Monotonicity and error bounds for networks of Erlang loss queues

Networks of Erlang loss queues naturally arise when modelling finite communication systems without delays, among which, most notably are (i) classical circuit switch telephone networks (loss networks) and (ii) present-day wireless mobile networks. Performance measures of interest such as loss probabilities or throughputs can be obtained from the steady state distribution. However, while this steady state distribution has a closed product form expression in the first case (loss networks), it does not have one in the second case due to blocked (and lost) handovers. Product form approximations are therefore suggested. These approximations are obtained by a combined modification of both the state space (by a hypercubic expansion) and the transition rates (by extra redial rates). It will be shown that these product form approximations lead to (1) upper bounds for loss probabilities and \ud (2) analytic error bounds for the accuracy of the approximation for various performance measures.\ud The proofs of these results rely upon both monotonicity results and an analytic error bound method as based on Markov reward theory. This combination and its technicalities are of interest by themselves. The technical conditions are worked out and verified for two specific applications:\ud (1)• pure loss networks as under (2)• GSM networks with fixed channel allocation as under.\ud The results are of practical interest for computational simplifications and, particularly, to guarantee that blocking probabilities do not exceed a given threshold such as for network dimensioning

Kronecker representation and decompositional analysis of closed queueing networks with phase-type service distributions and arbitrary buffer sizes

Two approximative fixed-point iterative methods based on decomposition for closed queueing networks with Coxian service distributions and arbitrary buffer sizes are extended to include phase-type service distributions. The irreducible Markov chain associated with each subnetwork in the respective decompositions is represented hierarchically using Kronecker products. The two methods are implemented in a software tool capable of computing the steady-state probability vector of each subnetwork by a multilevel method at each fixed-point iteration and are compared with other methods for accuracy and efficiency. Numerical results indicate that there is a niche filled by the two approximative methods

Performance Evaluation in Single or Multi-Cluster C-RAN Supporting Quasi-Random Traffic

In this paper, a cloud radio access network (C-RAN) is considered where the remote radio heads (RRHs) are separated from the baseband units (BBUs). The RRHs in the C-RAN are grouped in different clusters according to their capacity while the BBUs form a centralized pool of computational resource units. Each RRH services a finite number of mobile users, i.e., the call arrival process is the quasi-random process. A new call of a single service-class requires a radio and a computational resource unit in order to be accepted in the C-RAN for a generally distributed service time. If these resource units are unavailable, then the call is blocked and lost. To analyze the multi-cluster C-RAN, we model it as a single-rate loss system, show that a product form solution exists for the steady state probabilities and propose a convolution algorithm for the accurate determination of congestion probabilities. The accuracy of this algorithm is verified via simulation. The proposed model generalizes our recent model where the RRHs in the C-RAN are grouped in a single cluster and each RRH accommodates quasi-random traffic

Queueing networks: solutions and applications

During the pasttwo decades queueing network models have proven to be a versatile tool for computer system and computer communication system performance evaluation. This chapter provides a survey of th field with a particular emphasis on applications. We start with a brief historical retrospective which also servesto introduce the majr issues and application areas. Formal results for product form queuenig networks are reviewed with particular emphasis on the implications for computer systems modeling. Computation algorithms, sensitivity analysis and optimization techniques are among the topics covered. Many of the important applicationsof queueing networks are not amenableto exact analysis and an (often confusing) array of approximation methods have been developed over the years. A taxonomy of approximation methods is given and used as the basis for for surveing the major approximation methods that have been studied. The application of queueing network to a number of areas is surveyed, including computer system cpacity planning, packet switching networks, parallel processing, database systems and availability modeling.Durante as últimas duas décadas modelos de redes de filas provaram ser uma ferramenta versátil para avaliação de desempenho de sistemas de computação e sistemas de comunicação. Este capítulo faz um apanhado geral da área, com ênfase em aplicações. Começamos com uma breve retrospectiva histórica que serve também para introduzir os pontos mais importantes e as áreas de aplicação. Resultados formais para redes de filas em forma de produto são revisados com ênfase na modelagem de sistemas de computação. Algoritmos de computação, análise de sensibilidade e técnicas de otimização estão entre os tópicos revistos. Muitas dentre importantes aplicações de redes de filas não são tratáveis por análise exata e uma série (frequentemente confusa) de métodos de aproximação tem sido desenvolvida. Uma taxonomia de métodos de aproximação é dada e usada como base para revisão dos mais importantes métodos de aproximação propostos. Uma revisão das aplicações de redes de filas em um número de áreas é feita, incluindo planejamento de capacidade de sistemas de computação, redes de comunicação por chaveamento de pacotes, processamento paralelo, sistemas de bancos de dados e modelagem de confiabilidade

Throughput Analysis of Manual Order Picking Systems with Congestion Consideration

Throughput in manual order picking systems with narrow aisles suffers from congestion as pickers cannot pass each other. Only few models incorporate congestion but they have very strict assumptions. In this work, queueing theory is used to analyze systems with traversal routing as well as different storage policies. The models are able to estimate throughput for many alternative designs in a relatively short amount of time. New guidelines for narrow-­aisle order picking systems are introduced

Queueing Networks With Blocking.

The area of classical (product form) queueing networks is briefly discussed. The principal results for classical queueing networks are summarized. The transfer, service and rejection blocking policies are defined, and their use in queueing network models are presented. An overview of the literature in the area of queueing networks with blocking is given, and the relations between the three blocking policies is discussed in general. Duality theorems for open and closed queueing networks with rejection blocking and a single job class are proved. Using a duality theorem, an exact solution is found for closed blocking networks which contain so many jobs that if one station is empty all other stations are full. Algorithms to compute performance measures, in particular throughputs, follow from the way the solution is obtained. It is then proved that for open, mixed and closed networks with rejection blocking, multiple job classes, general service time distributions and reversible routing the equilibrium state probabilities have product form. The reversed process for these networks is examined, and it is proved that it represents a network of the same type. Formulas for throughputs are derived, and algorithms to compute performance measures are outlined. Finally, closed central server models with state-dependent routing, multiple job classes and rejection blocking are investigated. The equilibrium state probabilities have a modified product form, and the reversed process is a network of the same type. Formulas for performance measures are derived for this model and algorithms to compute them are outlined

Modeling Conveyor Merges in Zone Picking Systems

In many order picking and sorting systems conveyors are used to transport products through the system and to merge multiple flows of products into one single flow. In practice, conveyor merges are potential points of congestion, and consequently can lead to a reduced throughput. In this paper, we study merges in a zone picking system. The performance of a zone picking system is, for a large part, determined by the performance of the merge locations. We model the system as a closed queueing network that describes the conveyor, the pick zones, and the merge locations. The resulting model does not have a product-form stationary queue-length distribution. This makes exact analysis practically infeasible. Therefore, we approximate the behavior of the model using the aggregation technique, where the resulting subnetworks are solved using matrix-geometric methods. We show that the approximation model allows us to determine very accurate estimates of the throughput when compared with simulation. Furthermore, our model is in particular well suited to evaluate many design alternatives, in terms of number of zones, zone buffer lengths, and maximum number of totes in the systems. It also can be used to determine the maximum throughput capability of the system and, if needed, modify the system in order to meet target performance levels