Armin Heindl

A Markovian canonical form of second-order matrix-exponential processes

Besides the fact that - by definition - matrix-exponential processes (MEPs) are more general than Markovian arrival processes (MAPs), only very little is known about the precise relationship of these processes in matrix notation. For the first time, this paper proves the persistent conjecture that - in two dimensions - the respective sets, MAP(2) and MEP(2), are indeed identical with respect to the stationary behavior. Furthermore, this equivalence extends to acyclic MAPs, i.e., AMAP(2), so that AMAP(2)[reverse not equivalent]MAP(2)[reverse not equivalent]MEP(2). For higher orders, these equivalences do not hold. The second-order equivalence is established via a novel canonical form for the (correlated) processes. An explicit moment/correlation-matching procedure to construct the canonical form from the first three moments of the interarrival time distribution and the lag-1 correlation coefficient shows how these compact processes may conveniently serve as input models for arrival/service processes in applications.

Performance modeling of IEEE 802.11 wireless LANs with stochastic Petri nets

Abstract In 1997, IEEE standardized the physical layers and the medium access for wireless local area networks. This paper presents a performance study of the distributed coordination function, the fundamental contention-based access mechanism. Most performance studies adopt unchecked simplifying assumptions or do not reveal all details of the simulation model. We develop a stochastic Petri net model, which captures all relevant system aspects in a concise way. Simulation allows to quantify the influence of many mandatory features of the standard on performance, especially the backoff procedure, extended interframe spaces, and the timing synchronization function. We identify conditions when simplifying assumptions commonly used in analytical modeling are justified. Applying these conditions, we derive a more compact and analytically tractable model from the detailed model.

Decomposition of general tandem queueing networks with MMPP input

For tandem queueing networks with generally distributed service times, decomposition often is the only feasible solution method besides simulation. The network is partitioned into individual nodes which are analysed in isolation. In existing decomposition algorithms for continuous-time networks, the output of a queue is usually approximated as a renewal process, which serves as the arrival process to the next queue. In this paper, the internal traffic processes are described as semi-Markov processes (SMPs) and Markov modulated Poisson processes (MMPPs). Thus, correlations in the traffic streams, which are known to have a considerable impact on performance, are taken into account to some extent. A two-state MMPP, which arises frequently in communications modeling, serves as input to the first queue of the tandem network. For tandem networks with infinite or finite buffers, stationary mean queue lengths at arbitrary time computed quasi-promptly by the decomposition component of the tool TimeNET are compared to simulation.

Decomposition of general queueing networks with MMPP inputs and customer losses

For nontrivial general (open) queueing networks, decomposition often represents the only feasible solution method besides simulation. The network is partitioned into individual nodes which are analyzed in isolation with respect to approxamate internal traffic representations. The quality of the quickly obtained results very much depends on the descriptors for the traffic processes within the network, which may be split and merged before traversing the next queue. Recently, one of the existing decomposition formalisms based on renewal processes as traffic descriptors has been extended in order to include semi-Markov processes (SMPs) mad Markov-modulated Poisson processes (MMPPs) with two states. However, due to the restriction to tandem networks, no operations for the splitting and and merging were provided.The numerical procedures for the splitting of SMPs and the superposition of MMPPs proposed in this paper render the extended decomposition framework available for general queueing networks. The correlations in the traffic processes, which are known to have a considerable impact on performance measures, are taken into account to some extent. Moreover, in addition to renewal processes, MMPP inputs--as they frequently arise in computer communication modeling--increase the range of applications of general queueing networks.Numerical experiments on different network configurations illustrate the capabilities and accuracy of the extended decomposition framework.

Correlation bounds for second-order MAPs with application to queueing network decomposition

Tools for performance evaluation often require techniques to match moments to continuous distributions or moments and correlation data to correlated processes. With respect to efficiency in applications, one is interested in low-dimensional (matrix) representations. For phase-type distributions (or matrix exponentials) of second order, analytic bounds could be derived earlier, which specify the space of permissible moments. In this paper, we add a correlation parameter to the first three moments of the marginal distribution to construct a Markovian arrival process of second order (MAP(2)). Exploiting the equivalence of correlated matrix-exponential sequences and MAPs in two dimensions, we present an algorithm that decides whether the correlation parameter is permissible with respect to the three moments and - if so - delivers a valid MAP(2) which matches the four parameters.We also investigate the restrictions imposed on the correlation structure of MAP(2)s with hyperexponential marginals. Analytic bounds for the envelope correlation region (i.e., for arbitrary third moment) and for the specific correlation region (i.e., for fixed third moment) are given.When there is no need for a MAP(2) representation (as in linear-algebraic queueing theory), the proposed procedure serves to check the validity of the constructed correlated matrix-exponential sequence.Numerical examples indicate how these results can be used to efficiently decompose queueing networks.

Characterizing the BMAP/MAP/1 Departure Process via the ETAQA Truncation

Abstract We propose a family of finite approximations for the departure process of a BMAP/MAP/1 queue. The departure process approximations are derived via an exact aggregate solution technique (called ETAQA) applied to M/G/1-type Markov processes. The proposed approximations are indexed by a parameter n(n > 1), which determines the size of the output model as n + 1 block levels of the M/G/1-type process. This output approximation preserves exactly the marginal distribution of the true departure process and the lag correlations of the interdeparture times up to lag n − 2. Experimental results support the applicability of the proposed approximation in traffic-based decomposition of queueing networks.

Performance evaluation of IEEE 802.11 wireless LANs with stochastic Petri nets

IEEE has recently standardized the physical layer and the medium access mechanism for wireless local area networks. This paper presents a performance study of the distributed coordination function, the fundamental contention based access mechanism. Stochastic Petri nets are used as a modeling formalism. A detailed model captures all relevant system aspects in a concise way. The detailed model is evaluated by simulation and also used to derive two more compact models which are analytically tractable. All models are used to investigate different physical layer options and the influence of several system parameters.

The impact of backoff, EIFS, and beacons on the performance of IEEE 802.11 wireless LANs

In late 1997, IEEE standardized the physical layers and the medium access for wireless local area networks. This paper presents a performance study of the distributed coordination function, the fundamental contention based access mechanism. Most performance studies adopt unchecked simplifying assumptions or do not reveal all details of the simulation model. We develop a stochastic Petri net model, which captures all relevant system aspects in a concise way. Simulation allows to quantify the influence of many mandatory features of the standard on performance, especially the backoff procedure, extended interframe spaces, and the timing synchronisation function. This study also serves to identify conditions when simplifying assumptions commonly used in analytical modeling are justified.

Output models of MAP/PH/1(/K) queues for an efficient network decomposition

For non-trivial (open) queueing networks, traffic-based decomposition often represents the only feasible--and at the same time fast--solution method besides simulation. The network is partitioned into individual nodes which are analyzed in isolation with respect to approximate internal traffic representations. Since the correlations of network traffic may have a considerable impact on performance measures, they must be captured to some extent by the employed traffic descriptors. The decomposition methodology presented in this paper is based on Markovian arrival processes (MAPs), whose correlation structure is determined from the busy-period behavior of the upstream queues. The resulting compact MAPs in connection with sophisticated moment-matching techniques allow an efficient decomposition of large queueing networks. Numerical experiments demonstrate the substantially enhanced precision due to improved output models compared with the experiments of Heindl and Telek [MAP-based decomposition of tandem networks of ./PH/1(/K) queues with MAP input, in: Proceedings of the 11th GI/ITG Conference on Measuring, Modelling and Evaluation of Computer and Communication Systems, Aachen, Germany, 2001, pp. 179-194].

ETAQA truncation models for the MAP/MAP/1 departure process

We propose a family of finite approximations for the departure process of a MAP/MAP/1 queue. The departure process approximations are derived via an exact aggregate solution technique (called ETAQA) applied to quasi-birth-death processes (QBDs) and require only the computation of the frequently sparse fundamental-period matrix G. The approximations are indexed by a parameter n, which determines the size of the output model as n-1 QBD levels. The marginal distribution of the true departure process and the lag correlations of the interdeparture times up to lag n+1 are preserved exactly. Via experimentation we show the applicability of the proposed approximation in traffic-based decomposition of queueing networks and investigate how correlation propagates through tandem queues.