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We characterize heavy-traffic process and steady-state limits for systems staffed according to the square-root safety rule, when the service requirements of the customers are perfectly correlated with their individual patience for waiting in queue. Under the usual many-server diffusion scaling, we show that the system is asymptotically equivalent to a system with no abandonment. In particular, the limit is the Halfin-Whitt diffusion for the [Formula: see text] queue when the traffic intensity approaches its critical value 1 from below, and is otherwise a transient diffusion, despite the fact that the prelimit is positive recurrent. To obtain a refined measure of the congestion due to the correlation, we characterize a lower-order fluid (LOF) limit for the case in which the diffusion limit is transient, demonstrating that the queue in this case scales like [Formula: see text]. Under both the diffusion and LOF scalings, we show that the stationary distributions converge weakly to the time-limiting behavior of the corresponding process limit. Funding: This work was supported by the National Natural Science Foundation of China [Grant 72188101] and the Division of Civil, Mechanical and Manufacturing Innovation [Grants 1763100 and 2006350]. 

As empirically observed in restaurants, call centers, and intensive care units, service times needed by customers are often related to the delay they experience in queue. Two forms of dependence mechanisms in service systems with customer abandonment immediately come to mind: First, the service requirement of a customer may evolve while waiting in queue, in which case the service time of each customer is endogenously determined by the system’s dynamics. Second, customers may arrive (exogenously) to the system with a service and patience time that are stochastically dependent, so that the service-time distribution of the customers that end up in service is different than that of the entire customer population. We refer to the former type of dependence as endogenous and to the latter as exogenous. Because either dependence mechanism can have significant impacts on a system’s performance, it should be identified and taken into consideration for performance-evaluation and decision-making purposes. However, identifying the source of dependence from observed data is hard because both the service times and patience times are censored due to customer abandonment. Further, even if the dependence is known to be exogenous, there remains the difficult problem of fitting a joint service-patience times distribution to the censored data. We address these two problems and provide a solution to the corresponding statistical challenges by proving that both problems can be avoided. We show that, for any exogenous dependence, there exists a corresponding endogenous dependence, such that the queuing dynamics under either dependence have the same law. We also prove that there exist endogenous dependencies for which no equivalent exogenous dependence exists. Therefore, the endogenous dependence can be considered as a generalization of the exogenous dependence. As a result, if dependence is observed in data, one can always consider the system as having an endogenous dependence, regardless of the true underlying dependence mechanism. Because estimating the structure of an endogenous dependence is substantially easier than estimating a joint service-patience distribution from censored data, our approach facilitates statistical estimations considerably. Funding: C. A. Wu received financial support from the Hong Kong Research Grant Council [Early Career Scheme, Project 26206419]. A. Bassamboo and O. Perry received partial financial support from the National Science Foundation [Grant CMMI 2006350]. 

The fundamental problem in the study of parallel-server systems is that of finding and analyzing routing policies of arriving jobs to the servers that efficiently balance the load on the servers. The most well-studied policies are (in decreasing order of efficiency) join the shortest workload (JSW), which assigns arrivals to the server with the least workload; join the shortest queue (JSQ), which assigns arrivals to the smallest queue; the power-of-[Formula: see text] (PW([Formula: see text])), which assigns arrivals to the shortest among [Formula: see text] queues that are sampled from the total of [Formula: see text] queues uniformly at random; and uniform routing, under which arrivals are routed to one of the [Formula: see text] queues uniformly at random. In this paper we study the stability problem of parallel-server systems, assuming that routing errors may occur, so that arrivals may be routed to the wrong queue (not the smallest among the relevant queues) with a positive probability. We treat this routing mechanism as a probabilistic routing policy, named a [Formula: see text]-allocation policy, that generalizes the PW([Formula: see text]) policy, and thus also the JSQ and uniform routing, where [Formula: see text] is an [Formula: see text]-dimensional vector whose components are the routing probabilities. Our goal is to study the (in)stability problem of the system under this routing mechanism, and under its “nonidling” version, which assigns new arrivals to an idle server, if such a server is available, and otherwise routes according to the [Formula: see text]-allocation rule. We characterize a sufficient condition for stability, and prove that the stability region, as a function of the system’s primitives and [Formula: see text], is in general smaller than the set [Formula: see text]. Our analyses build on representing the queue process as a continuous-time Markov chain in an ordered space of [Formula: see text]-dimensional real-valued vectors, and using a generalized form of the Schur-convex order.