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Application of a simple analytical model of capacity requirements

Application of a simple analytical model of capacity requirements. Martin Utley Steve Gallivan, Mark Jit. Clinical Operational Research Unit University College London. Outline. A simple model for estimating capacity requirements for hospital environments Applications

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Application of a simple analytical model of capacity requirements

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  1. Application of a simple analytical model of capacity requirements Martin Utley Steve Gallivan, Mark Jit Clinical Operational Research Unit University College London

  2. Outline • A simple model for estimating capacity requirements for hospital environments • Applications Evaluation of Treatment Centres Paediatric intensive care unit

  3. A hospital environment with unlimited capacity Variable admissions Variable demand for beds ? ? Variable length of stay ?

  4. A hospital environment with unlimited capacity Variable admissions Variable demand for beds ? ? Variable length of stay ?

  5. Probability N Y 0 1 2 3 4 5 Contribution to demand Still in hospital? Number of admissions Calculate demand for beds on day of interest Contribution to bed demand from admissions on an earlier day Start

  6. Contribution to bed demand from admissions on an earlier day Probability N Y 0 1 2 3 4 5 Contribution to demand Still in hospital? Number of admissions Booked admissions (incorporating non-attendance) + emergencies... ...may follow weekly cycle, seasonal trends etc.

  7. Contribution to bed demand from admissions on a previous day Probability N Y 0 1 2 3 4 5 Contribution to demand Still in hospital? Number of admissions Reflects variability in length of stay No need to parameterise length of stay distribution

  8. Probability N Y 0 1 2 3 4 5 Contribution to demand Still in hospital? Number of admissions Contribution to bed demand from admissions on an earlier day Start Calculating the probability distribution of this contribution is relatively straightforward

  9. Contributions to bed demand from admissions on a sequence of days

  10. Probability Y Still in hospital? Contribution to demand Number of admissions Contributions to bed demand from admissions on a sequence of days

  11. TOTAL DEMAND Contribution to bed demand from a sequence of days

  12. Having calculated distribution of demand, one can estimate capacity required to meet demand on a given % of days. This depends on both average demand and the degree of variability Capacity required to meet demand on 95% of days

  13. The impact of variability Average occupancy 10 patients Capacity required to meet 95% of demand 14 beds* 95% Demand

  14. The impact of variability Average occupancy 10 patients Capacity required to meet 95% of demand 15 beds* 95% Demand

  15. The impact of variability Average occupancy 10 patients Capacity required to meet 95% of demand 17 beds* 95% Demand

  16. Hospital 1 Hospital 2 Hospital 1 Hospital 2 TC Application: UK Treatment Centres Intention is to separate routine elective cases from complex & emergency cases

  17. Application: UK Treatment Centres In Prague we presented plans for a 3 yr project to evaluate UK NHS Treatment Centres in terms of: • Organisational Development; • Knowledge Management; • Throughput and the efficient use of capacity. Here, I’ll briefly present our methods and findings.

  18. How could a TC improve efficiency? Genuinely reducing length of stay Management of patients Genuinely reducing variability in length of stay Gains in efficiency for whole system? Economies of Scale Structure and organisation of service Reducing variability in length of stay through patient selection Note: our work was limited to study effects associated with organisation of services

  19. Hospital 1 Hospital 2 Hospital 1 Hospital 2 TC Compare capacity requirements We evaluated a large number of hypothetical scenarios... ...to identify circumstances in which a TC might be an efficient use of capacity

  20. Capacity requirements* of  = TC Capacity requirements* of Hospital 1 Hospital 2 Hospital 1 Hospital 2 Compare capacity requirements  < 1 The system is more efficient with a TC * In order to meet demand on 95% of days

  21. What factors to consider? Level of emergency admissions Number of elective admissions Impact of TC () Number of Non-TC hospitals Success of patient selection

  22. Success of patient selection How to quantify success of patient selection? Level of emergency admissions Number of elective admissions Impact of TC () Number of Non-TC hospitals

  23. Worst Case Frequency Refer to TC Frequency Length of stay (days) Frequency Refer to hospital Length of stay (days) Length of stay (days)

  24. Best Case Frequency Refer to TC Frequency Length of stay (days) Frequency Refer to hospital Length of stay (days) Length of stay (days)

  25. Modelling patient selection Use mathematical function with parameter to separate length of stay distribution a

  26. Modelling patient selection a = 0

  27. Modelling patient selection a = 1

  28. Modelling patient selection a = 3

  29. Modelling patient selection a = 30

  30. RESULTS

  31. Example patient population Probability

  32. 1 3 5 9 11 13 15 17 19 21 23 N (number booked admissions per day) Efficiency of capacity use if a TC is introduced 20 18 Worse r > 1 16 14 Marginally better 0.975 < r≤ 1 12 Better 0.95 < r ≤ 0.975 10 8 Much better r < 0.95 6 4 2 a (degree of success in identifying shorter stay patients)

  33. Urology patients Probability Data: Elective urology inpatients to an English NHS Trust Jan 01 – Oct 04

  34. Level of emergencies 0% 10% 20% Worse Marginally better Better Much better

  35. Findings • In many circumstances, there is no theoretical benefit associated with treatment centres*. • Theoretical benefits exist if there is successful identification of shorter-stay patients and a number of participating non-TC hospitals. Benefits rely on cooperation between providers *in terms of the efficient use of capacity

  36. It has come to this... Payment by Results Is cooperation between providers likely? Fixed Tariffs Patient Choice

  37. George Box said... “All models are wrong... ...some are useful”

  38. Simple model of capacity requirements All models arewrong ...some areuseful

  39. Summary • Simple analytical model very useful for examining the interplay of key factors in a large number of hypothetical scenarios (1800 overall). • Also has application in practical planning problems: • admissions to paediatric intensive care • restructuring services for anxiety and depression?

  40. END

  41. Modelling patient selection % referred to hospital 100% We used the Michaelis-Menten function to represent the degree to which intelligent selection of patients can be achieved. 50% Refer to TC Refer to hospital Frequency Length of stay Length of stay

  42. Reminder: generating functions Let y be a positive integer valued random variable where Prob(y = i) = ri. The generating function that describes the probability distribution of y, Y(s), is defined as 0 < s  1 Expectation and variance of y, E(y) and Var(y) given by and The parameter s is a dummy variable used only to define the generating function and has no physical significance.

  43. Xb(s) Ad(s) A single hospital environment with unlimited capacity Based on Ad(s), the generating function for the number of patients entering on day d, and Xb(s) , that for the number of beds occupied by a single patient b days after entering, the distribution of ud, the occupancy on day d, is given by the generating function

  44. Xb(s) Ad(s) A single hospital environment with unlimited capacity Based on Ad(s), the generating function for the number of patients entering on day d, and Xb(s) , that for the number of beds occupied by a single patient b days after entering, (1) the contribution from admissions on day (d-b)

  45. Xb(s) Ad(s) A single hospital environment with unlimited capacity Based on Ad(s), the generating function for the number of patients entering on day d, and Xb(s) , that for the number of beds occupied by a single patient b days after entering, (2) summed over all days from d backwards

  46. Example 1 Ad(s) = sN N booked admissions per day that all attend Xb(s) = (1-pb) + pbs bed occupied b days after admission with probability pb, maximumvalue of b = M Gallivan et al, BMJ, 2002

  47. Example 2 N booked admissions that attend with probability (1-n) and a prob qi of i emergencies Xb(s) = (1-pb) + pbs bed occupied b days after admission with probability pb, maximumvalue of b = M Utley et al, HCMS, 2003

  48. TC 1 Non-TC 1 Non-TC + 1 TC Non-TC

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