulty of traditional mean-field models, this paper takes a different point of view on mean-field models. Instead of regarding a mean-field model as the limiting system of large-scale stochastic system, it views the equilibrium point of the mean-field model, called a mean-field solution, simply as an approximation of the stationary distribution of the finite-size system. Therefore both mean-field models and solutions can be functions of N. The proposed method focuses on quantifying the approximation error. If the approximation error is small (as we will show in two applications), then we can conclude that the mean-field solution is a good approximation of the stationary distribution.

Does the Past Predict the Future? The Case of Delay Announcements in Service Systems

Motivated by the recent interest in making delay announcements in large service systems, such as call centers, we investigate the accuracy of announcing the waiting time of the last customer to enter service (LES). In practice, customers typically respond to delay announcements by either balking or by becoming more or less impatient, and their response alters system performance. We study the accuracy of the LES announcement in single-class, multiserver Markovian queueing models with announcement-dependent customer behavior. We show that, interestingly, even in this stylized setting, the LES announcement may not always be accurate. This motivates the need to study its accuracy carefully and to determine conditions under which it is accurate. Since the direct analysis of the system with customer response is prohibitively difficult, we focus on many-server, heavy-traffic analysis instead. We consider the quality-and-efficiency-driven and efficiency-driven many-server, heavy-traffic regimes and prove, under both regimes, that the LES prediction is asymptotically accurate if and only if asymptotic fluctuations in the queue length process are small as long as some regulatory conditions apply. This result provides an easy check for the accuracy of LES announcements in practice. We supplement our theoretical results with an extensive simulation study to generate practical managerial insights.Data, as supplemental material, are available at http://dx.doi.org/10.1287/mnsc.2016.2425 .This paper was accepted by Yossi Aviv, operations management.

Wireless Scheduling Design for Optimizing Both Service Regularity and Mean Delay in Heavy-Traffic Regimes

We consider the design of throughput-optimal scheduling policies in multihop wireless networks that also possess good mean delay performance and provide regular service for all links—critical metrics for real-time applications. To that end, we study a parametric class of maximum-weight-type scheduling policies, called Regular Service Guarantee (RSG) Algorithm, where each link weight consists of its own queue length and a counter that tracks the time since the last service, namely Time-Since-Last-Service (TSLS). The RSG Algorithm not only is throughput-optimal, but also achieves a tradeoff between the service regularity performance and the mean delay, i.e., the service regularity performance of the RSG Algorithm improves at the cost of increasing mean delay. This motivates us to investigate whether satisfactory service regularity and low mean-delay can be simultaneously achieved by the RSG Algorithm by carefully selecting its design parameter. To that end, we perform a novel Lyapunov-drift-based analysis of the steady-state behavior of the stochastic network. Our analysis reveals that the RSG Algorithm can minimize the total mean queue length to establish mean delay optimality under heavily loaded conditions as long as the design parameter weighting for the TSLS scales no faster than the order of $ {{1}\over {\root{5}\of {\epsilon }}}$ , where $\epsilon $ measures the closeness of the network load to the boundary of the capacity region. To the best of our knowledge, this is the first work that provides regular service to all links while also achieving heavy-traffic optimality in mean queue lengths.

Beyond Heavy-Traffic Regimes: Universal Bounds and Controls for the Single-Server Queue

Brownian approximations are widely studied because of their tractability relative to the original queueing models. Their stationary distributions are used as proxies for those of the original queues that they model and the convergence of suitably scaled and centered processes provides mathematical support for the use of these Brownian models. To establish convergence, one must impose assumptions directly on the primitives or, indirectly, on the parameters of a related optimization problem. These assumptions reflect an interpretation of the underlying parameters -- a classification into so-called heavy-traffic regimes that specify a scaling relationship between the utilization and the arrival rate. From a heuristic point of view, though, there is an almost immediate Brownian analogue of the queueing model that is derived directly from the primitives and requires no (limit) interpretation of the parameters. In this paper we prove that for the fundamental M/GI/1 GI queue, the direct intuitive (limitless approach) in fact works. The Brownian model is universally (i.e., across regimes and patience distributions) accurate. It maintains the tractability and appeal of the limit approximations while avoiding many of the assumptions that facilitate them. In the process of building mathematical support for the accuracy of this model, we introduce a framework built around queue families that allows us to treat various patience distributions simultaneously, and uncovers the role of a concentration property of the queue.

A note on non-existence of diffusion limits for serve-the-longest-queue when the buffers are equal in size

We consider the serve-the-longest-queue discipline for a multiclass queue with buffers of equal size, operating under (i) the conventional and (ii) the Halfin-Whitt heavy traffic regimes, and show that while the queue length process’ scaling limits are fully determined by the first and second order data in case (i), they depend on finer properties in case (ii). The proof of the latter relies on the construction of a deterministic arrival pattern.

Asymptotically optimal index policies for an abandonment queue with convex holding cost

We investigate a resource allocation problem in a multi-class server with convex holding costs and user impatience under the average cost criterion. In general, the optimal policy has a complex dependency on all the input parameters and state information. Our main contribution is to derive index policies that can serve as heuristics and are shown to give good performance. Our index policy attributes to each class an index, which depends on the number of customers currently present in that class. The index values are obtained by solving a relaxed version of the optimal stochastic control problem and combining results from restless multi-armed bandits and queueing theory. They can be expressed as a function of the steady-state distribution probabilities of a one-dimensional birth-and-death process. For linear holding cost, the index can be calculated in closed-form and turns out to be independent of the arrival rates and the number of customers present. In the case of no abandonments and linear holding cost, our index coincides with the $$c\mu $$cμ-rule, which is known to be optimal in this simple setting. For general convex holding cost, we derive properties of the index value in limiting regimes: we consider the behavior of the index (i) as the number of customers in a class grows large, which allows us to derive the asymptotic structure of the index policies, (ii) as the abandonment rate vanishes, which allows us to retrieve an index policy proposed for the multi-class M/M/1 queue with convex holding cost and no abandonments, and (iii) as the arrival rate goes to either 0 or $$\infty $$?, representing light-traffic and heavy-traffic regimes, respectively. We show that Whittle's index policy is asymptotically optimal in both light-traffic and heavy-traffic regimes. To obtain further insights into the index policy, we consider the fluid version of the relaxed problem and derive a closed-form expression for the fluid index. The latter is shown to coincide with the index values for the stochastic model in asymptotic regimes. For arbitrary convex holding cost the fluid index can be seen as the $$Gc\mu /\theta $$Gcμ/?-rule; that is, including abandonments into the generalized $$c\mu $$cμ-rule ($$Gc\mu $$Gcμ-rule). Numerical experiments for a wide range of parameters have shown that the Whittle index policy and the fluid index policy perform very well for a broad range of parameters.

Novel Heavy-Traffic Regimes for Large-Scale Service Systems

We introduce a family of heavy-traffic regimes for large-scale service systems, presenting a range of scalings that include both moderate and extreme heavy traffic, as compared to classical heavy traffic. The heavy-traffic regimes can be translated into capacity sizing rules that lead to economies-of-scales, so that the system utilization approaches 100% while congestion remains limited. We obtain heavy-traffic approximations for stationary performance measures in terms of asymptotic expansions, using a nonstandard saddle point method, tailored to the specific form of integral expressions for the performance measures, in combination with the heavy-traffic regimes.

Excursion-Based Universal Approximations for the Erlang-A Queue in Steady-State

We revisit many-server approximations for the well-studied Erlang-A queue. This is a system with a single pool of i.i.d. servers that serve one class of impatient i.i.d. customers. Arrivals follow a Poisson process and service times are exponentially distributed as are the customers' patience times. We propose a diffusion approximation that applies simultaneously to all existing many-server heavy-traffic regimes: quality and efficiency driven, efficiency driven, quality driven, and nondegenerate slowdown. We prove that the approximation provides accurate estimates for a broad family of steady-state metrics. Our approach is “metric-free” in that we do not use the specific formulas for the steady-state distribution of the Erlang-A queue. Rather, we study excursions of the underlying birth-and-death process and couple these to properly defined excursions of the corresponding diffusion process. Regenerative process and martingale arguments, together with derivative bounds for solutions to certain ordinary differential equations, allow us to control the accuracy of the approximation. We demonstrate the appeal of universal approximation by studying two staffing optimization problems of practical interest.

On light and heavy traffic approximations of balanced fairness

Flow level analysis of communication networks with multiple shared resources is generally difficult. A recently introduced sharing scheme called balanced fairness has brought these systems within the realm of tractability. While straightforward in principle, the numerical evaluation of practically interesting performance metrics like per-flow throughput is feasible for limited state spaces only, besides some specific networks where the results are explicit. In the present paper, we study the behaviour of balanced fairness in light and heavy traffic regimes and show how the corresponding performance results can be used to approximate the flow throughput over the whole load range. The results apply to any network, with a state space of arbitrary dimension. A few examples are explicitly worked out to illustrate the concepts.

Deferred Assignment Scheduling in Cluster-Based Servers

This paper proposes a new scheduling policy for cluster-based servers called DAS (Deferred Assignment Scheduling). The main idea in DAS is to defer scheduling as much as possible in order to make better use of the accumulated information on job sizes. In broad outline, DAS operates as follows: (1) incoming jobs are held by the dispatcher in a buffer; (2) the dispatcher monitors the number of jobs being processed by each server; (3) when the number of jobs at a server queue drops below a prescribed threshold, the dispatcher sends to it the shortest job in its buffer. To gauge the efficacy of DAS, the paper presents simulation studies, using various data traces. The studies collected response times and slowdowns for two cluster configurations under multi-threaded and multi-process back-end server architectures. The experimental results show that in both architectures, DAS outperforms the Round-Robin policy in all traffic regimes, and the JSQ (Join Shortest Queue) policy in medium and heavy traffic regimes.