120 The UMAP Journal 28.2 (2007) For the rate constant A, we use the number of organ applicants in a given year,AA(number of new applicants)/365.25. o In phase I, we add cadaver organs to the leaf nodes. As with patients,we model cadaver arrivals as a Poisson process, with rate the average number of cadaver organs added in a given year. o In phase Ill, we allocate organs based on bottom-up priority rules. A bottom- up priority rule is a recursive allocation process propagated up from the bottom of the tree, which requires any organ-patient match to meet some minimum priority standard. For example, for kidney allocation, the first priority rule is to allocate kidneys to patients who match the blood type and HLa profile exactly. within this restriction, oPtN dictates that kidneys be allocated locally first, then regionally, then nationally. In our model, this corresponds to moving from the- leaves up the tree Matched organ patient pairs undergo transplantation, which has a success rate dependent on the quality of the match. (In later sections, we explore the success, rate as also a function of the experience of the doctors at the center and the quality of the kidney. o In phase iv, we simulate the death of patients on the waiting list. We treat the death rate k of a patient as a linear function aT+b of the persons wait timeT.Hence,calculating from time 0, a persons chance of survival to time Tis e-AT=e-(aT'+b)T Under this mathematical model, our problem becomes finding a good tree structure and an appropriate set of bottom-up priority rules Simulation To study this model, we average results sover na ny simulations of kidney transplant network. Our simulation works as follows: At every time round, in phase l, we generate a number according to the Poisson distribution of the numberofnew candidates. Foreach new patient added, we randomly generate the person's race and age according to data on race and age distributions Usingthe srace, we generate the person'sblood type and HLA makeup, according to known distributions, and the patient's PRA, based on probabilities published by the OPtN Similarly, in Phase I, we generate a list of donor organs according to known distributions of blood type and HLA makeup. Moreover, we record where the rgan was generated, so we can study the effect of having to move the organ before transplantation, the time for which lowers its quality In Phase Ill, we implement recursive routines that traverse the tree from he bottom up, following the OPTN system for kidneys. To model the success rate of an operation, we use the statistics published by the OPTN; our main method of determining whether an operation is successful is the number of120 The UMAP Journal 28.2 (2007) For the rate constant A, we use the number of organ applicants in a given year, A : (number of new applicants)/365.25. "In phase II, we add cadaver organs to the leaf nodes. As with patients, we model cadaver arrivals as a Poisson process, with rate the average number of cadaver organs added in a given year. " In phase RiL, we allocate organs based on bottom-up prioritj rules. A bottom￾up priority rule is a recursive allocation process propagated up from the bottom of the tree, which requires any organ-patient match to meet some minimum priority standard. For example, for kidney allocation, the first priority rule is to allocate kidneys to patients who match the blood type and HLA profile exactly. Within this restriction, OPTN dictates that kidneys be allocated locally first, then regionally, then nationally. In our model, this corresponds to moving from the-leaves up the tree. Matched organ-patient pairs undergo transplantation, which has a success rate dependent on the quality of the match. (In later sections, we explore the success.rate as also a function of the experience of the doctors at the center and the quality of the kidney.) " In phase IV, we simulate the death of patients on the waiting list. We treat the death rate k of a patient as a linear function aT + b of the person's wait time T. Hence, calculating from time 0, a person's chance of survival to time T is e-kT = e-(aT+b)T. Under this mathematical model, our problem becomes finding a good tree structure and an appropriate set of bottom-up priority rules. Simulation To study this model, we average results over many simulations of the kidney transplant network. Odr simulation works as follows: At every time round, in phase I, we generate a number according to the Poisson distribution of the number of new candidates. For each new patient added, we randomly generate the person's race and age according to data on race and age distributions. Using the person's race, we generate the person's blood type and I-ILA makeup, according to known distributions, and the patient's PRA, based on probabilities published by the OPTN. Similarly, in Phase HI, we generate a list of donor organs according to known distributions of blood type and HLA makeup. Moreover, we record where the organ was generated, so we can study the effect of having to move the organ before transplantation, the time for which lowers its quality. In Phase 1II, we implement recursive routines that traverse the tree from the bottom up, following the OPTN system for kidneys. To model the success rate of an operation, we use the statistics published by the OPTN; our main method of determining whether an operation is successful is the number of