We study a restricted version of the Erlang-R queueing model that has both server and space constraints, in which the space restriction is administered by two types of admission policies: blocking or holding in a pre-entrant queue. This model, in which customers occupying spaces may be either “needy” (requiring attention from servers) or “content,” captures a wide range of service environments, such as medical units, contact centers, and courts. We study the interplay between the two resource constraints and the reentrant queueing structure. We analyze the connection between staffing, space, and policy decisions and the system performance as a tool for optimizing resource utilization and in determining stability and performance. We develop product form and many-server approximations of performance measures when the blocking policy applies, and then utilize these results to determine approximations for the holding policy as well. We show that system dynamics are captured by a single measure of the fraction time a customer is “needy” in the network, and that returning customers should be explicitly accounted for both in steady-state and transient environments. We show that capacity allocation of both resource constraints (servers and space) should be considered jointly, and derive methodology to determine it. We show that applying a holding policy requires more resources than a blocking policy and demonstrate the application of our policies in two case studies of resource allocation in hospitals.
Semimartingale reflecting Brownian motions (SRBMs) are an integral part of queueing theory. Fluid paths are a standard tool for demonstrating positive recurrence of SRBMs, and their convergence provides both necessary and sufficient conditions for three- and fewer-dimensional SRBMs. Previous examples have been given in the literature for six- and greater-dimensional positive recurrent SRBMs with divergent linear fluid paths. Using predator–prey models, this article provides a family of four-dimensional positive recurrent SRBMs with divergent linear fluid paths, and a family of five-dimensional positive recurrent SRBMs with divergent linear fluid paths and identity covariance matrix. This closes the gap on the number of dimensions under which the positive recurrence of an SRBM does and does not imply convergence of the associated fluid paths.
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.
