Stochastic modeling for clinical scheduling
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Stochastic Modeling for Clinical Scheduling. by Ji Lin Reference: Muthuraman, K., and Lawley, M. A Stochastic Overbooking Model for Outpatient Clinical Scheduling with No-shows, submitted. Outline. Introduction to Clinical Scheduling Probability model Different policies

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Stochastic Modeling for Clinical Scheduling

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Stochastic Modeling for Clinical Scheduling


Ji Lin

Reference: Muthuraman, K., and Lawley, M. A Stochastic Overbooking Model for Outpatient Clinical Scheduling with No-shows, submitted


  • Introduction to Clinical Scheduling

  • Probability model

  • Different policies

  • Results and discussions

  • Recent work

Traditional appointment scheduling vs. Open access scheduling

  • Traditional appointment scheduling

  • - A patient is scheduled for a future appointment time

  • - lead time can be very long

  • - In some clinics, up to 42% of scheduled patients fail to show up for pre-booked appointments

  • Open access scheduling

  • - Patients get an appointment time within a day or two of their call in.

  • - see doctor soon when needed

  • - More reliable no-show predictions

Overbooking strategy

  • Airline industry

    • Fixed cost, capacity limits and fares on different class seats,

    • A low marginal cost of carrying additional passengers.

    • Either reserves or refuses a passenger.

    • System dynamics keeps the same for overshow situations (financial penalty)

Overbooking strategy 2

  • Clinical scheduling

    • Stochastic patient waiting time and staff overtime

    • The scheduler must search for an optimal appointment time

    • System dynamics changes (longer patient waiting times and excessive workload)

Model and Assumptions

  • Single server

  • A single service period is partitioned into time slots of equal length.

  • Patients call-in before the first slot

  • Once an appointment is made, it cannot be changed.

  • Patients have no show probabilities and are independent from each other

  • All arrived patients need to be served.

  • Service times are exponentially distributed

Call-in Procedure

Choose a slot or refuse to schedule

No Show Estimation


Service system

  • Xi - The number of patients arriving for slot i

  • Yi - The number of patients overflowing from slot i into slot i+1

  • Li - The number of services that would have been completed provided the queue does not empty

  • min(Li,Yi−1+Xi) - The actual number of services completed.


  • Minimize

    • Patient waiting times

    • Stuff overtime

  • Maximize

    • Resource Utilization

Weighted Profit Function

  • r – reward for each patient served

  • ci – cost for over flow from slot i to slot i+1

  • Q – arrival probability matrix

  • R – over-flow probability matrix

Attributes of this Appointment Scheduling

  • Static - Appointments made before the start of a session

  • Performance measure - Time based

  • Multiple block/Fixed-interval

  • Analytical Probability Modeling

Scheduling policies

  • Round Robin

  • Myopic Optimal policy

  • Non Myopic Optimal policy

Round Robin

  • assigns the ith customer to slot ((i−1) mod 8)+1.

Myopic policy


  • Call-in process simulation


  • Scheduled service simulation

Results: The schedule and expected profit evolution

Expected overflow from last slot

Effect of Call-in Sequence


  • Myopic policy improved the max profit by approx. 30% (compare with Round Robin)

  • Myopic policy is not optimal, but it provides solutions within a few percent of the optimal sequential

  • The probability model is readily extendable easily.

    • Patient type need not to be finite.

    • Walk-in can be added into the model (only Q matrix will change)

    • The restriction of exponential service time can be eliminate by conditioning our expectation.

Theory vs. Practice

  • Huge gap - Real clinic is much more complicated

    • More than one server

    • Registration, pre-exam, checkout, etc.

    • Physician's Restrictions

  • Probability model vs. simulation

    • The relaxed exponential service time within slots

  • Robustness of the policies

Recent extend on optimal policy – Dynamic Programming approach

Profit Function

  • Profit function is determined by current status and current time.

Example of 2 patients and 2 call-in time periods


  • Optimal Policy is not stationary

  • For M call-in time periods and N Slots, There are final statuses

  • When M>>N, the Complexity is closed to (M+N)!, which is NP-hard, and not computable for large cases.

Thank you!!


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