Browsing by Author "Gun, Levent"
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Item An Algorithmic Analysis of the MMPP/G/1 Queue.(1988) Gun, Levent; ISRA single server queue with general service time distribution is considered when the input is a Markov modulated Poisson process (MMPP). An algorithmic solution to the transform of the stationary delay and queue length distributions is summarized, and recursive closed-form expressions are obtained for the moments of these distributions. The numerical implementation of these results is discussed in detail with particular reference to an algorithm due to Lucantoni and Ramaswami [11] and its accelerated version due to Ramaswami [19]. This algorithm is shown to be an efficient tool in the matrix-analytic solution of many stochastic models, as various steps for saving considerable amounts of unnecessary computations are identified. A special case of the model where the service time distribution is of phase type is discussed and the stationary queue length distribution at arbitrary times is obtained in matriz-geometric form. Finally, the matrix-geometric and the M/G/I approaches are compared through this special case.Item Annotated Bibliography of Blocking Systems.(1987) Gun, Levent; ISRQueueing systems subject to blocking have been studied by researchers from different research communities. Due to the wide applicability of these models there is an exhaustive list of related papers. This annotated bibliography summarizes 55 of these papers by briefly describing the model, the performance measure(s), the basic methodology used and/or the results obtained in each paper.Item An Approximation Method for General Tandem Queueing Systems Subject to Blocking.(1987) Gun, Levent; Makowski, Armand M.; ISRAn iterative approximation algorithm is presented for calculating the stationary queue size probabilities of tandem queueing systems subject to blocking. The algorithm combines a decomposition/aggregation technique with exact analysis for two node systems. It applies to tandem lines under various blocking disciplines where failure type servers with phase-type service and repair time distributions are in attendance.Item Convexity Results for Parallel Queues with Bernoulli Routing(1990) Gun, Levent; Jean-Marie, Alain; Makowski, Armand M.; Tedijanto, T.; ISRIn this note, we derive various convexity and monotonicity properties of performance measures in parallel GI/GI/1 queues with Bernoulli routing. We then use these results to establish that the equilikely assignment is optimal among all Bernoulli routing. The main tools are notions of stochastic convexity recently introduced by Shaked and Shanthikumar. The optimality results are couched in terms of the stochastic orderings st and icx.Item Matrix-Geometric Solution for Finite Capacity Queues with Phase- Type Distributions.(1987) Gun, Levent; Makowski, Armand M.; ISRThis paper presents a class of Quasi-Birth-and-Death processes with finite state space for which the invariant probability vector is found to admit a matrix-geometric representation. The corresponding rate matrix is given explicitly in terms of the model parameters, and the resulting closed-form expression is proposed as a basis for efficient calculation of the invariant probability vector. The framework presented in this paper provides a unified approach to the study of several well-known queueing system.Item Matrix-Geometric Solution for Two Node Tandem Queueing Systems with Phase-Type Servers Subject to Blocking and Failures.(1987) Gun, Levent; Makowski, Armand M.; ISRA two node tandem queueing system with phase-type servers and Bernoulli arrivals is considered in discrete-time when servers are subject to bloclring and failuree. The invariant probability vector of the underlying finite state Quasi-Birth-and-Death process is shown to admit a matrix-geometric representation for all values of the arrival rate A. The corresponding rate matrix is given explicitly in terms of the model parameters and the resulting closed-form expression provides the basis for an efficient calculation of the invariant probability vector. The cases LAMBDA = 1 and LAMBDA < 1 are studied separately and the irreducibility of the underlying Markov chain is discussed for each case. The continuous-time formulation is briefly discussed and only major differences with the discrete-time results are pointed out. Some numerical examples are also provided.Item Optimal Production Strategies for Discrete Time Machines Subject to Failures and Breakdown.(1986) Gun, Levent; Makowski, Armand M.; ISRIn this paper, discrete-time versions of a model by Akella and Kumar [1] are presented for the production of a single commodity on a machine subject to random failures and breakdowns. The successive up and down times of the machines form an alternating renewal sequence, with arbitrary distributions, while the demands are i.i.d. over slots. The infinite-horizon discounted cost problem is investigated with a cost-per-stage which is a convex function of the inventory/backlog level. Under the assumptions made, the convexity of the value function is established by elementary arguments, and as in [1], the form of the optimal production strategy is shown to be characterized by a critical inventory level function. The results are specialized to the cases where the up time distribution is geometric and the demand is constant over time. Easy extensions are briefly discussed.Item Performance Evaluation and Optimization of Parallel Systems with Synchronization(1989) Gun, Levent; Makowski, A.; ISRThis thesis considers synchronization issues such as resequencing and fork/join in parallel architectures. The discussion is carried out in the context of K parallel single server queues with general servers where jobs are subject to resequencing. Both performance evaluation and optimal routing problems are addressed for such systems. In the first part, Poisson arrivals are assumed to be randomly allocated to different queues according to a Bernoulli switch. The distributions of the various delays in the system are obtained by sample path arguments. The problem of choosing the switching probabilities that minimize the average end-to-end delay is considered. In addition to obtaining exact results in some cases, simple but accurate approximations are provided when the service time distributions are exponential. The simple form of these approximations is then utilized to solve the optimization problem in the case when the service parameters are unknown, and a simple stochastic approximation algorithm is proposed. When the servers are all identical, several useful asymptotic results are obtained as K increases to infinity. Various stochastic monotonicity and convexity results are also provided for this parallel system. In the second part, the dynamic optimization of the same model is investigated under more general assumptions for the arrival process. The resequencing problem is combined with a fork/join problem, where the incoming packets are broken into smaller subpackets for processing at different queues. The problem of finding the optimal allocation policy that minimizes the average discounted and the long-run average costs is formulated as a Markov Decision problem, where the cost-per-stage is taken as the end-to-end delay of each packet. In both cases, the optimal policy is identified as the one that drives the workload in each queue to a balanced configuration as quickly as feasible.Item Tandem Queueing Systems Subject to Blocking with Phase Type Servers: Analytic Solutions and Approximations(1987) Gun, Levent; Makowski, A.; ISRAbstract's technical formulas were not compatible with the database used to create this booklet. To obtain a copy, please contact Maggie Virkus at (301) 454-6167, maggie@ra.src.umd.edu or write at to the Center's address listed in the front of this booklet.