Search Results (41)

This paper analyzes a single-server queueing system with infinite capacity, where arrivals follow a Markovian arrival process and service and repair times are modeled by phase-type distributions. The service mechanism is two-tier: every customer undergoes a mandatory primary service, after which an optional secondary service is available upon request. When the system is empty, the server initiates a closedown process before taking successive multiple vacations; upon return, the server goes through a setup process before beginning service again. Service can be interrupted by random breakdowns in either mode, triggering a phase-type repair. Matrix-analytic methods are used for the steady-state analysis, yielding the stability condition, stationary probability vectors, busy period analysis and key performance measures. A cost analysis framework is also developed. Numerical experiments validate the analytical results and illustrate the practical applicability of the model. 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

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

We analyze a model of the packet buffer in which a new packet can be discarded with a probability connected to the buffer occupancy through an arbitrary dropping function. Crucially, it is assumed that packet lengths can be correlated in any way and that the interarrival time has a general distribution. From an engineering perspective, such a model constitutes a generalization of many active buffer management algorithms proposed for Internet routers. From a theoretical perspective, it generalizes a class of finite-buffer models with the tail-drop discarding policy. The contributions include formulae for the distribution of buffer occupancy and the average buffer occupancy, at arbitrary times and also in steady state. The formulae are illustrated with numerical calculations performed for various dropping functions. The formulae are also validated via discrete-event simulations. Full article

Packet losses cause a decline in the performance of packet networks, and this decline is related not only to the percentage of losses but also to the clustering of them together in series. We study how the correlation of packet sizes influences this clustering when the losses are caused by buffer overflows. Specifically, for a model of a buffer with correlated packet sizes, we derive the burst ratio parameter, an intuitive metric for the inclination of losses to cluster. In addition to the burst ratio, we obtain the sequential losses distribution in the first, k-th, and stationary overflow periods. The Markovian Service Process (MSP) used by the model empowers it to mimic arbitrary packet size distributions and arbitrary correlation strengths. Using numeric examples, the impact of packet size correlation, buffer size, and traffic intensity on the burst ratio is showcased and discussed. 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

An $MAP/PH/N$-type queuing system functioning within a finite-state Markovian random environment is studied. The random environment’s state impacts the number of available servers, the underlying processes of customer arrivals and service, and the impatience rate of customers. The impact on the state space of the underlying processes of customer arrivals and of the more general, as compared to exponential, service time distribution defines the novelty of the model. The behavior of the system is described by a multidimensional Markov chain that belongs to the classes of the level-independent quasi-birth-and-death processes or asymptotically quasi-Toeplitz Markov chains, depending on whether or not the customers are absolutely patient in all states of the random environment or are impatient in at least one state of the random environment. Using the tools of the corresponding processes or chains, a stationary analysis of the system is implemented. In particular, it is shown that the system is always ergodic if customers are impatient in at least one state of the random environment. Expressions for the computation of the basic performance measures of the system are presented. Examples of their computation for the system with three states of the random environment are presented as 3-D surfaces. The results can be useful for the analysis of a variety of real-world systems with parameters that may randomly change during system operation. In particular, they can be used for optimally matching the number of active servers and the bandwidth used by the transmission channels to the current rate of arrivals, and vice versa. 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

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 $K\ge 2$ 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 (${\beta }_k$, $S_k$), $1\le k\le K$. 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. Full article

Process algebra can be considered one of the most practical formal methods for modeling Smart IoT Systems in Digital Twin, since each IoT device in the systems can be considered as a process. Further, some of the algebras are applied to predict the behavior of the systems. For example, PALOMA (Process Algebra for Located Markovian Agents) and PACSR (Probabilistic Algebra of Communicating Shared Resources) process algebras are designed to predict the behavior of IoT Systems with probability on choice operations. However, there is a lack of analytical methods in the algebras to predict the nondeterministic behavior of the systems. Further, there is no control mechanism to handle undesirable nondeterministic behavior of the systems. In order to overcome these limitations, this paper proposes a new process algebra, called dTP-Calculus, which can be used (1) to specify the nondeterministic behavior of the systems with static probability, (2) verify the safety and security requirements of the nondeterministic behavior with probability requirements, and (3) control undesirable nondeterministic behavior with dynamic probability. To demonstrate the feasibility and practicality of the approach, the SAVE (Specification, Analysis, Verification, Evaluation) tool has been developed on the ADOxx Meta-Modeling Platform and applied to a SEMS (Smart Emergency Medical Service) example. In addition, a miniature digital twin system for the SEMS example was constructed and applied to the SAVE tool as a proof of concept for Digital Twin. It shows that the approach with dTP-Calculus on the tool can be very efficient and effective for Smart IoT Systems in Digital Twin. 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

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. Full article

In this paper, we consider two queuing models. Model 1 considers a single-server working vacation queuing system with interdependent arrival and service processes. The arrival and service processes evolve by transitions on the product space of two Markovian chains. The transitions in the two Markov chains in the product space are governed by a semi-Markov rule, with sojourn times in states governed by the exponential distribution. In contrast, in the second model, we consider independent arrival and service processes following phase-type distributions with representation $(\mathbf{\alpha },T)$ of order m and $(\mathbf{\beta },S)$ of order n, respectively. The service time during normal working is the above indicated phase-type distribution whereas that during working vacation is a phase-type distribution with representation $(\mathbf{\beta },\theta S)$, $0<\theta <1$. The duration of the latter is exponentially distributed. The latter model is already present in the literature and will be briefly described. The main objective is to make a theoretical comparison between the two. Numerical illustrations for the first model are provided. Full article