1. An Accident and Emergency Case Study
Dave Worthington (Adrian Fletcher and Navid Izady)
Management Science Dept.
Lancaster University

2. Outline Time-dependent queueing networks
A ‘generic’ A&E simulation model
A queueing ‘theory’ based contribution
Preliminary Results
Theory meets practice
Where next?

3. A time-dependent queueing network:

4. Time-dependent queueing networks in healthcare:4
A&E,
walk-in centres,
NHS Direct,
polyclinics,
whole hospitals,
patient pathway planning,
ambulance services
and so on ………

5. e.g. Typical A&E arrival patterns

6. A ‘generic’ A&E queueing network model
Simulation model built by Adrian Fletcher (& co.) at DoH:
to inform the national policy team of significant barriers to achieving the national 4h target.
ALSO used later as ‘generic’ consultancy tool to aid struggling hospital trusts to improve their A&E departments

8. A&E model Inputs Demand: Process Times: Staff:
patient type, date and time of arrival, routing probabilities
Process Times:
minimum, average and maximum operation times for each process.
Staff:
numbers of ENPs, junior and senior doctors working in A&E at each hour of the day

9. Typical (baseline) results
Also available:
Staff utilisations by time of day
Breach rate by time of day

10. Objectives achieved at National level
Visual & analytical representation of typical A&E dept
Illustration of the impact of variability of demand and process times in A&E;
A facilitation tool to focus national resources onto the key issues;
The ability to run ‘what if’ scenarios both visually and analytically;
The ability to illustrate what success would look like:
e.g. What service levels would be required at each process to meet 98% overall?

11. Scenarios investigated relative to the 4 hour target:
The effect of streaming the minors into a separate stream with dedicated (nurse staffing)
& the number of nurses required to run a minor stream
The effect of potential reductions in assessment times
The effect of deflecting demand from A&E
Better matching of staffing to demand by time of day

12. Can we better understand ‘matching of staffing to demand by time of day’?
a proper understanding needs to consider staffing of different types at different nodes;
this could be investigated by trial and error using the simulation model, ……
….. but
so it would be better if there were some ‘queueing theory’ to help find good solutions.

13. Navid Izady’s work (PhD thesis, Aug 2010)
On Queues with Time-varying Demand
Single facility delay systems
Networks of services
Loss queues and networks of loss queues
Staffing Time-Dependent Queueing Networks: The Case of A&E Departments

14. Queueing Problem Definition:
What is the minimum number of staff (Doctors and Nurses) that should be working during each hour of a day at each node of the network for achieving the following Government target :
98% of patients finish their journey ( discharged or be admitted to hospital) in 4 hours

15. Results to date for queueing networks
Steady-state:
Jackson networks
Various decomposition approximations
Infinite server systems, i.e. uncapacitated networks
Time-dependent:
Some investigations of decomposition methods, but little reported success
Numerical methods, e.g. DTM, Uniformization or Runge Kutta
Fluid models for oversaturated networks ( rho >> 1)

16. A queueing theory based approach
A staffing algorithm is proposed which uses a combination of infinite server networks, heavy traffic limits theorems and simulation to arrive at a good staffing profile for all nodes.
The algorithm tries to find the staffing levels such that quality of service remains approximately the same over all periods of time across all nodes of the network.
The algorithm finds the staffing levels for achieving 98% target after around simulation experiments.
It does not guarantee that the proposed staffing is the minimum possible, but experiments show significant improvements against existing staffing levels.

17. Staffing Time-Dependent Queues
1. Treat as Network of M(t)/G/∞ systems, based on:
and extensions thereof.
2. Set staffing levels to allow for variation about this mean using a square root staffing equation:
where b is a no. of standard deviations above the mean to allow for random variation.

18. Algorithm
Calculate Offered Load (E[N(t)]) by time of day (t) at each node of the simplified network using uncapacitated network results,
Start with a low grade of service (which is specified in terms of b)
Calculate staffing levels by time of day at each node using Square Root Staffing equation in terms of offered load and grade of service.
Use simulation with the proposed staffing pattern to estimate percentage finished in 4 hours
If 98% target is not hit, increase/decrease b and go to 3, otherwise finish.

19. An Example of Algorithm Optimal Staffing

20. An Example of Algorithm Optimal Staffing
First Assessment
Blood Test
X-Ray
Second Assessment
Treatment 3 2 4 1 5 6 7 96 78 58 54
114
Total
400

21. Preliminary Results for simplified network: Algorithm v Existing
Doctors (First and Second Units)
Nurses (First and Second Units)
AlgorithmArithm
Existing
Total Staff : 100 hours (68 Doctors, 32 Nurses)
Total Staff: 101 hours(75 Doctors, 26 Nurses)
% finished in 4 hours: 95%
% finished in 4 hours: 90%
Average Waiting in First Assessment: 16 mins
Average Waiting in First Assessment: 28 mins
Average Waiting in Second Assess’t: 18 mins
Average Waiting in Second Assess’t: 24 mins

22. Preliminary Results for simplified network: Algorithm v Existing
Doctors (First and Second Units)
Nurses (First and Second Units)
AlgorithmArithm
Existing
Total Staff : 100 hours (68 Doctors, 32 Nurses)
Total Staff: 101 hours(75 Doctors, 26 Nurses)
% finished in 4 hours: 95%
% finished in 4 hours: 90%
Average Waiting in First Assessment: 16 mins
Average Waiting in First Assessment: 28 mins
Average Waiting in Second Assess’t: 18 mins
Average Waiting in Second Assess’t: 24 mins

24. Tentative Conclusions
The proposed staffing algorithm is simple and has been implemented in Excel. It covers staffing of many time dependent stochastic networks.
The algorithm helps to reduce number of simulation experiments required significantly.
Stabilizing performance measure over time seems to be a good insight & a desirable practical measure.
This combination of queue modelling approaches (in this case: uncapacitated networks and heavy traffic behaviour) can be used to help solve real (‘generic’?) time dependent queue network problems.

25. Theory meets practice Sojourn Time Distributions Hospital 1 Hospital 2
(From Murat Gunal, PhD thesis, Lancaster University, 2009)

26. Hospital 3

27. Important Performance Indicators
Sojourn Time : Total Time in the A&E from Arrival to Discharge/Admission
Government Target: 98% <= 4 Hours
FED Time : Time to First Encounter with a Doctor
Notional Target: 80% <= 1 Hour

28. Hospital 3: Actual performance
Percentage Discharged/Admitted in 4 Hours: 98%
Total Time in System (Sojourn Time) :
Mean: 172 min
Median: 190 min
Time to Encounter with a Doctor (FED) :
Mean : 100 min
Median : 100 min
Percentage Seen by a Doctor in 1 Hour: 30%

29. Hypothesis
RIGHT number of staff at WRONG times ?

30. Arrival Patterns

31. Simulation Results - Current Staffing: Sojourn time
Scenario
Data
Simulation
Mean Time in System
172
Median Time in System
190
160
% Left within 4 hours 98 82

32. Simulation Results - Current Staffing: FED
Scenario
Data
Simulation
Mean Time to See a Doctor
100 62
% Seen by a Doctor in 1 Hour 30 63

33. Staffing Results – Sojourn Time
Scenario
Data
Simulation-
Current Staffing
New staffing
Sojourn Time Mean
172
143
Sojourn Time Median
190
160
140
% Left within 4 hours 98 82 92

34. Simulation Results- FED
Scenario
Data
Simulation
Current Staffing
New Staffing
FED Mean
100 62 43
FED Median 40 30
% Seen by a Doctor in 1 Hour 63 71

35. Ongoing work with hospital 3
Staff levels needed to achieve targets
Staff scheduling issues
Future workload levels
General resistance to change
Now asked us to go back, recalibrate and recalculate using latest flow data

36. References
Fletcher A, Halsall D, Huxham S and Worthington D, 2007, ‘The DH Accident and Emergency Department model: a national generic model used locally’, Journal of the Operational Research Society, vol 58, pp1554 – 1562.
Fletcher A and Worthington D, 2009, ‘What is a ‘generic’ hospital model? – A comparison of ‘generic’ and ‘specific’ hospital models of emergency flow patients’, Health Care Management Science, vol 12(4), pp
Izady N and Worthington D, 2012, ‘Setting Staffing Requirements for Time Dependent Queueing Networks: The Case of Accident and Emergency Departments’, European Journal of Operational Research, vol 219(3), pp
Izady N and Worthington D, 2011, ‘Approximate Analysis of Non- stationary Loss Queues and Networks of Loss Queues with General Service Time Distributions’, European Journal of Operational Research, vol 213(3), pp