1. Appointment Systems - a Stochastic and Fluid Approach Michal Penn The William Davidson Faculty of Industrial Engineering and Management Technion - Israel Institute of Technology Joint work with Yossi Luzon and Avishai Mandelbaum January 23, 2008

2. Appointment Systems Airline Services Airline Services Why do Appointment Systems exist? Health Care

3. Long service times Why do appointment systems exist? Economically efficient Uncertainty Quality of care

4. Macro View scheduling appointments service stochastic customers requests for service Customers departure tool: appointment book Waiting time at server

5. Stochastic arrivals

6. Difficulties Appointment system scheduling problems are: Large Dynamic Combinatorial Stochastic To overcome the problem’s complexity we suggest using fluid approximation. Novelty: using fluid approximation in the context of appointment systems.

7. General framework Appointment book Combinatorial scheduling problem Scheduling customers based on a.b. Service; Stochastic service time Imitation of the fluid solution Heterogeneous stochastic arrivals Deterministic Fluid approximation

8. General framework Appointment book Combinatorial scheduling problem Scheduling customers based on a.b. service Imitation of the fluid solution Heterogeneous stochastic arrivals Deterministic Fluid approximation Based on Expectation (ignore variance) Objective function Slots based on service times Aim: asymptotically optimal

9. Finite time horizon History repeats itself Days Weeks Months… 0T Cyclic nature of history + solution for finite time horizon solution to the problem Finite time horizon Periodicity of customers behavior

10. Single server – minimum waiting time Fluid model solved – rule Discrete problem NP hard

11. Two servers – minimum makespan Appointment books

12. Two servers – minimum makespan Fluid model solved - work conserving - Proportion devoted By to customer 2

13. Single Server – Minimum Waiting Time We solved this system and found optimal -s

14. Single Server – Minimum Waiting Time

15. Algorithm: General idea Fluid Based Dispatching Rule Assume F is a feasible solution for a given fluid appointment system with its given time dependent expected arrival rates. In the discrete appointment system, if server i is idle at time t and there is a customer type available, then assign the next slot to the customer type with the largest deviation from its fluid solution at time t.

16. Constructing Optimal Control Optimal control - What do we have so far? Single Server – Minimum Waiting Time Single Server – Minimum Waiting Time Tandem Network of Two Servers – Minimum Makespan Tandem Network of Two Servers – Minimum Makespan These are special cases of…

17. The General Network

18. The Fluid Control Optimization Problem (minimum makespan)

19. Aims: 1.Develop appointment books that are near optimal. 2.Prove theoretically the quality of our procedure. 3.Demonstrate by simulation the quality of our procedure.

20. Literature Review Appointment Systems – Related Work. Time Dependent Stochastic Networks and Fluid Control Scheduling via Fluid Approximations

21. 1. Appointment Systems – Related Work. Performance Analysis and Optimization  Bailey (1952), Jackson (1964) : Appointment intervals, worked on balancing a trade-off between server idle times and patient waiting times. Used simulation.  Peterson-Bertsimas-Odoni (1995) : Aircraft landings, used a Markov/semi-Markov model for the changes in weather. Computed moments of queues.  Bosch-Van den-Dietz-Simeoni (2000) : Outpatient systems, worked on minimizing operating costs of wait and overtime. Offered a scheduling algorithm, used submodularity.  Wang (1993) : AS of a single server, computed the expected customers delay time recursively, used stochastic decreasing convexity.  Patrick-Puterman-Queyranne (2007 under review) : Public health care, worked on dynamically scheduling multi-priority patients. Used MDPs to allocate available capacity to incoming demand so that waiting time targets are achieved.

22. 2.Time Dependent Stochastic Networks and Fluid Control Performance AnalysisPerformance Analysis  Approximations: Newell, Keller, Massey, Dai,...  strong approximations: Mandelbaum-Massey,...  alternating load: Harchol-Balter,... ControlControl  multi-class, static overload: Avram-Bertsimas-Ricard, Kelly, Weiss, …  multi-class, transient overload: Chang-Ayhan-Dai-Xia 3.Scheduling via Fluid Approximations Job ShopJob Shop  Makespan: Bertsimas-Gammarnik, Bertsimas-Sethurman, Boudoukh-Penn-Weiss, …  Holding cost: Bertsimas-Gammarnik-Sethurman,…

23. Single Server – Minimum Waiting Time