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

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

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

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

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

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

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

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

12. Capacity required to meet demand on 95% of days
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. Application: UK Treatment Centres
Intention is to separate routine elective cases
from complex & emergency cases
Hospital 1
Hospital 1 TC
Hospital 2
Hospital 2

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
organisation of service
Structure and
Reducing variability in length of stay through patient selection
Note: our work was limited to study effects
associated with organisation of services

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

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

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

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

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

24. Best Case Refer to TC Refer to hospital Frequency 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

27. Modelling patient selection

28. Modelling patient selection

29. Modelling patient selection

30. RESULTS

31. Example patient population
Probability

32. Efficiency of capacity use
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 1 3 5 9 11 13 15 17 19 21 23
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. Is cooperation between providers likely?
Payment by Results It
has
come to
this...
Fixed Tariffs
Patient Choice

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

38. All models are wrong ...some are useful
Simple model of capacity requirements
All models are wrong
...some are useful

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) A single hospital environment with unlimited capacity Ad(s)
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,
Xb(s)
Ad(s)
the distribution of ud, the occupancy on day d, is given by the generating function

44. Xb(s) A single hospital environment with unlimited capacity Ad(s)
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,
Xb(s)
Ad(s)
(1) the contribution from admissions on day (d-b)

45. Xb(s) A single hospital environment with unlimited capacity Ad(s)
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,
Xb(s)
Ad(s)
(2) summed over all days from d backwards

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

47. Example 2 Xb(s) = (1-pb) + pbs Utley et al, HCMS, 2003
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, maximum value of b = M
Utley et al, HCMS, 2003

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