Queueing Network Approach to the Analysis of Healthcare Systems

1. Queueing Network Approach to the Analysis of Healthcare Systems H. Xie, T.J. Chaussalet and P.H. Millard Health and Social Care Modelling Group (HSCMG) University of Westminster, London, UK

2. Outline • Patient flow • Queueing network models for patient flow • Application • Conclusion and future works

3. Introduction • Patient flow • how patients move through a healthcare system • a major factor in improving efficiency • Patient regarded as progressing through a set of logical stages in the process of care • diagnosis, treatment, rehabilitation, long-stay care, etc. • In general, progressig through a set of (conceptual) stages, called phases health care system phase M phase 1 phase 2 discharge

4. Introduction (cont’d) • Compartmental models are developed to explain patient flow through departments (e.g. geriatric medicine) • compartments representing different phases of care • study the long-run behaviour of a system e.g. expected number of patients in system, etc. • usually with no bed capacity constraints e.g. the system has as many beds available as required at any time • Clearly, adding bed capacity constraint will provide a more realistic representation of the real world situation.

5. Queueing network models • Queueing network models are widely used to study computer and communication systems • typically, network consists of many service nodes • for a service node • service rate: how quickly jobs are dealt with • arrival rate: how quickly jobs are arriving • We model a healthcare system as a queueing network • a phase (or stage) is treated as a service node • the servers at a service node are all hospital beds performing the corresponding task at the time • the number of servers at a service node varies as patients move through care phases • small number of nodes with a variable number of servers

6. Queueing network models (cont’d) hospital ward S S S M S S M S M L hospital ward M S L

7. Queueing network models (cont’d) • Why queueing network? • enable access to a range of established methods dealing with queueing network models • a natural platform to incorporate a bed capacity constraint • direct calculation to stable-state solution (if exists) • Bed capacity constraint • in total, K beds available in the system • patients will be refused admission if all beds are occupied • Semi-open queueing network model • patients arrive following a Poisson process • system remains “open” when there is bed available • new arrivals are blocked (or lost) when capacity is reached

8. ( ) M M ¸ l H K 1 ( ) Q = P ¸ l l l l ! f l K i · e i ¼ ½ o r = M i i 1 2 ½ i ( ) i i = 1 1 H K ; ; : : : ; ; ; i = = ¹ i Queueing network models (cont’d) • Basic quantity • For a M phase system, the joint probability distribution of the number of patients in each phase (i.e. a service node) is where , patient arrival rate, and is the normalisation constant. • Derived quantities, e.g. • Expected number of patient in system

9. Queueing network models (cont’d) • What is the use of such a model? • calculate the expected (and variance of) number of patients in system, i.e. occupancy level • how the beds are distributed among the phases • calculate the probability that the system is closed to new arrival, i.e. the loss probability • calculate the effective admission rate • how these measurements change with respect to changes in bed capacity, patient management policies

10. Queueing network models (cont’d) • Special cases of the semi-open network • CLOSED network: system is always full • the number of patients in each phase jointly follow a mutlinomial distribution • OPEN network: system has infinite capacity to admit patients • the number of patients in each phase are independent Poisson

11. Queueing network models (cont’d) • Model parameters • model parameters are • arrival rate to system • number of phases • service rate (or average LOS) at each phase • transition probability between phases • Estimate parameters from LOS data • LOS follows Coxian distribution • Coxian distribution can approximate any distribution with non-negative support arbitrarily well • fit Coxian distribution to LOS data with increasing number of phases by maximum likelihood • the “best” number of phases suggested by Bayesian Information Criteria (BIC) • Parameters can come from literature

12. geriatric department K = 473 rehabilitation (67 days) acute care (9 days) 7% 17% long-stay care (863 days) new arrival (19 patients/day) 93% 83% Application • Model parameters from literature (Parry, 1996) • analysing data from a geriatric department in England • three phases (stages) were identified • acute care, rehabilitation and long-stay care Parry A. (1996) An age-related service revisited. In: Millard PH, McClean SI (editors) Go with the Flow. Royal Society of Medicine Press. pp127-30.

13. Application (cont’d) expected number of patients in system vs. bed capacity expected occupancy level vs. bed capacity 455.2 447.8 94.7%

14. Application (cont’d) expected occupancy level vs. arrival rate  bed capacity

15. Application (cont’d) loss probability vs. bed capacity loss probability vs. arrival rate 1.6%

16. Application (cont’d) loss probability vs. arrival rate  bed capacity

17. Application (cont’d) expected occupancy level vs. bed capacity loss probability vs. bed capacity

18. Application (cont’d) effective admission rate to the system vs. bed capacity 18.7

19. Application (cont’d) expected number of patient in system vs. % change in service rates loss probability vs. % change in service rates

20. Application (cont’d) expected number of patient in each phase vs. % change in service rates

21. Conclusion and future works • Queueing network models are suitable for modelling patient flow • access to well established methods • natural platform for capacity constraint • quite general • phases (conceptual states) are derived from data • Future works • different types of patients (mutli-class model) • more than one system • queues, blocking