Discrete Event vs. DiffEq Models

["Timothy Quinn"]

Here is the problem that got me into the field of System Dynamics. A common measure of outpatient clinic access in the healthcare industry is Time to Third Available Appointment; that is, how many days from today--when you call your doctor to schedule an appointment--is the third open appointment slot on your doctors calendar. Poorly run systems have TTAA in months, good ones have TTAA in days. Clinics want to know what their appointment template should look like, given the acutity profile of their physician panels (stocks of patients that each doctor takes care of). Sicker patients require more time (longer service time in the sysem), as do new patients (because an H&P must be taken for the first time). Emergent cases need to be seen immediately, usually by overbooking appointment slots at the last minute (like the airlines do). My discrete event model took a schedule template as input, replicated every week for as long as the simulation time horizon. Patients would arrive according to a Poisson process. Each patient was randomly assigned an acuity level, based on the functional health status distribution of the panel, which was translated into how many 15-minute blocks of service time were required for each patient. I assigned patients to appointment slots with different policies in two sets of simulation runs: (1) give each patient the first available appointment slot of appropriate length, or (2) give each patient the first available appointment slot of appropriate length after a stochastically determined (lognormal distribution) number of days (assuming that patients like to schedule in advance). Emergent cases were overbooked, but no more than 2 patients could be overbooked in the same slot(s). If an appointment time was not available before a certain threshold, the patient balks and goes elsewhere. Using the existing template, the clinic was interested in what the TTAA graph looked like over time. In particular, how long before most patients are balking and going elsewhere? Next, the clinic wanted to know if certain appointment slots should be reserved for emergent cases. Here is the challenge: Im not sure how one would model assigning patients to appointment slots given that the time order and vacancy of those slots is important for calculating a running measure of TTAA. Thanks, Tim Quinn System Dynamics Group Massachusetts Institute of Technology Sloan School of Management 30 Wadsworth Street Bldg E53, Rm 358A Cambridge, MA 02142 Telephone: 617-258-5585 Email: tdquinn@mit.edu

[=?iso-8859-1?Q?Jean-Jacques_Laub]

Hi Timothy I did not understand if you did model correctly your problem, and what method or software you used, or what method is supposed to be used if you did not . I have the same kind of problem than yours. I have a rental car company, and I try to model the way people reserve cars, depending on the availability of cars, the need of the client and of course the price given etc... I am still not sure if it is a continuous SD method or a discrete method that will do the job better Regards. From: =?iso-8859-1?Q?Jean-Jacques_Laubl=E9?= <JEAN-JACQUES.LAUBLE@WANADOO.FR>

["Keith Linard"]

This problem is virtually identical to the one I routinely set my 3rd year Civil Engineering students (as part of a 27 hr course - about 10% of semester workload). I am happy to make the models, built in Powersim, available on request. NOTE: This model combines low level mechanistic control engineering elements as well as high level policy feedback levers. My aim was to make SDM seem relevant to hard systems civil engineers. It worked. The student problem related to perceived productivity shortfalls in a Concrete Batching Plant. Key actors included shareholders, management, competing unions and workers (clerical, batching plant, truck drivers). Student tasks included: 1. Cognitive mapping / group decision exercise: OBJECTIVE: to understand the soft system: worldview(s), owners, clients, actors, transformation processes and environmental constraints; system boundaries; decision levers within control of owners / management. 2. Build System Dynamics Flight Simulator OBJECTIVE: Understand stock-flow dynamics ; understand system interrelationships; create a flight simulator which illustrated the consequences of feedback effects of key decision levers; learn the power and flexibility of the SD software (Powersim). 3. Components of the System Dynamics Model (The bare bones - ignoring the key policy feedback levers). Time step: 1 minute over 1 week operation. a. Build a minute-hour-day clock which controlled client order times, batch plant working hours and driver working hours. b. Build a log-normal probability module to drive order inter-arrival times. c. Build random probability modules to drive order size and delivery time-distance (both based on data) d. Build the order arrival module, office-telephonist module and transformation of potential order into logged order taking into account telephonist availability with 1 or 2 staff (key decision variable: orders lost if staff on rostered breaks or unscheduled toilet breaks). e. Build batch plant module: plant operates if: staff available & plant free & empty truck available. Plant takes variable time to fill order. f. Build trip module: trucks change state between available, at Batch Plant or En Route. g. Build Loose Customer Module: If order not acted on (dispatched to Batch Plant) within 1 hour, loose order. h. Build Order prioritising module: Orders placed previous day get first priority next day. Whats lacking in this compared with the challenge. 1. No queue jumping by urgent cases: But the order prioritising module is entirely flexible (based on manipulating the index number of an order in a time queue) and could readily be used to ramp up priority of randomly assigned high value customers. 2. Time slot reservation: But dead easy to implement given the above structures. 3. The model is essentially continuous (in that orders are undifferentiated in the various order-states), except for unactioned orders which go into a 60 level array to determine lost orders at the end of 1 hour; and priority queue for next day actioning. Models made available on the understanding that they are not fully bug-proofed (after all theyre only Civil Engineering undergraduates; it was their lowest credit unit; and it wasnt perceived as real engineering - especially by my academic colleagues). Paper to Engineering Education Coference on this course is also available. Keith Linard Visiting Fellow University of New South Wales (Australian Defence Force Academy) Phone: -61 (0)3-9747-6682 Fax: -61 (0)3-9747-6697 Mobile: 0412-376-317 Email: k.linard@adfa.edu.au Home Address: 5 Blackwood Drive Melton South Vic 3338

["geoff coyle"]

1 This is a common class of problem that arises in many domains - the workload in a machine shop is but one of many cases. It is usually addressed, as you have done, by discrete event simulation (DES) using any one of a number of DES languages. (CSL was a good one but Im not sure what is now available). The usual output is a distribution of such variables as queue length at a work station, time to go through the system etc. The characteristics of that distribution, such as its mean, mode, variance, probability that a given value will be exceeded, and its shape are indicators of the relative merits of different operating rules. For example, in a machine shop (which is a case I have dealt with) is it better to take the largest job first, the smallest first, the one that has waited longest, or whatever? The other input is, of course, the numbers of machines at each manufacturing stage, size of storage areas etc. All these can be taken into the simulation to see that happens to the distribution. It would be unusual to be interested in the time profile of, say, queue length; what normally matters is the nature of the performance over some period of time, as measured by the distribution. For your TTTA problem, you seem to be on the right track with distributions of patient needs and arrivals and your two decision rules on patient allocations. Other factors are, of course, the numbers of doctors (and nurses) available, the size of the facility in terms of consulting rooms and beds, the capacity of the support services such as patient records and no doubt other factors. You could simulate this over a year, say, to allow for winter epidemics and so forth. You could even separate out the different seasons and look at how the numbers of available resources might need to vary over the year. Im not at all sure that SD is an appropriate methodology for this class of problem and Id be interested in knowing why you think that it is. Of course, SD is great for all sorts of problems and it is not hard to represent discrete events within an SD model, when it is appropriate to do so (see the literature) but SD does not do everything and I always urge students (force is closer to it) to justify the choice of a methodology with respect to the type of problem being addressed. I hope that helps. Regards, Geoff From: "geoff coyle" <geoff.coyle@btinternet.com> Professor Geoff Coyle Visiting Professor of Strategic Analysis, University of Bath. Telephone 44 (0) 1793 782817 Fax 44 (0) 1793 783188

[Bill Braun]

5 >My discrete event model took a schedule template as input, replicated every >week for as long as the simulation time horizon. Patients would arrive >according to a Poisson process. Tim, Do you mean the patients actually arrive at the clinic according to a Poisson distribution, or their request for service and their respective severity of illness (which, I assume, correlates to length of service requested) is according to a Poisson distribution, and appointments are schedule accordingly? If it is the first, are you assuming that the clinic does not require an appointment in order to request service at the clinic? Bill Braun From: Bill Braun <medprac@hlthsys.com>