Quasi-birth-and-death Research Articles

Matched queues with matching batch pair [formula omitted]

In this paper, we develop the matrix-analytic method to discuss an interesting but challenging bilateral stochastic matching problem: A matched queue with matching batch pair (m,n) and two types of impatient customers, where the two types of customers arrive according to two independent Poisson processes. Once m A-customers and n B-customers are matched as a group, the m+n customers immediately leave the system. We show that this matched queue can be expressed as a novel bidirectional level-dependent quasi-birth-and-death (QBD) process whose analysis has its own interests, and specifically, computing the maximal non-positive inverse matrices of bidirectional infinite sizes by using the RG-factorizations. Based on this, we can provide an effective matrix-analytic method to deal with this matched queue, including the system stability, the average stationary queue lengths, the average sojourn times, and the departure process. We believe that the methodology and results developed in this paper can be applicable to studying more general matched queueing systems, which are widely encountered in many practical areas.

Analysis of heterogeneous two server queueing system with multiple working vacations and server breakdowns

The steady state analysis and the time-dependent behaviour of a heterogeneous two-server multiple vacation queueing model is examined in this research work. The model also takes into account the possibility of a breakdown happening when server 1 is busy while server 2 is on working vacation and both servers are busy, with the restoration procedure starting right away. The stability condition is established through the use of matrix geometric technique, and the steady-state probability vector of the system?s number of customers has been evaluated as a quasi-birth and death (QBD) process. Some system performance measures are obtained. Numerical analysis for both steady-state and transient analysis are given to our proposed model.

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/ .

QBD Processes Associated with Jacobi–Koornwinder Bivariate Polynomials and Urn Models

We study a family of quasi-birth-and-death (QBD) processes associated with the so-called first family of Jacobi–Koornwinder bivariate polynomials. These polynomials are orthogonal on a bounded region typically known as the swallow tail. We will explicitly compute the coefficients of the three-term recurrence relations generated by these QBD polynomials and study the conditions under we can produce families of discrete-time QBD processes. Finally, we show an urn model associated with one special case of these QBD processes.

Markov modeling and performance analysis of infectious diseases with asymptomatic patients.

After over three years of COVID-19, it has become clear that infectious diseases are difficult to eradicate, and humans remain vulnerable under their influence in a long period. The presence of presymptomatic and asymptomatic patients is a significant obstacle to preventing and eliminating infectious diseases. However, the long-term transmission of infectious diseases involving asymptomatic patients still remains unclear. To address this issue, this paper develops a novel Markov process for infectious diseases with asymptomatic patients by means of a continuous-time level-dependent quasi-birth-and-death (QBD) process. The model accurately captures the transmission of infectious diseases by specifying several key parameters (or factors). To analyze the role of asymptomatic and symptomatic patients in the infectious disease transmission process, a simple sufficient condition for the stability of the Markov process of infectious diseases is derived using the mean drift technique. Then, the stationary probability vector of the QBD process is obtained by using RG-factorizations. A method of using the stationary probability vector is provided to obtain important performance measures of the model. Finally, some numerical experiments are presented to demonstrate the model's feasibility through analyzing COVID-19 as an example. The impact of key parameters on the system performance evaluation and the infectious disease transmission process are analyzed. The methodology and results of this paper can provide theoretical and technical support for the scientific control of the long-term transmission of infectious diseases, and we believe that they can serve as a foundation for developing more general models of infectious disease transmission.

Stochastic performance modeling for practical byzantine fault tolerance consensus in the blockchain

The practical Byzantine fault tolerant (PBFT) consensus mechanism is one of the most basic consensus algorithms (or protocols) in blockchain technologies. Thus its performance evaluation is an interesting and challenging topic due to the higher complexity of its consensus work in a peer-to-peer network. This study describes a simple stochastic performance model of the PBFT consensus mechanism. This model is refined not only as a queuing system with complicated service times but also as a level-independent quasi-birth-and-death (QBD) process. With regard to the level-independent QBD process, we apply the matrix-geometric solution to obtain the necessary and sufficient condition under which the PBFT consensus system is stable and then numerically compute the stationary probability vector of the QBD process. Thus, we provide four useful performance measures for the PBFT consensus mechanism, and we can numerically calculate these performance measures. Finally, we use numerical examples to verify the validity of our theoretical results and demonstrate how the four performance measures are influenced by certain key parameters of the PBFT consensus. Considering theory of multi-dimensional Markov processes, we are optimistic that the methodology and results presented in this study are applicable to a wide range of PBFT consensus mechanism and even other types of consensus mechanisms.

Sensitivity analysis for a Markov regenerative software rejuvenation model

This article considers the parametric sensitivity of a software rejuvenation model for transaction systems whose system behavior is described by a quasi-birth-and-death (QBD) process with Markovian arrivals, and the software rejuvenation model is represented by a Markov regenerative process (MRGP). The stationary analysis of the MRGP model is based on the embedding Markov chain (EMC) approach. Then the sensitivities of the stationary solution and the performance criteria of interest, i.e., the loss probability of transactions, with respect to all model parameters, are formulated. Finally, a numerical experiment was conducted to illustrate the critical parameters for a transaction system with software rejuvenation. The numerical results showed that the proposed approach effectively revealed critical model parameters and provided significant insights into performance optimization and parameter estimation.

On a many-to-many matched queueing system with flexible matching mechanism and impatient customers

In this paper, we focus on a kind of many-to-many matching system which is applicable to many real-life matching systems. We use customers and servers to represent the inputs at both ends. To portray the inputs’ behavioral features in many actual matching service systems, we introduce the customer impatience and flexible matching mechanism. Considering the possibility of customer reneging from the system, an infinite-dimensional non-homogeneous QBD (quasi-birth-and-death) process is constructed to depict the process of queue lengths at both ends. Methodologically, we capture the joint stationary distribution of the queue lengths at both sides by truncating the Markov chain. In the above procedure, the matrix G at any truncation point is obtained, and two detailed algorithms are provided to determine the truncation point and the approximate stationary distribution. For the system without customers’ impatience, the stationary distribution of the queue lengths is also analyzed by using matrix analytic method. Subsequently, several important performance measures (e.g., the loss probability of customers) are obtained. In addition, by reconstructing the three-dimensional and two-dimensional absorbing Markov chains, we investigate the distributions of customer’s waiting time for matching and conditional clearing time of the system, and apply RG-factorizations to calculate the specific results. Finally, numerical examples are used to reveal the effects of different parameters on the performance measures.

Double-ended queues with non-Poisson inputs and their effective algorithms

It is interesting and challenging to study double-ended queues with First-Come-First-Match discipline under customers’ impatient behavior and non-Poisson inputs. The system stability can be guaranteed by the customers’ impatient behavior, while the existence of impatient customers makes analysis of such double-ended queues more difficult or even impossible to find an explicitly analytic solution, thus it becomes more and more important to develop effective numerical methods in a variety of practical matching problems. This paper studies a block-structured double-ended queue, whose block structure comes from two independent Markovian arrival processes (MAPs), which are non-Poisson inputs. We show that such a queue can be expressed as a new bilateral quasi birth-and-death (QBD) process which has its own interest. Based on this, we provide a detailed analysis for both the bilateral QBD process and the double-ended queue, including the system stability, the queue size distributions, the average stationary queue lengths, and the sojourn time of any arriving customers. Furthermore, we develop three effective algorithms for computing the performance measures (i.e., the probabilities of stationary queue lengths, the average stationary queue lengths, and the average sojourn times) of the double-ended queue with non-Poisson inputs. Finally, we use some numerical examples in tabular and graphical to illustrate how the performance measures are influenced by some key system parameters. We believe that the methodology and results described in this paper can be applicable to deal with more general double-ended queues in practice, and develop some effective algorithms for the purpose of many actual uses.

A service system where junior servers approach a senior server on behalf of customers

Many service systems in various type of industry operate under a policy according to which customers are not allowed to directly approach the senior server of the system. If the senior server's intervention in the service is required, the junior server who handles the request approaches the senior server on behalf of the customer. Thus, the service is divided into two phases, where the first phase (e.g., obtaining information or performing an initial service) is conducted by one of the junior servers, after which the junior server approaches the senior server and they together complete the service. However, the senior server may be occupied by another junior server, in which case the junior server joins a queue of such servers who are waiting for the senior server to become available. That is, the servers of the first phase (junior servers) act as customers in the second phase. We formulate and analyze this stochastic queueing system as a two-dimensional quasi-birth-and-death (QBD) process and derive its steady-state probabilities using matrix geometric methods. By means of economic analysis, we provide a scheme for deriving an optimal work division policy, which dictates how much work should be assigned to each phase in order to minimize the system's overall cost or maximize the system's utility. The economic analysis is carried out with an eye to three practical examples: (i) a bank office model; (ii) an app development model; and (iii) a system subject to the constraint that a certain amount of work is assigned to the senior server. Sensitivity analyses are conducted for all three models.

A mean-field matrix-analytic method for bike sharing systems under Markovian environment

This paper proposes a novel mean-field matrix-analytic method in the study of bike sharing systems, in which a Markovian environment is constructed to express time-inhomogeneity and asymmetry of processes that customers rent and return bikes. To achieve effective computability of this mean-field method, this study provides a unified framework through the following three basic steps. The first one is to deal with a major challenge encountered in setting up mean-field block-structured equations in general bike sharing systems. Accordingly, we provide an effective technique to establish a necessary reference system, which is a time-inhomogeneous queue with block structures. The second one is to prove asymptotic independence (or propagation of chaos) in terms of martingale limits. Note that asymptotic independence ensures and supports that we can construct a nonlinear quasi-birth-and-death (QBD) process, such that the stationary probability of problematic stations can be computed under a unified nonlinear QBD framework. Lastly, in the third step, we use some numerical examples to show the effectiveness and computability of the mean-field matrix-analytic method, and also to provide valuable observation of the influence of some key parameters on system performance. We are optimistic that the methodology and results given in this paper are applicable in the study of general large-scale bike sharing systems.

Quasi-birth-and-death processes and multivariate orthogonal polynomials

The aim of this paper is to study some models of quasi-birth-and-death (QBD) processes arising from the theory of bivariate orthogonal polynomials. First we will see how to perform the spectral analysis in the general setting as well as to obtain results about recurrence and the invariant measure of these processes in terms of the spectral measure supported on some domain Ω⊂Rd. Afterwards, we will apply our results to several examples of bivariate orthogonal polynomials, namely product orthogonal polynomials, orthogonal polynomials on a parabolic domain and orthogonal polynomials on the triangle. We will focus on linear combinations of the Jacobi matrices generated by these polynomials and produce families of either continuous or discrete-time QBD processes. Finally, we show some urn models associated with these QBD processes.

Tighter Lyapunov Truncation for Multi-Dimensional Continuous Time Markov Chains with Known Moments

Continuous Time Markov chains (CTMCs) are widely used to model and analyze networked systems. A common analysis approach is to solve the system of balance equations governing the state transitions of a CTMC to obtain its steadystate probability distribution, and use the state probabilities to derive or compute various performance measures. In many systems, the state space of the underlying CTMC is infinite and multi-dimensional with state-dependent transitions; exact analysis of such models is challenging. For example, the exact probability distribution of the number of jobs in the Discriminatory Processor Sharing (DPS) system, first proposed by Kleinrock in 1967 [4], is still an open challenge. Likewise, obtaining the exact state probabilities of quasi-birth-and-death (QBD) processes with leveldependent transitions is known to be challenging [1]; QBDs are infinite state space multi-dimensional Markov chains in which states are organized into levels and transitions are skip-free between the levels.

Exact analysis of summary statistics for continuous-time discrete-state Markov processes on networks using graph-automorphism lumping

We propose a unified framework to represent a wide range of continuous-time discrete-state Markov processes on networks, and show how many network dynamics models in the literature can be represented in this unified framework. We show how a particular sub-set of these models, referred to here as single-vertex-transition (SVT) processes, lead to the analysis of quasi-birth-and-death (QBD) processes in the theory of continuous-time Markov chains. We illustrate how to analyse a number of summary statistics for these processes, such as absorption probabilities and first-passage times. We extend the graph-automorphism lumping approach [Kiss, Miller, Simon, Mathematics of Epidemics on Networks, 2017; Simon, Taylor, Kiss, J. Math. Bio. 62(4), 2011], by providing a matrix-oriented representation of this technique, and show how it can be applied to a very wide range of dynamical processes on networks. This approach can be used not only to solve the master equation of the system, but also to analyse the summary statistics of interest. We also show the interplay between the graph-automorphism lumping approach and the QBD structures when dealing with SVT processes. Finally, we illustrate our theoretical results with examples from the areas of opinion dynamics and mathematical epidemiology.

The Role of Information in System Stability with Partially Observable Servers

We present a methodology for analyzing the role of information on system stability. For this we consider a simple discrete-time controlled queueing system, where the controller has a choice of which server to use at each time slot and server performance varies according to a Markov modulated random environment. At the extreme cases of information availability, that is when there is either full information or no information, stability regions and maximally stabilizing policies are trivial. But in the more realistic cases where only the environment state of the selected server is observed, only the service successes are observed or only queue length is observed, finding throughput maximizing control laws is a challenge. To handle these situations, we devise a Partially Observable Markov Decision Process (POMDP) formulation of the problem and illustrate properties of its solution. We further model the system under given decision rules, using Quasi-Birth-and-Death (QBD) structure to find a matrix analytic expression for the stability bound. We use this formulation to illustrate how the stability region grows as the number of controller belief states increases. The example that we consider in this paper is a case of two servers where the environment of each is modulated like a Gilbert-Elliot channel. As simple as this case seems, there appear to be no closed form descriptions of the stability region under the various regimes considered. However, the numerical approximations to the POMDP Bellman equations together with the numerical solutions of the QBDs, both of which are in agreement, hint at a variety of structural results.

On the ergodicity of a class of level-dependent quasi-birth-and-death processes

AbstractWe examine necessary and sufficient conditions for recurrence and positive recurrence of a class of irreducible, level-dependent quasi-birth-and-death (LDQBD) processes with a block tridiagonal structure that exhibits asymptotic convergence in the rows as the level tends to infinity. These conditions are obtained by exploiting a multi-dimensional Lyapunov drift approach, along with the theory of generalized Markov group inverses. Additionally, we highlight analogies to well-known average drift results for level-independent quasi-birth-and-death (QBD) processes.

A Quasi Birth-and-Death model for tumor recurrence

A major cause of chemoresistance and recurrence in tumors is the presence of dormant tumor foci that survive chemotherapy and can eventually transition to active growth to regenerate the cancer. In this paper, we propose a Quasi Birth-and-Death (QBD) model for the dynamics of tumor growth and recurrence/remission of the cancer. Starting from a discrete-state master equation that describes the time-dependent transition probabilities between states with different numbers of dormant and active tumor foci, we develop a framework based on a continuum-limit approach to determine the time-dependent probability that an undetectable residual tumor will become large enough to be detectable. We derive an exact formula for the probability of recurrence at large times and show that it displays a phase transition as a function of the ratio of the death rate μA of an active tumor focus to its doubling rate λ. We also derive forward and backward Kolmogorov equations for the transition probability density in the continuum limit and, using a first-passage time formalism, we obtain a drift-diffusion equation for the mean recurrence time and solve it analytically to leading order for a large detectable tumor size N. We show that simulations of the discrete-state model agree with the analytical results, except for O(1/N) corrections. As an example of the use of our model in a clinical setting, we show that a range of model parameters can fit Kaplan–Meier recurrence-free survival data for ovarian cancer. Finally, we show in simulations that extending the duration of chemotherapy increases both the mean recurrence time and the asymptotic (large-time) probability of no recurrence. Our results should be useful in planning optimized chemotherapy dosing and duration for cancer treatment, especially in cancer types for which no targeted therapy is available.

Modelling and optimisation of a two-server queue with multiple vacations and working breakdowns

This paper presents a steady-state analysis of an M/M/2 queue with heterogeneous servers (Server 1 and Server 2). Server 1 is reliable and may leave for a vacation when the system becomes empty. Sever 2 is unreliable and may break down while serving customers. When a breakdown occurs, Server 2 reduces the service rate rather than halting service. We formulate this queueing model as a quasi birth-and-death (QBD) process, using the matrix geometric method to compute the stationary distribution of system size. We also develop several measures to evaluate the performance of the system. A cost model based on system performance measures is formulated as a heuristic cost optimisation problem subject to stability conditions. A canonical particle swarm optimisation algorithm is used to obtain numerical solutions for the approximate optimal service rates of Server 1 and Server 2. Moreover, we present numerical results showing the effects of various parameters on the approximate optimal service rates as well as a practical example illustrating the application of the proposed model.

The time-dependent expected reward and deviation matrix of a finite QBD process

Deriving the time-dependent expected reward function associated with a continuous-time Markov chain involves the computation of its transient deviation matrix. In this paper we focus on the special case of a finite quasi-birth-and-death (QBD) process, motivated by the desire to compute the expected revenue lost in a MAP/PH/1/C queue.We use two different approaches in this context. The first is based on the solution of a finite system of matrix difference equations; it provides an expression for the blocks of the expected reward vector, the deviation matrix, and the mean first passage time matrix. The second approach, based on some results in the perturbation theory of Markov chains, leads to a recursive method to compute the full deviation matrix of a finite QBD process. We compare the two approaches using some numerical examples.

An imperfect sensing-based channel reservation strategy in CRNs and its performance evaluation

Channel reservation strategy in CRNs is an effective technology for conserving communication resources. In this paper, using the imperfect sensing of secondary user (SU) packets, and considering the patience degree of SU packets, we propose a channel reservation strategy in a CRN. Aligned with the proposed channel reservation strategy, we establish a continuous-time Markov chain model to capture the stochastic behavior of the two types of user packets. Then, in order to obtain the steady-state probability distribution for the system model, we present a new algorithm for solving the quasi-birth-and-death (QBD) process. At last, based on the energy detection method, we evaluate the system performance in terms of the throughput of SU packets, the average latency of SU packets, the switching rate of SU packets and the channel utilization in relation to the energy detection threshold and the number of reserved channels.