Sojourn Times in A Queueing System with Breakdowns and General Retrial Times

This paper centers on a discrete-time retrial queue where the server experiences breakdowns and repairs when arriving customers may opt to follow a discipline of a last-come, first-served (LCFS)-type or to join the orbit. We focused on the extensive analysis of the system, and we obtained the stationary distributions of the number of customers in the orbit and in the system by applying the generation function (GF). We provide the stochastic decomposition law and the application bounds for the proximity between the steady-state distributions for the queueing system under consideration and its corresponding standard system. We developed recursive formulae aimed at the calculation of the steady-state of the orbit and the system. We proved that our discrete-time system approximates M/G/1 with breakdowns and repairs. We analyzed the busy period of an auxiliary system, the objective of which was to study the customer’s delay. The stationary distribution of a customer’s sojourn in the orbit and in the system was the object of a thorough and complete study. Finally, we provide numerical examples that outline the effect of the parameters on several performance characteristics and a conclusions section resuming the main research contributions of the paper.

Metaheuristics and Transmission Expansion Planning: A Comparative Case Study

Transmission expansion planning (TEP), the determination of new transmission lines to be added to an existing power network, is a key element in power system planning. Using classical optimization to define the most suitable reinforcements is the most desirable alternative. However, the extent of the under-study problems is growing, because of the uncertainties introduced by renewable generation or electric vehicles (EVs) and the larger sizes under consideration given the trends for higher renewable shares and stronger market integration. This means that classical optimization, even using efficient techniques, such as stochastic decomposition, can have issues when solving large-sized problems. This is compounded by the fact that, in many cases, it is necessary to solve a large number of instances of a problem in order to incorporate further considerations. Thus, it can be interesting to resort to metaheuristics, which can offer quick solutions at the expense of an optimality guarantee. Metaheuristics can even be combined with classical optimization to try to extract the best of both worlds. There is a vast literature that tests individual metaheuristics on specific case studies, but wide comparisons are missing. In this paper, a genetic algorithm (GA), orthogonal crossover based differential evolution (OXDE), grey wolf optimizer (GWO), moth–flame optimization (MFO), exchange market algorithm (EMA), sine cosine algorithm (SCA) optimization and imperialistic competitive algorithm (ICA) are tested and compared. The algorithms are applied to the standard test systems of IEEE 24, and 118 buses. Results indicate that, although all metaheuristics are effective, they have diverging profiles in terms of computational time and finding optimal plans for TEP.

Discussion on the transient behavior of single server Markovian multiple variant vacation queues

In this paper, we consider an M/M/1 queue where beneficiary visits occur singly. Once the beneficiary level in the system becomes zero, the server takes a vacation immediately. If the server finds no beneficiaries in the system, then the server is allowed to take another vacation after the return from the vacation. This process continues until the server has exhaustively taken all the J vacations. The closed form transient solution of the considered model and some important time dependent performance measures are obtained. Further, the steady state system size distribution is obtained from the time-dependent solution. A stochastic decomposition structure of waiting time distribution and expression for the additional waiting time due to the presence of server vacations are studied. Numerical assessments are presented.

Synchronized abandonment in discrete-time renewal input queues with vacations

<p style='text-indent:20px;'>We consider a discrete-time infinite buffer renewal input queue with multiple vacations and synchronized abandonment. Waiting customers get impatient during the server's vacation and decide whether to take service or abandon simultaneously at the vacation completion instants. Using the supplementary variable technique and difference operator method, we obtain an explicit expression to find the steady-state system-length distributions at pre-arrival, random, and outside observer's observation epochs. We provide the stochastic decomposition structure for the number of customers and discuss the various performance measures. With the help of numerical experiments, we show that the method formulated in this work is analytically elegant and computationally tractable. The results are appropriate for light-tailed inter-arrival distributions and can also be leveraged to find heavy-tailed inter-arrival distributions.</p>

A Lévy-Driven Stochastic Queueing System with Server Breakdowns and Vacations

Motivated by modelling the data transmission in computer communication networks, we study a Lévy-driven stochastic fluid queueing system where the server may subject to breakdowns and repairs. In addition, the server will leave for a vacation each time when the system is empty. We cast the workload process as a Lévy process modified to have random jumps at two classes of stopping times. By using the properties of Lévy processes and Kella–Whitt martingale method, we derive the limiting distribution of the workload process. Moreover, we investigate the busy period and the correlation structure. Finally, we prove that the stochastic decomposition properties also hold for fluid queues with Lévy input.

Stochastic Decomposition for Two-Stage Stochastic Linear Programs with Random Cost Coefficients

Stochastic decomposition (SD) has been a computationally effective approach to solve large-scale stochastic programming (SP) problems arising in practical applications. By using incremental sampling, this approach is designed to discover an appropriate sample size for a given SP instance, thus precluding the need for either scenario reduction or arbitrary sample sizes to create sample average approximations (SAA). When compared with the solutions obtained using the SAA procedure, SD provides solutions of similar quality in far less computational time using ordinarily available computational resources. However, previous versions of SD were not applicable to problems with randomness in second-stage cost coefficients. In this paper, we extend its capabilities by relaxing this assumption on cost coefficients in the second stage. In addition to the algorithmic enhancements necessary to achieve this, we also present the details of implementing these extensions, which preserve the computational edge of SD. Finally, we illustrate the computational results obtained from the latest implementation of SD on a variety of test instances generated for problems from the literature. We compare these results with those obtained from the regularized L-shaped method applied to the SAA function of these problems with different sample sizes.

Impatient customers in Markovian queue with Bernoulli feedback and waiting server under variant working vacation policy

This paper deals with customers’ impatience behaviour for single server Markovian queueing system under K-variant working vacation policy, waiting server, Bernoulli feedback, balking, reneging, and retention of reneged customers. Using probability generating function (PGF) technique, we obtain the steady-state solution of the system. In addition, we prove the stochastic decomposition properties. Useful performance measures of the considered queueing system are derived. A cost model is developed. Then, the parameter optimisation is carried out numerically, using quadratic fit search method (QFSM). Finally, numerical examples are provided in order to visualize the analytical results.