Steady-state Queue Research Articles

Multi-dimensional State Space Collapse in Non-complete Resource Pooling Scenarios

We establish an explicit multi-dimensional state space collapse (SSC) for parallel-processing systems with arbitrary compatibility constraints between servers and job types. This breaks major new ground beyond the SSC results and queue length asymptotics in the literature which are largely restricted to complete resource pooling (CRP) scenarios where the steady-state queue length vector concentrates around a line in heavy traffic. The multi-dimensional SSC that we establish reveals heavy-traffic behavior which is also far more tractable than the pre-limit queue length distribution, yet exhibits a fundamentally more intricate structure than in the one-dimensional case.

Multi-dimensional State Space Collapse in Non-complete Resource Pooling Scenarios

The present paper establishes an explicit multi-dimensional state space collapse (SSC) for parallel-processing systems with arbitrary compatibility constraints between servers and job types. This breaks major new ground beyond the SSC results and queue length asymptotics in the literature which are largely restricted to complete resource pooling (CRP) scenarios where the steady-state queue length vector concentrates around a line in heavy traffic. The multi-dimensional SSC that we establish reveals heavy-traffic behavior which is also far more tractable than the pre-limit queue length distribution, yet exhibits a fundamentally more intricate structure than in the one-dimensional case, providing useful insight into the system dynamics. In particular, we prove that the limiting queue length vector lives in a K-dimensional cone of which the set of spanning vectors is random in general, capturing the delicate interplay between the various job types and servers. For a broad class of systems we provide a further simplification which shows that the collection of random cones constitutes a fixed K-dimensional cone, resulting in a K-dimensional SSC. The dimension~K represents the number of critically loaded subsystems, or equivalently, capacity bottlenecks in heavy-traffic, with K=1 corresponding to conventional CRP scenarios. Our approach leverages probability generating function (PGF) expressions for Markovian systems operating under redundancy policies.

Simple and Explicit Bounds for Multiserver Queues with 11−ρ Scaling

We consider the first-come-first-serve (FCFS) [Formula: see text] queue and prove the first simple and explicit bounds that scale as [Formula: see text] under only the assumption that interarrival times have finite second moment, and service times have finite [Formula: see text] moment for some [Formula: see text]. Here, ρ denotes the corresponding traffic intensity. Conceptually, our results can be viewed as a multiserver analogue of Kingman’s bound. Our main results are bounds for the tail of the steady-state queue length and the steady-state probability of delay. The strength of our bounds (e.g., in the form of tail decay rate) is a function of how many moments of the service distribution are assumed finite. Our bounds scale gracefully, even when the number of servers grows large and the traffic intensity converges to unity simultaneously, as in the Halfin-Whitt scaling regime. Some of our bounds scale better than [Formula: see text] in certain asymptotic regimes. In these same asymptotic regimes, we also prove bounds for the tail of the steady-state number in service. Our main proofs proceed by explicitly analyzing the bounding process that arises in the stochastic comparison bounds of Gamarnik and Goldberg for multiserver queues. Along the way, we derive several novel results for suprema of random walks and pooled renewal processes, which may be of independent interest. We also prove several additional bounds using drift arguments (which have much smaller prefactors) and point out a conjecture that would imply further related bounds and generalizations. We also show that when all moments of the service distribution are finite and satisfy a mild growth rate assumption, our bounds can be strengthened to yield explicit tail estimates decaying as [Formula: see text], with [Formula: see text], depending on the growth rate of these moments. Funding: Financial support from the National Science Foundation [Grant 1333457] is gratefully acknowledged. Supplemental Material: The supplemental appendix is available at https://doi.org/10.1287/moor.2022.0131 .

Optimization of Inventory and Capacity in Large-Scale Assembly Systems Using Extreme-Value Theory

High-tech systems are typically produced in two stages: (1) production of components using specialized equipment and staff and (2) system assembly/integration. Component production capacity is subject to fluctuations, causing a high risk of shortages of at least one component, which results in costly delays. Companies hedge this risk by strategic investments in excess production capacity and in buffer inventories of components. To optimize these, it is crucial to characterize the relation between component shortage risk and capacity and inventory investments. We suppose that component production capacity and produce demand are normally distributed over finite time intervals, and we accordingly model the production system as a symmetric fork-join queueing network with N statistically identical queues with a common arrival process and independent service processes. Assuming a symmetric cost structure, we subsequently apply extreme value theory to gain analytic insights into this optimization problem. We derive several new results for this queueing network, notably that the scaled maximum of N steady-state queue lengths converges in distribution to a Gaussian random variable. These results translate into asymptotically optimal methods to dimension the system. Tests on a range of problems reveal that these methods typically work well for systems of moderate size. Funding: This work is part of the research program Complexity in High-Tech Manufacturing, (partly) financed by the Dutch Research Council (NWO) [Grant 438.16.121]. The research is also supported by the NWO programs MEERVOUD to M. Vlasiou [Grant 632.003.002] and Talent VICI to B. Zwart [Grant 639.033.413].

Performance Modeling of Hyperledger Fabric 2.0: A Queuing Theory-Based Approach

Hyperledger Fabric (shortened to Fabric) is an open-source, enterprise-level, permissioned distributed ledger technology platform with a highly modular, configurable architecture. It supports writing smart contracts in general-purpose programing languages and has become the preferred choice for enterprise-level blockchain applications. However, the transaction throughput of the Fabric system remains a critical factor that restricts the further application of this technology in various fields. Therefore, it is necessary to evaluate and optimize the performance of the Fabric blockchain platform. Existing performance modeling methods need to be improved in terms of compatibility and effectiveness. To address this, we propose a performance-compatible modeling method for Fabric using queuing theory, which considers the limited transaction pool and the situation where node groups are attacked. Using the Fabric 2.0 version as an example, we have established a model of the transaction process in the Fabric network. By analyzing the model’s continuous 3D time Markov process, we solved the system stationary equation and obtained analytical expressions for performance indicators such as system throughput, system steady-state queue length, and system average response time. We conducted extensive analyses and simulations to verify the models’ and formulations’ accuracy and validity. We believe this approach can be extended to various scenarios in other blockchain systems.

Performance analysis and ANFIS computing of an unreliable Markovian feedback queueing model under a hybrid vacation policy

This study covers the stationary analysis of an unreliable Markovian queueing system with feedback based on a hybrid vacation policy. Customers arrive at the station at random using a Poisson process. Service time is distributed exponentially. The necessary and sufficient requirements for system stability have been demonstrated. Many performance metrics of this queueing system, such as the stationary queue length distribution, have been obtained from the steady-state probability distribution of the queueing model and the steady-state queue length using the matrix geometric method. To establish the model’s performance indicators, a few formulas have also been developed. The Adaptive Neuro-Fuzzy Inference System (ANFIS) was used to develop rules from the input datasets and generate different performance indices. Finally, the impact of system parameters and cost analysis was studied using numerical examples.

Queueing modeling and optimization of a fault-tolerant system with reboot, recovery, and vacationing server operating under admission control policy

This article proposes a finite population Markovian queueing model with discouragement and vacationing server to examine the performance of the fault-tolerant system in which admission of failed machines for repair jobs is controlled according to F-policy. When the number of broken-down machines in such systems reaches to its maximum capacity, it does not allow the failed machines to enter in the system until the counts of broken-down machines drop to a pre-specified threshold level ‘F’. The feature of vacationing server along with set up, reboot and recovery are incorporated to deal with real-time scenarios of fault-tolerant systems. By considering the birth–death process, the governing equations are constructed using appropriate transition rates of the concerned system. The steady-state queue size distribution and various system metrics are evaluated using a recursive approach. To illustrate the usage of analytical findings, sensitivity analysis of the system metrics with regard to specific parameters is presented. The particle swarm optimization and harmony search optimization techniques are employed to determine the optimal decision parameters and optimum total cost.

Clearing function in the context of the invariant manifold method

Clearing functions (CFs), which express a mathematical relationship between the expected throughput of a production facility in a planning period and its workload (or work-inprogress, WIP) in that period have shown considerable promise for modeling WIP-dependent cycle times in production planning. While steady-state queueing models are commonly used to derive analytic expressions for CFs, the finite length of planning periods calls their validity into question. We apply a different approach to propose a mechanistic model for one-resource, one-product factory shop based on the analogy between the operation of machine and enzyme molecule. The model is reduced to a singularly perturbed system of two differential equations for slow (WIP) and fast (busy machines) variables, respectively. The analysis of this slow-fast system finds that CF is nothing but a result of the asymptotic expansion of the slow invariant manifold. The validity of CF is ultimately determined by how small is the parameter multiplying the derivative of the fast variable. It is shown that sufficiently small characteristic ratio ’working machines: WIP’ guarantees the applicability of CF approximation in unsteady-state operation.

Back-Pressure traffic Signal Control in the Presence of Noisy Queue Information*

In this paper, we consider centralized traffic signal control policies using the max-weight algorithm when the queue size measurement is noisy. We first show analytically that the standard max-weight algorithm is throughput optimal even under noisy queue measurements. However, the average steady-state queue lengths and subsequently the average delays are increased. In order to alleviate the effect of these noisy measurements we add filtering to the max-weight algorithm; more specifically, we propose the Filtered-max-weight algorithm, which is based on particle filtering. We demonstrate via simulations that the Filtered-max-weight algorithm performs better than the standard max-weight algorithm in the presence of noisy measurements.

Low-complexity scheduling algorithms with constant queue length and throughput guarantees

Distributed scheduling algorithms based on carrier sense multiple access (CSMA) are optimal in terms of the throughput and the steady-state queue lengths. However, they take a prohibitively long time to reach the steady-state, often exponential in the network size. Therefore for large networks that operate over a finite time horizon, apart from the guarantees on the steady-state queue lengths, performance guarantees on the short-term (i.e., transient) queuing behaviour are also required. To that end, we propose distributed scheduling algorithms that are guaranteed to have O(1) expected queue lengths not just in the steady-state but at every time instant, where O(⋅) is with respect to the network size. Further, our algorithms have O(1) complexity and support a constant fraction of the maximum throughput for typical wireless topologies. The central idea of our algorithms is to resolve collisions among pairs of conflicting nodes by assigning a master–follower hierarchy. The master–follower hierarchy can either be chosen randomly or based on the topology of the conflict graph, leading to different performance guarantees.In addition to these hierarchical collision resolution algorithms, which are primarily designed for the conflict graph-based interference model, we also propose an Aloha-based algorithm for the K-neighbour collision tolerance interference model, which is a generalization of the conflict graph model. We show that the proposed Aloha-based algorithm supports a constant fraction of the maximum throughput for typical wireless topologies.

Queue Size Distribution on a New ND Policy Geo/G/1 Queue and Its Computation Designs

This paper examines a new ND policy in the discrete-time Geo/G/1 queue. Under this ND policy, the idle server restarts its service when the N and D policies are simultaneously satisfied. By two classifications of the customers, the probability-generating function and the probabilistic analysis, the steady-state queue size distributions at a departure time and an arbitrary time t + are studied. Finally, the theoretical results are applied to the power-saving problem of a wireless sensor network. To improve model universality and numerical slowness, some computation designs are carried out. Under the N, D, and two ND policies, the numerical experiments are presented to obtain the optimal policy thresholds and the corresponding minimum power consumptions are compared.

Heavy traffic queue length scaling in switches with reconfiguration delay

The Adaptive MaxWeight policy achieves optimal throughput for switches with nonzero reconfiguration delay and has been shown to have good delay performance in simulation. In this paper, we analyze the queue length behavior of a switch with nonzero reconfiguration delay operating under the Adaptive MaxWeight. We first show that the Adaptive MaxWeight policy exhibits a weak state space collapse behavior in steady state, which can be viewed as an inheritance of a similar property of the MaxWeight policy in a switch with zero reconfiguration delay. The weak state space collapse result is then utilized to obtain an asymptotically tight bound on an expression involving the steady-state queue length and the probability of reconfiguration for the Adaptive MaxWeight policy in the heavy traffic regime. We then derive the relation between the expected schedule duration and the steady-state queue length through Lyapunov drift analysis and characterize bounds for the expected steady-state queue length. While the resulting queue length bounds are not asymptotically tight, they suggest an approximate queue length scaling behavior, which approaches the optimal scaling with respect to the traffic load and the reconfiguration delay when the hysteresis function of the Adaptive MaxWeight policy approaches a linear function.

Large deviations for acyclic networks of queues with correlated Gaussian inputs

We consider an acyclic network of single-server queues with heterogeneous processing rates. It is assumed that each queue is fed by the superposition of a large number of i.i.d. Gaussian processes with stationary increments and positive drifts, which can be correlated across different queues. The flow of work departing from each server is split deterministically and routed to its neighbors according to a fixed routing matrix, with a fraction of it leaving the network altogether. We study the exponential decay rate of the probability that the steady-state queue length at any given node in the network is above any fixed threshold, also referred to as the ‘overflow probability’. In particular, we first leverage Schilder’s sample-path large deviations theorem to obtain a general lower bound for the limit of this exponential decay rate, as the number of Gaussian processes goes to infinity. Then, we show that this lower bound is tight under additional technical conditions. Finally, we show that if the input processes to the different queues are nonnegatively correlated, non-short-range dependent fractional Brownian motions, and if the processing rates are large enough, then the asymptotic exponential decay rates of the queues coincide with the ones of isolated queues with appropriate Gaussian inputs.

Optimal Profit Analysis of Machine Repair Problem with Repair in Phases and Organizational Delay

In this article, we study machine repair problems (MRP) consisting of the finite number of operating machines with the provisioning of the finite number of warm standby machines under the care of a single unreliable server. For the machining system’s uninterrupted functioning, an operating machine is immediately replaced with the available warm standby machine in negligible switchover time whenever it fails. The concept of threshold vacation policy: N-policy is also considered. Under this vacation policy, the server starts to serve the failed machines on the accumulation of a pre-specified number of failed machines in the system. The server continues until the system is empty from the failed machines; after that, the server goes for vacation. The notion of an organizational delay, server breakdown, and repair in multiple phases is also conceptualized to build the studied model more realistic. The recursive matrix method is used to find steady-state queue size distribution, and subsequently, various system performance measures are also developed to validate the studied model. The optimal analysis has been performed to identify the critical design parameters for the governing model. The state-of-the-art of the present study is its mathematical modeling of the multi-machine stochastic problem with varied limitations and strategies. The methodology to obtain queue size distribution, optimal design parameters, is beneficial for dealing with other complex and sophisticated real-time machining problems in the service system, computer and communication system, manufacturing and production system, etc. The present problem is limited to fewer machines, which can be extended to more machines with different topologies with high computational facilities.

The Effect of Flexible Service Rates in Appointment Scheduling Systems

Recent research offers overwhelming evidence that service workers, including healthcare providers in ambulatory care settings, exercise flexible service rates. While theoretical models of service rate control in steady-state queues explain service rate changes in response to queue length, there are no comparable models explaining the drivers of optimal service rate in appointment systems. In this paper, we analyze service rate control when the provider is responsible for serving a finite number of patients who arrive by prior appointments in a session of finite duration. Under general assumptions on random service times, we show that a provider should optimally increase the service rate in response to the queue length and the delay in starting a patient's service. In addition to the novel analytical finding on delay-driven speed-up behavior, we show that both congestion-driven and delay-driven speed-up behaviors persist even when the patient wait is costless. We also show that the provider may speed-up or slow-down while serving sequential arrivals in the appointment schedule, depending on the relative magnitudes of the cost of patient waiting and the cost of the provider's time. Finally, we explore implications of flexible service rates on the operational performance of appointment systems, and find that flexible service rates result in shorter, fairer, and less uncertain waiting times for patients in most situations.

Optimal Control F-policy for M/M/R/K Queue with an Additional Server and Balking

The multi-server finite queueing model with balking under admission control of customers based upon F-policy is investigated. In order to reduce the balking behavior of the customers, there is the provision of one additional server so as to shorten the queue length formed by the customers in rush hour. The noble realistic feature of the proposed Markov model is to control the admission of customers based on the threshold level ‘F’ of the queue size. According to ‘F’ policy, no more customers can be accommodated once the system is full and then after the further admission of the customers is not acceptable until the queue size drops to the predefined threshold level ‘F’. Chapman–Kolmogorov equations have been framed and are solved by using the recursive technique to establish the steady-state queue size probability distribution of the customers present in the system. A numerical example is taken to explore the various system indices. Adaptive neuro-fuzzy inference system (ANFIS) approach is applied to compare the soft computing results obtained by ANFIS with the results computed by the analytical method. The cost function is framed to determine the optimal control parameters namely threshold value ‘F’, the capacity of the system, the number of servers and service rate.

A Discrete-Time $$GI^{X}/Geo^{Y}/1$$ Queue with Early Arrival System

A discrete-time batch service queue with batch renewal input and random serving capacity rule under the late arrival delayed access system, has recently appeared in the literature (Barbhuiya and Gupta in Queueing Syst 91(3):347–365, 2019b). In this paper, we consider the same model under the early arrival system, since it is more applicable in telecommunication systems where an arriving batch of packets needs to be transmitted in the same slot in which it has arrived. In doing so, we derive the steady-state queue length distributions at various epochs, and show that in limiting case the result gets converted to the continuous-time queue (Barbhuiya and Gupta in J Differ Equ Appl 25(2):1–10, 2019a). We discuss few numerical results as well.

An N-policy discrete-time Geo/G/1 queue with modified multiple server vacations and Bernoulli feedback*

This paper deals with a single-server discrete-time Geo/G/1 queueing model with Bernoulli feedback and N-policy where the server leaves for modified multiple vacations once the system becomes empty. Applying the law of probability decomposition, the renewal theory and the probability generating function technique, we explicitly derive the transient queue length distribution as well as the recursive expressions of the steady-state queue length distribution. Especially, some corresponding results under special cases are directly obtained. Furthermore, some numerical results are provided for illustrative purposes. Finally, a cost optimization problem is numerically analyzed under a given cost structure.

Pollaczek contour integrals for the fixed-cycle traffic-light queue

The fixed-cycle traffic-light (FCTL) queue is the standard model for intersections with static signaling, where vehicles arrive, form a queue and depart during cycles controlled by a traffic light. Classical analysis of the FCTL queue based on transform methods requires a computationally challenging step of finding the complex-valued roots of some characteristic equation. Building on the recent work of Oblakova et al. (Exact expected delay and distribution for the fixed-cycle traffic-light model and similar systems in explicit form, 2016), we obtain a contour-integral expression, reminiscent of Pollaczek integrals for bulk-service queues, for the probability generating function of the steady-state FCTL queue. We also show that similar contour integrals arise for generalizations of the FCTL queue introduced in Oblakova et al. (2016) that relax some of the classical assumptions. Our results allow us to compute the queue-length distribution and all its moments using algorithms that rely on contour integrals and avoid root-finding procedures.

Equilibrium joining strategies in the Mn/G/1 queue with server breakdowns and repairs

This paper considers an almost observable unreliable $$M_{n}{/}G{/}1$$ queueing system in which the arriving customers can observe the queue length upon their arrivals but not the state of the server. The arrival rates are state-dependent and the server is subject to breakdowns when it works. The lifetime of the server and the repair time are independent, and they follow two different general distributions. To obtain the steady-state queue length distribution, we present an auxiliary system called modified $$M_{n}{/}G{/}1$$ queueing system. Comparing the unreliable system with the modified system, we derive the steady-state queue length distributions at the arrival instant of a tagged customer. Moreover, we study customers’ equilibrium joining strategies based on a nonlinear waiting cost function. These results provide managerial insights into strategic behaviors of customers.