B. Van Houdt

Structured Markov chains solver: software tools

The package SMC-Solver for solving structured Markov chains is presented. It contains the most advanced algorithms for solving QBD, M/G/1 and G/M/1 problems. The package is provided in two versions: a Matlab toolbox and a Fortran 95 version with a user-friendly graphical interface.

Structured Markov chains solver: algorithms

We analyze the problem of the numerical solution of structured Markov chains encountered in queuing models: we describe the main computational problems and present the most advanced algorithms currently available for their solutions.

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.

A unified model for synchronous and asynchronous FDL buffers allowing closed-form solution

Novel switching approaches like Optical Burst/Packet Switching have buffering implemented with Fiber Delay Lines (FDLs). Previous performance models of the resulting buffer only allowed for solution by numerical means, and only for one time setting: continuous, or discrete. With a Markov chain approach, we constructed a generic framework that encompasses both time settings. The output includes closed-form expressions of loss probabilities and waiting times for a rather realistic setting. This allows for exact performance comparison of the classic M/D/1 buffer and FDL M/D/1 buffer, revealing that waiting times are (more than) doubled in the case of FDL buffering.

Response time in a tandem queue with blocking, Markovian arrivals and phase-type services

A novel approach for obtaining the response time in a discrete-time tandem-queue with blocking is presented. The approach constructs a Markov chain based on the age of the leading customer in the first queue. We also provide a stability condition and carry out several numerical examples.

Q-MAM: a tool for solving infinite queues using matrix-analytic methods

In this paper we propose a novel MATLAB tool, called Q-MAM, to compute queue length, waiting time and so-journ time distributions of various discrete and continuous time queuing systems with an underlying structured Markov chain/process. The underlying paradigms include M/G/1-and GI/M/1-type, quasi-birth-death and non-skip-free Markov chains (implemented by the SMCSolver tool), as well as Markov processes with a matrix exponential distribution. We consider various single server queueing systems with phase-type, matrix exponential, Markovian, rational and semi-Markovian arrival and service processes; queues with multiple customer types, where the service depends on the customer type and where consecutive customer types may be correlated; and queues with multiple servers for which the typical dimensionality problem can be avoided. Apart from implementing various classical and more advanced solution techniques, the tool also extends and improves some of the existing solution techniques in a number of cases.

A multiaccess tree algorithm with free access, interference cancellation and single signal memory requirements

Tree algorithms are a well studied class of collision resolution algorithms for solving multiple access control problems. Successive interference cancellation, which allows one to recover additional information from otherwise lost collision signals, has recently been combined with tree algorithms with blocked access [Y. Yu, G.B. Giannakis, SICTA: A 0.693 contention tree algorithm using successive interference cancellation, in: INFOCOM 2005. 24th Annual Joint Conference of the IEEE Computer and Communications Societies, Miami, USA, 2005, pp. 1908-1916], providing a substantially higher maximum stable throughput (MST): 0.693 for Poisson arrivals, given an infinite number of memory locations for storing signals. We propose a novel tree algorithm for a similar problem, but with two relaxed model assumptions: free access is supported and a single signal memory location suffices. A study of the maximal stable throughput of this algorithm is provided using matrix analytical methods; as a result, an MST of 0.5698 for Poisson arrivals is achieved. Our methodology also allows us to investigate the MST when the multiple access channel is subject to Markovian arrival processes.

Approximated Transient Queue Length and Waiting Time Distributions via Steady State Analysis

Abstract We propose a method to approximate the transient performance measures of a discrete time queueing system via a steady state analysis. The main idea is to approximate the system state at time slot t or on the n-th arrival–-depending on whether we are studying the transient queue length or waiting time distribution–-by the system state after a negative binomially distributed number of slots or arrivals. By increasing the number of phases k of the negative binomial distribution, an accurate approximation of the transient distribution of interest can be obtained. In order to efficiently obtain the system state after a negative binomially distributed number of slots or arrivals, we introduce so-called reset Markov chains, by inserting reset events into the evolution of the queueing system under consideration. When computing the steady state vector of such a reset Markov chain, we exploit the block triangular block Toeplitz structure of the transition matrices involved and we directly obtain the approximation from its steady state vector. The concept of the reset Markov chains can be applied to a broad class of queueing systems and is demonstrated in full detail on a discrete-time queue with Markovian arrivals and phase-type services (i.e., the D-MAP/PH/1 queue). We focus on the queue length distribution at time t and the waiting time distribution of the n-th customer. Other distributions, e.g., the amount of work left behind by the n-th customer, that can be acquired in a similar way, are briefly touched upon. Using various numerical examples, it is shown that the method provides good to excellent approximations at low computational costs–-as opposed to a recursive algorithm or a numerical inversion of the Laplace transform or generating function involved–-offering new perspectives to the transient analysis of practical queueing systems.

Queues with Correlated Service and Inter-Arrival Times and Their Application to Optical Buffers

This paper presents an algorithmic procedure to calculate the queue length and delay distribution of customers in a discrete time D-MAP/PH/1 queue, where the service time distribution of a customer depends on the inter-arrival time between himself and his predecessor. Setting up a Markov chain that keeps track of the contents of such a queue will result in a state space explosion as the inter-arrival times of all customers present in the system must be remembered. We avoid these difficulties by making use of the age process, a process that keeps track of the “age” of the customer in the service facility. From this process, which we solve by means of matrix analytic methods, we compute the queue length and sojourn time distribution by means of a simple formula and obtain an expression for the stability of the system. We also demonstrate that the D-MAP arrival process can be easily replaced by the more general semi-Markovian arrival process, without any additional computational costs. Queueing systems of this type arise in the domain of synchronous optical buffers. Based on the numerical analysis of such a queueing system, some guidelines for the design of optical buffers are presented. We also show the impact on the numerical results when the cross-correlation that exists between the service and inter-arrival times is neglected.

The Waiting Time Distribution of a Type k Customer in a Discrete-Time MMAP[K]/PH[K]/c (c = 1, 2) Queue Using QBDs

Abstract This paper presents an improved method to calculate the delay distribution of a type k customer in a first-come-first-serve (FCFS) discrete-time queueing system with multiple types of customers, where each type has different service requirements, and c servers, with c = 1, 2 (the MMAP[K]/PH[K]/c queue). The first algorithms to compute this delay distribution, using the GI/M/1 paradigm, were presented by Van Houdt and Blondia [Van Houdt, B.; Blondia, C. The delay distribution of a type k customer in a first come first served MMAP[K]/PH[K]/1 queue. J. Appl. Probab. 2002, 39 (1), 213–222; The waiting time distribution of a type k customer in a FCFS MMAP[K]/PH[K]/2 queue. Technical Report; 2002]. The two most limiting properties of these algorithms are: (i) the computation of the rate matrix R related to the GI/M/1 type Markov chain, (ii) the amount of memory needed to store the transition matrices A l and B l . In this paper we demonstrate that each of the three GI/M/1 type Markov chains used to develop the algorithms in the above articles can be reduced to a QBD with a block size which is only marginally larger than that of its corresponding GI/M/1 type Markov chain. As a result, the two major limiting factors of each of these algorithms are drastically reduced to computing the G matrix of the QBD and storing the 6 matrices that characterize the QBD. Moreover, these algorithms are easier to implement, especially for the system with c = 2 servers. We also include some numerical examples that further demonstrate the reduction in computational resources.