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Randomized load balancing greatly improves the sharing of resources while being simple to implement. In one such model, jobs arrive according to a rate-αN Poisson process, with α < 1, in a system of N rate-1 exponential server queues. In Vvedenskaya et al. [19], it was shown that when each arriving job is assigned to the shortest of D, D ≥ 2, randomly chosen queues, the equilibrium queue sizes decay doubly exponentially in the limit as N → ∞. This is a substantial improvement over the case D = 1, where queue sizes decay exponentially.The reasoning in [19] does not easily generalize to jobs with nonexponential service time distributions. A modularized program for treating randomized load balancing problems with general service time distributions was introduced in Bramson et al. [5]. The program relies on an ansatz that asserts that, for a randomized load balancing scheme in equilibrium, any fixed number of queues become independent of one another as N → ∞. This allows computation of queue size distributions and other performance measures of interest.In this article, we demonstrate the ansatz in several settings. We consider the least loaded balancing problem, where an arriving job is assigned to the queue with the smallest workload. We also consider the more difficult prob- lem, where an arriving job is assigned to the queue with the fewest jobs, and demonstrate the ansatz when the service discipline is FIFO and the service time distribution has a decreasing hazard rate. Last, we show the ansatz always holds for a sufficiently small arrival rate, as long as the service distribution has 2 moments.

Randomized load balancing greatly improves the sharing of resources in a number of applications while being simple to implement. One model that has been extensively used to study randomized load balancing schemes is the supermarket model. In this model, jobs arrive according to a rate-nλ Poisson process at a bank of n rate-1 exponential server queues. A notable result, due to Vvedenskaya et.al. (1996), showed that when each arriving job is assigned to the shortest of d ≥ 2 randomly chosen queues, the equilibrium queue sizes decay doubly exponentially in the limit as n → ∞. This is a substantial improvement over the case d = 1, where queue sizes decay exponentially.The method of analysis used in the above paper and in the subsequent literature applies to jobs with exponential service time distributions and does not easily generalize. It is desirable to study load balancing models with more general, especially heavy-tailed, service time distributions since such service times occur widely in practice.This paper describes a modularized program for treating randomized load balancing problems with general service time distributions and service disciplines. The program relies on an ansatz which asserts that any finite set of queues in a randomized load balancing scheme becomes independent as n → ∞. This allows one to derive queue size distributions and other performance measures of interest. We establish the ansatz when the service discipline is FIFO and the service time distribution has a decreasing hazard rate (this includes heavy-tailed service times). Assuming the ansatz, we also obtain the following results: (i) as n → ∞, the process of job arrivals at any fixed queue tends to a Poisson process whose rate depends on the size of the queue, (ii) when the service discipline at each server is processor sharing or LIFO with preemptive resume, the distribution of the number of jobs is insensitive to the service distribution, and (iii) the tail behavior of the queue-size distribution in terms of the service distribution for the FIFO service discipline.

Background: Racial inequities for patients with heart failure (HF) have been widely documented. HF patients who receive cardiology care during a hospital admission have better outcomes. It is unknown whether there are differences in admission to a cardiology or general medicine service by race. This study examined the relationship between race and admission service, and its effect on 30-day readmission and mortality Methods: We performed a retrospective cohort study from September 2008 to November 2017 at a single large urban academic referral center of all patients self-referred to the emergency department and admitted to either the cardiology or general medicine service with a principal diagnosis of HF, who self-identified as white, black, or Latinx. We used multivariable generalized estimating equation models to assess the relationship between race and admission to the cardiology service. We used Cox regression to assess the association between race, admission service, and 30-day readmission and mortality. Results: Among 1967 unique patients (66.7% white, 23.6% black, and 9.7% Latinx), black and Latinx patients had lower rates of admission to the cardiology service than white patients (adjusted rate ratio, 0.91; 95% CI, 0.84–0.98, for black; adjusted rate ratio, 0.83; 95% CI, 0.72–0.97 for Latinx). Female sex and age >75 years were also independently associated with lower rates of admission to the cardiology service. Admission to the cardiology service was independently associated with decreased readmission within 30 days, independent of race. Conclusions: Black and Latinx patients were less likely to be admitted to cardiology for HF care. This inequity may, in part, drive racial inequities in HF outcomes.

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