Delays and Waiting in Healthcare

Queueing Systems in Healthcare• Many healthcare related systems have important queueing

subsystems that must be managed– ED and OB are important customer entry and contact points for hospitals– call centers such as centralized appt scheduling, Dial-a-Nurse, main

hospital operators, physician referral are all important customer contact points

– access to clinic appointments, surgical schedules, therapeutic and diagnostic equipment is important dimension of patient satisfaction

– turnaround times of ancillary services such as lab, pharmacy, radiology, transcription can affect inpatient length of stay and outpatient satisfaction

– cost of capacity in terms of staff must be minimized while still meeting service level targets related to waiting

• Institute of Medicine in “Crossing the Quality Chasm” has identified “timeliness” as a major area for improvement in hour healthcare system

Wait Register Complete HHQ WaitStart/ntrVitals/

AssessmentWait

ProviderContactExam

WaitDiagnostic/Intervention

WaitProviderContact/Results

Wait Discharge

CollectionsMCHC

PharmacyWait

Leave

OutsidePharmacy

Wait

Start/Enter

Finish

An Urgent Care Clinic

Patients visit a series of queueing systems in series

The Registration Queue

Wait RegisterStart/Enter

• Random arrivals

• Average arrival rate depends on time of day and day of week

• # of registration staff varies by TOD/DOW

• The amount of time it takes to register varies from patient to patient

• Patient waits for next available registration staff

• Long delays may cause patient to “balk” or “renege”

• FCFS and/or priority queueing discipline

• Wait times play major role as customer dissatisfier

BalkRenege

Essential Features of Queuing Systems

DepartureQueue

discipline

Arrival process

Queueconfiguration

ServiceProcess

& capacity

Renege

Balk

Callingpopulation

No futureneed for service

0

10

20

30

40

50

15 30 45 60 75 90

Upper end of category

Freq

uenc

y

0

10

20

30

40

50

60

15 30 45 60 75 90

Upper end of category

Freq

uenc

y

Interarrival time distribution Service time distribution

Important Definitions and Relationships

• a=avg arrival rate (customers/hour)• b=avg service time (hours/customer)• c=# of servers

= avg server utilization

1ab

c for queue to be stable

Coefficient of Variation (C)• Coefficient of variation applies to probability

distributions and gives a sense of the magnitude of variability in the distribution

• It’s just the ratio of the standard deviation and the mean

C

Distribution Mean (m) Std Dev (s) CConstant 0 0Exponential 1Normal 10 2 0.2

Corrupting Influence of Variability

• Queues form due to variability in– time between arrivals– duration of service process– along with lack of synchronization between arrivals and service

• Queues also form due to highly utilized capacity subject to random demands for service

• Reducing variability in arrival and/or service process tends to improve performance.

• Since service cannot be provided from “stock”, safety capacity must be provided to cover for variability.

• Tradeoff is between cost of waiting, lost revenue, and cost of capacity.

• Pooling servers improves performance.– large pools of servers (staff, beds, etc.) can run at higher utilization levels than smaller

pools for the SAME level of customer service

• Subsystems with a queueing component must be treated appropriately in the broader context of staffing

– i.e. you can’t just add up the average work and divide by the available staff hours

(1) WhyQueue.xls(2) SimpleClinic.igx

Many Managerial Responsibilities and Levers

• The Input Stream (IHI Concepts 12-20, MBPF Ch 8, 10)– predicting and shaping demand

• The Waiting Experience (MBPF Ch 8)– where, what happens, how long, value?– Psychology of waiting

• The Service Experience (IHI Concepts 1-11)– designing processes and systems

• The Capacity (IHI Concepts 21-27, MBPF Ch 8)– matching capacity to demand

• Overall System Performance (IHI, MBPF Ch 8)– cost– customer wait, satisfaction, and outcomes

IHI = Reducing Delays and Waiting Times book, MBPF = Anupindi book

Many Challenges in Managing Queueing Systems in Healthcare

• The Input Stream (IHI Concepts 12-20, MBPF Ch 8)

– often demand difficult to predict or to influence– different urgency levels of demand

• The Waiting Experience (MBPF Ch 8)

– waits in healthcare are rampant, patients compare waits when possible, waiting areas often unpleasant, lost demand, suboptimal care

• The Service Experience (IHI Concepts 1-11)

– patient participates in the process– complex technology and highly variable processes– potential for tragic consequences

• The Capacity (IHI Concepts 21-27, MBPF Ch 8)

– TOD/DOW fluctuations in demand make matching capacity difficult– often labor is specialized, expensive and highly skilled– cost of insufficient capacity can be very high

• Overall System Performance (IHI, FF MBPF Ch 8)

– difficult tradeoffs between capacity cost, patient wait and satisfaction, and patient outcomes

– waits and delays are often highly visible to patients, staff, the public

Queueing Models• Given assumptions about system inputs

– arrival patterns (distribution of time between arrivals)– service time distribution– number of servers (beds, staff, machines)– service discipline (FCFS, priority)

• Mathematical models that allow us to predict system performance measures such as:– probability of waiting to be served– average time spent waiting– server (e.g. bed or staff) utilization

• Unlike simple Poisson occupancy model, queueing models let us model explicit consequences of not having enough capacity

• Some queueing models are “simple”, others are horribly complex

The Single Server Queue

Customers in Queue

Server

Applications?

M/M/1 queueing system

Example_7-11.xls

Elements of Queueing Systems• Arrival processes

– interarrival time distribution• fixed – appointment like• exponential – random arrivals

– single vs batch arrivals• patients• patients with families (waiting room sizing)

– single or multiple classes of customers• patient types, acuity levels, demand types

• Service Process– service time distribution– how many servers?

• staffing level

0

10

20

30

40

50

60

15 30 45 60 75 90

Upper end of category

Freq

uenc

y

0

10

20

30

40

50

15 30 45 60 75 90

Upper end of category

Freq

uenc

y

Elements of Queueing Systems• Service Discipline

– first come first served (FIFO)– last come first served (LIFO)– priority service (triage)– served in random order – balking, reneging, jockeying (real life)

• Service and Queue configuration– single stage– queues in parallel– queues in series – queueing network

patient works his/her waythrough various ancillarydepartments

Ironic email from some hospitalReceived Days Before Session on Queueing Models

How are you??? Life here at Hospital X is OK. We have had a few changes within the department and a few leadership changes in the System recently...  BLAH BLAH … Personal update … BLAH BLAH I actually do have a work related question for you (hopefully you don't mind!!)...I assume you recall the phone model you completed for us...is it possible to use that same model (either as is or with a few tweaks) to look at CSR staffing at the front desks? A co-worker and I are looking for a way to get decent numbers for both the phone rooms and the front desk personnel and believe this kind of a model (we don't know if this exact one would work) may give us our best answer. What do you think??

A Capacity Planning Workhorse: The M/M/c queue “How Many Beds?” – Green (2003)

Servers (Beds)Patients in Queue

The M/M/c/infinity Model

randomarrivals

LOS assumedto be exponentially

distributed

cbeds

unlimitedwaitingspace

Q’ng model shorthand

Random Arrivals

arrivals / service time / # servers / # servers + Q size

Sidebar: Data

collection in clinics

Excel based manual logs

Infrared tracking system

Process Time Distributions in a Primary Care Clinic

Time In Vital Signs Area

711

842

1361

1604

834

380

235

102 73 48135

0

200

400

600

800

1000

1200

1400

1600

1800

0 to

0.9

1 to

1.9

2 to

2.9

3 to

3.9

4 to

4.9

5 to

5.9

6 to

6.9

7 to

7.9

8 to

8.9

9 to

9.9

10+

Duration in Minutes

Nu

mb

er o

f P

atie

nts

Exam Duration

685

898

569

267

9649 21 14 7 6 2 1 4

0

100

200

300

400

500

600

700

800

900

1000

0 to

4.9

5 to

9.9

10 to

14.

9

15 to

19.

9

20 to

24.

9

25 to

29.

9

30 to

34.

9

35 to

39.

9

40 to

44.

9

45 to

49.

9

50 to

54.

9

55 to

59.

9

60+

Duration in Minutes

Nu

mb

er o

f P

atie

nts

Provider Documentation Time

605

296

168 157113

77 67 56 43 31 28 32 18 20 10 13 5 8 5 544

0

100

200

300

400

500

600

700

0 to

0.9

1 to

1.9

2 to

2.9

3 to

3.9

4 to

4.9

5 to

5.9

6 to

6.9

7 to

7.9

8 to

8.9

9 to

9.9

10 to

10.

9

11 to

11.

9

12 to

12.

9

13 to

13.

9

14 to

14.

9

15 to

15.

9

16 to

16.

9

17 to

17.

9

18 to

18.

9

19 to

19.

9

20+

Duration in Minutes

Nu

mb

er o

f P

atie

nts

Post Exam Wait Time

481

319

388

450

369

260

191

141107

78 8054 47 41 28 25 24 15 7 8

50

0

50

100

150

200

250

300

350

400

450

500

0 to

4.9

9

5 to

9.9

9

10 to

14.

9

15 to

19.

9

20 to

24.

9

25 to

29.

9

30 to

34.

9

35 to

39.

9

40 to

44.

9

45 to

49.

9

50 to

54.

9

55 to

59.

9

60 to

64.

9

65 to

69.

9

70 to

74.

9

75 to

79.

9

80 to

84.

9

85 to

89.

9

90 to

94.

9

95 to

99.

9

100+

Duration in Minutes

Nu

mb

er o

f P

atie

nts

Ward, T.J., Isken, M.W., and Minds, D. (2003) Automated Data Collection in a Primary Care Clinic , INFORMS Annual Conference, Atlanta, GA.

M/M/c Basics - MMs-Template-HCM540.xls

Parameter Units SymbolPatient Arrival Rate pats/min aAvg Registration time min/pat b# of Reg Staff staff c

Mathematical equations

(2) Queueing Model(s)(1) Inputs

(3) Outputs or Performance Measures Formula

Utilization = c

ab

Probability of Waiting in Queue (pn=probability of n patients in the

system) Dp =

1

0

1c

nnp

Expected Wait Time in Queue QW =

b

cpD

)1(

Probability of Waiting in Queue less than t mins

t

QWP = t

b

c

eD

p)1(

Call Center “What if” Examples1. Given a=40 calls/hr, b=15 mins/call and c=12

customer service representatives (CSR), what is the expected time customers will spend on hold (E[Wq]) ? What percentage will wait at all?

2. What is the % utilization for the servers?

3. If a increases to 45 calls/hr, how will E[Wq] and the percentage that wait change?

4. If a increases to 45 calls/hr but we can decrease b to 12 mins/call, how does E[Wq] change?

5. For a=45 and b=12, how much can we reduce c (# of staff) down to before E[Wq] exceeds 5 minutes?

MMS-Template-HCM540.xls

PhoneModel-HCM540.xls

Based on actual model used in practice. Day is

divided up into hours and arrival rates and staffing can differ by

hour.

A simple template for exploring various wait time

performance measures for a given arrival rate, mean

service time and # of servers.

Relationship Between Utilization, Capacity and Overflow Probability or Wait Time in Queue

• Economies of scale – larger server pools can run at higher utilizations for given service level

• Non-linear increase in overflow or waiting as capacity utilization approaches 100%

• Decreasing marginal return of adding capacity

Utilization and Economies of ScaleAvg Wait in Queue - Numer of Server Effect

0.00

20.00

40.00

60.00

80.00

100.00

120.00

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 0.99

Server Utilization

Avg

Tim

e i

n Q

ueu

e

1

2

3

4

5

10

50

100

Number of ServersNotice that for the same utilization, more servers leads to a smaller average number in queue. This is an example of risk, or variance, pooling. Large server pools can operate at very high utilization levels while still meeting service level objectives. Small pools must operate at lower utilization levels to meet similar service level goals. This is why organizations create large centralized call centers.

Decreasing Marginal Improvements of Additional Capacity

121314151617

Avg Wait Time in Queue(a=4.5 customers per hour, b=1 hour)

0.0000

0.2000

0.4000

0.6000

0.8000

1.0000

1.2000

1.4000

1.6000

1.8000

5 6 7 8 9 10 11 12 13 14 15 16 17

Number of Staff

Av

g T

ime

in

Qu

eu

e (

ho

urs

)

Avg Wait

When adding servers to a system (while holding arrival rate and service time fixed) you get decreasing absolute returns. You get a big improvement with the addition of the first server, and each additional server buys you an ever decreasing level of improvement. Obviously, you eventually drive the average wait time in queue to 0 and no more improvement is possible.

Queuing can fool your intuitionState Unemployment Office - Processing application forms for new companies

InputsBase worker processing rate 4 forms per week

1992 1993 % IncreaseForm arrival rate (per week) 1.80 3.90 117%Average Delay (weeks) 0.45 5.00 1011%

OutputsWorker Utilization 45% 98%

Predicted average forms in queue 0.37 38.03Predicted average time in queue (weeks) 0.20 9.75Predicted average time in system (weeks) 0.45 10.00

Implied worker processing rate 4.00 4.10Delay at implied rate 0.45 5.00

Mark Isken:YOWZA, asking for trouble!

Mark Isken:So, the worker must have actually been working harder than before to keep the backlog at 5! See D19 for actual implied service rate.

The employee was fired due to this increase in delay time. He sued and court ruled company was justified since demand doubled and so, the delay time should have only doubled as well.

IHI Change Concepts: Shaping Demand(see Breakthrough Guide for details)

• Eliminate things that aren’t used– Standard drug formularies

• Insert an “informative delay”– Patient education during waits

• Combine services– Group appointments

• Standardize and automate– Telephone or internet based FAQs

• Triage– “express” system within ED for simple problems

IHI Change Concepts: Shaping Demand

• Extinguish demand for ineffective care– Evidence based medicine

• Relocate the demand– Immunizations at school

• Anticipate demand– Planning for post-discharge care

• Promote self-care– Diagnostic testing at home

IHI Change Concepts: Matching Capacity to Demand

• Improve predictions– Analysis of historical

data (e.g. Hillmaker)– explanatory models

• Smooth the work flow– Appointment scheduling– inform patients of

current wait times

• Adjust to peak demand– Flexible staff scheduling – “open access” in clinics

• Identify and manage the constraint– Provider, support staff, or

exam rooms? • Work down the backlog

– Preparing for “open access” by increasing capacity in the short term

• Balance centralized and decentralized capacity– Staffing pools

• Use contingency plans– What can be done to cope

with short term demand spikes?

Staffing a Centralized Appointment Scheduling System in Lourdes Hospital

• Very nice application of a simple queueing model to appt center staffing

• Advantages of centralized scheduling?• Service dissatisfiers? Impacts?• Prior emphasis on “high staff utilization” was the wrong goal• Well accepted approach of using M/M/c queueing model with time

of day specific arrival rates– found service time were NOT exponential but that M/M/c worked very

well anyway (insensitive to actual distribution of call time)

• Created staffing tables to facilitate managerial use (see Table 2)• Used heuristic (common sense and trial and error) approach to

adjust staff schedules to implement new staffing patterns with no staff adds

Interfaces 21:5 Sept-Oct 1991 (pp. 1-11) See WebCT