Analysis of a batch-service queue with variable service capacity, correlated customer types and generally distributed class-dependent service times

Queueing models with batch service have been studied frequently, for instance in the domain of telecommunications or manufacturing. Although the batch server's capacity may be variable in practice, only a few authors have included variable capacity in their models. We analyse a batch server with multiple customer classes and a variable service capacity that depends on both the number of waiting customers and their classes. The service times are generally distributed and class-dependent. These features complicate the analysis in a non-trivial way. We tackle it by examining the system state at embedded points, and studying the resulting Markov Chain. We first establish the joint probability generating function (pgf) of the service capacity and the number of customers left behind in the queue immediately after service initiation epochs. From this joint pgf, we extract the pgf for the number of customers in the queue and in the system respectively at service initiation epochs and departure epochs, and the pgf of the actual server capacity. Combined with additional techniques, we also obtain the pgf of the queue and system content at customer arrival epochs and random slot boundaries, and the pgf of the delay of a random customer. In the numerical experiments, we focus on the impact of correlation between the classes of consecutive customers, and on the influence of different service time distributions on the system performance. (C) 2019 Elsevier B.V. All rights reserved

Delay analysis of a two-class batch-service queue with class-dependent variable server capacity

In this paper, we analyse the delay of a random customer in a two-class batch-service queueing model with variable server capacity, where all customers are accommodated in a common single-server first-come-first-served queue. The server can only process customers that belong to the same class, so that the size of a batch is determined by the length of a sequence of same-class customers. This type of batch server can be found in telecommunications systems and production environments. We first determine the steady state partial probability generating function of the queue occupancy at customer arrival epochs. Using a spectral decomposition technique, we obtain the steady state probability generating function of the delay of a random customer. We also show that the distribution of the delay of a random customer corresponds to a phase-type distribution. Finally, some numerical examples are given that provide further insight in the impact of asymmetry and variance in the arrival process on the number of customers in the system and the delay of a random customer

An analysis of a batch server with variable and class-dependent service capacity

In many studies on batch service queueing systems, the service capacity is assumed to be constant. However, this service capacity often depends on the content of the queue. In this paper, we analyse a discrete-time single server batch server queue with general inde- pendent arrivals. We distinguish two dierent classes in the arrival stream and products of both classes are added to the tail of a single queue. The single batch server can group all waiting customers at the head of the queue that belong to the same product class up to a certain class-dependent maximum capacity. This results in a stochastic service capacity that depends on both the number of customers in the queue and their respective classes. Since it is clear that the length of a sequence of same-class customers will have a signicant impact on the performance of the system, we also include correlation between the classes of consecutive customers. Applications of this type of batch server can, for instance, be found in the pacemaker loop of a Lean manufacturing system. In the course of the analysis, we calculate the probability generating function of the system occupancy at service initi- ation opportunities. In the numerical experiments, we will look at the impact of dierent parameters on both the mean system occupancy and the probability that the server is idle at a random service initiation opportunity. We also provide a number of guidelines to pick between the exact solution and an approximated approach with unlimited service capacities, by looking at the trade-o between accuracy and computational complexity

Delay analysis of a variable-capacity batch-server queue with general class-dependent service times

In manufacturing, a batch server groups multiple customers that require the same type of service based on a specific characteristic, such as temperature or destination. In this paper, we extend previous work with the analysis of the delay in a variable-capacity batch-service queueing system with general class-dependent service times and customer-based correlation in the arrival process. The impact of asymmetry and correlation in the arrival process on the mean delay of a random customer and the tail distribution of the delay is investigated as well

Assessing financial and flexibility incentives for integrating wind energy in the grid via agent-based modeling

This article provides an agent-based model of a hypothetical standalone electricity network to identify how the feed-in tariffs and the installed capacity of wind power, calculated in percentage of total system demand, affect the electricity consumption from renewables. It includes the mechanism of electricity pricing on the Day Ahead Market (DAM) and the Imbalance Market (IM). The extra production volumes of Electricity from Renewable Energy Sources (RES-E) and the flexibility of electrical consumption of industries is provided as reserves on the IM. Five thousand simulations were run by using the agent-based model to gather data that were then fit in linear regression models. This helped to quantify the effect of feed-in tariffs and installed capacity of wind power on the consumption from renewable energy and market prices. The consumption from renewable sources, expressed as percentage of total system consumption, increased by 8.17% for every 10% increase in installed capacity of wind power. The sharpest increase in renewable energy consumption is observed when a feed-in tariff of 0.04 €/kWh is provided to the wind farm owners, resulting in an average increase of 9.1% and 5.1% in the consumption from renewable sources while the maximum installed capacity of wind power is 35% and 100%, respectively. The regression model for the annualized DAM prices showed an increase by 0.01 €cents/kWh in the DAM prices for every 10% increase in the installed wind power capacity. With every increase of 0.01 €/kWh in the value of feed-in tariffs, the mean DAM price is lowered as compared to the previous value of the feed-in tariff. DAM prices only decrease with increasing installed wind capacity when a feed-in tariff of 0.04 €/kWh is provided. This is observed because all wind power being traded on DAM at a very cheap price. Hence, no volume of electricity is being stored for availability on IM. The regression models for predicting IM prices show that, with every 10% increase in installed capacity of wind power, the annualized IM price decreases by 0.031 and 0.34 €cents/kWh, when installed capacity of wind power is between 0 and 25%, and between 25 and 100%, respectively. The models also showed that, until the maximum installed capacity of wind power is less than 25%, the IM prices increase when the value of feed-in tariff is 0.01 and 0.04 €/kWh, but decrease for a feed-in tariff of 0.02 and 0.03 €/kWh. When installed capacity of wind power is between 25 and 100%, increasing feed-in tariffs to the value of 0.03 €/kWh result in lowering the mean IM price. However, at 0.04 €/kWh, the mean IM price is higher, showing the effect of no storage reserves being available on IM and more expensive reserves being engaged on the IM. The study concludes that the effect of increasing installed capacity of wind power is more significant on increasing consumption of renewable energy and decreasing the DAM and IM prices than the effect of feed-in tariffs. However, the effect of increasing values of both factors on the profit of RES-E producers with storage facilities is not positive, pointing to the need for customized rules and incentives to encourage their market participation and investment in storage facilities

System occupancy in a multiclass batch-service queueing system with limited variable service capacity

In multi-class telecommunications or manufacturing systems, customers belonging to the same class can often be processed together. This results in a service capacity that depends on the classes of the customers in the queue. In this paper, we analyse a discrete-time batch- service queue with two customer classes. The single batch server can group all same-class customers at the head of the queue up to a constant class-dependent maximum service capac- ity. We focus on the analysis of the system occupancy at service initiation opportunities, and also compute both a light- and heavy traffic approximation in order to reduce the numeri- cal complexity introduced by the maximum service capacities. Additionally, we propose a method for interpolating between these approximations in order to study the behaviour in the intermediate region. We also deduce the system occupancy and its approximations at random slot boundaries. In the numerical experiments, we examine the conditions under which these proposed approximations are accurate

Variable and class-dependent service capacity with a multi-class arrival process

In manufacturing, a single batch server can often group a number of customers that require the same type of service. In this paper, a shared queue without customer reordering is used in order to reduce the variability of throughput time of material throughout the manufacturing process which guarantees a global First-Come-First-Served (FCFS) service discipline. This is a significant difference with the more common polling systems where each type of customer has a dedicated queue. The batch server in this paper has a variable service capacity that depends on the classes of the customers in the queue. This paper extends previous work by considering a general number of N customer classes. During the analysis, we focus on the system occupancy of this system at random slot boundaries