We develop an unobservable M/M/1 queueing model to analyze advance selling and spot selling under uncertainty. Customers are heterogeneous in waiting costs, drawn from a uniform distribution. We disentangle two fundamental sources of uncertainty: customer valuation and service capacity. We derive the respective revenue functions for both advance and spot selling strategies and compare their performance. Our analysis reveals that advance selling is more profitable when customer valuations are uncertain. Conversely, spot selling emerges as the more profitable strategy when uncertainty originates from service capacity. When both sources are present, our numerical experiments suggest that the preferred regime depends on the relative dispersion of the two uncertainties: advance selling tends to be favored when valuation uncertainty is sufficiently pronounced relative to capacity uncertainty, whereas spot selling tends to be favored when capacity uncertainty is more pronounced. These results highlight how the origin and magnitude of uncertainty shape the optimal selling strategy in service queues and offer practical guidance for platforms and service providers deciding between advance and spot selling.
Markov-modulated fluid queues whose modulating chain is a finite birth–death process often exhibit a structured rate pattern in practice: the interior of the state space is homogeneous, while only a small number of boundary states deviate from uniform behaviour. This paper introduces the class of interior-homogeneous birth–death chains and develops an exact structural analysis for the associated fluid queues. The symmetrized fluid operator is shown to decompose into a uniform tridiagonal Toeplitz component plus a low-rank symmetric perturbation supported on the boundary. This structure reduces the classical eigenvalue problem to a fixed-size secular equation independent of system capacity and yields explicit interlacing-based localization of all eigenvalues. From a probabilistic perspective, a pathwise coupling argument combined with a symmetric-pencil perturbation identity establishes that strict monotonicity of service rates propagates to strict acceleration of the stationary tail decay. This leads to a coherent stochastic ordering framework for interior-homogeneous fluid queues. Numerical examples illustrate the decomposition and its interlacing brackets, and confirm the results of eigenvalue hierarchy for the M/M/c/K special case.
This paper investigates several queueing-related quantities associated with workload processes driven by spectrally one-sided Lévy inputs. We derive explicit expressions for the workload correlation function, the busy-period survival function, and the distribution of the running infimum of the workload process. The analysis is based on an exact expression for the transient workload distribution, which also enables a direct study of the structural properties of the correlation function. In the spectrally negative case, we further derive the joint distribution of the stationary workloads in a Lévy-driven tandem queue. The results are illustrated through explicit examples, with particular emphasis on Brownian motion with drift and compound Poisson input with exponential jumps.
In this work, we study an M/M/1 ticket queue with a clearance time for balking customers that follows the exponential distribution. The customers’ join/balk decisions depend on the total queue length that they observe upon arrival. The latter leads to a queue with two types of customers, with the type being decided according to the queue length upon arrival. We obtain the steady-state distribution applying when customers use a threshold strategy. We show that the conditional distributions of the tail of the queue given its head can be constructed as mixtures and sums of geometric distributions. We analyze the individual best response and check whether a threshold strategy is an equilibrium, for given values of the clearance rate.
Motivated by the challenges inherent in high-frequency appointment scheduling systems, we explore the scaling behavior of customer unpunctuality in arrival patterns and its impact on service dynamics. In particular, we characterize the asymptotic limits of unpunctual arrivals within the appointment system and establish corresponding limiting results for the associated single-server queueing process. Our findings reveal that, while the limiting arrival process is Poisson and the prelimit process is well approximated by a Poisson process, the Poisson approximation can fail dramatically at the queueing level.
We investigate optimal dynamic lead-time quotes for a service provider by considering the reference effects of quotes on customers, both ex-ante, when customers form expectations and decide whether to join based on the quote and their prior information about waits, and ex-post, when customers compare their realized waiting times to the quote to assess satisfaction, resulting in goodwill loss for the provider. We adopt a behavioral model in which customers respond to quoted lead times but do not interact strategically. We model the system as an unobservable queue. Customers decide whether to join based on their expected waiting cost, a function of both the quote and their expected sojourn time. The provider chooses lead-time quotes to maximize its expected revenues minus a loss of goodwill associated with customer dissatisfaction, a function of the quotes and the customers’ actual sojourn times. We prove that a threshold-like policy is optimal for the provider. In the presence of a piecewise linear loss-of-goodwill function, we characterize the optimal threshold and the quotes for each number of customers in the system. We also consider endogenous arrival rates driven by goodwill and show the existence and properties of a steady-state solution. These policies can inform the design of lead-time quotation for service providers, especially small and medium-sized enterprises (SMEs).
Long waiting times in service systems reduce customer satisfaction, increase abandonment rates, and harm provider performance. To address these challenges, we study a multi-server, multi-class queueing system in which the operator can jointly control scheduling and implement two types of rejections: immediate rejections at arrival and delayed rejections during the waiting period. The objective is to minimize total operational costs, including waiting costs and rejection penalties. Using a fluid approximation, we characterize the structure of the optimal policy and develop an index-based control rule—the ℒμ rule—that jointly governs scheduling and rejection decisions. Under the ℒμ rule, each customer class operates in one of two distinct queueing regimes: an Erlang-B regime, where customers are rejected immediately if capacity is unavailable, or an Erlang-A regime, where customers are admitted and may be rejected after waiting. We further extend the model to incorporate class-specific service-level (SL) requirements. In this setting, we show that the index structure adjusts to ensure SL compliance, relaxing strict scheduling priorities and enabling partial capacity sharing across classes. Delayed abandonments, which play a limited role in the basic model, become effective under the adjusted policy due to the presence of SL constraints. Numerical simulation experiments illustrate the effectiveness of the proposed policies.
We study two-station closed queueing network models where jobs cyclically visit a non-preemptive last-come first-serve (LCFS-NP) station and a last-come first-serve preemptive-resume (LCFS-PR) station. Jobs belong to multiple classes and receive at both stations exponential service times with arbitrary means. Even though multiclass LCFS-NP stations are not quasi-reversible, we show that the considered models still admit a product-form solution. A feature of the new product-form expression is to include factors that relate the mean service times at the LCFS-NP queue with the positions occupied by the jobs at both stations. To account for job positions, we propose a strategy to compute the normalizing constant of the state probabilities using permanents which, for a fixed number of classes, solves the model exactly in polynomial time as the total number of jobs grows. A mean-value analysis algorithm is also derived, using a recursion on networks where the job holding the last position at the LCFS-PR queue is removed from the model.
We consider a perishable inventory system (PIS) in which demands for items arrive according to a Poisson process and items according to a renewal process. Stored items have a deterministic maximum lifetime ‘on the shelf.’ Exploiting a relation between the so-called virtual outdating time (VOT) process of this PIS and the workload process of the M/G/1+D queue, we prove a decomposition property of each of these two processes. Subsequently we analyze two generalizations of the above PIS, where the quality of items on the shelf is not constant. In the first one, there are two types of items, with different maximum lifetimes. In the second, the quality of an item gradually deteriorates with age.
In this study, we analyze the reneging behavior of customers in a Markovian M/M/1 queue with an alternating service process, representing a normal operation mode and a working repair mode with reduced service speed. The customers are strategic and, as they continuously observe the system’s state, they decide whether to stay or to renege. For this model, we derive equilibrium customer reneging strategies and the corresponding performance measures of the system, considering both the baseline model with discrete units of customers and its fluid counterpart. To capture the effect of reneging, we compare their performance with similar systems, operating under a no-reneging policy. Both theoretical findings, as well as numerical experiments, reveal that allowing reneging significantly affects system performance. Finally, to assess the quality of the fluid approximation we conduct a numerical comparison between the fluid and the baseline version of the model.
In Naor’s model (Econometrica 37:15–24, 1969), customers decide whether or not to join a queue after observing its length. This work considers a variation in which customers are heterogeneous in their service value (reward) R from completed service and homogeneous in the cost of staying in the system per unit of time. It is assumed that the values of customers are independent random variables generated from a common parametric distribution. The manager observes the queue length process, but not the balking customers. Assuming that the distribution of R admits a known parametric form, a Maximum Likelihood Estimator based on the queue length data is constructed for the underlying parameters of R. We provide verifiable conditions for which the estimator is consistent and asymptotically normal. The estimation procedure is further leveraged to construt a dynamic pricing scheme that estimates the revenue maximizing admission price by iteratively updating the price using the estimated parameters. The performance of the estimator and the pricing algorithm are studied through a series of simulation experiments.
The study of Markov-modulated fluid models with upward jumps and phase transitions shows that the joint distribution is governed by a non-homogeneous linear differential system with specific boundary conditions. Spectral analysis technique was used to solve this system in which the unique solution is expressed by components that are calculated through the resolution of a linear system. The purpose of this paper is to propose a novel methodology that computes numerically the moments for these models. Our approach is recursive: the moment is obtained from the preceding moment via a linear system which in turn is expressed via the components derived for the joint distribution of the fluid level and the modulating process. Numerical illustrations are presented and analyzed.
We consider the ergodic risk-sensitive admission control problem for a Markovian multi-server queueing system with abandonment, where costs are incurred for server idleness, customer abandonment, and rejecting incoming arrivals. We first derive the Bellman optimality equation for this problem and show that a threshold policy—one that rejects incoming arrivals whenever the system-size exceeds a threshold—is optimal among all admissible control policies. We then propose a policy iteration algorithm to identify the optimal threshold, where we prove, under certain conditions on the problem’s parameters, that the algorithm will terminate at the optimal threshold level. We also characterize the effect of risk sensitivity on the optimal threshold, proving that this threshold monotonically decreases with respect to the sensitivity parameter and converges to the average cost optimal threshold from below as the sensitivity parameter tends to zero.
In this paper, we analyze a single-server Markovian queue with customer abandonment. We use confluent hypergeometric functions to derive exact expressions for the probability mass function (pmf), cumulative distribution function (cdf), probability generating function, and moments of the queue length. We also derive the Laplace-Stieltjes transform (LST) of the steady-state queue length and waiting time distributions and show how to compute all moments as simple integrals. Finally, we derive new formulas for the conditional and unconditional probability of abandonment. Thus, our work provides a complete analysis of the single-server Markovian queue with customer abandonment.
In stochastic service systems, growing customer demands and high service volatility combine to challenge the efficient operation of service disciplines in multiple dimensions. Priority mechanisms offer an effective approach for managing demand heterogeneity and congestion, thereby generating revenue and improving service quality. However, both preemptive priority and non-preemptive priority schemes exhibit inherent limitations. To address the deficiencies arising from implementing either scheme independently, this study develops a switching mechanism between these priority types and proposes a fully observable mixed priority model. In such a model, the priority customer (type-I customer) has non-preemptive priority over the ordinary customer (type-II customer) when the priority queue length is not greater than a fixed value L. In turn, the priority mechanism will be transferred from non-preemptive to preemptive once the length of the priority queue exceeds L. The expected delays for both priority and ordinary customers are derived in closed form. The expected wait time of a priority customer is revealed to not necessarily increase with the length of the priority queue under the L-policy, as the future-arriving priority customers may generate a positive external effect. In addition, considering that priority and ordinary customers are two stakeholders with different “market power,” a noncooperative game is established by capturing the noncooperative interactions among them. The equilibrium strategic behaviors of both kinds of customers are investigated, which are shown to be state-dependent. Finally, the sensitivity of customers’ strategic behavior on key system parameters is explored by implementing numerical experiments.
When a large crowd forms in a service system, customers’ service experience and service providers’ operating expenses will be greatly degraded by the negative effect of mass gathering. We examine a queueing system where customers exhibit gathering aversion, and consider customers’ overlapping number and time during their sojourn time as metrics to characterize the negative effects of gathering. We develop a mathematical framework to evaluate these two metrics, allowing us to characterize equilibrium balking thresholds and their bounds. Closed-form expressions for key system performance metrics such as average overlapping number and time in equilibrium are derived to assess the overall gathering levels of customers in the system, and their tradeoffs with other system performance metrics like throughput and social welfare are analyzed. Our findings reveal that increasing staffing levels does not always reduce gathering, as lower traffic can attract more customers to enter, leading to even increased gathering. As a result, increasing the staffing level does not necessarily enhance social welfare as customer gathering increases. We also analyze the strategy of pooling queues, finding that although the pooled system has a united service space, its advantage over the dedicated system is significant in reducing customers’ gathering when their arrival rate is large, due to the pooled space that discourages gathering-averse customers from entering. Furthermore, we evaluate the impact of shortening opening hours, showing that it may not discourage customers from entering. Instead, it can exacerbate congestion when the arrival rate is low due to the condensed arrival process. Lastly, we explore how gathering sensitivity influences system performance, finding that greater aversion to gathering will always degrade throughput. However, customers’ social welfare may increase as fewer of them choose to enter the system, thereby reducing the wait times and gathering levels. We also find that imposing an additional admission fee can help reduce congestion in the system and improve overall social welfare. These insights provide practical guidance for designing queueing systems that account for customer behavior and operational efficiency.
In this paper, we analyze a single-server Markovian queue with customer abandonment. We use confluent hypergeometric functions to derive exact expressions for the probability mass function (pmf), cumulative distribution function (cdf), probability generating function, and moments of the queue length. We also derive the Laplace-Stieltjes transform (LST) of the steady-state queue length and waiting time distributions and show how to compute all moments as simple integrals. Finally, we derive new formulas for the conditional and unconditional probability of abandonment. Thus, our work provides a complete analysis of the single-server Markovian queue with customer abandonment.
