In this paper we propose a prototype model for the problem of managing waiting lists for organ transplantations. Our model captures the double-queue nature of the problem: there is a queue of patients, but also a queue of organs. Both may suffer from "impatience": the health of a patient may deteriorate, and organs cannot be preserved longer than a certain amount of time. Using advanced tools from queueing theory, we derive explicit results for key performance criteria: the rate of unsatisfied demands and of organ outdatings, the steady-state distribution of the number of organs on the shelf, the waiting time of a patient, and the long-run fraction of time during which the shelf is empty of organs.
We consider a time-dependent stopping problem and its application to the decision-making process associated with transplanting a live organ. “Offers” (e.g., kidneys for transplant) become available from time to time. The values of the offers constitute a sequence of independent identically distributed positive random variables. When an offer arrives, a decision is made whether to accept it. If it is accepted, the process terminates. Otherwise, the offer is lost and the process continues until the next arrival, or until a moment when the process terminates by itself. Self-termination depends on an underlying lifetime distribution (which in the application corresponds to that of the candidate for a transplant). When the underlying process has an increasing failure rate, and the arrivals form a renewal process, we show that the control-limit type policy that maximizes the expected reward is a nonincreasing function of time. For non-homogeneous Poisson arrivals, we derive a first-order differential equation for the control-limit function. This equation is explicitly solved for the case of discrete-valued offers, homogeneous Poisson arrivals, and Gamma distributed lifetime. We use the solution to analyze a detailed numerical example based on actual kidney transplant data.
We consider a sequential matching problem where M offers arrive in a random stream and are to be sequentially assigned to N waiting candidates. Each candidate, as well as each offer, is characterized by a random attribute drawn from a known discrete-valued probability distribution function. An assignment of an offer to a candidate yields a (nominal) reward R > 0 if they match, and a smaller reward r ≤ R if they do not. Future rewards are discounted at a rate 0 ≤ α ≤ 1. We study several cases with various assumptions on the problem parameters and on the assignment regime and derive optimal policies that maximize the total (discounted) reward. The model is related to the problem of donor-recipient assignment in live organ transplants, studied in an earlier work.
An infinite random stream of ordered pairs arrives sequentially in discrete time. A pair consists of a “candidate” and an “offer,” each of which is either of type I (with probability p) or of type II (with probability q = 1 – p). Offers are to be assigned to candidates, yielding a reward R > 0 if they match in type, or a smaller reward 0 ≤ r ≤ R if not. An arriving candidate resides in the system until it is assigned, whereas an arriving offer is either assigned immediately to one of the waiting candidates or lost forever. We show that the optimal long-term average reward is R, independent of the population proportion p and the “second prize” r, and that the optimal average reward policy is to assign only a match. Optimal policies for discounted and finite horizon models are also derived.
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