Quasi-birth-death Research Articles

QBD approximations of a call center queueing model with general patience distribution

This paper investigates a computationally practical way for analyzing a call center queueing model, i.e., a finite-capacity, multi-server queueing model, where each server goes on a single vacation. Poisson arrival process and exponential service and vacation times are assumed. We also assume that each customer may leave the queue due to impatience. Customers’ patience times are i.i.d. random variables with a general distribution. Level-dependent finite QBD (quasi-birth–death) processes are employed to approximate such a queueing model. Two approaches are considered. The first one uses the phase-type (PH) distribution to approximate the general patience distribution, whereas the second one is based on the idea of replacing the eventual reneging of customers with balking. We find that the first approach is almost impossible to compute numerically due to the exponential growth of the size of the block matrices in a level-dependent finite QBD. We examine the validity and applicability of the approximation based on the second approach and show that it gives us a practical way to obtain performance measures of call center systems in practical scale with sufficiently reasonable accuracy.

State-space reduction in the model of Access Control Lists in IP routers

This document provides an analytic performance model of routers using Access Control Lists (ACLs). For mathematical modelling, a Markovian process, the Discrete-Time Quasi Birth-Death (DQBD) process is used. ACLs in IP routers provide access control and protection for network segments and hosts. The mathematical model of ACLs that also includes input and output interface models is also introduced here, as well as an idea to reduce the state-space of the model, together with the corresponding proof.

An auxiliary server threshold queueing model and its application to the operational characteristics of web server system

This paper analyzes an auxiliary server queueing model in which secondary servers are added to the main server(s) when the queue length exceeds some predetermined threshold value at the main queue. We model this system by the level-dependent quasi-birth-death (QBD) process and develop computation algorithms. We apply this model to the web-server system to explore some specific operational characteristics and draw some useful conclusions.

Response Time Distribution in a D-MAP/PH/1 Queue with General Customer Impatience

ABSTRACT This paper presents two methods to calculate the response time distribution of impatient customers in a discrete-time queue with Markovian arrivals and phase-type services, in which the customers’ patience is generally distributed (i.e., the D-MAP/PH/1 queue). The first approach uses a GI/M/1 type Markov chain and may be regarded as a generalization of the procedure presented in Van Houdt[14] for the D-MAP/PH/1 queue, where every customer has the same amount of patience. The key construction in order to obtain the response time distribution is to set up a Markov chain based on the age of the customer being served, together with the state of the D-MAP process immediately after the arrival of this customer. As a by-product, we can also easily obtain the queue length distribution from the steady state of this Markov chain. We consider three different situations: (i) customers leave the system due to impatience regardless of whether they are being served or not, possibly wasting some service capacity, (ii) a customer is only allowed to enter the server if he is able to complete his service before reaching his critical age and (iii) customers become patient as soon as they are allowed to enter the server. In the second part of the paper, we reduce the GI/M/1 type Markov chain to a Quasi-Birth-Death (QBD) process. As a result, the time needed, in general, to calculate the response time distribution is reduced significantly, while only a relatively small amount of additional memory is needed in comparison with the GI/M/1 approach. We also include some numerical examples in which we apply the procedures being discussed.

A New Policy Evaluation Algorithm for Markov Decision Processes with Quasi Birth-Death Structure

ABSTRACT This paper describes a new algorithm for policy evaluation for Markov decision processes (MDP) that possess a quasi birth-death structure. The proposed algorithm is based on matrix analytic methods which use probabilistic concepts associated with restricting the underlying Markov process to certain state subsets. A telecommunications application example shows that the method offers significant computational reduction compared to a standard MDP policy evaluation approach.

JOIN MINIMUM COST QUEUE FOR MULTICLASS CUSTOMERS: STABILITY AND PERFORMANCE BOUNDS

We consider a system of K parallel queues providing different grades of service through each of the queues and serving a multiclass customer population. Service differentiation is achieved by specifying different join prices to the queues. Customers of class j define a cost function ψij(ci,xi) for taking service from queue i when the join price for queue i is ci and congestion in queue i is xi and join the queue that minimizes ψij(·,·). Such a queuing system will be called the “join minimum cost queue” (JMCQ) and is a generalization of the join shortest queue (JSQ) system. Non-work-conserving (called Paris Metro pricing system) and work-conserving (called the Tirupati system) versions of the JMCQ are analyzed when the cost to an arrival of joining a queue is a convex combination of the join price for that queue and the expected waiting time in that queue at the arrival epoch. Our main results are for a two-queue system.We obtain stability conditions and performance bounds. To obtain the lower and upper performance bounds, we propose two quasi-birth–death (QBD) processes that are derived from the original systems by suitably truncating the state space. The state space truncation in the non-work-conserving JMCQ follows the method of van Houtum and colleagues. We then show that this method is not applicable to the work-conserving JMCQ and provide sample-path-based proofs to show that the number in each queue is bounded by the number in the corresponding queues of these QBD processes. These sample-path proof techniques might also be of independent interest. We then show that the performance measures like mean queue length and revenue rate of the system are also bounded by the corresponding quantities of these QBD processes. Numerical examples show that these bounds are fairly tight. Finally, we generalize some of these results to systems with more queues.

Finding equilibrium probabilities of QBD processes by spectral methods when eigenvalues vanish

In this paper, we discuss the use of spectral or eigenvalue methods for finding the equilibrium probabilities of quasi-birth–death processes for the case where some eigenvalues are zero. Since this leads to multiple eigenvalues at zero, a difficult problem to analyze, we suggest to eliminate such eigenvalues. To accomplish this, the dimension of the largest Jordan block must be established, and some initial equations must be eliminated. The method is demonstrated by two examples, one dealing with a tandem queue, the other one with a shorter queue problem.

Etaqa-MG1: an efficient technique for the analysis of a class of M/G/1-type processes by aggregation

We extend the Etaqa approach, initially proposed for the efficient numerical solution of a class of quasi-birth–death processes, to a more complex class of M/G/1-type Markov processes where arbitrary forward transitions are allowed but backward transitions must be to a single state to the previous level. The new technique reduces the exact solution of this class of M/G/1-type models to that of a finite linear system. We demonstrate the utility of our method by describing the exact computation of an extensive class of Markov reward functions that include the expected queue length or its higher moments. We also provide an algorithm that finds an appropriate state reordering satisfying our applicability conditions, if one such order exists. We illustrate the method, discuss its complexity and numerical stability, and present comparisons with other traditional techniques.

Tree Structured QBD Markov Chains and Tree‐Like QBD Processes

In this paper, we show that an arbitrary tree structured quasi‐birth–death (QBD) Markov chain can be embedded in a tree‐like QBD process with a special structure. Moreover, we present an algebraic proof that applying the natural fixed point iteration (FPI) to the nonlinear matrix equation V = B + ∑ s=1 d U s (I − V)−1 D s that solves the tree‐like QBD process, is equivalent to the more complicated iterative algorithm presented by Yeung and Alfa (1996).

Comments on a Shifted Cyclic Reduction Algorithm for Quasi-Birth-Death Problems

A shifted cyclic reduction algorithm has been proposed by He, Meini, and Rhee [SIAM J. Matrix Anal. Appl., 23 (2001), pp. 673--691] for finding the stochastic matrix G associated with discrete-time quasi-birth-death (QBD) processes. We point out that the algorithm has quadratic convergence even for null recurrent QBDs. We also note that the approximations (to the matrix G) obtained by their algorithm are always stochastic when they are nonnegative.

동적인 고정 및 가변 전송을 서비스를 위한 다중 서비스 네트워크 스위치의 성능 분석

본 논문에서는 고정 전송율 서비스 및 가변 전송율 서비스 호들이 동적으로 네트워크에 들어와서 데이터를 전송하고, 데이터를 모두 전송하면 네트워크에서 사라지는 동적인 다중 서비스 네트워크에서, 고정 전송율 및 가변 전송율 서비스의 요구 성능을 만족하기 위한 네트워크 스위치(링크)의 성능 분석 및 설계 방법을 제안한다. 먼저 동적인 다중 서비스 네트워크의 링크를 Harkov 체인을 이용해 모델하고, QBD (Quasi-Birth-Death) 프로세스와 행렬 방정식을 이용해 가변 전송율 서비스의 호들에 할당될 전송율의 성능을 구한다. 즉, 가변 전송율 서비스의 평균 호수를 추정하고 이를 바탕으로, 호당 평균 전송율 및 평균 전송 지연 시간을 구한다. 고정 전송율 서비스의 성능은 고정 전송율 서비스 호들의 차단 확률(blocking probability)로 구해진다. 이렇게 해서 구해진 고정 전송율 및 가변 전송율 서비스의 성능을 이용해서, 고정 전송율 및 가변 전송율 서비스 호들의 요구 성능(전송율, 전송 지연 시간, 호의 차단 확률 등)을 만족하기 위한 네트워크 링크의 용량을 설계하는 방법을 제안한다. 이러한 방법을 데이터 및 음성(비디오)을 동시에 지원하는 네트워크 스위치의 성능 분석 및 용량 설계에 이용할 수 있다. We consider design of multiservice network link, in which connections of constant- and adaptive-rate services arrive and leave dynamically. We propose performance analysis and design methods of these dynamic multiservice networks. A multiservice network link is modeled by a Markov chain, and data rates for adaptive-rate connections are derived using QBD (Quasi-Birth-Death) processes and matrix-geometric equations. We estimate average number of adaptive-rate connections, average data rate and average connection delay. The performance of constant-rate connections is determined from the blocking probability of the connections. Based on the performance of constant-and adaptive- rate connections, we propose a design methods of a network link to satisfy performance requirements of constant- and adaptive-rate connections (data rates, delay, blocking probability). Our methods can be used for the analysis and design of network switch supporting dynamic data and voice (video) traffic simultaneously.

A Shifted Cyclic Reduction Algorithm for Quasi-Birth-Death Problems

The problem of the computation of the stochastic matrix G associated with discrete-time quasi-birth-death (QBD) Markov chains is analyzed. We present a shifted cyclic reduction algorithm and show that the speed of convergence of the latter modified algorithm is always faster than that of the original cyclic reduction.

STABILITY AND PERFORMANCE OF STACK ALGORITHMS FOR RANDOM ACCESS COMMUNICATION MODELED AS A TREE STRUCTURED QBD MARKOV CHAIN

In this paper, we introduce an analytical model to study the stability and the main performance measures of a binary stack algorithm for random multiple access communication. The input traffic is a discrete time Batch Markovian Arrival Process (D-BMAP). The analytical model is nearly exact (one minor approximation is required) and the analysis is based on recent results obtained from tree structured Quasi-Birth-Death (QBD) Markov chains. Apart from studying the stability of the protocol, we are also able to calculate the mean delay and other important performance measures. The method deployed in this paper can also be extended to evaluate other medium access control (MAC) protocols with an underlying stack structure.

Performance evaluation of multiple input-queued ATM switches with PIM scheduling under bursty traffic

In this letter, we analyze the performance of multiple input-queued asynchronous transfer mode (ATM) switches that use parallel iterative matching (PIM) for scheduling the transmission of head-of-line cells in the input queues. A queueing model of the switch is developed under independently, identically distributed, two-state Markov modulated Bernoulli processes bursty traffic. The underlying Markov chain of the queueing model is a quasi-birth-death (QBD) chain. The QBD chain is solved using an iterative computing method. Interesting performance metrics of the ATM switch such as the throughput, the mean cell delay, and the cell loss probability can be derived from the model. Numerical results from both the analytical model and simulation are presented, and the accuracy of the analysis is briefly discussed.

A general theory on spectral properties of state-homogeneous finite-state quasi-birth–death processes

In this paper a spectral theory pertaining to Quasi-Birth–Death Processes (QBDs) is presented. The QBD, which is a generalization of the birth–death process, is a powerful tool that can be utilized in modeling many stochastic phenomena. Our theory is based on the application of a matrix polynomial method to obtain the steady-state probabilities in state-homogeneous finite-state QBDs. The method is based on finding the eigenvalue–eigenvector pairs that solve a matrix polynomial equation. Since the computational effort in the solution procedure is independent of the cardinality of the counting set, it has an immediate advantage over other solution procedures. We present and prove different properties relating the quantities that arise in the solution procedure. By also compiling and formalizing the previously known properties, we present a formal unified theory on the spectral properties of QBDs, which furnishes a formal framework to embody much of the previous work. This framework carries the prospect of furthering our understanding of the behavior the modeled systems manifest.

Numerical analysis of queues with batch arrivals

The steady state distribution of quasi-birth–death processes can be efficiently obtained by matrix geometric (MG) methods. Since a number of telecommunication problems are modelled by processes with batch arrivals, the extension of MG methods for these processes has practical importance. This paper presents an extension of MG methods which is effective for the analysis of quasi-birth–death processes with batch arrivals. The proposed method is compared with one of the well-known methods.

Sojourn-time analysis on nodal congestion in broadband networks

In this paper, we study the sojourn-time statistics and the temporal behavior of nodal congestion in integrated broadband networks. The node is modeled as a finite quasi-birth-death (QBD) process with level-dependent transitions. By formulating the problem as one which is amenable to the generalized folding algorithm (GFA), we are, for the first time, able to analyze realistic systems with large buffer and complex input traffic. The dynamics of the system under realistic traffic environment and different operating regimes are studied. The effects of various modeling artifacts such as fluid-flow and infinite-buffer approximations are also investigated. The potential of statistical multiplexing in reducing bursty cell loss is demonstrated. The trade-offs between different system design alternatives, e.g., buffering vs. statistical multiplexing, are discussed. We also investigate the controlling effect of preemptive cell discarding on steady-state and transient system performance. Both single-level and two-level overload control with hysteresial-switching mechanisms are considered. The use of sojourn-time based QoS metrics to supplement long-term steady-state metrics is also discussed.

Computing approximate stationary distributions for discrete Markov processes with banded infinitesimal generators

We develop an algorithm for computing approximations to the stationary distribution of a discrete birth-and-death process, provided that the infinitesimal generator is a banded matrix. We begin by computing stationary distributions for processes whose infinitesimal generators are Hessenberg. Our derivation in this special case is different from the classical case but it leads to the same result. We then show how to extend these ideas to processes where the infinitesimal generator is banded (or half-banded) and to quasi-birth–death processes. Finally, we give an example of the application of this method to a nearly completely decomposable Markov chain to demonstrate the general applicability of the technique.

Computing approximate stationary distributions for discrete Markov processes with banded infinitesimal generators

We develop an algorithm for computing approximations to the stationary distribution of a discrete birth-and-death process, provided that the infinitesimal generator is a banded matrix. We begin by computing stationary distributions for processes whose infinitesimal generators are Hessenberg. Our derivation in this special case is different from the classical case but it leads to the same result. We then show how to extend these ideas to processes where the infinitesimal generator is banded (or half-banded) and to quasi-birth–death processes. Finally, we give an example of the application of this method to a nearly completely decomposable Markov chain to demonstrate the general applicability of the technique.

Computing approximate stationary distributions for discrete Markov processes with banded infinitesimal generators

We develop an algorithm for computing approximations to the stationary distribution of a discrete birth-and-death process, provided that the infinitesimal generator is a banded matrix. We begin by computing stationary distributions for processes whose infinitesimal generators are Hessenberg. Our derivation in this special case is different from the classical case but it leads to the same result. We then show how to extend these ideas to processes where the infinitesimal generator is banded (or half-banded) and to quasi-birth–death processes. Finally, we give an example of the application of this method to a nearly completely decomposable Markov chain to demonstrate the general applicability of the technique.