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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

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queueing systems 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
An Urgent Care Clinic





Complete HHQ


























Patients visit a series of queueing systems in series

the registration queue
The Registration Queue






  • Random arrivals
  • Average arrival rate depends on time of day and day of week
  • 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
  • # of registration staff varies by TOD/DOW
  • The amount of time it takes to register varies from patient to patient
essential features of queuing systems







Process & capacity





No future

need for



Essential Features of Queuing Systems

Service time distribution

Interarrival time distribution

important definitions and relationships
Important Definitions and Relationships
  • a=avg arrival rate (customers/hour)
  • b=avg service time (hours/customer)
  • c=# of servers

r = avg server utilization

for queue to be stable

coefficient of variation c
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
corrupting influence of variability
Corrupting Influence of Variability

(1) WhyQueue.xls

(2) SimpleClinic.igx

  • 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
many managerial responsibilities and levers
Many Managerial Responsibilities and Levers

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

  • 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
many challenges in managing queueing systems in healthcare
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
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
The Single Server Queue

M/M/1 queueing system

Customers in Queue




elements of queueing systems
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
elements of queueing systems1
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 way

through various ancillary


ironic email from some hospital received days before session on queueing models
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 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
A Capacity Planning Workhorse: The M/M/c queue“How Many Beds?” – Green (2003)

Servers (Beds)

Patients in Queue

Random Arrivals

Q’ng model shorthand

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

The M/M/c/infinity Model






LOS assumed

to be exponentially




process time distributions in a primary care clinic
Process Time Distributions in a Primary Care Clinic

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

(1) Inputs

(2) Queueing Model(s)

Mathematical equations

call center what if examples
Call Center “What if” Examples
  • 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?
  • What is the % utilization for the servers?
  • If a increases to 45 calls/hr, how will E[Wq] and the percentage that waitchange?
  • If a increases to 45 calls/hr but we can decrease b to 12 mins/call, how does E[Wq] change?
  • For a=45 and b=12, how much can we reduce c (# of staff) down to before E[Wq] exceeds 5 minutes?

A simple template for exploring various wait time performance measures for a given arrival rate, mean service time and # of servers.


Based on actual model used in practice. Day is divided up into hours and arrival rates and staffing can differ by hour.

relationship between utilization capacity and overflow probability or wait time in queue
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
ihi change concepts shaping demand see breakthrough guide for details
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
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
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
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

See WebCT

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