Service Time Distribution Research Articles

Prioritized Retrial Queueing Systems with Randomized Push-Out Mechanism

The article is focused on a single-channel preemptive queuing system. Two stationary Poisson flows of customers are incoming to the system. The first flow has an absolute priority over the second one: a new high-priority customer from the first flow displaces a low-priority one from the service channel and takes its place. The capacity of the system is limited to k customers. There is a probabilistic push-out mechanism in the system: if a new high-priority customer finds that all the places in the queue are occupied, then it has the right to displace one low-priority customer from the queue with probability a. Both types of customers have the same exponentially distributed service times. Customers who failed to enter the system due to the limited size of the queue, as well as those expelled from the queue or service channel when the push-out mechanism is triggered, are not lost immediately, but they are sent to a special part of the system called the orbit and designed to store repeated customers. In orbit, there are two separate unlimited queues, consisting of low-priority and high-priority repeated customers, respectively. If there are no free places in the system, new customers with a probability q are added to the corresponding orbital queue. The waiting time of repeated customers in orbit is distributed according to an exponential law. The parameter of this law may differ for different types of customers. After waiting in orbit, secondary customers try to re-enter the system. The probabilistic characteristics of the described queuing system are calculated by the method of generating functions, previously proposed by the authors for calculating a similar system without repeated customers. This method allows finding the main probabilistic characteristics of distributions for both types of customers. Particular attention is paid to the study of the dependence of the loss probabilities for both types of customers on the parameters of the system, primarily on the push-out probability a, the capacity of the system k, and the probability of repeated circulation (probability of persistence) q. It is shown that the effect of blocking the system and the effect of the linear law of customers’ losses, previously identified in similar problems without repeated customers, remain valid even in the presence of secondary repeated customers. The theoretical results are proved by numerical calculations. The blocking area for the second type of customers was calculated along with the area of linear loss law for both types of customers. We studied the influence of the probability of repeated circulation q on the shape of these areas and on the dependence of the loss probabilities for both types of customers on the push-out probability a.

Analysis of Tandem Queue with Multi-Server Stages and Group Service at the Second Stage

In this paper, we consider a tandem dual queuing system consisting of multi-server stages. Stage 1 is characterized by an infinite buffer, one-by-one service of customers, and an exponential distribution of service times. Stage 2 is characterized by a finite buffer and a phase-type distribution of service times. Service at Stage 2 is provided to groups of customers. The service time of a group depends on the size of the group. The size is restricted by two thresholds. The waiting time of a customer at each stage is limited by a random variable with an exponential distribution, with the parameter depending on the stage. After service at Stage 1, a customer can depart from the system or try to enter Stage 2. If the buffer at this stage is full, the customer is either lost or returns for service at Stage 1. Customer arrivals are described by the versatile Markov arrival process. The system is studied via consideration of a multi-dimensional continuous-time Markov chain. Numerical examples, which highlight the influence of the thresholds on the system performance measures, are presented. The possibility of solving optimization problems is illustrated.

Response Time of Queueing Mechanisms

We study response time, a key performance characteristic of queueing mechanisms. The studied model incorporates both active and passive queue management, arbitrary service time distribution, as well as a complex model of arrivals. Therefore, the obtained formulas can be used to calculate the response time of many real queueing mechanisms with different features, by parameterizing adequately the general model considered here. The paper consists of two parts. In the first, mathematical part, we derive the distribution function for the response time, its density, and the mean value. This is done by constructing two systems of integral equations, for the distribution function and the mean value, respectively, and solving these systems with transform techniques. All the characteristics are derived both in the time-dependent and steady-state cases. In the second part, we present numerical values of the response time for a few system parameterizations and point out several of its properties, some rather counterintuitive.

Investigation of the Fork–Join System with Markovian Arrival Process Arrivals and Phase-Type Service Time Distribution Using Machine Learning Methods

This paper presents a study of fork–join systems. The fork–join system breaks down each customer into numerous tasks and processes them on separate servers. Once all tasks are finished, the customer is considered completed. This design enables the efficient handling of customers. The customers enter the system in a MAP flow. This helps create a more realistic and flexible representation of how customers arrive. It is important for modeling various real-life scenarios. Customers are divided into K≥2 tasks and assigned to different subsystems. The number of tasks matches the number of subsystems. Each subsystem has a server that processes tasks, and a buffer that temporarily stores tasks waiting to be processed. The service time of a task by the k-th server follows a PH (phase-type) distribution with an irreducible representation (βk, Sk), 1≤k≤K. An analytical solution was derived for the case of K=2 when the input MAP flow and service time follow a PH distribution. We have efficient algorithms to calculate the stationary distribution and performance characteristics of the fork–join system for this case. In general cases, this paper suggests using a combination of Monte Carlo and machine learning methods to study the performance of fork–join systems. In this paper, we present the results of our numerical experiments.

The Saturated Multiserver Job Queuing Model with Two Classes of Jobs: Exact and Approximate Results

We study a Multiserver Job Queuing Model (MJQM), i.e., a FIFO (First-In-First-Out) queue in which jobs request varying numbers of servers. Our investigation focuses on the saturation analysis of MJQM with two job classes. Jobs in the first class need only one server for an exponentially distributed service time, while jobs in the second class require either all servers or a fraction of them. We provide exact results for the scenario in which jobs in the second class require all servers and have generally distributed service times. Additionally, we develop an accurate approximate model of the MJQM behavior when jobs in the second class require a fraction of the available servers, and have exponentially distributed service times.

Whittle index approach to multiserver scheduling with impatient customers and DHR service times

We consider the optimal scheduling problem in a multiserver queue with impatient customers belonging to multiple classes. We assume that each customer has a random abandonment time, after which the customer leaves the system if its service has not been completed before that. In addition, we assume that the scheduler is not able to anticipate the expiration of the abandonment times but only knows their distributions and how long each customer has been in the system. Many papers consider this scheduling problem under Poisson arrivals and linear holding costs assuming further that both the service times and the abandonment times have exponential distributions. Even with these additional assumptions, the exact solution is known only in very few special cases. To tackle this tricky problem, we apply the Whittle index approach. Unlike the earlier papers, which were restricted to exponential service times, we allow the service time distributions for which the hazard rate is decreasing. The Whittle index approach is applied to the discrete-time multiserver queueing problem with discounted costs. As our main theoretical result, we prove that the related relaxed optimization problem is indexable and derive the corresponding Whittle index explicitly. Based on this discrete-time result, we develop a reasonable heuristic for the original continuous-time multiserver scheduling problem. The performance of the resulting policy is evaluated in the M/G/M setup by numerical simulations, which demonstrate that it, indeed, gives better performance than the other policies included in the comparison.

Seismic resilience assessment of extended pile shaft supported coastal bridges considering scour and uniform corrosion effects

The resilience-informed assessment for coastal highway bridges exposed to multiple hazards plays a crucial role in the decision-making process for their operation, maintenance, and management during the bridge’s long-term service period. This study performs a seismic resilience assessment for a coastal reinforced concrete (RC) bridge under the combined effects of scour and uniform corrosion. Toward this goal, an extended pile shaft supported coastal RC bridge is taken as the benchmark and fifteen scenarios are generated to cover three regular scour conditions and five different corrosion levels. To inform the bridge management, the probabilistic distribution of the bridge service time corresponding to different corrosion levels is developed. Subsequently, the finite element model (FEM) for each scenario of the benchmark coastal highway RC bridge is developed, involving the soil-pile interaction and considering the uncertainty of structural and geotechnical parameters, as well as earthquake ground motions. The failure mode of the corroded extended pile shaft is identified as a flexural failure by comparing their flexural and shear capacities. Finally, a resilience loss ratio is proposed in this study for the first time to quantify the impacts of corrosion level and scour depth changes on seismic resilience across different ground motion intensities. Results indicate that both the seismic fragility and resilience decrease as the corrosion intensifies over time. Specifically, at a given intensity measure (IM) level, the resilience loss ratio increases as the corrosion level intensifies. For the studied benchmark bridge, the scour significantly increases the structural fundamental period, consequently reducing the curvature demand on the pile shaft. As a result, the scour generally decreases the failure probability and the earthquake-induced functionality loss for the extended pile shaft. As scour depth increases, the resilience loss of the bridge at a given corrosion level decreases.

Appointment Scheduling with Delay Tolerance Heterogeneity

In this study, we investigate an appointment sequencing and scheduling problem with heterogeneous user delay tolerances under service time uncertainty. We aim to capture the delay tolerance effect with heterogeneity, in an operationally effective and computationally tractable fashion, for the appointment scheduling problem. To this end, we first propose a Tolerance-Aware Delay (TAD) index that incorporates explicitly the user tolerance information in delay evaluation. We show that the TAD index enjoys decision-theoretical rationale in terms of Tolerance sensitivity, monotonicity, and convexity and positive homogeneity, which enables it to incorporate the frequency and intensity of delays over the tolerance in a coherent manner. Specifically, the convexity of TAD index ensures a tractable modeling of the collective delay dissatisfaction in the appointment scheduling problem. Using the TAD index, we then develop an appointment model with known empirical service time distribution that minimizes the overall tolerance-aware delays of all users. We analyze the impact of delay tolerance on the sequence and schedule decisions and show that the resultant TAD appointment model can be reformulated as a mixed-integer linear program (MILP). Furthermore, we extend the TAD appointment model by considering service time ambiguity. In particular, we encode into the TAD index a moment ambiguity set and a Wasserstein ambiguity set, respectively. The former captures effectively the correlation among service times across positions and user types, whereas the latter captures directly the service time data information. We show that both of the resultant TAD models under ambiguity can be reformulated as polynomial-sized, mixed-integer conic programs (MICPs). Finally, we compare our TAD models with some existing counterpart approaches and the current practice using synthetic data and a case of real hospital data, respectively. Our results demonstrate the effectiveness of the TAD appointment models in capturing the user delay tolerance with heterogeneity and mitigating the worst-case delays. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. Funding: S. Wang was supported by the National Natural Science Foundation of China [Grants 71922020, 72171221, and 71988101, entitled “Econometric Modeling and Economic Policy Studies”], the Fundamental Research Funds for the Central Universities [Grant UCAS-E2ET0808X2], and the Major Program of National Natural Science Foundation of China [Grant 72192843]. S. Wang was also supported by a grant from MOE Social Science Laboratory of Digital Economic Forecasts and Policy Simulation at UCAS. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0025 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0025 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Scheduling in the High-Uncertainty Heavy Traffic Regime

We propose a model uncertainty approach to heavy traffic asymptotics that allows for a high level of uncertainty. That is, the uncertainty classes of underlying distributions accommodate disturbances that are of order 1 at the usual diffusion scale as opposed to asymptotically vanishing disturbances studied previously in relation to heavy traffic. A main advantage of the approach is that the invariance principle underlying diffusion limits makes it possible to define uncertainty classes in terms of the first two moments only. The model we consider is a single-server queue with multiple job types. The problem is formulated as a zero sum stochastic game played between the system controller, who determines scheduling and attempts to minimize an expected linear holding cost, and an adversary, who dynamically controls the service time distributions of arriving jobs and attempts to maximize the cost. The heavy traffic asymptotics of the game are fully solved. It is shown that an asymptotically optimal policy for the system controller is to prioritize according to an index rule, and for the adversary, it is to select distributions based on the system’s current workload. The workload-to-distribution feedback mapping is determined by a Hamilton–Jacobi–Bellman equation, which also characterizes the game’s limit value. Unlike in the vast majority of results in the heavy traffic theory and as a direct consequence of the diffusive size disturbances, the limiting dynamics under asymptotically optimal play are captured by a stochastic differential equation where both the drift and the diffusion coefficients may be discontinuous. Funding: R. Atar is supported by the Israeli Science Foundation [Grant 1035/20].

Practice Summary: Henkel Uses Analytics to Improve Call Center Performance

The development of a decision support tool to assist with call center agent scheduling at Henkel is described. Call center demand varies by hour, and agents have dissimilar service time distributions. Target agent utilizations are calculated using a metamodel of a simulation that finds the elbow of the utilization to waiting time curve.

Large-System Insensitivity of Zero-Waiting Load Balancing Algorithms

This paper studies the sensitivity (or insensitivity) of a class of load balancing algorithms that achieve asymptotic zero-waiting in the sub-Halfin-Whitt regime, named LB-zero. Most existing results on zero-waiting load balancing algorithms assume the service time distribution is exponential. This paper establishes the large-system insensitivity of LB-zero for jobs whose service time follows a Coxian distribution with a finite number of phases. This result justifies that LB-zero achieves asymptotic zero-waiting for a large class of service time distributions as the Coxian family is dense in the class of positive-valued distributions. To prove this result, this paper develops a new technique, called “iterative state-space peeling” (ISSP). ISSP first identifies an iterative relation between the upper and lower bounds on the queue states and then proves that the system lives near the fixed point of the iterative bounds with a high probability. Based on ISSP, the steady-state distribution of the queue length is further analyzed by applying Stein’s method in the neighborhood of the fixed point. ISSP, like state-space collapse in heavy-traffic analysis, is a general approach that may be used to study other complex stochastic systems.

Transient Behavior of a Two-Phase Queuing System with a Limitation on the Total Queue Size

The transient mode of a two-phase queuing system with a Poisson input flow, exponential distribution of service time in each phase, and a limitation on the total buffer size of the two phases is considered. Nonstationary probabilities of system states are found using the Laplace transform. A numerical calculation and analysis of the system performance characteristics in transient mode with parameters corresponding to new-generation optical networks were carried out.

Supervised ML for Solving the GI/GI/1 Queue

We apply supervised learning to a general problem in queueing theory: using a neural net, we develop a fast and accurate predictor of the stationary system-length distribution of a GI/GI/1 queue—a fundamental queueing model for which no analytical solutions are available. To this end, we must overcome three main challenges: (i) generating a large library of training instances that cover a wide range of arbitrary interarrival and service time distributions, (ii) labeling the training instances, and (iii) providing continuous arrival and service distributions as inputs to the neural net. To overcome (i), we develop an algorithm to sample phase-type interarrival and service time distributions with complex transition structures. We demonstrate that our distribution-generating algorithm indeed covers a wide range of possible positive-valued distributions. For (ii), we label our training instances via quasi-birth-and-death(QBD) that was used to approximate PH/PH/1 (with phase-type arrival and service process) as labels for the training data. For (iii), we find that using only the first five moments of both the interarrival and service times distribution as inputs is sufficient to train the neural net. Our empirical results show that our neural model can estimate the stationary behavior of the GI/GI/1—far exceeding other available methods in terms of both accuracy and runtimes. History: Ram Ramesh, Area Editor for Data Science and Machine Learning. Funding: O. Baron received financial support from the Natural Sciences and Engineering Research Council of Canada (NERC) [Grant 458051]. D. Krass received financial support from the NERC [Grant 458395]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0263 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0263 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Determination of probabilistic-time characteristics when studying load properties in computer networks

The article is devoted to the determination of probabilistic-time characteristics when studying the properties of load in computer networks. The proposed approach to determining these characteristics is based on consideration of the probabilistic-time characteristics of a single-line queuing system with a geometric input flow and an exponential distribution of service time with a limited queue (g/M/1/N). The work takes into account that the flow of messages at the input of a subscriber station, hub or net-work controller serving a multipoint network with a “ring” or “tree” structure differs from Poisson. The proposed analysis method (G/M/1/N type system), called the parameter method, is illustrated with examples of various types of queuing systems.

Structural properties of optimal fidelity selection policies for human-in-the-loop queues

We study optimal fidelity selection for a human operator servicing a queue of homogeneous tasks. The agent can service a task with a normal or high fidelity level, where fidelity refers to the degree of exactness and precision while servicing the task. Therefore, high-fidelity servicing results in higher-quality service but leads to larger service times and increased operator tiredness. We treat the human cognitive state as a lumped parameter that captures psychological factors such as workload and fatigue. The operator’s service time distribution depends on her cognitive dynamics and the fidelity level selected for servicing the task. Her cognitive dynamics evolve as a Markov chain in which the cognitive state increases with high probability whenever she is busy and decreases while resting. The tasks arrive according to a Poisson process and the operator is penalized at a fixed rate for each task waiting in the queue. We address the trade-off between high-quality service of the task and consequent penalty due to a subsequent increase in queue length using a discrete-time Semi-Markov Decision Process framework. We numerically determine an optimal policy and the corresponding optimal value function. Finally, we establish the structural properties of an optimal fidelity policy and provide conditions under which the optimal policy is a threshold-based policy.

Fluid Limits for Longest Remaining Time First Queues

A single-server queue with renewal arrivals and generally distributed independent and identically distributed service times is considered. Customers are served using the longest remaining time first scheduling algorithm. In case of a tie, processor sharing is utilized. We introduce a fluid model for the evolution of a measure-valued state descriptor of this queue, and we investigate its properties. We also prove a fluid limit theorem justifying our fluid model as the first-order approximation of the queueing system under consideration.

Analysis of Age of Information in Dual Updating Systems

We study the average Age of Information (AoI) and peak AoI (PAoI) of a dual-queue status update system that monitors a common stochastic process through two independent channels. Although the double queue parallel transmission is instrumental in reducing AoI, the out of order of data arrivals also imposes a significant challenge to the performance analysis. We consider two settings: the M-M system where the service time of two servers is exponentially distributed; the M-D system in which the service time of one server is exponentially distributed and that of the other is deterministic. For the two dual-queue systems, closed-form expressions of average AoI and PAoI are derived by resorting to the graphic method and state flow graph analysis method. Our analysis reveals that when the two servers have the same service rate, compared with the single-queue system with an exponentially distributed service time, the average PAoI and the average AoI of the M-M system decrease by 33.3% and 37.5%, respectively, and those of the M-D system decrease by 27.7% and 39.7%, respectively. Numerical results show that the two dual-queue systems also outperform the M/M/2 single queue dual-server system with optimized arrival rate in terms of average AoI and PAoI.

Lower and upper stochastic bounds for the joint stationary distribution of a non-preemptive priority retrial queueing system

Consider a single-server retrial queueing system with non-preemptive priority service, where customers arrive in a Poisson process with a rate of $\lambda_1$ for high-priority customers (class 1) and $\lambda_2$ for low-priority customers (class 2). If a high-priority customer is blocked, they are queued, while a low-priority customer must leave the service area and return after some random period of time to try again. In contrast with existing literature, we assume different service time distributions for the two customer classes. This investigation proposes a stochastic comparison method based on the general theory of stochastic orders to obtain lower and upper bounds for the joint stationary distribution of the number of customers at departure epochs in the considered model. Specifically, we discuss the stochastic monotonicity of the embedded Markov queue-length process in terms of both the usual stochastic and convex orders. We also perform a numerical sensitivity analysis to study the effect of the arrival rate of high-priority customers on system performance measures.

Approximation Method for a Non-preemptive Multiserver Queue with Quasi-Poisson Arrivals

We consider a non-preemptive multiserver queue with multiple priority classes. We assume distinct exponentially distributed service times and separate quasi-Poisson arrival processes with a predefined maximum number of requests that can be present in the system for each class. We present an approximation method to obtain the steady-state probabilities for the number of requests of each class in our system. In our method, the priority levels (classes) are solved “nearly separately,” linked only by certain conditional probabilities determined approximately from the solution of other priority levels. Several numerical examples illustrate the accuracy of our approximate solution. The proposed approach significantly reduces the complexity of the problem while featuring generally good accuracy.

Open networks of infinite server queues with non-homogeneous multivariate batch Poisson arrivals

In this paper, we consider the occupancy distribution for an open network of infinite server queues with multivariate batch arrivals following a non-homogeneous Poisson process, and general service time distributions. We derive a probability generating function for the transient occupancy distribution of the network and prove that it is necessary and sufficient for ergodicity that the expected occupancy time for each batch be finite. Further, we recover recurrence relations for the transient probability mass function formulated in terms of a distribution obtained by compounding the batch size with a multinomial distribution.