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Journal of Simulation

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Discrete-event simulation and scheduling for Mohsmicrographic surgery

Patrick Burns, Sailesh Konda & Michelle Alvarado

To cite this article: Patrick Burns, Sailesh Konda & Michelle Alvarado (2020): Discrete-event simulation and scheduling for Mohs micrographic surgery, Journal of Simulation, DOI:10.1080/17477778.2020.1750315

To link to this article: https://doi.org/10.1080/17477778.2020.1750315

Published online: 14 May 2020.

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Discrete-event simulation and scheduling for Mohs micrographic surgeryPatrick Burnsa, Sailesh Kondab and Michelle Alvarado c

aIndustrial Engineering, Northrop Grumman, Baltimore, MD, USA; bCollege of Medicine, University of Florida, Gainesville, FL, USA; cHerbertWortheim College of Engineering, University of Florida, Gainesville, FL, USA

ABSTRACTMohs Micrographic Surgery (MMS) is a layer-by-layer, skin-saving surgical method for excisingskin cancer. The number of excised layers is stochastic, which creates operational challenges forthe clinic. We develop a discrete-event simulation of a MMS surgical clinic to investigate howappointment schedules impact clinic operations. The model simulates the patient flow for asingle day with physician, procedure rooms, and histotechnicians as limiting resources. Theprocess times, rate of no-shows, and number of excised layers are stochastic model inputs. Thevalue of the MMS simulation is demonstrated through analysis of five scheduling methods,thirteen schedule templates, and three performance measures: clinic throughput, patientwaiting time, and clinic overtime. The results show that the number of patients scheduledchanges the ideal spacing between appointments. MMS clinics can benefit from using thissimulation model to explore new scheduling templates, especially when reduced patientwaiting time and clinic overtime is a priority.

ARTICLE HISTORYReceived 29 August 2019Accepted 30 March 2020

KEYWORDSModelling; discrete-eventsimulation; appointmentscheduling; mohsmicrographic surgery; skincancer; outpatient clinic

1. Introduction

Nonmelanoma skin cancers NMSC) account for 40% ofall cancer diagnoses (Cakir et al., 2012), including2.8 million basal cell carcinomas and 700,000 squamouscell carcinomas diagnosed every year (Paoli et al., 2011).NMSCs are localised cancers that rarely metastasise orskip areas. However, without proper treatment, NMSCscan still be locally aggressive (Finley, 2003). MohsMicrographic Surgery (MMS) is a layer-by-layer surgicalmethod used for the excision of NMSC with each layerbeing examined for the presence of cancerous cells untila cancer-free layer is identified. MMS is used to treat skincancers that have a high risk of recurrence, have hard todefine borders, or are located in areas where preservationof healthy tissue is critical, such as the head, neck, hands,feet, anterior lower legs, and genitals. Larger and aggres-sive skin cancers on the trunk and extremities may alsobe candidates for MMS (Finley, 2003). MMS has beenquite successful and has low recurrence rates accordingto the Skin Cancer Foundation (https://www.skincancer.org/treatment-resources/mohs-surgery/). However,MMS is also heavily associated with long patient waittimes because of the stochastic nature of the re-entranceprocess for additional layers of tissue removal.Consequently, patients sometimes spend a large portionof their day waiting at the clinic for results, wound repair,and discharge.

The entire healthcare system is moving towardspatient-centric care and has been pushing for qualitymeasures based on the patient experience and thepatient’s perceptions of their healthcare service.

Consequently, one important component is patientwaiting times at the clinic. The availability of theMMS physician, the pathology lab, and procedurerooms can all impact patient waiting times. The timingand arrival of patient appointments can help balanceworkloads and keep a smooth operational flow. EachMMS clinic uses their own style of scheduling.However, one commonality among MMS schedulingmethods is that patients are scheduled throughout themorning while the afternoon is reserved for overflow.MMS clinics operate on multiple patients per day andseek to avoid clinic overtime. However, this is notalways possible due to the unpredictable nature of theiterative layered skin removal process. The challenge ofresource availability, clinic capacity, and uncertainty allcreate a challenging environment to evaluate best sche-duling practices for reduced patient waiting times andclinic overtime, but with acceptable throughput levels.Thus, a discrete-event simulation model of a MMSclinic has been developed to capture the complexities,dynamics, and uncertainties of clinic operations.

The objectives of this paper are to 1) establisha process flow map of the MMS procedure; 2) design,implement, verify, and validate a discrete-event simu-lation model of a MMS clinic; 3) demonstrate thevalue of the MMS simulation by evaluating schedulingtemplates derived from five scheduling methods; 4)identify preliminary managerial insights for the sche-duling of MMS appointments. The MMS schedulingproblem is interesting and challenging due to thestochastic re-entrance process and processing times.Publishing the specific process flow and characteristics

CONTACT Michelle Alvarado alvarado.m@ufl.edu Herbert Wortheim College of Engineering, University of Florida, Gainesville, FL, 32611

JOURNAL OF SIMULATIONhttps://doi.org/10.1080/17477778.2020.1750315

© Operational Research Society 2020.

of this unique problem will provide opportunity forothers within the operations research and manage-ment science community to investigate new MMSscheduling techniques.

The theoretical simulation model of the MMS sur-gical clinic presented in this paper is critical foradvancements in MMS scheduling and operations.The verified and validated model is parameterised byvalues from the literature and will enable futureresearch studies to compare different scheduling tem-plates and operational policies based on performancemeasures for throughput, overtime, and patient wait-ing times. By increasing throughput, but reducingwaiting times without sacrificing clinic overtime,clinics can have greater capacity for MMS procedures,thus enabling the highly successful procedure tobecome more accessible for patients. Since MMS is97% to 99.8% successful in removing skin cancer(Mohs & Mikhail, 1991), there will be less recurrencesfor certain types of skin cancers. Additionally, MMSclinics may be able to see more patients and providea better work environment for their employeesthrough balanced workloads and reduced overtimecosts. One important finding in this paper is that thethe number of patients scheduled impacts the optimalschedule designs. Adding and subtracting patients toa schedule shifts the ideal spacing between appoint-ments in a meaningful way. Thus, the experimentalanalysis of the simulation model will demonstrate howand why clinics can benefit from changing their sche-duling templates, rather than rigidly adhering toa one-size fits all scheduling policy.

The next section discusses closely related work forMMS operations planning and modelling. Section 3describes theMMS clinic setting while Section 4 presentsthe abstraction of the clinic as a simulation model.Section 5 defines the model’s input parameters duringverification and validation, then Section 6 presents resultsfor the simulation model verification and validation anddefines numerical experiments, scheduling methods, andresulting MMS templates. Finally, Section 7 presentscomputational results while Section 8 summarises thefindings and discusses future research directions.

2. Literature review

Discrete-event simulation is a well-established methodthat has been successfully used to model various pro-cesses across industries, including healthcare (Günal &Pidd, 2010). Simulation is particularly valuable whenimplemented for smaller healthcare units to analyseprocesses in high patient volume situations like emer-gency rooms (Komashie & Mousavi, 2005; McKinleyet al., 2020), maternity wards (Johnson, 1998), surgicalunits (Reese et al., 2020), and outpatient clinics (Weng& Houshmand, 1999). Within clinics, the focus iscommonly on micro-waiting times, which is the

period of time the patient waits for a physician toarrive and will vary by minutes rather than hours ordays (Günal & Pidd, 2010). Simulation models aredesigned based on the units flow path and are para-meterised to be reflective of many different clinics,often with strong results (Swisher et al., 2001). Werefer the reader to a recent systematic review of dis-crete-event simulation in health care (Zhang, 2018) formore information.

MMS has been studied extensively from a medicaldecision-making perspective for determining the size ofthe excision, preparing the tissue for analysis, and iden-tifying cancerous cells (Snow&Mikhail, 2004). However,there have only been two publications on schedulingMMS procedures (Steidle, 2013; Walker et al., 2020).One of the articles introduced a MMS surgery scoringsystem to streamline scheduling and resource allocation(Walker et al., 2020), while the other utilised an analyticalsimulation approach (Steidle, 2013). The MMS simula-tion model developed in this paper uses discrete-eventsimulation to capture multiple clinic resources (physi-cians, surgery rooms, histotechnicians), queues, andmulti-step processing times. Through observations ata MMS clinic over a two month period, the simulationincludes additional complexities, more accurate inputdata, and detailed process flow maps.

Other outpatient clinics have been analysedthrough simulation models to find the optimal sche-duling of patients during a day. Specific clinic settingsinclude mammography clinics (Coelli et al., 2007),oncology clinics (Alvarado et al., 2018), and emer-gency medical service routing (Bogle et al., 2017).There have also be been several papers publishedthat consider overbooking and no-shows withinclinics (Muthuraman & Lawley, 2008). These modelscan be viewed as physician-centric because they placepriority on maximising the number of patients sched-uled each day rather than considering patient waittime (Huang et al., 2012). Many outpatient clinicsallow walk-in patients to balance overbooking, butMMS clinics work on referrals for patients so thisstrategy is not viable for this setting (Kim &Giachetti, 2006). Overbooking has been a successfuland robust tactic in outpatient scheduling where no-shows occur, but careful implementation is needed toavoid negative consequences such as high waitingtimes (El-Sharo et al., 2015; Lee et al., 2013;Muthuraman & Lawley, 2008). Overbooking is parti-cularly challenging in the MMS setting because thesource of uncertainty is not just from patient no-shows, but from the high variance in operation timesbased on number of layers removed and congestion inthe pathology lab. To the best of our knowledge, thereare no published simulations of MMS clinic opera-tions. Thus, the paper establishes the foundation forunderstanding MMS clinic operations to develop andtest innovative MMS scheduling methods.

2 P. BURNS ET AL.

3. MMS clinic setting

We now describe process flow for a patient with n layersremoved, as depicted in Figure A1. Upon arrival ata MMS clinic, patients check-in and are escorted totheir own room. In this model, we assume the patientswill occupy the same room throughout their visit includ-ing waiting time; however, this is not true for all clinicswhich transition patients to a waiting room betweensteps in the procedure. When a room is available, thenurse brings the patient into the room, asks standardhealth questions, and performs health diagnostics (e.g.,asking about current medications and taking blood pres-sure). While following the surgical process flow path,patients have a unique Mohs physician who will exciseall layers required for the procedure. The physician willdiscuss the operation and any associated risks with thepatient before applying anaesthesia and removing thefirst layer from the patient (1a). The wound is coveredwith a bandage and the patient begins waiting for theresults. The skin sample will be sent to pathology (1b)where it will be prepared on slides by the histotechnician.

After preparation, the slides will be inspected by thephysician for a few minutes to determine if the sampleis clear of skin cancer (termed negative), or if anotherlayer of removal is necessary (termed positive). Thepathology and analysis processes is generally the bot-tleneck of the system because it is highly variable andhas limited resources. If the results are positive for skincancer, the physician will re-enter the patients room(2a) to remove another layer. Additional layers takeabout half as much time as the first because the pre-paration and consent process do not have to berepeated. However, pathology still takes the sameamount of time. The process repeats until all n layersare removed and the margins are “negative”. If thesample is negative for skin cancer, the physician willre-enter the patients room and begin repairing thewound. Afterwards, the physician is free to attendthe next patient while the nurse discharges the patient.

MMS scheduling is challenging because it is diffi-cult to predict how many layers of skin removala patient will require to become cancer-free. The can-cer is often hidden underneath the skin and can havetendrils that go deep into the skin tissue. Thus, theappearance of the cancer on the skin’s surface is nota reliable prediction method for how many tissueslayers will need to be removed. Because MMS is a skin-saving technique, the goal is to excise small layers andfollow the tendrils of the tumour with each additionallayer, rather than taking a large piece that will takeexcessive amounts of healthy skin. The average num-ber of layers removed is 1.74 layers (Krishnan et al.,2017). Some days, all patients will only have one layerexcised, whereas other days will require physicians towork well into the afternoon and evening to finishmultiple layers for several patients. This paper’s

investigations have unveiled that as many as 7–13layers is possible, but rare. The process becomes taxingon both the patients and the clinic staff as they wait forall patients to be cleared and wound repairs to becompleted. The wound is left open between successivelayers and concludes with wound repair. The vastmajority of wound repairs happen in clinic, thoughoccasionally the location is in a sensitive area (e.g.,eyelid) or so invasive that the patient is scheduled forsurgery outside of the dermatology clinic for woundrepair. Finding the right scheduling template willstreamline the process, decrease unnecessary down-time for physicians and histotechnicians, while alsoensuring patient wait time is low.

4. Model design

The discrete-event simulation model was designed tosimulate a single day at a MMS clinic. The purpose ofthe model is to evaluate the impact of the appointmentschedule on the clinic operations. The primary perfor-mancemeasures observed are patient waiting time, clinicovertime, and throughput. The complete model is shownin Figure A2. The arrows describe patient flow and theskin excision (or sample) flow through the system. Thesimulation model was implemented in Arena SimulationSoftware, Version 14.5 (https://www.arenasimulation.com/)1. For simplicity, we will now explain the modelin four stages: 1) Patient Generation, 2) Skin Excision, 3)Pathology, and 4) Repair and Discharge.

4.1. Patient generation

In the Patient Generation stage of the model, all sched-uled patient entities are created using a “Create” blockat time 0 and are assigned a patient ID, appointmenttime, and physician from the scheduling file. Thepatients pass through a “Delay” block which delays thepatient from entering the clinic until their assignedappointment time. The model does not accommodateearly arrivals, but does permit the delay to extendbeyond the appointment time to simulate late arrivalsaccording to an input distribution for punctuality. Afterthe delay, the patient will pass through the “Decide”block (shown as a green diamond), which serves asa decision point to determine if the patient is a “No-show”. Show-ups are determined by an input probabil-ity (e.g., 90.5%) and will be eliminated from the systemin a “Dispose” block. Next, the process flow moves tothe Skin Excision stage.

4.2. Skin excision

In the Skin Excision stage of the model, the patiententity first enters the Skin Excision stage and entersa “Seize” block and seizes an available room resourcein the system. This room will remain seized by the

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patient until the Repair and Discharge stage because weassume patients wait in the procedure room for results.Regardless of which layer the patient is on, the processmoves through a secondary “Seize” block and seizes thepreviously assigned physician once the physicianbecomes free and will be delayed for their excision bypassing through the “Delay” block. Once the excision iscompleted, the patient passes through the “Release”block and the physician is released. Note, that theroom is not released because we assume patients waitin their surgery room for pathology results. Next, theprocess flow moves to the Pathology stage.

4.3. Pathology

In the Pathology stage of the model, the histotechnicianfirst prepares the excised skin on slides for analysis. Thehistotechnician is seized in the “Seize” block, followed bya “Delay” block to create slides, and is released in the“Release” block. The histotechnician is then free to beseized by the next excision in the queue. The patient willthen move to a “Seize” block to seize a physician, thena “Delay” block to determine if the layer is negative or ifanother layer is required. The physician is then releasedby a “Release” block. Next, a “Decide” block usesa probability of re-entrance to determine if the mostrecent skin excision is negative for skin cancer or not.Details of the probability of re-entrance is explained laterin Section 5.3. If positive, the patient will return foranother layer back in the Skin Excision stage. If thelayer is negative, the patient will move to the Repairand Discharge stage.

4.4. Repair and discharge

Finally, in the Repair and Discharge stage of the model,when a patient entity leaves the Pathology stage, theywill seize their assigned physician for wound repair. Thephysician is seized in the “Seize” block, delayed in the“Delay” block for the extent of the repair, and releasedin the “Release” block. The patients room is thenreleased in the “Release” block and the patient leavesthe system through the “Dispose” block.

5. Model inputs and parameter values

In this section, we describe the model inputs andparameter values for the resources, schedules, andrandom variables.

5.1. Clinic characteristics

The simulation model requires several Input para-meters that define the clinic size, including resourcecapacities and the normal clinic operating hours.

Input parameters for M physicians, T histotechni-cians, and R surgery rooms specify the resourcecapacity. The overtime period begins after H normalclinic operating hours. Furthermore, the simulationmodel assumes that the physicians and histotechni-cians work in a First In, First Out (FIFO) system.The physicians each have their own patients and arenot interchangeable, whereas the histotechnician andsurgery rooms are shared, interchangeable resources.In Section 6, for the model validation and experi-ments, we assume M ¼ 2 physicians, T ¼ 1 histo-technicians, and R ¼ 4 surgery rooms, and H ¼ 8normal operating hours.

5.2. Scheduling templates

The simulation also requires an appointment scheduleand physician assignment for each patient. Section 6.3and the Appendix provides details on how to generatescheduling templates (e.g., wave, batch, etc.). The out-put of these functions will be input for the simulation.The methods build a two-dimensional matrix definingthe the appointment time (in minutes) of the ith

patient assigned to physician j.

5.3. Distributions

The final input for the simulation model is the theo-retical distributions and parameters for the randomvariables in the system. This was non-trivial becausemost values found in the literature only specifieda simple statistic (e.g., mean value, range) and didnot provide sufficient data describing the underlyingdistributions for the parameter values. Thus, we con-ducted an observational time study at a MMS clinic.Samples were collected by observing 100 patientappointments over a 12-day period. The time studywas crucial for obtaining seven theoretical distribu-tions and two empirical distributions. These distribu-tions were identified using Minitab 17 and werechosen based on p-values. While the distributiontype was identified from the time study at a MMSclinic, the specific parameter values were obtainedfrom the literature. This was done for two reasons: 1)to respect the privacy of the clinic, 2) to utilise moregeneral MSM clinic settings since this manuscript isnot focusing on a specific clinic case study. All of theparameter values were taken from publications onMMS clinics (if available) or similar clinic settings(e.g., outpatient dermatology clinics). Table A1 liststhe simulation model parameter names in the firstcolumn, and the second column gives the distributiontype and parameter values used in the simulation.Columns 3–5 of Table A1 list the literature values,sources, and clinic setting.

4 P. BURNS ET AL.

The next challenge in fitting the distributions and inputparameters for the simulation model was that the para-meter values in the outpatient clinic literature were notnecessarily in the right format for the distribution format.For example, only a mean value appeared in the literaturefor the repair time, but parameterising the normal distri-bution requires a mean and standard deviation. Thus, wenow elaborate on the following manipulations:

● Any distributions or probabilities identified in theliterature that had the correct parameters were usedwithout manipulation, as was the case forProbability of Showing Up and AppointmentPunctuality

● Probability of Re-entrance for Excision n. Weassume a geometric distribution for estimatingthe probability of re-entrance r. Under thisassumption, the average number of layers, E½N�is determined by:

E½N� ¼ 1þX1

k¼1rk�ðkþ1Þ=2: (1)

From the literature sources, we assume the averagenumber of layers E½N� ¼ 1:74 (Krishnan et al.,2017). Solving equation (1), r ¼ 52:87%, meaningthat the probability of re-entrance for excision nis 52:87n�1%.

● Time between Initial Room Seize and PhysicianSeize, Initial Excision Time, Total PathologyTime: For gamma distributions Γðα; βÞ withmean μ ¼ α � β, we had to identify the newshape and scale parameters that yield the desiredmean μ0 from the literature sources. The timestudy data provided the numerical baselineshape α and scale β parameters. Keeping propor-tions between the shape and scale parameters

equal, we identified a ρ value such that μ0 ¼ αρ �

βρ and the new distribution became Γðαρ ; βρÞ.Example: If the time study identifieda distribution of Γð2; 6Þ with μ ¼ 12 and μ’needed to be 3, then ρ ¼ 2 and the new distribu-tion would be Γð1; 3Þ.

● Repair Time, Discharge Time: For normal dis-tributions Nðμ; σÞ, we were able to source thenew μ’ values from the literature, but sometimesneeded to identify the new standard deviation σ’.

This was done in equal proportion using λ ¼ μ0

μ ,

such that Nðμ; σÞ became Nðμ0; σ0Þ =Nðλ � μ; λ � σÞ. Example: If the literature cited,Nð5; 3Þ and μ0 ¼ 15 was desired, then λ ¼ 3defines the new distribution ,Nð15; 9Þ.

● Secondary excision time: the gamma distributionwas scaled to be 50% of the initial excision time’smean by adjusting the scale parameter β by 50%.

6. Model validation and scheduling methods

6.1. Model verification and validation

Before further experimentation, we need to verify andvalidate the simulation model. To do so, we reliedupon the MMS clinic where we conducted the timestudy. We implemented this clinic’s current practicescheduling template and set the simulation parametervalues to the original values from the time study clinicdata. The clinic’s current practice scheduling templateequally spaces two patients (one per physician) at thebeginning of each hour for a total of 8 patients.

For model verification, the MMS simulation modelwas checked to confirm that it was working as expected.The approach was to examine the outputs in twoways: 1)comparing direct model outputs between the simulatedclinic and the real clinic, and 2) performing a systemcheck to ensure the systems was stable. Direct outputsrefer to simulation outputs that that were explicitlydefined as a random variable in the simulation model,such as initial excision time, secondary excision time,discharge time, and repair time. Next, we checked theresource utilisation and performance measures to ensurethat the system was working correctly. To preserve theprivacy of the clinic, we cannot reveal the actual valuesfor the direct outputs. However, Table A2 summarisesthe percent difference in the average of the actual clinicdata and the average of the MMS simulation directmodel output for 100 replications. If the simulation out-put value was X and the raw data reported Y , then wecomputed the percent difference as X�Y

Y . For the directoutputs, the percent difference was less than 3.36% for allfour processing times, which was a good result. For thesystem check, we monitored several statistics and perfor-mance measures to ensure the model was reportingreasonable values. Our clinic partner also served as anexpert opinion to confirm that the results were in linewith the clinic’s performance outcomes. For example,specific outputs that were cross-checked include resourceutilisation, queue sizes, throughput, overtime, and wait-ing time. The team felt confident the model was operat-ing as intended. Thus, we were ready to move into themore rigorous and precise validation step.

For model validation of the MMS simulationmodel, we analysed the indirect model outputs.Indirect model outputs refer to simulation outputsthat were a result of the system input parameters andmodelling logic, such as average number of layers,time in system, waiting time for a room, and waitingtime for a physician. For, example, the probability ofre-entrance (model input) would impact the averagenumber of layers (model output). Similarly the totalsystem time is a consequence of multiple input para-meters and the model logic. Table A2 also givesthe percent difference for the indirect output in thevalidation step. The MMS simulation overestimated

JOURNAL OF SIMULATION 5

the average number of layers and the waiting time forthe physician by 2.19% and 1.52%, respectively. Thetotal time the patient was in the system, or clinic, wasunderestimated by 1.26% and the waiting time for theroom was underestimated by 1.40%. Since all gapswere less than 2.2%, the research team was satisfiedwith the validation results.

6.2. Design of experiments

Several scheduling templates were analysed using theMMS simulation model to observe the impact on theperformance measures. The templates were designed toevaluate different clustering and spacing options for thescheduled patient appointments. The goal is to maximisethroughput while minimising patient waiting time andclinic overtime. To evaluate the scheduling templates, wedefined the following objective function:

F ¼ wt � ðTPÞ � ww � ðWTÞ � wo � ðOTÞ; (2)

based on Throughput (TP), Wait Time (WT), andOvertime (OT), which are weighted by the parameterswt , ww, and wo, respectively. Throughput is the numberof patients that show-up and are discharged.Wait Time isthe cumulative patient waiting time to seize a room, toreceive pathology results, and to seize a physician forexcision or repair. Note that from the patients perspective,the pathology time is also a waiting period, hence we haveincluded it in our model as such. Overtime is the amountof time the clinicmust stay open past its operating hours toservice and discharge all patients that showed up for theirappointment. The parameter wt is the weight per unit ofthroughput (patients served), while ww and wo are thepenalties for waiting time per patient and clinic overtime,respectively.

The simulation was run with different weights foreach parameter to find the optimal value for clinicswith different priorities: increasing throughput (orpatient access to care), reducing waiting times, andreducing clinic overtime. Table A3 summarises theselected weights for each competing priority in theexperimental design. We chose the throughput weightwt as a baseline. The waiting time value ww is 10-50%in comparison because it is an estimator of patientsatisfaction, which is important, but relatively lessimportant than actually performing the procedurefor patients via high throughput. The weight of theovertime value wo was selected in two extremes: 1) low(e.g., 10%) compared to throughput, or 2) much larger(e.g., 200%) to reward schedules that minimise clinicovertime.

6.3. Scheduling methods and templates

To generate the different scheduling templates, weimplemented five scheduling methods: traditional,staggered, alternating, batch, and wave. Table A4

provides a summary of the five scheduling methodsbased on the interval sizes T and R, batch size B,number of waves W, and wave spacing A. It is impor-tant to note that each method schedules P patients/dayusing M ¼ 2 physicians. The patients are alwaysevenly split such that template results in P/2 patientsper day for each physician. Next, we describe the fivescheduling methods and pseudocode for each can befound in the Appendix:

● The traditional scheduling method generatesscheduling templates with one patient (e.g.,B ¼ 1) arriving at a fixed interval size T foreach physician and assumes the M physicianshave identical schedules. Patients for each physi-cian will arrive at t ¼ f0;T; 2T; 3T; :::g.

● The staggered scheduling method generatestemplates with one patient arrival at a fixed inter-val size T for each physician, but the patientarrivals are equally staggered throughout the Tinterval for each physician. The first physician,M1, begins appointments at time t ¼ 0 andthe second physician, M2, begins appointmentsat t ¼ T=M. Patients will arrive at t ¼f0;T; 2T; :::g for the first physician, and at t ¼fT=m; 3T=m; 5T=m; :::g for physician m.

● The alternating scheduling method generatestemplates where patients are scheduled at alter-nating interval sizes of T and R. Patients fora physician will arrive att ¼ f0;T;T þ R; 2T þ R; 2T þ 2R; :::g. Asidefrom this, alternating scheduling is identical tothe traditional scheduling method.

● The batch scheduling method is similar to tradi-tional scheduling where patients arrive at intervalsof T but each physician receives their patients inbatches of size B rather than the standard size(B ¼ 1). For a batch size B ¼ 2, patients for eachphysician will arrive at t ¼ f0; 0;T;T; 2T; 2T; :::g.

● The wave scheduling method generates Wave-Tscheduling templates where each physician oper-ates a simultaneous schedule of up to � ½P=M�patients and the patients arrive in W wavesspaced A hours apart with T interval spacing. Inthe templates for W ¼ 2 waves at A ¼ 2 hoursapart, each physician would have a wave startingat t ¼ 0 and t ¼ Awith P/4 patients in each wave.The first wave for each physician would have t ¼f0;T; 2T; :::g minutes and the second wavewould have t ¼ fA;Aþ T;Aþ 2T; :::g minutesfor each physician.

Scheduling templates were generated for P ¼ 8, 10,or 12 patients using various interval size (e.g., T ¼ 15,30, 45, 60, 75, 90). The Wave-T and Batch-T schedulesfor P ¼ 10 patients were omitted because the waves/batches did not divide into equal sizes for the two

6 P. BURNS ET AL.

physicians. For example, with batches of size 2 withP ¼ 10 patients, one physician would have had threebatches and the other would have had 2 batches. Formore examples, Table A5 contains examples of eachscheduling method for P ¼ 8 patients and M ¼ 2physicians.

7. Simulation results

The results of the scheduling methods and templatesfor 8, 10, and 12 patients are visually graphed inFigures A3–A5 for the objectives of prioritisingthroughput, wait time, and overtime respectively.The figures depict the average value over 100 replica-tions. The 95% confidence intervals are given in TableA6 and the performance measure outcomes for theobjective function are in Table A7.

The scheduling template with the best objectivefunction varies based on the number of patients sched-uled and the priority weights of the clinic. EvenThroughput and Wait Time prioritisation wentthrough very noticeable and unique changes depend-ing on the number of patients scheduled. This indi-cates adding and subtracting patients can shift theideal spacing in a meaningful way, and means thatclinics can benefit from changing their schedulingtemplates based on the number of patients scheduledeach day, rather than rigidly adhering to a one-size fitsall template.

As would be expected, prioritising throughput ismostly dependent on the number of patients arriving.Consequently, a throughput driven system producesrelatively uniform results with the 12 patient schedulesoutperforming the 8 and 10 patient schedules. Thespecific scheduling method had little impact on the 8patient schedule throughput-prioritised objectives.However, the 10 and 12 patient schedules were slightlybetter with 60-minute intervals and worst with 15-minute intervals. Adding more patients will likelycause throughput to continue to outperform the nega-tives of wait time and overtime with considerablymore patients simply based on the linear trend of theobjective function values.

Increasing the penalties for Overtime and WaitTime brings in new effects on the performance of thescheduling templates. Naturally, scheduling patientsearly and close together decreases Overtime, whichincreases the objective function. In contrast, creatinglarge time intervals between appointments is a betterapproach for minimisingWait Time. Interestingly, the12 patient schedules continue to dominate the 8 and10 patient schedules under the Overtime prioritisationfor intervals < 45 minutes, and for the alternating,batch, and wave scheduling methods. The 12 patientschedules also do well for Wait Time prioritisationwith longer time intervals (> 45 minutes). Thisshows the importance of having the right scheduling

method based on the number of patients that thatclinic wants to treat daily. Poor scheduling policiesfor a large number of patients can cause problems inone or more performance areas.

Under the Overtime prioritisation approach, thealternating, batch, and wave-scheduling approachesresulted in larger objective function values than thetraditional and staggered schedules. Among the tradi-tional and staggered schedules, the shorter time inter-vals were best. However, under the Wait Timeprioritisation, the traditional and staggered scheduleswith large time intervals were the best approaches,especially for intervals larger than 60 minutes. Thisindicates that there are several conflicting factors inplay while the simulation runs and shows the need forthe MMS simulation model to analyse specific MMSclinic settings, which may have different resource avail-ability (e.g., rooms, histotechnicians, etc.) than theexample clinic depicted in this analysis. These findingsreinforce the contributions of this paper because thesimulation model can be customised to different MMSclinics that can evaluate their own scheduling practicesand test new scheduling strategies.

8. Conclusions

This research established the process flow and devel-ops the theoretical framework for a discrete-eventsimulation model of a MMS clinic. The model wasverified and validated using time study data and expertopinion at an MMS clinic. We also identified para-meters from the literature to create a more generalMMS clinic setting. Multiple simulation runs of thegeneralised model revealed that scheduling based onthroughput optimisation was trivial in this setting.However, prioritising overtime or patient waitingtime is dependent on the template structure andrequires more creativity in design.

The analysis results show that the generalised MMSsimulation model is an effective tool for simulatingMMS clinics and comparing different scheduling tem-plates via an objective function. Clinics can easilyadjust the resources, scheduling policies, and prioriti-sation to fit their specific situation and use the modelto determine a scheduling strategy, or even adjustbased on the daily demand.

From the MMS scheduling and operations per-spective, the field is ripe with opportunity to developnew stochastic scheduling models, medically-basedpredictive models for the number of required exci-sions per patient, and even operational policies forhow physicians should prioritise among patientswaiting at different stages of the MMS process.Because there are no other fully defined models thatfocus on MMS scheduling and there is limitedresearch on optimising clinics with the stochasticpatient re-entrance feature, this detailed and

JOURNAL OF SIMULATION 7

validated simulation model can serve as a test bed forother MMS decision models (e.g., scheduling, capa-city, etc.). By using the MMS simulation model,clinics can identify scheduling strategies that reduceovertime costs, possibly increase throughput, and/orinvestigate the impact of scheduling policies onpatient waiting times. Ultimately, research in thisdomain will also bring value to patients throughdecreased wait times at MMS clinics and increasedaccessibility to MMS appointments.

Future research will focus on developing an optimalscheduling strategy using a slot model. The MMSsimulation will serve as a practical evaluation tool forthe new scheduling models. The model can even serveas an operational tool to help the physicians prioritisetheir tasks among the different job tasks (e.g., exci-sions, slide review, repair, etc.)

Note

1. Arena model files available by email request to alvar-ado.m@ufl.edu.

Disclosure statement

The authors do not have any financial interest or benefitfrom the direct application of this research.

Funding

This work was supported by the Department of Industrialand Systems Engineering in the Herbert Wortheim Collegeof Engineering at the University of Florida.

ORCID

Michelle Alvarado http://orcid.org/0000-0001-9649-214X

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Appendix. Pseduocode functions forscheduling methods

The appendix contains pseudocode functions for the fivescheduling methods. The functions populate a two-

dimensional matrix appt time, where appt timeðj; iÞ is theappointment time (in minutes) of the ith patient assigned tophysician j. The variable numsched is a counter. Additionally,the scheduling functions rely on two other methods, zerosð�Þand templateð�Þ. The zerosðm; nÞ function creates a matrixwith m rows and n columns filled with the value zero (0).The templateð�Þ function converts the appt time matrix toa format with m rows for each physician, but now eachcolumn corresponds to an appointment slot containing thenumber of patients scheduled.

1: function TRADITIONAL ðP;M;TÞ2: numsched 0, appt time zerosðM; P=Md eÞ3: for i :¼ 1 : P=Md e do4: for j :¼ 1 : M do5: if numsched< P then6: appt timeðj; iÞ ¼ T � ði� 1Þ7: numsched ¼ numschedþ 18: end if9: end for10: end for11: return template(appt time)12: end function

1: function STAGGERED ðP;M;TÞ2: numsched 0, appt time zerosðM; P=Md eÞ3: for i :¼ 1 : P=Md e do4: for j :¼ 1 : M do5: if numsched< P then6: appt timeðj; iÞ ¼ T � ði� 1Þ þ ðT=MÞ�ðj� 1Þ7: numsched ¼ numschedþ 18: end if9: end for10: end for11: return template(appt time)12: end function

1: function ALTERNATING ðP;M;T;RÞ2: numsched 0, appt time zerosðM; P=Md eÞ3: for i :¼ 1 : P=Md e do4: for j :¼ 1 : M do5: if numsched< P then6: appt timeðj; iÞ ¼ T � i=2b c þ R � ði� 1Þ=2b c7: numsched ¼ numschedþ 18: end if9: end for10: end for11: return template(appt time)12: end function

1: function BATCH ðP;M;T;BÞ2: numsched 0, appt time zerosðM; P=Md eÞ3: for i :¼ 1 : P=Md e do4: for j :¼ 1 : M k :¼ 1 : B do5: if numsched< P then6: appt timeðj;B � ði� 1Þ þ kÞ ¼ T � ði� 1Þ7: numsched ¼ numschedþ 18: numsched = numsched + 19: end if10: end for11: end for12: end for13: return template(appt time)14: end function

JOURNAL OF SIMULATION 9

Table A1. Simulation model parameters.Simulation Literature Literature Literature

Parameters Input Input Source Clinic Setting

Probability of Showing Up (%) 90.5 90.5 (Faiz & Kristoffersen,2018)

Neurology

Appointment Punctuality (minutes) max½0;Nð�16:62; 27:07Þ� N ð�16:62; 27:07Þ (Cayirli et al., 2006) Primary CareProbability of Re-entrance for Excision n (%) 0:5287n�1 E½N� ¼ 1:74 (Krishnan et al.,

2017)MMS

Time between Initial Room Seize and Physician Seize(minutes)

Γð5:5; 2:2Þ Mean of 13.66 (Ahmad et al., 2017) Primary Care

Initial Excision Time (minutes) Γð1:1; 12:4Þ Mean of 15 min (Loven, 2020) MMSSecondary Excision Time (minutes) Γð1:1; 6:2Þ 50% of initial

excisionClinic observations n/a

Total Pathology Time (minutes) Γð1:7; 9:6Þ Mean of 22–43 min (Rajadhyaksha et al.,2001)

MMS

Repair Time (minutes) Nð21:64; 15:3Þ Mean of 21.64 (Rogers et al., 2010) MMSDischarge Time (minutes) Nð15:69; 8:4Þ Mean of 15.69 (Williams et al.,

2014)Pain Centre

Table A2. Model verification and validation results.Verification Validation

1) Direct Outputs Percent Difference Indirect Outputs Percent DifferencePrimary Excision Time 0.60% Average number of layers E½N� 2.19%Secondary Excision Time 3.36% Time in system −1.26%Repair Time 0.06% Waiting time for room −1.40%Discharge Time 2.77% Waiting time for physician 1.52%

2) System CheckResource Utilisation OverTimeQueue Sizes ThroughputWaiting Time

Table A3. Objective function priorities.Parameters

Description wt ww wo

Throughput Prioritised 100 10 25Wait Time Prioritised 100 50 25Overtime Prioritised 100 10 200

Table A4. Scheduling method key assuming M ¼ 2 physicians.Interval B W A Physicians

Primary Scheduling Size Batch Number Wave spacing Using Same

Key Method (minutes) size of waves (hours) Schedule?

Trad-T Traditional T 1 – – YesStag-T Staggered T 1 – – NoWave-T Wave T 1 2 2 YesBatch-T Batch T 2 – – YesAlt-T/R Alternating Alternates T & R 1 – – Yes

10 P. BURNS ET AL.

TableA5.

Exam

pleschedu

lingtemplates

forP¼

8patients,M¼

2ph

ysicians.

Time

Template

Patients

Physician

8:00

8:15

8:30

8:45

9:00

9:15

9:30

9:45

10:00

10:15

10:30

10:45

11:00

11:15

11:30

11:45

12:00

Trad-60

8M1

11

11

M2

11

11

Stag-60

8M1

11

11

M2

11

11

Alt-15/90

8M1

11

11

M2

11

11

Batch-60

8M1

22

M2

22

Wave-15

8M1

11

11

M2

11

11

JOURNAL OF SIMULATION 11

TableA6.

Objectivefunctio

nvalues

forthroug

hput

prioritisationwith

95%

confidenceintervals.

Throug

hput

Prioritisation

WaitTimePrioritisation

OverTimePrioritisation

Num

berof

Patients

Templates

810

128

1012

810

12

Trad-15

631.51

759.10

868.60

250.02

178.45

35.55

630.71

746.90

790.62

(617.83,645.19)

(744.16,774.03)

(841.77,895.43)

(228.09,271.94)

(148.03,208.86)

(0,81.39)

(616.88,644.54)

(733.55,760.25)

(777.00,804.24)

Trad-30

633.58

778.59

903.87

340.58

347.65

309.33

632.36

766.85

823.44

(617.30,649.85)

(758.84,798.34)

(877.57,930.18)

(317.41,363.75)

(318.4,376.89)

(269.13,349.54)

(615.65,649.06)

(749.56,784.14)

(805.06,841.83)

Stag-30

637.73

778.09

905.81

329.19

332.71

295.80

636.81

753.13

796.02

(622.97,652.50)

(760.19,795.99)

(878.72,932.91)

(295.96,362.42)

(303.92,361.49)

(258.13,333.47)

(622.02,651.60)

(738.21,768.04)

(779.63,812.40)

Trad-45

658.69

815.92

959.07

329.19

473.62

526.46

656.92

791.42

855.63

(645.38,672.00)

(795.63,836.21)

(933.11,985.03)

(314.34,344.04)

(447.56,499.69)

(496.84,556.07)

(643.47,670.38)

(776.34,806.49)

(837.94,873.32)

Trad-60

674.72

834.64

978.77

410.46

556.83

632.72

667.10

751.27

735.76

(660.45,688.98)

(808.54,860.74)

(952.5,1005.04)

(395.83,425.09)

(533.99,579.68)

(605.56,659.87)

(651.61,682.59)

(735.67,766.87)

(718.43,753.09)

Stag-60

673.70

824.88

968.15

469.94

557.42

633.38

661.44

704.11

680.10

(659.66,687.74)

(800.89,848.88)

(941.63,994.68)

(454.70,485.18)

(539.31,575.52)

(610.71,656.04)

(646.67,676.21)

(689.81,718.41)

(663.63,696.58)

Trad-75

666.64

814.12

951.35

490.50

589.88

678.63

629.32

605.34

521.02

(651.08,682.19)

(788.24,839.99)

(926.36,976.33)

(476.00,505.00)

(573.35,606.42)

(661.4,695.87)

(609.47,649.17)

(588.36,622.32)

(502.67,539.38)

Trad-90

674.84

806.58

935.02

506.33

596.62

688.78

562.64

443.04

344.04

(659.06,690.63)

(781.8,831.35)

(913.07,956.97)

(491.49,521.16)

(581.27,611.96)

(672.44,705.11)

(538.04,587.24)

(426.46,459.63)

(327.06,361.02)

Stag-90

644.56

778.91

937.69

483.32

579.92

712.34

468.63

335.55

342.90

(628.53,660.58)

(754.02,803.81)

(915.09,960.29)

(468.13,498.51)

(563.29,596.55)

(693.18,731.5)

(443.05,494.21)

(318.48,352.62)

(322.59,363.2)

Alt-15/90

666.74

822.76

966.01

422.49

522.27

566.92

665.31

800.96

859.46

(652.98,680.49)

(805.42,840.11)

(945.32,986.69)

(405.82,439.16)

(501.1,543.43)

(540.82,593.01)

(651.24,679.39)

(786.21,815.72)

(843.12,875.79)

Batch-60

642.81

–910.35

310.03

–257.17

642.81

–844.84

(628.88,656.73)

–(884.4,936.3)

(291.15,328.90)

–(216.58,297.75)

(628.88,656.73)

–(828.60.01,861.08)

Wave-10

668.26

–931.42

437.78

–344.30

667.43

–876.80

(651.74,684.78)

–(907.81,955.02)

(422.24,453.32)

–(311.96,376.63)

(650.90,683.96)

–(861.99,891.60)

Wave-15

663.62

–931.13

438.26

–321.30

711.88

–844.23

(648.82,678.43)

–(904.65,957.6)

(422.53,454.00)

–(282.4,360.2)

(696.01,727.75)

–(828.97,859.48)

12 P. BURNS ET AL.

Table A7. Mean values from 100 replications of the performance measures in the objective function.8 Patients 10 Patients 12 Patients

Templates TP WT OT TP WT OT TP WT OT

Trad-15 7.270 9.537 0.005 9.060 14.516 0.070 10.880 20.826 0.446Trad-30 7.070 7.325 0.007 8.880 10.774 0.067 10.640 14.864 0.460Stag-30 7.150 7.714 0.005 8.930 11.135 0.143 10.740 15.250 0.627Trad-45 7.210 6.206 0.010 9.050 8.558 0.140 10.820 10.815 0.591Trad-60 7.270 5.119 0.044 9.160 6.945 0.476 11.000 8.651 1.389Stag-60 7.250 4.955 0.070 9.090 6.687 0.690 10.930 8.369 1.646Trad-75 7.160 4.403 0.213 9.000 5.606 1.193 10.810 6.818 2.459Trad-90 7.330 4.213 0.641 9.110 5.249 2.077 10.810 6.156 3.377Stag-90 7.100 4.031 1.005 8.920 4.975 2.534 10.790 5.634 3.399Alt-15/90 7.280 6.106 0.008 9.010 7.512 0.125 10.810 9.977 0.609Batch-60 7.260 8.319 0.000 – – – 10.830 16.329 0.374Wave-10 7.260 5.762 0.005 – – – 10.860 14.678 0.312Wave-15 7.200 5.634 0.001 – – – 10.960 15.246 0.497

Figure A1. Process path of a MMS appointment with n layers removed.

Figure A2. Full simulation model flowchart.

JOURNAL OF SIMULATION 13

Figure A3. Throughput prioritised objective function values and 95% confidence intervals.

Figure A4. Wait time prioritised objective function values and 95% confidence intervals.

14 P. BURNS ET AL.

Figure A5. Overtime prioritised objective function values and 95% confidence intervals.

JOURNAL OF SIMULATION 15