Search Results (54)

This study focuses on forecasting and optimizing the performance of a real-world object modelled by a multi-server queueing system that processes two types of users: primary (new) users and repeating users. The repeating users are those who succeeded in entering processing upon arrival and then decided to repeat it. These users have privilege and can enter processing when they wish once at least one device is idle. The primary user is admitted to the system only if the number of occupied devices is less than some threshold value and the quantity of repeating users residing in the system does not exceed certain thresholds. Repeating users are impatient and non-persistent. Arrivals of primary users are described by the Markovian arrival process. Processing times of primary and repeating users have distinct phase-type distributions. Utilizing the concept of the generalized phase–time distributions, the dynamics of this queueing system are formally characterized by the multidimensional Markov chain, which is examined in this paper. The ergodicity condition is derived. The relation of the key performance characteristics of the system and the thresholds defining the policy of the primary user’s admission is numerically highlighted. Optimal threshold selection is demonstrated numerically. Full article

We study waiting times in a queue with active management and correlated job/packet sizes, which induce correlated service times. In the transient case, formulae for the distribution tail, the probability density, and the expected virtual waiting time at any time t, are found. Then, grounded on these results, stationary versions of the probability density and the expected value are obtained. The correlation of service times resulting from correlated job sizes is modeled through an MSP (Markovian service process). Theoretical results are reinforced by numerical examples, in which we examine the impact of symmetric positive and negative correlation of service times, and the impact of symmetric weak and strong active management, on transient and stationary waiting times. We also compare the effects of these factors on the waiting time with their effects on the queue length. In these examples, we can see a surprisingly large expected virtual waiting time, much greater than the product of the expected service time and the queue length. This effect is observed for both weak and strong management functions when the correlation is positive, but it vanishes when a symmetric negative correlation is applied. We also observe a weaker effect of active management on virtual waiting times than that of service time correlation, as well as a weaker impact of active management on virtual waiting time densities than on queue length distributions. Full article

Analysis of a crowdsourcing Markovian queue with phase-type service is considered in this paper. In this model, a customer not only receives service but also assists in delivery. In other words, in a retail environment, while some customers shop in-store, others place orders online or by phone and require home delivery. Store management can utilize online customers as couriers to complete these deliveries. However, because not every customer may agree to take part, a probabilistic element is included to capture the chances of their participation. The model also incorporates imperfect service, reflecting cases where deliveries may fail or require rework, and working breakdowns, representing partial disruptions in service capacity rather than complete stoppages. To analyze the system under steady-state conditions, matrix-analytic methods are applied. Numerical examples illustrate the significant benefits of incorporating these dynamics into traditional queueing models. Full article

The efficiency of intelligent urban mobility increasingly depends on adaptive mathematical models that can optimize multimodal transportation resources under stochastic and heterogeneous conditions. This study proposes a Markovian stochastic modeling and metaheuristic optimization framework for the adaptive management of bus lane capacity in mixed connected traffic environments. The heterogeneous vehicle arrivals are modeled using a Markov Arrival Process (MAP) to capture correlated and busty flow characteristics, while the system-level optimization aims to minimize total fuel consumption through discrete lane capacity allocation. To support real-time adaptation, a Hidden Markov Model (HMM) is integrated for queue-length estimation under partial observability. The resulting nonlinear and nonconvex optimization problem is solved using Genetic Algorithm (GA), Differential Evolution (DE), and Particle Swarm Optimization (PSO), ensuring robustness and convergence across diverse traffic scenarios. Numerical experiments demonstrate that the proposed stochastic–adaptive framework can reduce fuel consumption and vehicle delay by up to 68% and 65%, respectively, under high saturation and connected-vehicle penetration. The findings verify the effectiveness of coupling stochastic modeling with adaptive control, providing a transferable methodology for energy-efficient and data-driven lane management in smart and sustainable cities. Full article

This paper analyzes a finite-capacity Markovian queueing system with two customer types, each assigned to a separate buffer, and a single alternating server whose service priority is dynamically controlled by a queue-length-based threshold policy. The arrivals of both customer types follow independent Poisson processes, and the service times are generally distributed. The server alternates between the two buffers, granting service priority to buffer 1 when its queue length exceeds a specified threshold immediately after service completion; otherwise, buffer 2 receives priority. Once buffer 1 gains priority, it retains it until it becomes empty, with all priority transitions occurring non-preemptively. We develop an embedded Markov chain model to derive the joint queue length distribution at departure epochs and employ supplementary variable techniques to analyze the system performance at arbitrary times. This study provides explicit expressions for key performance measures, including blocking probabilities and average queue lengths, and demonstrates the effectiveness of threshold-based control in balancing service quality between customer classes. Numerical examples illustrate the impact of buffer capacities and threshold settings on system performance and offer practical insights into the design of adaptive scheduling policies in telecommunications, cloud computing, and healthcare systems. Full article

This study examines a single-server Markovian queueing system featuring continuous service and an infinite number of customers at both ends—namely, offline and online clients. Offline customers are conventional clients who arrive at the system following a Poisson process, while online customers are assumed to be endlessly present in the system. All service times are exponentially and identically distributed and independent. Utilizing generating functions and Laplace transform techniques, this study derives exact analytical expressions for the system size probabilities in both transient and steady states. Furthermore, it evaluates key performance measures for each state and provides graphical representations to illustrate the system’s dynamics, thereby enriching the understanding of its operational behavior. This work contributes to the advancement of priority-based queueing models and proposes a novel framework applicable to hybrid service architectures in contemporary digital ecosystems. Full article

This article focuses on both classical and Bayesian inference of traffic intensity in a single-server Markovian queueing model considering balking. To reflect real-world situations, the article introduces the concept of balking, where customers opt not to join the queue due to the perceived waiting time. The essence of this article involves a comprehensive analysis of different loss functions, namely, the squared error loss function (SELF) and the precautionary loss function (PLF), on the accuracy of the Bayesian estimation. To evaluate the performance of the Bayesian method with various priors such as inverted beta, gamma, and Jeffreys distributions, an assessment is performed using the Markov Chain Monte Carlo (MCMC) simulation technique. The efficacy of the Bayesian estimators is assessed by comparing the mean squared errors (MSEs). Full article

Background/Objectives: The COVID-19 pandemic underscored the urgent need for rapid, efficient testing methods at large-scale events to control virus spread. This study leverages queueing theory to explore how different floor plan configurations affect the efficiency of Rapid Antigen Diagnostic Test (RADT) centers at mass gatherings, aiming to enhance throughput and minimize wait times. Methods: Employing the MAP/PH/c model (Markovian Arrival Process/phase-type service distribution with c servers), this study compared the operational efficiency of RADT centers using U-shaped and straight-line floor plans. The research involved 500 healthy participants, who underwent the RADT process, including queue number issuance, registration, sample collection, sample mixing, and results dissemination. Agile management techniques were implemented to optimize operations. Results: The findings demonstrated that the U-shaped layout was more efficient than the straight-line configuration, reducing the average time from sample collection to results acquisition—1.6 minutes in the U-shaped layout versus 1.8 minutes in the straight-line layout. The efficiency of the U-shaped layout was particularly notable at the results stage, with statistically significant differences (p < 0.05) in reducing congestion and improving resource allocation. Conclusions: The study confirms the feasibility of implementing RADT procedures at mass gatherings and identifies the U-shaped floor plan as the optimal configuration. This layout significantly enhances testing efficiency and effectiveness, suggesting its suitability for future large-scale testing scenarios. The research contributes to optimizing mass testing strategies, vital for public health emergency management during pandemics. Full article

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. Full article

In non-Markovian tandem queueing networks the output process of one site, which is the input process to the next site, is not renewal. Consequently, the correlation analysis of that output processes is essential when studying such networks. A correlation analysis in the M/G/1 queue has been studied in the literature via derivation of the joint Laplace-Stieltjes transform (LST) of the sum of two consecutive inter-departure times. That LST is obtained by considering all possible cases at departure epochs. However, those epochs are expressed via dependent variables. In this paper, we first extend the analysis to the more general PH/G/1 queue, and investigate various queues, such as E₂/G/1 and C₂/C₂/1. Then, we consider the lag-n correlation, which requires derivation of the joint LST of sum of n + 1 consecutive inter-departure times. Yet, deriving this LST by the common approach becomes impractical for n + 1 ≥ 3, as the number of all possible cases at departure epochs increases significantly. To overcome this obstacle, we derive a corresponding single-parameter LST, which expresses the sum of n + 1 consecutive inter-departure times via the (n + 1)-st departure epoch only. Consequently, the latter LST is expressed via a much fewer number of possible cases, and not less important, as a function of independent variables only, eliminating the need to derive the corresponding joint density. Considering the M/G/1 and the E₂/G/1 queues, we demonstrate that the joint LST can be reconstructed directly via the corresponding single-parameter LST when n + 1 = 2. We further conjecture that the multi-parameter joint LST can be reconstructed from the corresponding single-parameter LST in more general queues and for values of n + 1 > 2. The conjecture is validated for various PH/G/1 queues and proved for n + 1 = 3 in the M/G/1 case. The new approach facilitates the calculation of lag-n correlation of the departure process from PH/G/1 queue for n + 1 ≥ 3. Our analysis illuminates the cases when using renewal approximation of the output process provides a proper approximation when studying non-Markovian stochastic networks. Full article

The mathematical framework presented in this article focuses on the controlled-transmission protocol’s asynchronous process of bandwidth allocation for the target virtual connection implemented under competition for communication resources. The studied process is formalized as a two-dimensional discrete Markovian chain, taking into account the distributions of queue lengths of TCP data fragments from competing client nodes. Such a chain describes the dynamics of filling the stack of transmitted but unacknowledged data fragments of the investigated end device. Distributions of the chain states were found for various ratios of the target virtual-connection bandwidth, transmission-protocol parameters, and communication-channel characteristics. Analytical dependencies for computing the performance of the target virtual connection for different operating modes were obtained. The results of experiments conducted based on the obtained analytical constructions showed that the performance of the virtual connection with a selective repeat mode is mainly determined by the data-loss intensity, the queue size distribution in transit nodes, and the ratio between the protocol window size and the route length. Full article

The paper presents a Markovian queueing model for assessing the performance of synchronisation between stations in a production system. The system at hand consists of K distinct buffers, each buffer storing an item that is needed for the next production stage. Departures are immediate when all items are present. Due to the presence of multiple buffers, there is no reasonably fast way to calculate the stationary distribution of the Markov chain. Therefore, we focused on the series expansion of the stationary distribution in terms of the arrival rate. We provide a fast algorithm for calculating these terms. Comparing our results with stochastic simulation, we show that the expansion approach converges to the simulated values for a wide range of arrival rates. Full article

Heterogeneous systems of limited capacity have general applications in manufacturing, but also in logistic or service systems due to the differences in server or workstation performance or work assignment; this is in close relationship with system flexibility, where saturation and blocking are ordinary situations of systems with high demand and limited capacity, and thus, accurate loss quantification is essential for performance evaluation. Multi-class systems of limited capacity have been studied much less than parallel homogeneous systems (Erlang models). In this context, accurate models for parallel heterogeneous ordered-entry systems were developed: without any prior queue, i.e., M/Mi/c/c, and with a k-capacity queue, i.e., M/Mi/c/c + k. These new matrix models gave an exact state formulation, and their accuracy was verified using discrete event simulation and comparison with literature results. Also, the effect of the queue capacity was studied in relationship to the pattern of service rates. Next, the heterogeneous recirculating system model was also developed with good approximation results. Finally, the proposed models were applied to evaluate systems with non-exponential service times using a new hybrid methodology by combining the Markovian model and the Monte Carlo method (MCM) for normal or lognormal service times, which also yielded useful good approximations to the simulated system. Full article

We discuss two queueing-inventory systems with catastrophes in the warehouse. Catastrophes occur according to the Poisson process and instantly destroy all items in the inventory. The arrivals of the consumer customers follow a Markovian arrival process and they can be queued in an infinite buffer. The service time of a consumer customer follows a phase-type distribution. The system receives negative customers which have Poisson flows and as soon as a negative customer comes into the system, he causes a consumer customer to leave the system, if any. One of two inventory policies is used in the systems: either $(s,S)$ or $(s,Q)$. If the inventory level is zero when a consumer customer arrives, then this customer is either lost (lost sale) or joins the queue (backorder sale). The system is formulated by a four-dimensional continuous-time Markov chain. Ergodicity condition for both systems is established and steady-state distribution is obtained using the matrix-geometric method. By numerical studies, the influence of the distributions of the arrival process and the service time and the system parameters on performance measures are deeply analyzed. Finally, an optimization study is presented in which the criterion is the minimization of expected total costs and the controlled parameter is warehouse capacity. Full article

In recent times, we have encountered new situations that have imposed restrictions on our ability to visit public places. These changes have affected various aspects of our lives, including limited access to supermarkets, vegetable shops, and other essential establishments. As a response to these circumstances, we have developed a continuous review retrial queueing–inventory system featuring a single server and controlled customer arrivals. In our system, customers arriving to procure a single item follow a Markovian Arrival Process, while the service time for each customer is modeled by an exponential distribution. Inventories are replenished according to the $(s,Q)$ reordering policy with exponentially distributed lead times. The system controls arrival in the waiting space with setup time. The customers who arrive at a not allowed situation decide to enter an orbit of infinite size with predefined probability. Orbiting customers make retrials to claim a place in the waiting space, and their inter-retrial times are exponentially distributed. The server may experience essential interruption (emergency situation) which arrives according to Poisson process. Then, the server goes for an emergency vacation of a random time which is exponentially distributed. In the steady-state case, the joint probability of the number of customers in orbit and the inventory level has been found, and the Matrix Geometric Method has been used to find the steady-state probability vector. In numerical calculations, the convexity of the system and the impact of F-policy and emergency vacation in the system are discussed. Full article