Model validation, 9,600 arrivals, two outpatient clinics

Mathematics research poster example

A mathematics research poster has to make an abstract object readable at two metres, which means stating the model as assumptions a passer by can check rather than as a page of notation. This board fits a standard M/M/c queue to 9,600 recorded clinic arrivals and puts the predicted waiting times next to the observed ones for every session of the clinic day.

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KU
Does a textbook queue model predict real clinic waits? An M/M/c fit to 9,600 arrivals
E. Varga ¹, T. Olamide ¹ ²
1 Department of Mathematics, Kelvedon University · 2 Health Operations Group, Kelvedon University
ABSTRACT
Waiting time models are taught with the M/M/c queue and then used in hospitals as though the assumptions held. This study fits the standard model to one outpatient service and compares what it predicts against what patients actually waited. Arrival and service times came from the booking system of two general outpatient clinics over 30 opening days, 10 hours a day, which gives 9,600 recorded arrivals and 9,600 measured waits. Arrivals ran from 11.4 an hour at opening to 21.8 in the late afternoon. Mean consultation time was 10.0 minutes, so the fitted service rate was 6.0 patients an hour for each rostered clinician, with 3 to 5 rostered by hour. Predicted mean wait sat within 2.3 minutes of the observed mean in 9 of the 10 hours of the clinic day, and the mean absolute error across the whole day was 1.1 minutes.
INTRODUCTION
The M/M/c queue rests on three statements, and each one is a claim about a real clinic that can be checked rather than assumed. First, arrivals are independent and their count in any hour follows a Poisson distribution, which is another way of saying that one patient walking in tells you nothing about when the next one will. Second, the time a clinician spends with a patient is exponential with a single rate, so short consultations are common, long ones are rare, and the average is the only parameter the model needs. Third, the c clinicians are interchangeable and take the next patient in order, with no reserved slot, no triage jump and no patient who leaves before being seen. None of the three is exactly true in an outpatient clinic. Booked appointments make arrivals more regular than Poisson, consultation length has a floor of about four minutes that an exponential distribution does not, and a nurse can and does move an unwell patient forward. The question is not whether the assumptions hold, because they do not. It is whether the wait the model predicts is close enough to the wait patients get that a roster can be planned with it.
METHODS
Setting. Two general outpatient clinics in one health service, both open 08:00 to 18:00 on weekdays, both using the same booking and check in system, observed over 30 opening days in one calendar quarter.
Data. 9,600 arrivals with a check in timestamp, a called timestamp and a consultation end timestamp. The wait is the gap between check in and called, so a patient who arrived early waits from the time they arrived.
Fitted rates. Arrival rate estimated for each hour of the day as the mean count over the 30 days, from 11.4 an hour at 08:00 to 21.8 an hour at 16:00. Service rate fitted from 9,600 consultations at 6.0 an hour per clinician.
Servers. The number of rostered clinicians was read from the roster file for each hour, not assumed. It runs from 3 in the first and last hour to 5 across the middle of the day, and it changes on the hour.
Comparison. Predicted mean wait from the M/M/c formula for each hour against the observed mean wait for the same hour, with the difference reported in minutes rather than as a percentage so the size is readable.
811131606121824
Figure 1. Observed and predicted mean wait by clinic session, labelled by first hour: 8 to 10, 11 to 12, 13 to 15, 16 to 17.
Source: 9,600 arrivals, two clinics, 30 opening days, in minutes.
RESULTS
The fitted parameters are three numbers. Arrival rate runs from 11.4 patients an hour at 08:00 to 21.8 an hour at 16:00, with a dip to 13.1 over the lunch hour. Service rate is 6.0 patients an hour for each rostered clinician, from a mean consultation of 10.0 minutes across 9,600 visits. Rostered clinicians run from 3 to 5 by hour of day, read from the roster file rather than fitted. With those three in place the model predicted a mean wait within 2.3 minutes of the observed mean in 9 of the 10 hours, and the mean absolute error over the day was 1.1 minutes. The error has no consistent sign: it runs slightly low in the early morning and slightly high in the afternoon. The 16:00 hour is the only miss worth naming, where the model predicted 32.4 minutes against an observed 29.8. That hour has the highest arrival rate and the tightest roster, which is where the queue is most sensitive to the assumption that clinicians are interchangeable.
DISCUSSION
A mean absolute error of 1.1 minutes is small enough that the model is useful for a roster decision and large enough that it should not be quoted to a patient. The practical reading is that the assumptions this model gets wrong mostly cancel. Booked arrivals are more regular than Poisson, which makes the real queue shorter than predicted; consultation times have a floor and a long tail that one exponential rate does not capture, which makes it longer. In the 16:00 hour the second effect stops cancelling the first, because the afternoon list carries the complex reviews the morning deferred just as the roster thins. Three limits sit on this result rather than under it. Both clinics book appointments, so nothing here transfers to a walk in service. Patients who left before being called are not in the data, and there were 141 of them. And a mean hides the tail that patients remember: the 90th percentile wait was 54 minutes.
REFERENCES
1.Halvorsen, I. and Mbeki, R. (2024) When does M/M/c hold in an appointment clinic? Journal of Applied Queueing, 22(3), 188 to 214.
2.Tanaka, S. (2025) Service time distributions in outpatient care: the four minute floor. Operations in Health Services, 11(2), 76 to 98.
3.Olamide, T. and Varga, E. (2026) Roster sensitivity in multi server clinic queues. Kelvedon Mathematics Working Paper 2026-04.

Block by block

What each block on the board is for, in the order a reader walks it.

Title band and authors
The title asks the question the board answers and names the sample size, with two authors over two affiliations. The layout gives the title a full width band and no ornament, which suits a study whose subject is a formula.
Abstract
Aim, data, fitted rates and the headline comparison in one block: 9,600 arrivals across 30 opening days, mean consultation 10.0 minutes, and a prediction that lands within 2.3 minutes of the observed mean in nine hours out of ten.
The model in three statements
The introduction block states the Poisson arrivals assumption, the exponential service assumption and the interchangeable servers assumption, then says which one each clinic practice breaks. It ends by framing the study as a question about usefulness rather than about truth.
Methods as a labelled protocol
Five labelled entries: the setting, the data and how a wait was defined, the fitted rates, where the server count came from, and how the comparison was made. The server entry matters most, because it was read from the roster rather than fitted.
Figure 1, predicted against observed
Four clinic sessions on the horizontal axis, observed and modelled mean wait as paired bars. Each session is labelled by its first hour and the caption names the hours it covers, which keeps the tick row clear of the series legend underneath.
Results and the fitted parameters
The three parameter values are written out in the results block, followed by the fit: a mean absolute error of 1.1 minutes across the day and one miss of 2.6 minutes at 16:00, which is the hour with the highest arrival rate and the thinnest roster.
Discussion and limits
Why the errors mostly cancel, why they stop cancelling in the late afternoon, and three limits stated in full: booked arrivals are not walk ins, 141 people left before being called, and a mean says nothing about the 54 minute ninetieth percentile.
References
Three numbered references in journal style at the foot of the right column, written to the three entries this layout carries rather than trimmed to fit afterwards.

What makes this board work

The assumptions are the introduction

Instead of deriving the formula, the second block states the three things the model assumes and says plainly which of them an outpatient clinic breaks. A reader who knows queueing theory can audit the study in twenty seconds, and a reader who does not still learns what was taken on faith.

Prediction and observation share one axis

Figure 1 puts the modelled and the measured mean wait in the same bars for four clinic sessions, and the hour by hour detail sits in the results block. The one 2.6 minute miss at 16:00 is named there instead of being averaged away.

Three fitted numbers, written out

Arrival rate, service rate and the number of rostered clinicians are the only parameters, and all three appear with their values in the results block. A maths poster that shows a fit without its parameters cannot be replicated by anyone standing at it.

Questions people ask

What goes on a mathematics research poster?

The question, the model stated as assumptions rather than derivation, the data the model was fitted to, the fitted parameter values, a figure comparing prediction with observation, and the limits. A maths board fails most often by printing the derivation, which nobody reads standing up, and omitting the parameters, which everybody wants.

How do you show a model on a poster without the algebra?

Write the assumptions as short statements and say which ones the real setting breaks. That is what this board does in its introduction block. The algebra belongs in the paper the printed link points to. A poster is read at two metres by someone who has ninety seconds, so the assumptions carry more than the notation.

Does a statistics poster need a chart?

It needs one figure that answers the question, and for a model validation that figure is prediction against observation. This board uses paired bars for four clinic sessions, with the hourly numbers in the results block. Ten chart kinds are available on a figure block, including line, area and waterfall, so the shape of the comparison can drive the choice.

How big is an A0 research poster?

A0 is 841 by 1189 millimetres. This board is A0 portrait, and the working canvas is 1656 by 2340 pixels. The export captures at twice that, so 3312 by 4680 pixels, which is about 100 dots per inch at full A0 size. There is no bleed setting, no crop marks and no CMYK conversion.

Portrait or landscape for a maths conference?

Check the call for posters first, because most conferences specify one and some set a maximum width. Portrait suits a board with several text blocks read in a column, which is the shape of a model validation. Both A0 orientations carry twenty layouts each.

How many references should a poster carry?

As many as the layout holds, because a reference block prints only the entries it has room for. This layout takes three. If your reading list is longer, cite the three a reviewer would check and leave the rest to the preprint.

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