An Accident and Emergency Case Study Dave Worthington (Adrian Fletcher and Navid Izady) Management Science Dept. Lancaster University d.worthington@lancaster.ac.uk

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

A time-dependent queueing network:

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 ………

e.g. Typical A&E arrival patterns

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

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

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

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?

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  

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.

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

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

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)

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 10-15 simulation experiments. It does not guarantee that the proposed staffing is the minimum possible, but experiments show significant improvements against existing staffing levels.

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.

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.

An Example of Algorithm Optimal Staffing

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

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

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

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.

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

Hospital 3

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

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%

Hypothesis RIGHT number of staff at WRONG times ?

Arrival Patterns

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

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

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

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

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

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), pp374-391 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), pp531- 540. 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), pp498-508..