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HCM Queueing

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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
   – 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
 Start/Enter                                                                                              Vitals/
Start/ntr               Wait                Register          Complete HHQ        Wait                                    Wait
                                                                                                        Assessment




          Provider                                                                Provider
                                               Diagnostic/
          Contact              Wait                                Wait           Contact/              Wait         Discharge
                                               Intervention
           Exam                                                                   Results




                                  MCHC                                                        Outside
         Collections                                   Wait
                                 Pharmacy                                                    Pharmacy


                                                                          Leave                                                  Finish




                     Patients visit a series of queueing
                             systems in series
          The Registration Queue
                                Renege
                 Balk


 Start/Enter                Wait                      Register


• Random           •    Patient waits for next   • # of registration
  arrivals              available registration     staff varies by
                        staff                      TOD/DOW
• Average          •    Long delays may
                                                 • The amount of
  arrival rate          cause patient to
                                                   time it takes to
  depends on            “balk” or “renege”
                                                   register varies
                   •    FCFS and/or priority
  time of day           queueing discipline
                                                   from patient to
  and day of                                       patient
                   •    Wait times play
  week                  major role as
                        customer dissatisfier
                                           Essential Features of Queuing
                                                      Systems
                                                                  Renege

                                                     Arrival                      Queue
                                                     process                                                        Service                    Departure
   Calling                                                          Queue        discipline                        Process &
   population                                                    configuration                                      capacity

                                                          Balk                                                                                        No future
                                                                                                                                                      need for
Interarrival time distribution                                                   Service time distribution
                                                                                                   60
                                                                                                                                                       service
                                                                                                   50




                                                                                       Frequency
              50                                                                                   40
              40                                                                                   30
Fre que ncy




                                                                                                   20
              30
                                                                                                   10
              20
                                                                                                   0
              10                                                                                        15   30     45     60     75      90
              0                                                                                                   Upper end of category
                   15   30     45     60     75      90
                             Upper end of category
Important Definitions and Relationships

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


       ab
    r    1                for queue to be stable

        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
                         
                      C
                         
  Distribution   Mean (m) Std Dev (s)       C
  Constant                    0            0
  Exponential                             1
  Normal            10         2           0.2
    Corrupting Influence of Variability
•    Queues form due to variability in                    (1) WhyQueue.xls
      – time between arrivals                            (2) SimpleClinic.igx
      – 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
 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
•   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
                        M/M/1 queueing system
 Customers in Queue




                       Server




Applications?         Example_7-11.xls
 Elements of Queueing Systems
• Arrival processes                                     50
                                                        40




                                          Fre que ncy
  – interarrival time distribution                      30
                                                        20

     • fixed – appointment like                         10


     • exponential – random arrivals
                                                        0
                                                             15   30     45     60     75      90

  – single vs batch arrivals                                           Upper end of category



     • patients
     • patients with families (waiting room sizing)
  – single or multiple classes of customers
     • patient types, acuity levels, demand types
                                                        60
                                                        50

• Service Process
                                          Frequency
                                                        40
                                                        30

  – service time distribution                           20
                                                        10


  – how many servers?
                                                         0
                                                             15   30     45     60     75      90


     • staffing level
                                                                       Upper end of category
 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        patient works his/her way
  –   queueing network        through various ancillary
                              departments
 Ironic email from some hospital
Received 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


Random
Arrivals


                Q’ng model shorthand
 arrivals / service time / # servers / # servers + Q size
         The M/M/c/infinity Model

                                               unlimited
   random      LOS assumed
                                       c        waiting
   arrivals to be exponentially
                                      beds       space
                 distributed
       Excel based
       manual logs




Sidebar:
  Data
collection
in clinics
    Infrared tracking
         system
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                                                                                                        Duration in Minutes
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                                                                                                                                                                                                                                                                                                                                                                                                                             Time In Vital Signs Area




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Clinic , INFORMS Annual Conference, Atlanta, GA.
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Ward, T.J., Isken, M.W., and Minds, D. (2003) Automated Data Collection in a Primary Care
    M/M/c Basics - MMs-Template-HCM540.xls
                (1) Inputs
Parameter                    Units        Symbol            (2) Queueing Model(s)
Patient Arrival Rate        pats/min        a                    Mathematical
Avg Registration time       min/pat         b                     equations
# of Reg Staff                staff         c
        (3) Outputs or Performance
        Measures                                                Formula
                                                                                ab
                      Utilization                     r       =                c
             Probability of Waiting in Queue                                     c 1
            (pn=probability of n patients in the       pD       =            1   pn
                                                                                 n 0
                         system)
                                                                                pD
         Expected Wait Time in Queue                  W         =                       c
                                                          Q                  (1  r )
                                                                                        b
            Probability of Waiting in                                                     c
                                                   P W  t    =                (1  r ) t
             Queue less than t mins                   Q
                                                           
                                                                    p       e           b
                                                                         D
Call Center “What if” Examples
1. 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

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



                                             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.
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 Scale
                                                                          Avg Wait in Queue - Numer of Server Effect

                    120.00




                    100.00



                                            Notice that for the same utilization, more servers leads to a smaller
                                                                                                                                                          Number of Servers
                                            average number in queue. This is an example of risk, or variance,
                     80.00                  pooling. Large server pools can operate at very high utilization levels                                             1
                                            while still meeting service level objectives. Small pools must operate at
                                                                                                                                                                2
                                            lower utilization levels to meet similar service level goals. This is why
Avg Time in Queue




                                            organizations create large centralized call centers.                                                                3
                                                                                                                                                                4
                     60.00                                                                                                                                      5
                                                                                                                                                                10
                                                                                                                                                                50
                                                                                                                                                                100
                     40.00




                     20.00




                      0.00
                             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
Decreasing Marginal Improvements
      of Additional Capacity
                                                 Avg Wait Time in Queue
                                          (a=4.5 customers per hour, b=1 hour)
                                                  When adding servers to a system (while holding arrival rate and service
                             1.8000               time fixed) you get decreasing absolute returns. You get a big
                                                  improvement with the addition of the first server, and each additional
                             1.6000               server buys you an ever decreasing level of improvement. Obviously, you
 Avg Time in Queue (hours)




                                                  eventually drive the average wait time in queue to 0 and no more
                             1.4000
                                                  improvement is possible.
                             1.2000
                                                                                                                            12
                             1.0000                                                                                         13
                                                                                                  Avg Wait                  14
                             0.8000                                                                                         15
                                                                                                                            16
                             0.6000                                                                                         17

                             0.4000
                             0.2000
                             0.0000
                                      5   6   7   8   9   10 11 12 13 14 15 16 17
                                                      Number of Staff
              Queuing can fool your intuition
State Unemployment Office - Processing application forms for new companies


          Inputs                                                                              The employee was fired due
          Base worker processing rate                   4     forms per week                  to this increase in delay time.
                                                                                              He sued and court ruled
                                                                                              company was justified since
                                                      1992      1993     % Increase           demand doubled and so, the
          Form arrival rate (per week)                1.80      3.90        117%              delay time should have only
          Average Delay (weeks)                       0.45      5.00       1011%              doubled as well.



          Outputs                                                              Mark Isken:
                                                                               YOWZA, asking for trouble!
          Worker Utilization                          45%       98%

          Predicted average forms in queue            0.37      38.03
          Predicted average time in queue (weeks)     0.20      9.75
          Predicted average time in system (weeks)    0.45      10.00
                                                                             Mark Isken:
                                                                             So, the worker must have actually
          Implied worker processing rate              4.00      4.10         been working harder than before
          Delay at implied rate                       0.45      5.00         to keep the backlog at 5! See
                                                                             D19 for actual implied service
                                                                             rate.
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             • Identify and manage the
   – Analysis of historical         constraint
     data (e.g. Hillmaker)           – Provider, support staff, or
   – explanatory models                exam rooms?
                                  • Work down the backlog
• Smooth the work flow               – Preparing for “open access”
                                       by increasing capacity in the
   – Appointment scheduling            short term
   – inform patients of current   • Balance centralized and
     wait times                     decentralized capacity
                                     – Staffing pools
• Adjust to peak demand           • Use contingency plans
   – Flexible staff scheduling       – What can be done to cope
   – “open access” in clinics          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

				
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