Retrial BMAP/PH/N Queueing System with a Threshold-Dependent Inter-Retrial Time Distribution

In this paper, we study a multi-server queueing system with retrials and an infinite orbit. The arrival of primary customers is described by a batch Markovian arrival process (BMAP), and the service times have a phase-type (PH) distribution. Previously, in the literature, such a system was mainly considered under the strict assumption that the intervals between the repeated attempts from the orbit have an exponential distribution. Only a few publications dealt with retrial queueing systems with non-exponential inter-retrial times. These publications assumed either the rate of retrials is constant regardless of the number of customers in the orbit or this rate is constant when the number of orbital customers exceeds a certain threshold. Such assumptions essentially simplify the mathematical analysis of the system, but do not reflect the nature of the majority of real-life retrial processes. The main feature of the model under study is that we considered the classical retrial strategy under which the retrial rate is proportional to the number of orbital customers. However, in this case, the assumption of the non-exponential distribution of inter-retrial times leads to insurmountable computational difficulties. To overcome these difficulties, we supposed that inter-retrial times have a phase-type distribution if the number of customers in the orbit is less than or equal to some non-negative integer (threshold) and have an exponential distribution in the contrary case. By appropriately choosing the threshold, one can obtain a sufficiently accurate approximation of the system with a PH distribution of the inter-retrial times. Thus, the model under study takes into account the realistic nature of the retrial process and, at the same time, does not resort to restrictions such as a constant retrial rate or to rough truncation methods often applied to the analysis of retrial queueing systems with an infinite orbit. We describe the behavior of the system by a multi-dimensional Markov chain, derive the stability condition, and calculate the steady-state distribution and the main performance indicators of the system. We made sure numerically that there was a reasonable value of the threshold under which our model can be served as a good approximation of the BMAP/PH/N queueing system with the PH distribution of inter-retrial times. We also numerically compared the system under consideration with the corresponding queueing system having exponentially distributed inter-retrial times and saw that the latter is a poor approximation of the system with the PH distribution of inter-retrial times. We present a number of illustrative numerical examples to analyze the behavior of the system performance indicators depending on the system parameters, the variance of inter-retrial times, and the correlation in the input flow.

On the stationary distributions in some queueing systems

В статье рассматриваются стационарные распределения числа требований в системах массового обслуживания $M_{\lambda}|G|n|\infty$ и $GI_{\lambda}^{\nu}|M_{\mu}|1|\infty$, и показывается, что введение в данные системы массового обслуживания вспомогательных распределений с понятным вероятностным смыслом вместе с их производящими функциями позволяет упростить как доказательство так и его восприятие, а также приводит к новой записи полученных результатов. В первой системе рассматривается усечённое распределение искомого стационарного распределения для вложенной цепи Маркова. Данное усечение связано с количеством каналов $n$ и описывает вероятностные веса состояний системы, когда существует хотя бы один незанятый канал. Во второй системе для описания результатов используется распределение, связанное с распределением количества заявок во входящей группе требований: определяются вероятности хвостов описанного распределения, а потом для получения вспомогательного вероятностного распределения берётся их удельный вес между собой. This paper deals with two queuing system: $M_{\lambda}|G|n|\infty$ and $GI_{\lambda}^{\nu}|M_{\mu}|1|\infty$. The purpose is to find the steady-state results in terms of the probability-generating functions.

Incentives for Shared Services: Multiserver Queueing Systems with Priorities

Problem definition: We study shared service whereby multiple independent service providers collaborate by pooling their resources into a shared service center (SSC). The SSC deploys an optimal priority scheduling policy for their customers collectively by accounting for their individual waiting costs and service-level requirements. We model the SSC as a multiclass [Formula: see text] queueing system subject to service-level constraints. Academic/practical relevance: Shared services are increasingly popular among firms for saving operational costs and improving service quality. One key issue in fostering collaboration is the allocation of costs among different firms. Methodology: To incentivize collaboration, we investigate cost allocation rules for the SSC by applying concepts from cooperative game theory. Results: To empower our analysis, we show that a cooperative game with polymatroid optimization can be analyzed via simple auxiliary games. By exploiting the polymatroidal structures of the multiclass queueing systems, we show when the games possess a core allocation. We explore the extent to which our results remain valid for some general cases. Managerial implications: We provide operational insights and guidelines on how to allocate costs for the SSC under the multiserver queueing context with priorities.

Estimating the true arrival, balking, and reneging processes from censored transactional data: a simulation-based approach

The transactional data typically collected/available on queueing systems are often subject to censoring as unsuccessful arrivals due to balking and/or unserved entities due to reneging are not recorded. In fact, in many situations, the true arrival, balking, and reneging events are unobservable, making it virtually impossible to collect data on these stochastic processes—information that is crucial for capacity planning and process improvement decisions. The objective of this paper is to estimate the true (latent) external arrival, balking, and reneging processes in queueing systems from such censored transactional data. The estimation problem is formulated as an optimization model and an iterative simulation-based inference approach is proposed to find appropriate input models for these stochastic processes. The proposed method is applicable in any complex queueing situation as long as it can be simulated. The problem is investigated under both known and unknown reneging distribution. Through extensive simulation experiments, general guidelines are provided for specifying the parameters of the proposed approach, namely, sample size and number of replications. The proposed approach is also validated through a real-world application in a call center, where it successfully estimates the underlying arrival, balking, and reneging distributions. Finally, to enable reproducibility and technology transfer, a working example, including all codes and sample data, are made available in an open online data repository associated with this paper.

Strong Embeddings for Transitory Queueing Models

In this paper, we establish strong embedding theorems, in the sense of the Komlós-Major-Tusnády framework, for the performance metrics of a general class of transitory queueing models of nonstationary queueing systems. The nonstationary and non-Markovian nature of these models makes the computation of performance metrics hard. The strong embeddings yield error bounds on sample path approximations by diffusion processes in the form of functional strong approximation theorems.

Queueing Systems with Rationally Inattentive Customers

Problem definition: Classical models of queueing systems with rational and strategic customers assume queues to be either fully visible or invisible, while service parameters are known with certainty. In practice, however, people only have “partial information” on the service environment, in the sense that they are not able to fully discern prevalent uncertainties. This is because assessing possible delays and rewards is costly, as it requires time, attention, and cognitive capacity, which are all limited. On the other hand, people are also adaptive and endogenously respond to information frictions. Methodology: We develop an equilibrium model for a single-server queueing system with customers having limited attention. Following the theory of rational inattention, we assume that customers optimize their learning strategies by deciding the type and amount of information to acquire and act accordingly while internalizing the associated costs. Results: We establish the existence and uniqueness of a customer equilibrium when customers allocate their attention to learn uncertain queue lengths and delineate the impact of service characteristics. We provide a complete spectrum of the impact of information costs on throughput and show numerically that throughput might be nonmonotone. This is also reflected in social welfare if the firm’s profit margin is high enough, although customer welfare always suffers from information costs. Managerial implications: We identify service settings where service firms and social planners should be most cautious for customers’ limited attention and translate our results to advisable strategies for information provision and service design. For example, we recommend firms to avoid partial hindrance of queue-length information when a low-demand service is not highly valued by customers. For a popular service that customers value reasonably highly, however, partial hindrance of information is particularly advisable. Academic/practical relevance: We propose a microfounded framework for strategic customer behavior in queues that links beliefs, rewards, and information costs. It offers a holistic perspective on the impact of information prevalence (and information frictions) on operational performance and can be extended to analyze richer customer behavior and complex queue structures, rendering it a valuable tool for service design.