1. Israel David and Michal Moatty-Assa A Stylistic Queueing-Like Model for the Allocation of Organs on the Public List

2. Supply-Demand Discrepancy Increasing shortage in kidneys for transplant 4,252 died waiting (2008)

3. (Kidney offers are thrown away) ~50% refuse 1 st kidney offered!

5. Who waits the longest? Whom do I best fit? Who’s the youngest? Objectives: Clinical Efficiency: QALY, % survival. Equity: in waiting, across social groups. Matching Criteria: ABO, HLA, PRA, Age, Waiting

6. pointsAgepointsPRA 418 – 0025% - 0% 240 – 19250% - 26% 160 – 41475% - 51% 0>606>75% points Waiting time (months) points HLA mismatches 0<244No MM 148 – 2531 MM 296 – 492No MM in DR 4>97 The Israeli “Point System” for kidney allocation

7. First In First Offered FIFOf – FIFO sorting for Offering simplifying assumptions, “stylistic” moel Decision rule Allocation rule

8. A continuous, time-dependent, full-info “Secretary”) ) The future arrival process How long do I wait? How good is this offer? my HLA, ABO population statistics by ABO, HLA donors arrival rate The decision of the single candidate

9. Model Assumptions Constant lifetime under dialysis ( T ) Poisson arrival of donor kidneys (rate ) Poisson arrival of patients "Aggregate HLA " – only one relevant genetic quality What is the compromising t? gain (life years) frequency in population kidney offer Rp a match r1-p a mismatch

10. First candidateSecond candidaten’th candidateSimulation

11. n = 1, basics – פונקציית המטרה, הרווח האופטימלי מהצעת כליה ברגע - מ"מ, רווח (שנות חיים) מהשתלת הכליה - תוחלת הרווח הצפויה מדחיית ההצעה ברגע t X – Offer random value;  = E[ X ] = Rp + r (1- p ) U ( t ) – expected optimal value assuming that at t an offer is pending V ( t ) – optimal value from t onwards (exclusive of t if an offer is pending); V(T) = 0. , T, R, r, p, ant 1

12. Dynamic Programming 1.U(t, x) = max{x, V(t)} 2.U(t) = E X [U(t, X)] 3.V(t) =

13. n = 1, depiction of V and U

14. n = 1, Explicit t *  = E[ X ] = Rp + r(1-p)

15. n = 1, Explicit solution of V(t), U(t).

16. (A solvable Volterra)

17. 0 effective n = 2, (approx.) outlook for the second candidate Non-hom.-Poisson stream with 3 stages

18. n = 2, conditional expected gains

19. n = 2, Explicit t*

20. n > 1, general specifics of cand. n specifics of cand.(n - 1 ) and t * n-1 optimizatio n optimal decision rule (t n * ) for cand. n inputoutput

21. still… n = 3 0 effective

22. n = 3 0 effective

23. The -recursion per sub-intervals for all Except for intersections with or where

24. leftmost V n ( t )’s for sub-intervals - optimal value for cand. n in rejecting at the beginning of sub-val l - arrival probability of an offer during sub-val l - conditional expected gain if during sub-val l

25. (explicit expressions for )

26. The critical subinterval and determining t n * is taken to be such that t is substituted for the beginning of subinterval

27. 0 blocking and releasing of simultaneous antigen currents

28. Simulation Measures Long-run proportion of "good" transplants Long-run death-rate Long-run Waiting Time for allocated candidate