Level-dependent Quasi-birth-and-death Research Articles

Modeling gene content across a phylogeny to determine when genes become associated

We consider a model for inferring functional links between genes. We begin with the simple case of two genes whose presence or absence evolves stochastically along a phylogenetic tree. We develop a hidden Markov model where the hidden states of the model correspond to whether or not the genes perform a joint function. In the case that two genes do perform a joint function, the rates of gain or loss of each gene depend on the presence or absence of the other gene. Otherwise, those two genes are assumed to be gained and lost independently. Using simulation, we investigate the conditions under which the package corHMM can infer the hidden state correctly, and we also investigate when the Akaike information criterion (AIC) has the power to reject the simpler model when it is incorrect. We find that we can more accurately determine the dependent and independent rate class regimes when the trees have more tips and when the differences in the two rate classes are larger. We find the accuracy of corHMM is not overly affected by whether or not there are multiple transitions between the rate classes or just a single transition. We show how the two-gene case can be extended to a more general n-gene model with a level-dependent quasi-birth-and-death (LD-QBD) framework. We assume that the level n of the QBD corresponds to the number of genes that are required for some beneficial function, and the phases within each level record the presence/absence of particular genes.

Performance Analysis of Multi-processor Two-Stage Tandem Call Center Retrial Queues with Non-Reliable Processors

We analyze a multi-processor two-stage tandem call center retrial queueing network in which the processors are subject to active breakdowns and repairs at stage-I. A level-dependent quasi-birth-and-death (LDQBD) process is formulated and a sufficient condition for ergodicity of the system is discussed. Under the stability condition, the stationary distribution of the number of calls in the system, the mean number of calls in the orbit, the mean waiting time of calls in the orbit and the mean busy period of the system along with other descriptors of the system are determined by using the matrix-analytic techniques. Besides, the availability analysis of the processors is also studied. Further, we have also discussed the characteristics of the first-passage time to reach the orbit critical level, the number of calls served in the call center during this period and their corresponding moments. Finally, extensive numerical results are presented to highlight the impact of the system parameters on the performance measures of the two-stage tandem call center retrial queueing system under investigation.

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.

Tandem queues with impatient customers

This paper studies a Markovian two-station tandem queueing network with impatient customers. Queueing networks with abandonment are common in many industries, e.g., call centers and healthcare. Therefore, their management has received much attention. The resulting model is a level-dependent quasi-birth-and-death (LDQBD) process. Such models are considered analytically intractable and require numerical methods for their solution. We study a specific type of LDQBD process, where the total abandonment rate increases with the number of waiting customers leading to the level-dependent feature. We analyze an equivalent last-come-first-serve system to develop a recursive relation in our LDQBD process, reducing the problem to solving quadratic matrix equations, for which efficient and exact numerical methods exist. We further simplify the analysis by combining the recursive renewal reward theorem with Queueing and Markov chain decomposition (QMCD), so that we only need to solve one quadratic matrix equation instead of infinite ones caused by the system’s level-dependent feature. We develop an exact numerical method to evaluate various performance measures of a tandem queueing network with abandonment. Our method is applicable to the analysis of queueing networks with abandonment under settings with diverse features and in various service disciplines.

Perturbation analysis in finite LD-QBD processes and applications to epidemic models.

SummaryIn this paper, our interest is in the perturbation analysis of level‐dependent quasi‐birth‐and‐death (LD‐QBD) processes, which constitute a wide class of structured Markov chains. An LD‐QBD process has the special feature that its space of states can be structured by levels (groups of states), so that a tridiagonal‐by‐blocks structure is obtained for its infinitesimal generator. For these processes, a number of algorithmic procedures exist in the literature in order to compute several performance measures while exploiting the underlying matrix structure; among others, these measures are related to first‐passage times to a certain level L(0) and hitting probabilities at this level, the maximum level visited by the process before reaching states of level L(0), and the stationary distribution. For the case of a finite number of states, our aim here is to develop analogous algorithms to the ones analyzing these measures, for their perturbation analysis. This approach uses matrix calculus and exploits the specific structure of the infinitesimal generator, which allows us to obtain additional information during the perturbation analysis of the LD‐QBD process by dealing with specific matrices carrying probabilistic insights of the dynamics of the process. We illustrate the approach by means of applying multitype versions of the susceptible‐infective (SI) and susceptible‐infective‐susceptible (SIS) epidemic models to the spread of antibiotic‐sensitive and antibiotic‐resistant bacterial strains in a hospital ward.

Staffing Tandem Queues with Impatient Customers Application in Financial Service Operations

This paper studies a Markovian two-station tandem queueing network with impatient customers. Queueing networks with abandonment are common in many industries, e.g., call centers and healthcare. Therefore, their management has received much attention. The resulting model is a level-dependent quasi-birth-and-death (LDQBD) process. Such models are considered analytically intractable and require numerical methods for their solution. We study a specific type of LDQBD process, where the total abandonment rate increases with the number of waiting customers leading to the level-dependent feature. We analyze an equivalent last-come-first-serve system to develop a recursive relation in our LDQBD process, reducing the problem to solving quadratic matrix equations, for which efficient and exact numerical methods exist. We further simplify the analysis by combining the recursive renewal reward theorem with Queueing and Markov chain decomposition (QMCD), so that we only need to solve one quadratic matrix equation instead of infinite ones caused by the system's level-dependent feature. We develop an exact numerical method to evaluate various performance measures of a tandem queueing network with abandonment. Our method is applicable to the analysis of queueing networks with abandonment under settings with diverse features and in various service disciplines.

Multiserver Queue with Guard Channel for Priority and Retrial Customers

This paper considers a retrial queueing model where a group of guard channels is reserved for priority and retrial customers. Priority and normal customers arrive at the system according to two distinct Poisson processes. Priority customers are accepted if there is an idle channel upon arrival while normal customers are accepted if and only if the number of idle channels is larger than the number of guard channels. Blocked customers (priority or normal) join a virtual orbit and repeat their attempts in a later time. Customers from the orbit (retrial customers) are accepted if there is an idle channel available upon arrival. We formulate the queueing system using a level dependent quasi-birth-and-death (QBD) process. We obtain a Taylor series expansion for the nonzero elements of the rate matrices of the level dependent QBD process. Using the expansion results, we obtain an asymptotic upper bound for the joint stationary distribution of the number of busy channels and that of customers in the orbit. Furthermore, we develop an efficient numerical algorithm to calculate the joint stationary distribution.

THROUGHPUT ANALYSIS OF TWO-STAGE MANUFACTURING SYSTEMS WITH MERGE AND BLOCKING

Parallel lines are often used to increase production rate in many manufacturing systems where the main line splits into several lines in parallel, and after some operations, they merge into a main line again. Queueing networks with finite buffers have been widely used for modeling and analyzing manufacturing systems. This paper provides an approximation technique for multi-server two-stage networks with merge configuration and blocking which will be a building block for analysis of general manufacturing systems with parallel lines and merge configuration. The main idea of the method is to decompose the original system into subsystems that have two service stations with multiple servers, two buffers and external arrivals to the second stage are allowed. The subsystems are modeled by level dependent quasi-birth-and-death (LDQBD) process.

Algorithmic approach to Markovian multi-server retrial queues with vacations

There are many practical situations that have both features of customer retrials and server vacations. The vacation policy is characterized by the vacation startup rules and vacation termination rules. The queueing system with retrials and vacations has been described in variety of ways according to the vacation rules and retrial policies such as constant retrial policy and linear retrial policy and analyzed for each specific model. In this paper, we model the Markovian multi-server queues with customers’ retrials and vacations with level dependent quasi-birth-and-death (LDQBD) process in a unified manner and present an algorithmic approach to compute the performance measures of the system.

Approximation of throughput in tandem queues with multiple servers and blocking

In this paper, we develop an approximation method for throughput in tandem queues with multiple independent reliable servers at each stage and finite buffers between service stations. We consider the blocking after service (BAS) blocking protocol of each service stage. The service time distribution of each server is exponential. The approximation is based on the decomposition of the system into a set of coupled subsystems which are modeled by two-stage tandem queue with two buffers and are analyzed by using the level dependent quasi-birth-and-death (LDQBD) process.

Discrete-time GI/G/1 retrial queues with time-controlled vacation policies

A discrete-time GI/G/1 retrial queue with Bernoulli retrials and time-controlled vacation policies is investigated in this paper. By representing the inter-arrival, service and vacation times using a Markov-based approach, we are able to analyze this model as a level-dependent quasi-birth-and-death (LDQBD) process which makes the model algorithmically tractable. Several performance measures such as the stationary probability distribution and the expected number of customers in the orbit have been discussed with two different policies: deterministic time-controlled system and random time-controlled system. To give a comparison with the known vacation policy in the literature, we present the exhaustive vacation policy as a contrast between these policies under the early arrival system (EAS) and the late arrival system with delayed access (LAS-DA). Significant difference between EAS and LAS-DA is illustrated by some numerical examples.

On the numerical solution of Kronecker-based infinite level-dependent QBD processes

Infinite level-dependent quasi-birth-and-death (LDQBD) processes can be used to model Markovian systems with countably infinite multidimensional state spaces. Recently it has been shown that sums of Kronecker products can be used to represent the nonzero blocks of the transition rate matrix underlying an LDQBD process for models from stochastic chemical kinetics. This paper extends the form of the transition rates used recently so that a larger class of models including those of call centers can be analyzed for their steady-state. The challenge in the matrix analytic solution then is to compute conditional expected sojourn time matrices of the LDQBD model under low memory and time requirements after truncating its countably infinite state space judiciously. Results of numerical experiments are presented using a Kronecker-based matrix-analytic solution on models with two or more countably infinite dimensions and rules of thumb regarding better implementations are derived. In doing this, a more recent approach that reduces memory requirements further by enabling the computation of steady-state expectations without having to obtain the steady-state distribution is also considered.

Infinite level-dependent QBD processes and matrix-analytic solutions for stochastic chemical kinetics

Systems of stochastic chemical kinetics are modeled as infinite level-dependent quasi-birth-and-death (LDQBD) processes. For these systems, in contrast to many other applications, levels have an increasing number of states as the level number increases and the probability mass may reside arbitrarily far away from lower levels. Ideas from Lyapunov theory are combined with existing matrix-analytic formulations to obtain accurate approximations to the stationary probability distribution when the infinite LDQBD process is ergodic. Results of numerical experiments on a set of problems are provided.

Infinite level-dependent QBD processes and matrix-analytic solutions for stochastic chemical kinetics

Systems of stochastic chemical kinetics are modeled as infinite level-dependent quasi-birth-and-death (LDQBD) processes. For these systems, in contrast to many other applications, levels have an increasing number of states as the level number increases and the probability mass may reside arbitrarily far away from lower levels. Ideas from Lyapunov theory are combined with existing matrix-analytic formulations to obtain accurate approximations to the stationary probability distribution when the infinite LDQBD process is ergodic. Results of numerical experiments on a set of problems are provided.

ALGORITHMIC SOLUTION FOR M/M/c RETRIAL QUEUE WITH PH_2-RETRIAL TIMES

We present an algorithmic solution for the stationary distribution of the M/M/c retrial queue in which the retrial times of each customer in orbit are of phase type distribution of order 2. The system is modeled by the level dependent quasi-birth-and-death (LDQBD) process.

Discriminatory Processor Sharing Queues and the DREB Method

We study a discriminatory processor-sharing (DPS) queue with a Markovian arrival process (MAP) and K classes of customers. This system can be modeled into a K-dimensional quasi-birth-and death (QBD) process by the standard matrix-analytical approach, but such a process is computationally difficult to handle. We present a different formulation by which the K-dimensional QBD process is reduced to a single-dimensional level-dependent QBD process with expanding blocks. We derive analytical expressions and use them to compute the sojourn time and joint queue length distributions efficiently. This method allows us to study numerically a wide range of important open queueing problems that are common in computer and communication systems, such as various priority queues, queueing networks, and other dependent queues.

On Matrix-Geometric Solution of Nested QBD Chains

In this paper, a generalization of the level dependent Quasi-Birth-and-Death (QBD) chains is presented. We analyze nested level dependent QBD chains and provide the complete characterization of their fundamental matrices in terms of minimal non-negative solutions of a number of matrix quadratic equations. Our results provide mixed matrix-geometric solution for the stationary distribution of nested QBD chains. Applications in overload control in communication networks are also discussed.

Stochastic block–monotonicity in the approximation of the stationary distribution of infinite markov chains

Markov chains with block-structured transition matrices find many applications in various areas. Such Markov chains are characterized by having a state space which is partitioned into levels, each level consisting of a number of stages. Examples include Markov chains of GI/M/1 type and M/G/l type, and, more generally, Markov chains of Toeplitz type. The level-dependent quasi-birth-and-death (LDQBD) process provides an additional example; the transition matrix does not have repeating blocks in this case. In the analysis of such Markov chains, a number of properties and/or measures which relate to transitions among levels play a dominant role, while transitions between stages within the same level are less important In this paper, we introduce the concept of block-monotonicity and apply this notion to the analysis of Markov chains possessing block structure. In particular, the problem of approximating the stationary probability vectors is successfully treated

Calculating the equilibrium distribution in level dependent quasi-birth-and-death processes

Quasi-Birth-and-Death (QBD) processes have been analysed in detail by many authors. This paper considers the calculation of equilibrium distributions in level dependent QBD processes which are an extension of the classical QBD process. In addition to the general case, we consider a number of special cases of level dependent QBD processes, presenting algorithms for computing the equilibrium distribution