Single-server Queue Research Articles

A Bayesian approach to determine sample size in Erlang single server queues

An essential aspect of the practical application of Erlang single-server queues is the statistical inference of their parameters, notably traffic intensity. This metric is crucial as it serves as a fundamental performance indicator, allowing the derivation of other significant measures, including the mean queue size and the expected number of customers in the system. Additionally, it provides the proportion of time the queue system is occupied. This article explores algorithms for calculating sample sizes for these estimations, based on the number of customers who arrived during service periods, a highly intuitive approach, allowing data collection without concerns about correlations among data points. For this purpose, we have considered two forms of informative priors: Gauss hypergeometric and beta priors. Additionally, a non conjugate and objective prior, known as Jeffreys prior, has been taken into account. We present tables with sample sizes for specific configurations and abacuses for more general configurations obtainable through approximate interpolations.

A Comprehensive examination of utilizing neutrosophic parameters in manufacturing industry through queueing approach

Queueing theory plays a pivotal role in analyzing the dynamics of manufacturing systems subject to random arrivals and service times. The M/G/1 queueing model, which represents a single-server queue with general distributions for inter-arrival and service times, is fundamental in this regard. However, traditional queueing models often fail to account for the inherent uncertainties encountered in real-world manufacturing scenarios. Neutrosophic theory provides a robust framework for modeling indeterminacy, ambiguity, and inconsistency in queueing parameters, thereby offering a more adaptable and precise depiction of system behavior. This research investigates the utilization of Neutrosophic sets in defining arrival rates, service times, and queue length distributions within the M/G/1 framework tailored to manufacturing settings. By employing numerical examples and comparative analyses, the study explores the effects of Neutrosophic parameters on key performance indicators such as mean waiting time, queue length distribution, and server utilization, considering distributions like Exponential, Erlang, and Deterministic. Additionally, it delves into the implications of integrating Neutrosophic parameters into queueing theory, providing valuable insights into improved decision-making and system optimization across various manufacturing operational contexts.

Stability of a Queue Fed by Scheduled Traffic at Critical Loading

Performance of a queueing system with scheduled arrivals A scheduled arrival sequence is one in which customers are scheduled to arrive at constant interarrival times, but each customer’s actual arrival time is perturbed from her scheduled arrival time by a random perturbation. In “Stability of a Queue Fed by Scheduled Traffic at Critical Loading”, V.F. Araman and P.W. Glynn consider a single server queue with deterministic service times in which customers arrive following a scheduled arrival process. Unlike a queue fed by renewal traffic, this queue is shown to be stable even when the utilization is equal to one. It is also shown that for finite mean perturbations, a necessary and sufficient condition for stability is when the positive part of the perturbation has bounded support, with no requirement on the negative part of the perturbation. Perhaps surprisingly, this criterion is not reversible, in the sense that such a queue can be stable for a scheduled traffic process in forward time, but unstable for the time-reversal of the same traffic process.

Sojourn Time Analysis of a Single-Server Queue with Single- and Batch-Service Customers

Sojourn Time Analysis of a Single-Server Queue with Single- and Batch-Service Customers

Ideal Arrival and Service Control Model Policy Based on Fuzzy Queuing System of Uncertainty Measure

In this work, we investigate how to optimize costs for a single-server queuing system operating in a fuzzy, unpredictable environment. Constructing the overall optimal cost and cost function of the queuing system under uncertainty in the fuzzy paradigm is the aim of the inquiry. The fuzzy analysis is carried out to offer a more practical answer to the issues at hand, as opposed to the model's usual crisp responses. The model's crisp and fuzzy systems have different theoretical advances that have been determined, and the estimated costs are easily verifiable and comparable. Lastly, sensitivity analysis has also been carried out utilising numerical analysis to evaluate the theoretical conclusions of the model that is being studied.

Regenerative Analysis and Approximation of Queueing Systems with Superposed Input Processes

A single-server queueing system with n classes of customers, stationary superposed input processes, and general class-dependent service times is considered. An exponential splitting is proposed to construct classical regeneration in this (originally non-regenerative) system, provided that the component processes have heavy-tailed interarrival times. In particular, we focus on input processes with Pareto interarrival times. Moreover, an approximating GI/G/1-type system is considered, in which the independent identically distributed interarrival times follow the stationary Palm distribution corresponding to the stationary superposed input process. Finally, Monte Carlo and regenerative simulation techniques are applied to estimate and compare the stationary waiting time of a customer in the original and in the approximating systems, as well as to derive additional information on the regeneration cycles’ structure.

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.

Single server queueing model with Poisson arrival and gamma service time distribution

Queueing theory is pivotal in understanding waiting lines and their pervasive role in everyday life. Its profound application spans across diverse sectors, including computer programming networks, healthcare, transportation, and so on. Researchers have applied many statistical distributions in analyzing queueing data. This article focuses on the analysis of a single server queueing system with Poisson input and service times distributed according to a two-parameter gamma distribution. The study elaborates on the construction and estimation of key queueing properties such as steady state equations, average number of customers in the system, average queue length, expected size of the non-empty queue, distribution of waiting time, average waiting time in the system, and average waiting time in the queue of the M/Gamma/1 queueing model using the two-parameter gamma distribution. Keywords: Gamma distribution; Poisson; service; single server; queuing model.

Message from the editor in Chief

The year 2024 promises many exciting developments for the Global Journal of Computer Sciences: Theory and Research (GJCS). In addition, every year, the number of citations continues to rise strongly in proportion to the increase in the number of articles published. In this respect, first of all, we would like to thank all our authors who have contributed to the success of our journal. While reviewing our new issue, we saw articles from different authors from different countries: Turkey, India The multiculturalism policy followed in our globally-focused journal makes us happy with every new issue. We examine the following research areas in the current issue; “Determination of reinforcement diameters of reinforced concrete deep beams with genetic algorithms”, “Single server queueing model with Poisson arrival and gamma service time distribution”, “Sustainable innovation policy in non-walled technoparks” . We will continue to contribute to literature within the scope of Computer Science with our strong team. Thanks to all our authors the contributors to this issue. Hope to see you in our new issue. Assoc. Prof. Dr. Ezgi Pelin YILDIZ Global Journal of Computer Sciences: Theory and Research (GJCS) CHEF EDITOR

An Encourage Arrival of Single-Server Queuing System with Defeat Balking and Feedback Customers

Abstract: In this paper, we develop a finite capacity Single-server Markovian queuing system with encourage arrival, defeat balking and feedback customers. The steady state solution and the system characteristic are derived for this model and various probabilistic and performance measures of the system are derived.

Study of a two types of general heterogeneous service queueing system in a single server with optional repeated service and feedback queue

This paper addresses a model on a single server queue and two service representatives. After a customer is served, he/she has the three options: opting for receive the same service again (re-service), joining as a new customer for another regular service (feedback), or leaving the service system altogether. To ensure the queueing system is Markovian, we introduce an additional variable (supplementary variable) and using this approach, we derive the explicit distribution of queue size at random and departure epochs. Additionally, we determine the distribution of response time, inter-departure time, and busy period. By using the embedded Markov chain technique we have also derived the queue size distribution at departure epoch. We have also presented the cost analysis of the model with some numerical examples. The numerical illustration validates our findings and provides valuable insights into the queuing system.

On the time-dependent behavior of preemptive single-server queueing systems with Poisson arrivals

On the time-dependent behavior of preemptive single-server queueing systems with Poisson arrivals

Heavy-traffic queue length behavior in a switch under Markovian arrivals

AbstractThis paper studies the input-queued switch operating under the MaxWeight algorithm when the arrivals are according to a Markovian process. We exactly characterize the heavy-traffic scaled mean sum queue length in the heavy-traffic limit, and show that it is within a factor of less than 2 from a universal lower bound. Moreover, we obtain lower and upper bounds that are applicable in all traffic regimes and become tight in the heavy-traffic regime.We obtain these results by generalizing the drift method recently developed for the case of independent and identically distributed arrivals to the case of Markovian arrivals. We illustrate this generalization by first obtaining the heavy-traffic mean queue length and its distribution in a single-server queue under Markovian arrivals and then applying it to the case of an input-queued switch.The key idea is to exploit the geometric mixing of finite-state Markov chains, and to work with a time horizon that is chosen so that the error due to mixing depends on the heavy-traffic parameter.

Performance of the Gittins Policy in the G/G/1 and G/G/k, With and Without Setup Times

We consider the classic problem of preemptively scheduling jobs in a queue to minimize mean number-in-system, or equivalently mean response time. Even in single-server queueing models, this can be a nontrivial problem whose answer depends on the information available to the scheduler. The simplest case is when the scheduler knows each job's size, for which the optimal policy is Shortest Remaining Processing Time (SRPT) [9]. In the more realistic case of scheduling with unknown or partially known job sizes, people consider the Gittins policy [1-3, 12]. Roughly speaking, Gittins assigns each job a scalar rank based on an estimate of its remaining work, then serves the job of least rank.

On a Single Server Vacation Queue with Two Types of Service and Two Types of Vacation

We study a single server queueing system that receives singly arriving customers according to a Poisson process. The server offers one of the two types of heterogeneous services. Before the beginning of a service, , the customer can choose an exponential service with probability p1 or a deterministic service with probability p2, where p1+p2=1 Immediately after a service is completed, the server has a choice of taking a vacation with probability δ, or, with probability 1-δ, the server may continue staying in the system. We further assume that if the server opts to take a vacation, then with probability α1, he may take a vacation of an exponential duration with mean vacation time 1/u (u>0) or with probability he may want to take a deterministic vacation with constant duration d>0, where α1+α2=1. After a vacation is complete, the server instantly starts providing service if there is at least one customer in the system or the server remains idle in the system till a new customer arrives for service. We find a steady state solution in terms of the generating function of the queue length as well as the steady state probabilities for all different states of the system.

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].

Analysis of single server repairable queueing system with many kinds of service, single working vacation and standby server

We consider a single server queueing model in which the customers arrive according to a Markovian Arrival Process (MAP). We modeled this queueing system with many kinds of service where the customers can choose any kind from the server. Each kind of service follows Phase-type (PH) distribution. We analyzed the model with breakdown, repair, single working vacation and standby server. A standby server works only when the main server is breakdown while serving the customer during a busy or vacation period. Using the matrix-geometric approach, we examined the model. A few real-world examples have been given, and several graphical and numerical examples have also been provided.

Transient Analysis of a Single Server Queue with Working Vacation Operating in a Multi-Level Environment Controlled by a Random Switch

A single server queueing system with working vacation operating in a multi-level environment controlled by a random switch is considered. The environment has [Formula: see text] levels and the random switch puts the server in one of the [Formula: see text] levels to perform a service. The server resides in a level and reports to the random switch immediately after completing the service of the last customer in the queue. The random switch initiates an assignment process immediately at the arrival of a customer. The assignment process takes a random time and immediately at the end of the assignment interval, the server is put to operate in any one of the [Formula: see text] levels with a positive probability. In each level [Formula: see text] of the environment, the queueing system behaves like an [Formula: see text] queue and switches to a working vacation at a random time and serves with a lesser rate [Formula: see text] During the working vacation, all newly arriving customers are lost. When the server is with the control switch, customers are permitted to join in the queue. For this model, time-dependent state probabilities are explicitly found and the corresponding steady-state probabilities are deduced. Some key performance measures are also obtained. A numerical study is also made.

A single server Non-Markovian with non-compulsory re-service and balking under Modified Bernoulli Vacation

In this paper, a single-server queue with Modified Bernoulli Vacation(MBV) has been considered. The server provides two kinds of services, Compulsory Service (CR) and Non-Compulsory Re-service (NCR). Any new customer who arrives and finds the server engaged, on breakdown, or on vacation will be placed on an orbit; or else if he finds the server free, he enters the service immediately. The server takes a vacation as soon as the orbit is empty at the usual service completion moment. The dissatisfied customer may re-enter the orbit after usual service is complete to receive another service. We have used SVM (Supplementary Variable Method) to derive PGF (Probability Generating Function) for the system. In order to demonstrate the influence that the system parameters have, numerical examples are discussed.

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.