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The -BMAP/G/1 Queueing Model: Queue Contents and Delay Analysis  [PDF]
Bart Steyaert,Joris Walraevens,Dieter Fiems,Herwig Bruneel
Mathematical Problems in Engineering , 2011, DOI: 10.1155/2011/401365
Abstract: We consider a single-server discrete-time queueing system with N sources, where each source is modelled as a correlated Markovian customer arrival process, and the customer service times are generally distributed. We focus on the analysis of the number of customers in the queue, the amount of work in the queue, and the customer delay. For each of these quantities, we will derive an expression for their steady-state probability generating function, and from these results, we derive closed-form expressions for key performance measures such as their mean value, variance, and tail distribution. A lot of emphasis is put on finding closed-form expressions for these quantities that reduce all numerical calculations to an absolute minimum. 1. Introduction In this paper, we analyze a queueing model with generally distributed customer service times and a correlated arrival process. This kind of model is, for instance, useful in assessing the performance of a packet-switched communication infrastructure, where messages carried by the network can have a variable transmission time, such as the current Internet, or a dedicated packet-based network carrying video-on-demand (VoD). Multiplexer queues and/or switches and routers in such a network can in general be modelled by means of a discrete-time queueing system, where new packets (i.e., “customers”) are generated by a superposition of individual traffic sources. The size of the packets is proportional to their transmission times, whose distribution depends on the specific application under consideration. Analyzing such a system is mandatory in the design and evaluation of these networks. However, this can be a difficult task because of the time-correlated behaviour that each of the individual sources may exhibit. To facilitate the queueing analysis, a source is often modelled as a discrete-time Markov modulated arrival process, such as the discrete-time batch Markovian arrival process (D-BMAP; [1, 2]) with the Markov modulated Bernoulli process (MMBP; [3–5]) and the switched batch bernoulli process (SBBP; [6, 7]) as frequently used special cases. As a consequence, the analysis of a queueing model with a specific SBBP [8] or MMBP [4, 9–11] source model or with a general D-BMAP (e.g., [1, 12–21], and the vast amount of references therein) has become the focus of many research papers. In addition to the characteristics of the arrival process, such queueing models can also be classified by means of the service process. For customer service times equal to one slot, single-server queueing models are considered in [4, 9,
Further Research on the M/G/1 Retrial Queueing Model with Server Breakdowns
Ehmet Kasim,Geni Gupur
Journal of Applied Mathematics , 2012, DOI: 10.1155/2012/890243
Abstract: We study spectral properties of the operator which corresponds to the M/G/1 retrial queueing model with server breakdowns and obtain that all points on the imaginary axis except zero belong to the resolvent set of the operator and 0 is not an eigenvalue of the operator. Our results show that the time-dependent solution of the model is probably strongly asymptotically stable.
On an M/G/1 Queueing Model with k-Phase Optional Services and Bernoulli Feedback  [PDF]
Sayeedeh Abdollahi, Mohammad Reza Salehi Rad
Journal of Service Science and Management (JSSM) , 2012, DOI: 10.4236/jssm.2012.53033
Abstract: In this article an M/G/1 queueing model with single server, Poisson input, k-phases of heterogeneous services and Bernoulli feedback design has been considered. For this model, we derive the steady-state probability generating function (PGF) of queue size at the random epoch and at the service completion epoch. Then, we derive the Laplace-Stieltjes Transform (LST) of the distribution of response time, the means of response time, number of customers in the system and busy period.
A queueing model for error control of partial buffer sharing in ATM
Ahn Boo Yong,Lee Ho Woo
Mathematical Problems in Engineering , 1999,
Abstract: We model the error control of the partial buffer sharing of ATM by a queueing system M 1 , M 2 / G / 1 / K + 1 with threshold and instantaneous Bernoulli feedback. We first derive the system equations and develop a recursive method to compute the loss probabilities at an arbitrary time epoch. We then build an approximation scheme to compute the mean waiting time of each class of cells. An algorithm is developed for finding the optimal threshold and queue capacity for a given quality of service.
Multiserver queueing networks and the tandem queue model  [cached]
Pierre Le Gall
International Journal of Stochastic Analysis , 1998, DOI: 10.1155/s1048953398000318
Abstract: Using a tandem queue model we evaluate the local endogenous (= internal) queueing delay in single server and multiserver queueing networks. The new concept of the apparent overall upstream queueing delay(as perceived by the downstream network) allows us to analyze the distribution of this local queue by interpolating between the distributions of the tandem queue (generated by a concentration tree) and the isolated G/G/squeue. The interpolation coefficients depend on the proportion of premature departures , typically interfering in the upstream stage and leaving the considered path without being offered to the considered local queue. On the other hand, local exogenous arrivals (from outside the network) require the introduction of the interference delay concept. Finally, in the case of single server queueing networks, we stress the need to extend the capacities of the buffers, by considering the worst case scenario and by using an equivalent tandem queue model.
Queueing Maximal Covering Location-Allocation Problem: An Extension with M/G/1 Queueing Systems
Foroogh Moeen Moghadas,Hossein Taghizadeh Kakhki
Advances in Decision Sciences , 2011, DOI: 10.1155/2011/605629
Abstract: We consider the queueing maximal covering location-allocation problem (QM-CLAP) with an M/G/1 queueing system. We first formulate the problem as a binary quadratic programming problem and then propose a new solution procedure based on decomposition of the problem into smaller binary quadratic sub-problems. The heuristic procedure GRASP is used to solve the sub-problems, as well as the entire model. Some computational results are also presented.
Reliability Analysis of the Repairable Queueing System GI/G(M/G)/1

Shi Dinghua,

自动化学报 , 1995,
Abstract: In this paper, we thoroughly investigate the structure of the repairable queueing system GI/G(M/G)/1 by renewal theory and the method of vector Markov process, and obtain its all interested. indices. The obtained results show that server's reliability indices of the repairable queueing system only depend on the busy, idle,and cycle time of the system.
A new queueing strategy for the Adversarial Queueing Theory  [PDF]
Michael Hilker,Christoph Schommer
Computer Science , 2008,
Abstract: In the today's Internet and TCP/IP-networks, the queueing of packets is commonly implemented using the protocol FIFO (First In First Out). Unfortunately, FIFO performs poorly in the Adversarial Queueing Theory. Other queueing strategies are researched in this model and better results are performed by alternative queueing strategies, e.g. LIS (Longest In System). This article introduces a new queueing protocol called interval-strategy that is concerned with the reduction from dynamic to static routing. We discuss the maximum system time for a packet and estimate with up-to-date results how this can be achieved. We figure out the maximum amount of time where a packet can spend in the network (i.e. worst case system time), and argue that the universal instability of the presented interval-strategy can be reached through these results. When a large group of queueing strategies is used for queueing, we prove that the interval-strategy will be universally unstable. Finally, we calculate the maximum time of the static routing to reach an universal stable and polynomial - in detail linear - bounded interval-strategy. Afterwards we close - in order to check this upper bound - with up-to-date results about the delivery times in static routing.
A queueing/inventory and an insurance risk model  [PDF]
Onno Boxma,Rim Essifi,Augustus J. E. M. Janssen
Mathematics , 2015,
Abstract: We study an M/G/1-type queueing model with the following additional feature. The server works continuously, at fixed speed, even if there are no service requirements. In the latter case, it is building up inventory, which can be interpreted as negative workload. At random times, with an intensity {\omega}(x) when the inventory is at level x > 0, the present inventory is removed, instantaneously reducing the inventory to zero. We study the steady-state distribution of the (positive and negative) workload levels for the cases {\omega}(x) is constant and {\omega}(x) = ax. The key tool is the Wiener-Hopf factorisation technique. When {\omega}(x) is constant, no specific assumptions will be made on the service requirement distribution. However, in the linear case, we need some algebraic hypotheses concerning the Laplace-Stieltjes transform of the service requirement distribution. Throughout the paper, we also study a closely related model coming from insurance risk theory. Keywords: M/G/1 queue, Cramer-Lundberg insurance risk model, workload, inventory, ruin probability, Wiener-Hopf technique. 2010 Mathematics Subject Classification: 60K25, 90B22, 91B30, 47A68.
Simple queueing approach to segregation dynamics in Schelling model  [PDF]
Pawel Sobkowicz
Physics , 2007,
Abstract: A simple queueing approach for segregation of agents in modified one dimensional Schelling segregation model is presented. The goal is to arrive at simple formula for the number of unhappy agents remaining after the segregation.
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