Abstract This paper concerns the recurrence structure of the infinite server queue, as viewed through the prism of the maximum dater sequence, namely the time to drain the current work in the system as seen at arrival epochs. Despite the importance of this model in queueing theory, we are aware of no complete analysis of the stability behavior of this model, especially in settings in which either or both the inter arrival and service time distributions have infinite mean. In this paper, we fully develop the analog of the Loynes construction of the stationary version in the context of stationary ergodic inputs, extending earlier work of E. Altman [1], and then classify the Markov chain when the inputs are independent and identically distributed. This allows us to classify the chain, according to transience, recurrence in the sense of Harris, and positive recurrence in the sense of Harris. We further go on to develop tail asymptotics for the stationary distribution of the maximum dater sequence, when the service times have tails that are asymptotically exponential or Pareto, and we contrast the stability theory for the infinite server queue relative to that for the single server queue.

Abstract Priority queues have long been used to increase revenue by exploiting the fact that time-sensitive customers are willing to pay for shorter waiting times. This fact begs the question: Can one make even more revenue by relaxing the strictness of the priority policy? This paper answers this question under the unobservable queue setting, where customers are heterogeneous in their time-sensitivity; specifically the time-sensitivity of customers is allowed to follow an arbitrary distribution. In this paper, we prove necessary and sufficient conditions under which partial priority can increase the revenue. Specifically, we find a surprising result: Although partial priority offers much more flexibility than strict priority, partial priority only increases revenue if there are two additional constraints on the service provider, one setting a maximum price and the other setting a maximum waiting time. In the absence of either of these constraints, we prove that strict priority maximizes revenue. Finally, in situations where partial priority increases the revenue, we analytically characterize the amount of improvement.

Abstract In this note, we revisit the fundamental question of the strong law of large numbers and central limit theorem for processes in continuous time with conditional stationary and independent increments. For convenience, we refer to them as Markov additive processes, or MAPs for short. Historically used in the setting of queuing theory, MAPs have often been written about when the underlying modulating process is an ergodic Markov chain on a finite state space, cf. (Asmussen in Applied probability and queues, Springer-Verlag, New York, 2003; Asmussen and Albrecher in Ruin probabilities, Hackensack, 2010), not to mention the classical contributions of Pacheco and Prabhu (Markov additive processes of arrivals, CRC, Boca Raton, 1995), Prabhu (Stochastic storage processes, Springer-Verlag, New York, 1998). Recent works have addressed the strong law of large numbers when the underlying modulating process is a general Markov processes; cf. as reported (Kyprianou et al. Entrance laws at the origin of self-similar Markov processes in high dimensions, 2019; Yaran and Çağlar in ALEA Lat Am J Probab Math Stat 22:991–1010, 2025) . We add to the latter with a different approach based on an ergodic theorem for additive functionals and on the semimartingale structure of the additive part. This approach also allows us to deal with the setting that the modulator of the MAP is either positive or null-recurrent. The methodology additionally inspires a CLT-type result.

Abstract Hybrid hospitals combine on-site hospitalization with remote care via telemedicine, requiring new operational policies to balance costs, efficiency, and patient well-being across both care modalities. We address two key questions: (i) how to assign patients to remote or on-site care based on individual characteristics and proximity and (ii) how to optimally allocate shared medical resources between care modes and patient types. We develop a stochastic model using Brownian motion to capture the randomness in recovery and travel-related risk during remote and on-site care. While the optimal call-in threshold is shaped by a cost-minimization objective, its behavior—specifically, its non-monotonicity in travel time and the narrowing of the effective distance range for more severe cases—is also driven by clinical constraints on allowable delays before hospital admission. These constraints, motivated by medical guidelines, limit the threshold and lead to cases where remote hospitalization becomes infeasible for very distant or severely ill patients. Under resource constraints, the optimal solution mirrors a simultaneous increase in remote and on-site costs relative to the abundant-resource case. For multiple patient types, we characterize how optimal thresholds shift with resource availability. Our findings indicate that distant patients may at times be better served by on-site care. This outcome arises not purely from economic trade-offs, but from the interplay between clinical constraints (e.g., safe limits on call-in delays), operational considerations, and treatment costs. These insights can help inform healthcare decision-makers and policymakers in designing hybrid care systems.

Abstract Bundle pricing is commonly adopted by service firms managing multiple congestion-prone service facilities. Under bundle pricing, the firm sells all services as a single package. This scheme is in contrast to à la carte pricing, whereby the firm sells each service separately. The existing theory generally sees bundling as being more lucrative when the marginal cost of production is low. However, little is known about how bundling compares to à la carte pricing in service systems with delay-sensitive customers, despite the prevalence of both practices. Our paper compares these two pricing schemes in congested service systems. We find that the classical prescription can be reversed in such congested service settings even in the absence of any marginal cost of service provision. Specifically, bundling generates less revenue than à la carte pricing when the potential arrival rate of customers is high relative to service capacity or when customers are highly delay-sensitive relative to their valuation of services. Moreover, the relative revenue difference between the two pricing schemes is non-monotone in either the potential arrival rate or delay sensitivity, with the percentage revenue loss from suboptimally practicing bundle pricing being the most substantial when the potential arrival rate or delay sensitivity is intermediate. From an operational perspective, bundle pricing results in higher (resp. lower) capacity utilization and thus more (resp. less) system congestion than à la carte pricing when the potential arrival rate is low (resp. high). For customers, bundling generates higher consumer surplus when the potential arrival rate is low or high, but may generate lower consumer surplus when the potential arrival rate is intermediate. Our results offer normative guidance to service firms considering these two pricing strategies and shed light on their operational and welfare implications.