Hendrik Baumann

Numerical solution of level dependent quasi-birth-and-death processes

We consider the numerical computation of stationary distributions for level dependent quasi-birth-and-death processes. An algorithm based on matrix continued fractions is presented and compared to standard solution techniques. Its computational efficiency and numerical stability is demonstrated by numerical examples.

Steady state analysis of level dependent quasi-birth-and-death processes with catastrophes

Quasi-birth-and-death processes, that is multi-dimensional Markov chains with block tridiagonal transition probability or generator matrices, are appropriate models for various types of queueing systems, amongst many other population dynamics. We consider continuous-time level dependent quasi-birth-and-death processes (LDQBDs) extended by catastrophes, which means that the transition rates are allowed to depend on the process level and additionally in each state the level component may drop to zero such that the generator matrix deviates from the block tridiagonal form in that the first block column is allowed to be completely occupied. A matrix analytic algorithm (MAA) for computing the stationary distribution of such processes is introduced that extends and generalizes similar algorithms for LDQBDs without catastrophes. The algorithm is applied in order to analyze M/M/c queues in random environment with catastrophes and state dependent rates. We present a detailed steady state analysis by computing the stationary distribution for different parameter sets, thereby focusing on the marginal probabilities of the level component which represents the number of customers. It turns out that the stationary marginal distribution is bimodal in the sense that it has two local modes that significantly depend on the specific parameters and rates. We also study the efficiency of our matrix analytic algorithm (MAA). Comparisons with standard solution algorithms for Markov chains demonstrate its superiority in terms of runtime and memory requirements.

Multi-server tandem queue with Markovian arrival process, phase-type service times, and finite buffers

We consider multi-server tandem queues where both stations have a finite buffer and all services times are phase-type distributed. Arriving customers enter the first queueing station if buffer space is available or get lost otherwise. After completing service in the first station customers proceed to the second station if buffer space is available, otherwise a server at the first station is blocked until buffer space becomes available at the second station. We provide an exact computational analysis of various steady-state performance measures such as loss and blocking probabilities, expectations and higher moments of numbers of customers in the queues and in the whole system by modeling the tandem queue as a level-dependent quasi-birth-and-death process and applying suitable matrix-analytic methods. Numerical results are presented for selected representative 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.

Structured Modeling and Analysis of Stochastic Epidemics with Immigration and Demographic Effects.

Stochastic epidemics with open populations of variable population sizes are considered where due to immigration and demographic effects the epidemic does not eventually die out forever. The underlying stochastic processes are ergodic multi-dimensional continuous-time Markov chains that possess unique equilibrium probability distributions. Modeling these epidemics as level-dependent quasi-birth-and-death processes enables efficient computations of the equilibrium distributions by matrix-analytic methods. Numerical examples for specific parameter sets are provided, which demonstrates that this approach is particularly well-suited for studying the impact of varying rates for immigration, births, deaths, infection, recovery from infection, and loss of immunity.

Markovian Modeling and Security Measure Analysis for Networks under Flooding DoS Attacks

Network flooding is among the most prevalent modes of denial-of-service (DoS) attacks. It can seriously degrade the network operation to the point of being unable to serve any legitimate user as intended, because all resources are occupied with serving malicious attack requests. We model flooding DoS attacks by a three-dimensional continuous-time Markov chain (CTMC) that accounts for the environment in which the network under attack operates and incorporates a random dropping policy as a potential defense mechanism. The state space is structured such that the generator matrix is block tridiagonal and the CTMC becomes numerically tractable by matrix analytic methods. This enables us to compute security measures accurately and efficiently. Numerical results for varying parameter settings are provided in order to study flooding DoS attacks.

Two-sided continued fractions in Banach algebras- A Sleszyński-Pringsheim-type convergence criterion and applications

We consider two-sided continued fractions in Banach algebras, that is K = b 0 + a 1 ( b 1 + a 2 ( b 2 + ? ) - 1 c 2 ) - 1 c 1 , where the coefficients b n , a n , c n are elements of some Banach algebra. We prove convergence criteria which are exact generalizations of the well-known Pringsheim-type convergence criteria for complex continued fractions, and state a result concerning the rate of convergence. Both convergence criteria and error bounds improve all known results.