Commuting matrices in the sojourn time analysis of MAP/MAP/1 queues

Queues with Markovian arrival and service processes, i.e., MAP/MAP/1 queues, have been useful in the analysis of computer and communication systems and different representations for their sojourn time distribution have been derived. More specifically, the class of MAP/MAP/1 queues lies at the intersection of the class of QBD queues and the class of semi-Markovian queues. While QBD queues have an order $N^2$ matrix exponential representation for their sojourn time distribution, where $N$ is the size of the background continuous time Markov chain, the sojourn time distribution of the latter class allows for a more compact representation of order $N$. In this paper we unify these two results and show that the key step exists in establishing the commutativity of some fundamental matrices involved in the analysis of the MAP/MAP/1 queue. We prove, using two different approaches, that the required matrices do commute and identify several other sets of commuting matrices. Finally, we generalize some of the results to queueing systems with batch arrivals and services

BMAP/G/c Queueing Model with Group Clearance Useful in Telecommunications Systems – A Simulation Approach

Queueing models in which customers or messages arrive in batches with inter-arrival times of batches possibly correlated and services rendered in batches of varying sizes play an important role in telecommunication systems. Recently queueing models of BMAP/G/1-type in which a new type of group clearance was studied using embedded Markov renewal process as well as continuous time Markov chain whose generator has a very special structure. In this paper, we generalize these models to multi-server systems through simulation approach. After validating the simulation model for the single server case, we report our simulated results for much more general situations

Extremes of Markov-additive processes with one-sided jumps, with queueing applications

Through Laplace transforms, we study the extremes of a continuous-time Markov-additive pro- cess with one-sided jumps and a finite-state background Markovian state-space, jointly with the epoch at which the extreme is ‘attained’. For this, we investigate discrete-time Markov-additive pro- cesses and use an embedding to relate these to the continuous-time setting. The resulting Laplace transforms are given in terms of two matrices, which can be determined either through solving a nonlinear matrix equation or through a spectral method. Our results on extremes are first applied to determine the steady-state buffer-content distribution of several single-station queueing systems. We show that our framework comprises many models dealt with earlier, but, importantly, it also enables us to derive various new results. At the same time, our setup offers interesting insights into the connections between the approaches developed so far, including matrix-analytic techniques, martingale methods, the rate-conservation approach, and the occupation-measure method. Then we turn to networks of fluid queues, and show how the results on single queues can be used to find the Laplace transform of the steady-state buffer-content vector; it has a matrix quasi-product form. Fluid-driven priority systems also have this property

Performance Modeling and Analysis of Wireless Local Area Networks with Bursty Traffic

The explosive increase in the use of mobile digital devices has posed great challenges in the design and implementation of Wireless Local Area Networks (WLANs). Ever-increasing demands for high-speed and ubiquitous digital communication have made WLANs an essential feature of everyday life. With audio and video forming the highest percentage of traffic generated by multimedia applications, a huge demand is placed for high speed WLANs that provide high Quality-of-Service (QoS) and can satisfy end user’s needs at a relatively low cost. Providing video and audio contents to end users at a satisfactory level with various channel quality and current battery capacities requires thorough studies on the properties of such traffic. In this regard, Medium Access Control (MAC) protocol of the 802.11 standard plays a vital role in the management and coordination of shared channel access and data transmission. Therefore, this research focuses on developing new efficient analytical models that evaluate the performance of WLANs and the MAC protocol in the presence of bursty, correlated and heterogeneous multimedia traffic using Batch Markovian Arrival Process (BMAP). BMAP can model the correlation between different packet size distributions and traffic rates while accurately modelling aggregated traffic which often possesses negative statistical properties. The research starts with developing an accurate traffic generator using BMAP to capture the existing correlations in multimedia traffics. For validation, the developed traffic generator is used as an arrival process to a queueing model and is analyzed based on average queue length and mean waiting time. The performance of BMAP/M/1 queue is studied under various number of states and maximum batch sizes of BMAP. The results clearly indicate that any increase in the number of states of the underlying Markov Chain of BMAP or maximum batch size, lead to higher burstiness and correlation of the arrival process, prompting the speed of the queue towards saturation. The developed traffic generator is then used to model traffic sources in IEEE 802.11 WLANs, measuring important QoS metrics of throughput, end-to-end delay, frame loss probability and energy consumption. Performance comparisons are conducted on WLANs under the influence of multimedia traffics modelled as BMAP, Markov Modulated Poisson Process and Poisson Process. The results clearly indicate that bursty traffics generated by BMAP demote network performance faster than other traffic sources under moderate to high loads. The model is also used to study WLANs with unsaturated, heterogeneous and bursty traffic sources. The effects of traffic load and network size on the performance of WLANs are investigated to demonstrate the importance of burstiness and heterogeneity of traffic on accurate evaluation of MAC protocol in wireless multimedia networks. The results of the thesis highlight the importance of taking into account the true characteristics of multimedia traffics for accurate evaluation of the MAC protocol in the design and analysis of wireless multimedia networks and technologies