Interarrival Time Distributions Research Articles

Transient GI/MSP/1/N Queue.

A non-zero correlation between service times can be encountered in many real queueing systems. An attractive model for correlated service times is the Markovian service process, because it offers powerful fitting capabilities combined with analytical tractability. In this paper, a transient study of the queue length in a model with MSP services and a general distribution of interarrival times is performed. In particular, two theorems are proven: one on the queue length distribution at a particular time t, where t can be arbitrarily small or large, and another on the mean queue length at t. In addition to the theorems, multiple numerical examples are provided. They illustrate the development over time of the mean queue length and the standard deviation, along with the complete distribution, depending on the service correlation strength, initial system conditions, and the interarrival time variance.

Transient analysis of a bulk stream queue with arbitrarily distributed arrival intervals

We study the classical infinite buffer single server queueing model with renewal input of customers in batches of random size, having arbitrarily distributed arrival intervals and exponentially distributed service times. Using the technique of supplementary variable and shift operator we derive closed form expression of the time dependent system content distribution in terms of its Laplace transform. The analysis is mainly based on the root-finding technique of the non-linear characteristic equation in terms of the Laplace transform variable. Additionally, using asymptotic properties of Laplace transform, we deduce the corresponding steady-state distribution. We discuss some special cases of the model, thus providing an alternative approach in deriving the transient distribution. We further evaluate certain performance measures and present extensive numerical examples in tabular and graphical form to illustrate the applicability of our theoretical work. The effect of system parameters, interarrival time distribution and traffic intensity on the system behavior is also demonstrated.

Parameter Estimation and Testing for the Doubly Geometric Process with Lognormal Distribution: Application to Bladder Cancer Patients’ Data

The geometric process (GP) has been widely utilized as a stochastic monotone model in the fields of probability and statistics. However, its practical application is often limited by certain assumptions. To address this, [Wu (2018). Doubly geometric process and applications. Journal of the Operational Research Society, 69(1), 66–67] introduced the doubly geometric process (DGP) as an extension of the GP model, relaxing some of its assumptions. Due to its ability to overcome the limitations of the GP model, the DGP has gained significant popularity in recent times. This study focuses on the parameter estimation problem for the DGP when the distribution of the first interarrival time follows a lognormal distribution with parameters [Formula: see text] and [Formula: see text]. We employ the maximum likelihood method to obtain estimates for both the model parameters and the distribution parameters. Additionally, we investigate the asymptotic joint distribution and statistical properties such as asymptotic unbiasedness and consistency of the estimators. Furthermore, we propose a novel test procedure to distinguish between the GP and DGP models. To assess the performance of the estimators and the proposed test procedure, we conduct a simulation study involving various sample sizes and parameter values. Finally, we present an application of the developed methods in fitting data from bladder cancer patients.

Taming the Elephants: Affordable Flow Length Prediction in the Data Plane

Machine Learning (ML) shows promising potential for enhancing networking tasks by providing early traffic predictions. However, implementing an ML-enabled system is a challenging task due to network devices limited resources. While previous works have shown the feasibility of running simple ML models in the data plane, integrating them into a practical end-to-end system is not an easy task. It requires addressing issues related to resource management and model maintenance to ensure that the performance improvement justifies the system overhead. In this work, we propose DUMBO, a versatile end-to-end system to generate and exploit early flow size predictions at line rate. Our system seamlessly integrates and maintains a simple ML model that offers early coarse-grain flow size prediction in the data plane. We evaluate the proposed system on flow scheduling, per-flow packet inter-arrival time distribution, and flow size estimation using real traffic traces, and perform experiments using an FPGA prototype running on an AMD(R)-Xilinx(R) Alveo U280 SmartNIC. Our results show that DUMBO outperforms traditional state-of-the-art approaches by equipping network devices data planes with a lightweight ML model. Code is available at https://github.com/cpt-harlock/DUMBO.

Energy-efficient production control of a make-to-stock system with buffer- and time-based policies

Increasing energy efficiency in manufacturing has significant environmental and cost benefits. Turning on or off a machine dynamically while considering the production rate requirements can offer substantial energy savings. In this work, we examine the optimal policies to control production and turn on and off a machine that operates in working, idle, off, and warmup modes for the case where demand inter-arrival, production, and warmup times have phase-type distributions. The optimal control problem that minimises the expected costs associated with the energy usage in different energy modes and the inventory and backlog costs is solved using a linear program associated with the underlying Markov Decision Process. We also present a matrix-geometric method to evaluate the steady-state performance of the system under a given threshold control policy. We show that when the inter-arrival time distribution is not exponential, the optimal control policy depends on both the current phase of the inter-arrival time and inventory position. The phase-dependent policy implemented by estimating the current phase based on the time elapsed since the last arrival yields a buffer- and time-based policy to control the energy mode and production. We show that policies that only use the inventory position information can be effective if the control parameters are chosen appropriately. However, the control policies that use both the inventory and time information further improve the performance.

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/ .

Probabilistic Tsunami Hazard Analysis for Vancouver Island Coast Using Stochastic Rupture Models for the Cascadia Subduction Earthquakes

Tsunami hazard analysis is an essential step for designing buildings and infrastructure and for safeguarding people and assets in coastal areas. Coastal communities on Vancouver Island are under threat from the Cascadia megathrust earthquakes and tsunamis. Due to the deterministic nature of current megathrust earthquake scenarios, probabilistic tsunami hazard analysis has not been conducted for the coast of Vancouver Island. To address this research gap, this study presents a new probabilistic tsunami hazard model for Vancouver Island from the Cascadia megathrust subduction events. To account for uncertainties of the possible rupture scenarios more comprehensively, time-dependent earthquake occurrence modeling and stochastic rupture modeling are integrated. The time-dependent earthquake model can capture a multi-modal distribution of inter-arrival time data on the Cascadia megathrust events. On the other hand, the stochastic rupture model can consider variable fault geometry, position, and earthquake slip distribution within the subduction zone. The results indicate that the consideration of different inter-arrival time distributions can result in noticeable differences in terms of site-specific tsunami hazard curves and uniform tsunami hazard curves at different return period levels. At present, the use of the one-component renewal model tends to overestimate the tsunami hazard values compared to the three-component Gaussian mixture model. With the increase in the elapsed time since the last event and the duration of tsunami hazard assessment, the differences tend to be smaller. Inspecting the regional variability of the tsunami hazards, specific segments of the Vancouver Island coast are likely to experience higher tsunami hazards due to the directed tsunami waves from the main subduction zone and due to the local underwater topography.

Optimal Scheduling in General Multi-Queue System by Combining Simulation and Neural Network Techniques

The problem of optimal scheduling in a system with parallel queues and a single server has been extensively studied in queueing theory. However, such systems have mostly been analysed by assuming homogeneous attributes of arrival and service processes, or Markov queueing models were usually assumed in heterogeneous cases. The calculation of the optimal scheduling policy in such a queueing system with switching costs and arbitrary inter-arrival and service time distributions is not a trivial task. In this paper, we propose to combine simulation and neural network techniques to solve this problem. The scheduling in this system is performed by means of a neural network informing the controller at a service completion epoch on a queue index which has to be serviced next. We adapt the simulated annealing algorithm to optimize the weights and the biases of the multi-layer neural network initially trained on some arbitrary heuristic control policy with the aim to minimize the average cost function which in turn can be calculated only via simulation. To verify the quality of the obtained optimal solutions, the optimal scheduling policy was calculated by solving a Markov decision problem formulated for the corresponding Markovian counterpart. The results of numerical analysis show the effectiveness of this approach to find the optimal deterministic control policy for the routing, scheduling or resource allocation in general queueing systems. Moreover, a comparison of the results obtained for different distributions illustrates statistical insensitivity of the optimal scheduling policy to the shape of inter-arrival and service time distributions for the same first moments.

An Analysis of the “Horizontal Y” Shaped Queuing Model to Assist in Health Care Institution

The “Horizontal Y” shaped queuing model in which there is bulk infinite arrival of patients, but available M services are limited. The discrete flow of patients in the system is reduced in continuous flow and a diffusion equation is used. In terms of means and variances of inter-arrival time distribution, this process of the number of patients and the number of servers is used, imposing reflecting boundaries. The inspiration for this analysis came from Aradhye and Kallurkar [2], and Armony [3], who explained that hospitals are complex systems whose problems can be resolved utilizing queuing theory. The present paper deals with a double-ended queue in which patients wait in a queue for services. The discrete distribution equations for queue size with various cases have been derived. Finally, the expected length of the queue, i.e., LC and the expected finite server, i.e., LS have been derived.

Design of fork-join networks of First-In-First-Out and infinite-server queues applied to clinical chemistry laboratories

This paper considers optimal design of queueing networks in which each node consists of a single-server FIFO queue and an infinite-server queue, which is referred to as incubation queue. Upon service completion at a FIFO queue, a job splits (forks) into two parts: the first part is routed to the next node on its route, and the second part is placed in the incubation queue. Routing of the jobs of multiple types is governed by a central decision maker that decides on the routes for each job type and aims to minimize the mean turnaround time of the jobs, i.e., the time spent in the system until service completion at the FIFO queue in the last node, and at all incubation queues on the job’s route, which may be viewed as a join operation. We provide explicit results for the turnaround time when all service and inter-arrival time distributions are exponential and invoke the Queueing Network Analyzer when these distributions are general. We then develop a Simulated Annealing approach to find the optimal routing configuration. We apply our approach to determine the optimal routing configuration in a chemistry analyzer line.

Remaining Loads in a PH/M/c Queue with Impatient Customers

There are three contributions to the literature in this note. Firstly, we point out that under some weak conditions the result in Swensen (1986) on the remaining loads in a GI/M/c queue with impatient customers, derived under the assumption that the distribution of inter-arrival times is Coxian, is also valid for the more general phase type distribution. From this result the distributions of the virtual and the actual waiting time can easily be obtained. Secondly, the relation to an alternative expression for the distribution of the virtual waiting time derived by Kawanishi and Takine (2015) is also clarified. Finally, we explain how the results on the remaining loads can be used to find distributions describing particular fixed servers and provide a couple of numerical examples of how this can be done.

Topology-Aware Resilient Routing Protocol for FANETs: An Adaptive Q-Learning Approach

Flying ad hoc networks (FANETs) play a crucial role in numerous military and civil applications since it shortens mission duration and enhances coverage significantly compared with a single unmanned aerial vehicle (UAV). Whereas, designing an energy-efficient FANETs routing protocol with a high packet delivery rate (PDR) and low delay is challenging owing to the dynamic topology changes. In this article, we propose a topology-aware resilient routing strategy based on adaptive <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q$ </tex-math></inline-formula> -learning (TARRAQ) to accurately capture topology changes with low overhead and make routing decisions in a distributed and autonomous way. First, we analyze the dynamic behavior of UAVs nodes via the queuing theory, and then the closed-form solutions of neighbors’ change rate (NCR) and neighbors’ change interarrival time (NCIT) distribution are derived. Based on the real-time NCR and NCIT, a resilient sensing interval (SI) is determined by defining the expected sensing delay of network events. Besides, we also present an adaptive <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q$ </tex-math></inline-formula> -learning approach that enables UAVs to make distributed, autonomous, and adaptive routing decisions, where the above SI ensures that the action space can be updated in time with low cost. The simulation results verify the accuracy of the topology dynamic analysis model, and also prove that our TARRAQ outperforms the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q$ </tex-math></inline-formula> -learning-based topology-aware routing (QTAR), mobility prediction-based virtual routing (MPVR), and greedy perimeter stateless routing based on energy-efficient hello (EE-Hello) in terms of 25.23%, 20.24%, and 13.73% lower overhead, 9.41%, 14.77%, and 16.70% higher PDR, and 5.12%, 15.65%, and 11.31% lower energy consumption, respectively.

Analytic and Computational Analysis of GI/Ma,b/c Queueing System

Bulk-service queueing systems have been widely applied in many areas in real life. While single-server queueing systems work in some cases, multi-servers can efficiently handle most complex applications. Bulk-service, multi-server queueing systems (compared to well-developed single-server queueing systems) are more complex and harder to deal with, especially when the inter-arrival time distributions are arbitrary. This paper deals with analytic and computational analyses of queue-length distributions for a complex bulk-service, multi-server queueing system GI/Ma,b/c, wherein inter-arrival times follow an arbitrary distribution, a is the quorum, and b is the capacity of each server; service times follow exponential distributions. The introduction of quorum a further increases the complexity of the model. In view of this, a two-dimensional Markov chain has to be involved. Currently, it appears that this system has not been addressed so far. An elegant analytic closed-form solution and an efficient algorithm to obtain the queue-length distributions at three different epochs, i.e., pre-arrival epoch (p.a.e.), random epoch (r.e.), and post-departure epoch (p.d.e.) are presented, when the servers are in busy and idle states, respectively.

Work-Conserving Disciplines Are Asymptotic Optimal in Completion Time Minimization

In this paper, we prove that in a stable multi-server system, where different machines are allowed to have different speeds, all work-conserving disciplines are asymptotic optimal for minimizing total completion time, if job size and interarrival time distributions have finite moments. As a byproduct of our analysis, we obtain tight upper bound on the competitive ratios of work-conserving disciplines on minimizing the metric of flow time.

Waiting time distribution in single-channel deterministic flow lines with discrete interarrival time distributions

Waiting time distribution in single-channel deterministic flow lines with discrete interarrival time distributions

Waiting time distribution in single-channel deterministic flow lines with discrete inter-arrival time distributions

Although previous studies proved that deterministic flow lines can be analysed separately by an exact decomposition method called a channel method and that the delay an item experiences in each machine possesses a Markovian property, the exact performance measures for the system with general arrival process cannot be obtained due to the lack of the results on relevant GI/D/1 queue. The solution obtained only for the system with specific arrival patterns such as geometric distribution or JIT arrivals considering setups. In this paper, exact decomposition methods are investigated through the systems consisting of one channel. Assuming general and discrete inter-arrival times, we introduce a procedure to calculate the equilibrium probabilities for waiting time distributions. It is demonstrated that probabilities can be computed by a matrix analytic method and recursions based on the decomposition methods. Inspired by the steel product manufacturing process, two-stage conveyor belt systems are analysed as an application. [Received: 28 July 2020; Accepted: 1 March 2021]

Age of Information in G/G/1/1 Systems: Age Expressions, Bounds, Special Cases, and Optimization

We consider the average age of information in G/G/1/1 systems under two service discipline models. In the first model, if a new update arrives when the service is busy, it is blocked; in the second model, a new update preempts the current update in service. For the blocking model, we first derive an exact age expression for G/G/1/1 systems. Then, using the age expression for G/G/1/1 systems, we calculate average age expressions for special cases, i.e., M/G/1/1 and G/M/1/1 systems. We observe that deterministic interarrivals minimize the average age of G/M/1/1 systems for a given mean interarrival time. Next, for the preemption in service model, we first derive an exact average age expression for G/G/1/1 systems. Then, similar to blocking discipline, using the age expression for G/G/1/1 systems, we calculate average age expressions for special cases, i.e., M/G/1/1 and G/M/1/1 systems. Average age for G/M/1/1 can be written as a summation of two terms, the first of which depends only on the first and second moments of interarrival times and the second of which depends only on the service rate. In other words, interarrival and service times are decoupled. We prove that deterministic interarrivals are optimum for G/M/1/1 systems for a given mean interarrival time. On the other hand, we observe for non-exponential service times that the optimal distribution of interarrival times depends on the relative values of the mean interarrival time and the mean service time. Finally, we propose a simple to calculate upper bound to the average age for the preemption in service discipline.

Optimal mission aborting in multistate systems with storage

Though intensive research efforts have been devoted to the study of mission aborting rules (MARs) for diverse types of systems, no previous work has discussed this problem for systems with product storage. This paper contributes by modeling and optimizing MARs for multistate systems with product storage, which may accumulate surplus product when the system performance surpasses the required demand and compensate product deficiency otherwise. The storage has limited capacity, uploading performance, and downloading performance. The system state may deteriorate due to random shocks, which have different severity and inter-arrival time distributions during primary mission (PM) and rescue procedure (RP). We put forward a numerical algorithm to evaluate the mission success probability (MSP) and damage avoidance probability (DAP) of the considered system with specified PM and RP demands and durations. Based on the MSP and DAP evaluation, we make a further contribution by formulating and solving a constrained optimization problem that finds the optimal MAR maximizing the MSP while meeting a desired level of DAP. Using a power system example, we also investigate influences of storage capacity, storage uploading/downloading performance constraints, and desired DAP level on the system performance metrics and on the MAR optimization solutions.

On the structure of data losses induced by an overflowed buffer

We deal with the statistical structure of data losses, when packetized data are transmitted through a channel with a buffering mechanism and may be subject to losses due to the buffer overflow. The main contribution is an explicit formula for the burst ratio parameter, which reflects the tendency of losses to cluster together, in long series. A general model of the arrival stream is used, which enables arbitrary shaping of the interarrival time distribution and the autocorrelation function of interarrival times. In numerical examples, the dependence of the burst ratio on various model parameters is shown, with a special attention to the impact of the autocorrelation of the arrival process. Some unexpected, counterintuitive observations are made.

Statistical inference for doubly geometric process with exponential distribution

The geometric process is widely applied as a stochastic monotone model in many practical applications since its introduction. However, it sometimes does not satisfy some requirements in the real-world applications due to model limitations. For this reason, it is proposed a new stochastic model which is called doubly geometric process. In the applications of the doubly geometric process, the estimation problem associated with the process arises naturally. In this study, the statistical inference problem for the doubly geometric process is considered by assuming that the distribution of the first interarrival time has an exponential distribution. The maximum likelihood method is used to estimate the model parameters of the doubly geometric process and the parameter of distribution. The joint distribution of the maximum likelihood estimators is obtained. A simulation study is presented to evaluate the small sample performance of the estimators with different parameter values. Finally, three real-world-data sets are used to illustrate the applicability of the methods.