  We study the problem of equitably and efficiently allocating an arriving
resource to multiple queues with customer abandonment. The problem is motivated
by the cadaveric liver allocation system of the United States, which includes a
large number of small-scale (in terms of yearly arrival intensities) patient
waitlists with the possibility of patients abandoning (due to death) until the
required service is completed (matched donor liver arrives). We model each
waitlist as a GI/MI/1+GI queue, in which a virtual server receives a donor
liver for the patient at the top of the waitlist, and patients may abandon
while waiting or during service. To evaluate the performance of each queue, we
develop a finite approximation technique as an alternative to fluid or
diffusion approximations, which are inaccurate unless the queue's arrival
intensity is large. This finite approximation for hundreds of queues is used
within an optimization model to optimally allocate donor livers to each
waitlist. A piecewise linear approximation of the optimization model is shown
to provide the desired accuracy. Computational results show that solutions
obtained in this way provide greater flexibility, and improve system
performance when compared to solutions from the fluid models. Importantly, we
find that appropriately increasing the proportion of livers allocated to
waitlists with small scales or high mortality risks improves the allocation
equity. This suggests a proportionately greater allocation of organs to smaller
transplant centers and/or those with more vulnerable populations in an
allocation policy. While our motivation is from liver allocation, the solution
approach developed in this paper is applicable in other operational contexts
with similar modeling frameworks.
