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In this article, we examine a second optional service queueing system using two types of servers, viz. the main operating server and a reliable standby server. All arriving customers receive the first essential service (FES), and only a few may thereafter request a second optional service (SOS) with some probability. During FES and SOS services, the primary operational server may break down. The server is promptly sent for repair if a break down arises and the standby server will be replaced only if the system size is q ([greater than or equal to] 1); otherwise, customers would queue up while the main server is being repaired and resumes the service. We also derive the necessary and sufficient condition for the system to be stable. The model's steady state solution is discovered using the matrix geometric approach. Further, multiple system performance measures are obtained and a cost optimization problem is taken into consideration. Graphs are used to display the numerical outcomes. Keyword: Queue, Standby server, Matrix geometric method, Breakdown and Repairs.

We study a type of correlated queue in which its inter-arrival and service times are correlated, with their joint distribution being characterized by a copula function. This characterization allows for very generally correlated inter-arrival and service times. For such a queue, we first derive an infinite system of linear equations for the moments of the waiting time, based on which we design an algorithm to calculate the moments of the waiting time. In contrast with traditional methods, our algorithm incorporates a recursive procedure which is computationally much more efficient. Furthermore, we show how the moments and covariances of the inter-departure times of the queue can be obtained based on the moments of the waiting time. Numerical experiments are provided to validate our method.

Motivated by queueing applications, we study various reflected autoregressive processes with dependencies. Among others, we study cases where the interarrival and service times are proportionally dependent on additive and/or subtracting delay, as well as cases where interarrival times depend on whether the service duration of the previous arrival exceeds or not a random threshold. Moreover, we study cases where the autoregressive parameter is constant as well as a discrete or a continuous random variable. More general dependence structures are also discussed. Our primary aim is to investigate a broad class of recursions of autoregressive type for which several independence assumptions are lifted and for which a detailed exact analysis is provided. We provide expressions for the Laplace transform of the waiting time distribution of a customer in the system in terms of an infinite sum of products of known Laplace transforms. An integer-valued reflected autoregressive process that can be used to model a novel retrial queueing system with impatient customers and a general dependence structure is also considered. For such a model, we provide expressions for the probability generating function of the stationary orbit queue length distribution in terms of an infinite sum of products of known generating functions. A first attempt towards a multidimensional setting is also considered.

We consider the optimal scheduling problem in a multiserver queue with impatient customers belonging to multiple classes. We assume that each customer has a random abandonment time, after which the customer leaves the system if its service has not been completed before that. In addition, we assume that the scheduler is not able to anticipate the expiration of the abandonment times but only knows their distributions and how long each customer has been in the system. Many papers consider this scheduling problem under Poisson arrivals and linear holding costs assuming further that both the service times and the abandonment times have exponential distributions. Even with these additional assumptions, the exact solution is known only in very few special cases. To tackle this tricky problem, we apply the Whittle index approach. Unlike the earlier papers, which were restricted to exponential service times, we allow the service time distributions for which the hazard rate is decreasing. The Whittle index approach is applied to the discrete-time multiserver queueing problem with discounted costs. As our main theoretical result, we prove that the related relaxed optimization problem is indexable and derive the corresponding Whittle index explicitly. Based on this discrete-time result, we develop a reasonable heuristic for the original continuous-time multiserver scheduling problem. The performance of the resulting policy is evaluated in the M/G/M setup by numerical simulations, which demonstrate that it, indeed, gives better performance than the other policies included in the comparison.

Dense network deployment is now being evaluated as one of the viable solutions to meet the capacity and connectivity requirements of the fifth-generation (5G) cellular system. The goal of 5G cellular networks is to offer clients with faster download speeds, lower latency, more dependability, broader network capacities, more accessibility, and a seamless client experience. However, one of the many obstacles that will need to be overcome in the 5G era is the issue of energy usage. For energy efficiency in 5G cellular networks, researchers have been studying at the sleeping strategy of base stations. In this regard, this study models a 5G BS as an M[X]/G/1 feedback retrial queue with a sleeping strategy to reduce average power consumption and conserve power in 5G mobile networks. The probability-generating functions and steady-state probabilities for various BS states were computed employing the supplementary variable approach. In addition, an extensive palette of performance metrics have been determined. Then, with the aid of graphs and tables, the resulting metrics are conceptualized and verified. Further, this research is accelerated in order to bring about the best possible (optimal) cost for the system by adopting a range of optimization approaches namely particle swarm optimization, artificial bee colony and genetic algorithm.

This paper analyses the )/1/ queuing model with various types of breakdowns. During peak times, the system has breakdown due to server unavailability. The model considers system with two types of breakdowns. The system may breakdown in two different ways during busy and working vacation stages. Customer arrives to the system with parameter which follows Poisson distribution and server provides service in regular busy period with parameter and under the multiple working vacations it provides service with parameter with exponential distribution. In this model batches of customers are served under General Bulk Service Rule. The system may breakdown at any time. The breakdown that occurs during working vacation is denoted as and the breakdown during busy state is denoted as . The steady-state equation, the performance of measures for the system and particular cases of described model are derived. Finally, a real life example demonstrated the validity of the model in a proper way.

This paper considers a capacity allocation problem in a two-channel service system. Customers can receive service from either a single-server queueing system, which serves the customers waiting in line one by one, or a clearing service system, which serves a fixed number of customers simultaneously according to its capacity. Customers who join the queueing system should wait till they receive service. In contrast, customers who join the clearing system face the risk of service denial when there are more customers than the clearing system’s capacity. The social planner aims to minimize the total expected cost of all customers by determining the capacities and the arrival rates for the two channels. There are two settings: an unobservable setting where only the expected waiting time information is available and an observable setting where real-time information about the exact workload of the queueing system is known. We also consider the same system under the same settings with strategic customers who choose one of the two channels strategically to minimize their costs. The planner still has the same objective but can now decide only on the capacity allocation. Comparing the performance of the resulting systems allows us to understand the value of coordination and information. Extensions of these systems that serve two customer types are also explored.

We consider a stochastic matching model with a general compatibility graph, as introduced in Mairesse and Moyal (J Appl Probab 53(4):1064–1077, 2016). We prove that most common matching policies (including fcfm, priorities and random) satisfy a particular sub-additive property, which we exploit to show in many cases, the coupling-from-the-past to the steady state, using a backwards scheme à la Loynes. We then use these results to explicitly construct perfect bi-infinite matchings, and to build a perfect simulation algorithm in the case where the buffer of the system is finite.

Timely status updating in mobile edge computing (MEC) systems has recently gained the utmost interest in internet of things (IoT) networks, where status updates may need higher computations to be interpreted. Moreover, in real-life situations, the status update streams may also be of different priority classes according to their importance and timeliness constraints. The classical disciplines used for priority service differentiation, preemptive and non-preemptive disciplines, pose a dilemma of information freshness dissatisfaction for the whole priority network. This work proposes a hybrid preemptive/non-preemptive discipline under an M/M/1/2 priority queueing model to regulate the priority-based contention of the status update streams in MEC systems. For this hybrid discipline, a probabilistic discretionary rule for preemption is deployed to govern the server and buffer access independently, introducing distinct probability parameters to control the system performance. The stochastic hybrid system approach is utilized to analyze the average age of information (AoI) along with its higher moments for any number of classes. Then, a numerical study on a three-class network is conducted by evaluating the average AoI performance and the corresponding dispersion. The numerical observations underpin the significance of the hybrid-discipline parameters in ensuring the reliability of the whole priority network. Hence, four different approaches are introduced to demonstrate the setting of these parameters. Under these approaches, some outstanding features are manifested: exploiting the buffering resources efficiently, conserving the aggregate sensing power, and optimizing the whole network satisfaction. For this last feature, a near-optimal low-complex heuristic method is proposed.