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a survey of Queuing Theory applications in 
healthcare 
samuel fomundam, Jeffrey herrmann
Isr TechnIcal reporT 2007-24
A SURVEY OF QUEUING THEORY APPLICATIONS IN HEALTHCARE 
Samuel F. Fomundam 
Department of Mechanical Engineering 
University of Maryland, College Park, MD 20742 
 
Jeffrey W. Herrmann  
Department of Mechanical Engineering and Institute for Systems Research 
University of Maryland, College Park, MD 20742 
Abstract 
This paper surveys the contributions and applications of queuing theory in the field of 
healthcare.  The paper summarizes a range of queuing theory results in the following areas: 
waiting time and utilization analysis, system design, and appointment systems.  The paper also 
considers results for systems at different scales, including individual departments (or units), 
healthcare facilities, and regional healthcare systems.  The goal is to provide sufficient 
information to analysts who are interested in using queuing theory to model a healthcare process 
and want to locate the details of relevant models.   
Introduction 
The organizations that care for persons who are ill and injured vary widely in scope and 
scale, from specialized outpatient clinics to large, urban hospitals to regional healthcare systems.  
Despite these differences, one can view the healthcare processes that these organizations provide 
as queuing systems in which patients arrive, wait for service, obtain service, and then depart.  
The healthcare processes also vary in complexity and scope, but they all consist of a set of 
activities and procedures (both medical and non-medical) that the patient must undergo in order 
to receive the needed treatment.  The resources (or servers) in these queuing systems are the 
trained personnel and specialized equipment that these activities and procedures require. 
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A considerable body of research has shown that queuing theory can be useful in real-
world healthcare situations, and some reviews of this work have appeared.  McClain (1976) 
reviews research on models for evaluating the impact of bed assignment policies on utilization, 
waiting time, and the probability of turning away patients.  Nosek and Wilson (2001) review the 
use of queuing theory in pharmacy applications with particular attention to improving customer 
satisfaction.  Customer satisfaction is improved by predicting and reducing waiting times and 
adjusting staffing.  Preater (2002) presents a brief history of the use of queuing theory in 
healthcare and points to an extensive bibliography of the research that lists many papers 
(however, it provides no description of the applications or results).  Green (2006a) presents the 
theory of queuing as applied in healthcare.  She discusses the relationship amongst delays, 
utilization and the number of servers; the basic M/M/s model, its assumptions and extensions; 
and the applications of the theory to determine the required number of servers.   
This paper surveys the contributions and applications of queuing theory in the field of 
healthcare.  The reviews mentioned above focus on presenting mathematical models or limit 
their scope to a single type of application.  This paper, however, seeks to show the applicability 
of queuing theory from the perspective of healthcare organizations.  Thus, this paper summarizes 
a range of queuing theory results in the following areas: waiting time and utilization analysis, 
system design, and appointment systems.  This covers processes that provide direct patient 
treatment and processes that provide auxiliary services such as pharmacy and medical laboratory 
processing.  The paper also considers results for systems at different scales, including individual 
departments (or units), healthcare facilities, and regional health systems.  The goal is to provide 
sufficient information to analysts who are interested in using queuing theory to model a 
 2
healthcare process and want to locate the details of relevant models.  We assume that the reader 
is familiar with healthcare organizations and the basic concepts of queuing theory. 
This survey covers analytical queuing theory models applied directly to healthcare 
systems.  It is reasonable for an analyst to understand, adapt, and apply such a model to his own 
situation.  Because they require specialized software and the details of the simulation model are 
usually unknown, this paper does not review simulation studies of healthcare processes. 
Queuing models and simulation models each have their advantages.  It is clear that 
queuing models are simpler, require less data, and provide more generic results than simulation 
(see also Green, 2006a).  However, discrete-event simulation permits modeling the details of 
complex patient flows.  Jacobson et al. (2006) present a list of steps that must be done carefully 
to model each healthcare scenario successfully using simulation and warn about the slim margins 
of tolerable error and the effects of such errors in lost lives.  Tucker et al. (1999) and Kao and 
Tung (1981) use simulation to validate, refine or otherwise complement the results obtained by 
queuing theory.  Albin et al. (1990) show how one can use queuing theory for get approximate 
results and then use simulation models to refine them.  We will not explore simulation studies 
further in this paper. 
Spreadsheets and software tools based on queuing theory research can automate the 
necessary calculations.  For example, Albin et al. (1990) use the QNA software, which calculates 
the time that patients are in a multi-node network, server utilization, the mean and variance of the 
number of customers at each node, the mean and variance of waiting time at each node, the mean 
and variance of the number of customers in the network, and the proportion of customers at each 
node that arrived from other nodes.  Aaby et al. (2006) describe the use of spreadsheets to 
implement queueing network models of mass vaccination and dispensing clinics.  However, the 
 3
authors are not aware of any other software that is specifically designed to analyze queuing 
models of healthcare processes. 
The next section (?Waiting time and Utilization Analysis?) is an overview of research 
into using queuing theory as an analytical tool to predict how particular healthcare configurations 
affect delay in patient service and healthcare resource utilization.  The ?System Design? section 
reviews research of a prescriptive nature that seeks to determine the optimal allocation of 
resources necessary to attain the goals determined by healthcare providers and decision makers.  
In ?Appointment Systems? we look at applications to appointment scheduling where the main 
challenge is reducing patient waiting without greatly increasing server idleness.  The ?System 
Size? section considers results for systems of different scales.  The paper ends with some 
observations about the use of queuing theory and suggestions for future research. 
Waiting Time and Utilization Analysis 
In a queuing system, minimizing the time that customers (in healthcare, patients) have to 
wait and maximizing the utilization of the servers or resources (in healthcare, doctors, nurses, 
hospital beds, e.g.) are conflicting goals.   
Reneging 
When a patient is waiting in a queue, he may decide to forgo the service because he does 
not wish to wait any longer.  This phenomenon, called reneging, is an important characteristic of 
many healthcare systems.  The probability that a patient reneges usually increases with the queue 
length and the patient?s estimate of how long he must wait to be served.  In systems where 
demand exceeds server capacity, reneging is the only way that a system attains a ?state of 
dysfunctional equilibrium? (Hall et al., 2006). 
 4
An important example of such a system is an emergency department.  Broyles and 
Cochran (2007) calculate the percentage of patients who leave an emergency department without 
getting help using arrival rate, service rate, utilization, capacity.  From this percentage, they 
determine the resulting revenue loss.   
It is possible to redesign a queuing system to reduce reneging.  A common approach is to 
separate patients by the type of service required.  Roche et al. (2007) find that the number of 
patients who leave an emergency department without being served is reduced by separating non-
acute patients and treating them in dedicated fast-track areas.  Most of their waiting would be for 
tests or test results after having first seen a doctor.  The paper also estimates the size of the 
waiting area for patients and those accompanying them. 
Variable Arrival Rate 
Although most analytical queuing models assume a constant customer arrival rate, many 
healthcare systems have a variable arrival rate.  In some cases, the arrival rate may depend upon 
time but be independent of the system state.  For instance, arrival rates change due to the time of 
day, the day of the week, or the season of the year.  In other cases, the arrival rate depends upon 
the state of the system.   
A system with congestion discourages arrivals.  Worthington (1991) suggests that 
increasing service capacity (the traditional method of attempting to reduce long queues) has little 
effect on queue length because as soon as patients realize that waiting times would reduce, the 
arrival rate increases, which increases the queue again.  Worthington (1987) presents an 
M(?q)/G/S model for service times of any fixed probability distribution and for arrival rates that 
decrease linearly with the queue length and the expected waiting time. 
 5
The arrival rate may increase over time due to population growth or other factors.  
Rosenquist (1987) studies how an increase in patient arrival rate affects waiting times and queue 
length for an emergency radiology service. 
Priority Queuing Discipline 
In most healthcare settings, unless an appointment system is in place, the queue discipline 
is either first-in-first-out or a set of patient classes that have different priorities (as in an 
emergency department, which treats patients with life-threatening injuries before others). 
McQuarrie (1983) shows that it is possible, when utilization is high, to minimize waiting 
times by giving priority to clients who require shorter service times.  This rule is a form of the 
shortest processing time rule that is known to minimize waiting times.  It is found infrequently in 
practice due to the perceived unfairness (unless that class of customers is given a dedicated 
server, as in supermarket check-out systems) and the difficulty of estimating service times 
accurately. 
When arriving patients are placed in different queues, each of which has a different 
service priority, the queue discipline may be preemptive or non-preemptive.  In the latter, low 
priority patients receive service only when no high priority patients are waiting, but the low 
priority patient who is receiving service is not interrupted if a high priority patient arrives and all 
servers are busy.  In the preemptive queue discipline, however, the service to a low priority 
patient is interrupted in this event.  Green (2006a) presents models for both queue disciplines. 
Siddhartan et al. (1996) analyze the effect on patient waiting times when primary care 
patients use the Emergency Department.  They propose a priority discipline for different 
categories of patients and then a first-in-first-out discipline for each category.  They find that the 
 6
priority discipline reduces the average wait time for all patients: however, while the wait time for 
higher priority patients reduces, lower priority patients endure a longer average waiting time. 
Haussmann (1970) investigates the relationship between the composition of prioritized 
queues and the number of nurses responding to inpatient demands.  The research finds that slight 
increases in the number of patients assigned to a nurse and/or a patient mix with more high-
priority demands result in very large waiting times for low priority patients. 
Worthington (1991) analyzes patient transfer from outpatient physicians to inpatient 
physicians.  The patient is assigned one of three priority levels.  Based on the priority level, there 
is a standard time period before which a referred patient should be scheduled to see the inpatient 
physician.  The model assumes sufficient in-patient capacity to treat the highest priority category 
within its standard time, and proposes sharing the remaining service capacity amongst the lower 
priority levels in such a manner that they each exceed their standard target times by the same 
percentage. 
Taylor et al. (1969) model an emergency anesthetic department operating with priority 
queuing discipline.  They are interested in the probability that a patient would have to wait more 
than a certain amount of time to be served. 
Fiems et al. (2007) investigate the effect of emergency requests on the waiting times of 
scheduled patients with deterministic processing times.  It is a preemptive repeat priority queuing 
system in which the emergency patients interrupt the scheduled patients and the latter?s service is 
restarted as opposed to being resumed.  This paper models a single server queue and divides time 
into equally long slots (discretizing time).  Periods of emergency interruptions are considered to 
have no server available from the point of view of the scheduled patients (vacation).  The result 
is a discrete-time queuing model with exhaustive vacations. 
 7
Blocking 
Blocking occurs when a queuing system places a limit on queue length.  For example, an 
outpatient clinic may turn away walk-in patients when its waiting room is full.  In a hospital, 
where in-patients can wait only in a bed, the limited number of beds may prevent a unit from 
accepting patients.  McManus et al. (2004) present a medical-surgical Intensive Care Unit where 
critically ill patients cannot be put in a queue and must be turned away when the facility is fully 
occupied.  This is a special case where the queue length cannot be greater than zero, which is 
called a pure loss model (see Green, 2006a, for more details). 
Koizumi et al. (2005) find that blocking in a chain of extended care, residential and 
assisted housing facilities results in upstream facilities holding patients longer than necessary.  
They analyze the effect of the capacity in downstream facilities on the queue lengths and waiting 
times of patients waiting to enter upstream facilities.  System-wide congestion could be caused 
by bottlenecks at only one downstream facility. 
System Design 
Because patient waiting is undesirable, limiting waiting times is an important objective 
when designing a healthcare system.  This section reviews work on determining system capacity 
based on desired system goals and requirements.  The variables of interest are usually staffing 
levels, beds, or other key resources.   
Bailey (1954) first establishes the existence in outpatient and inpatient clinics of a 
threshold capacity which occurs at the point where service supply equals demand.  When the 
number of servers is below this threshold, a clinic develops an infinite queue.  Slightly above this 
threshold, waiting time and queue length are low.  He argues that it is therefore sufficient to 
 8
design for a capacity that exceeds the expected demand (with stochastic error accounted for) by a 
value of 1 or 2.  Long waiting lists are most likely the result of accumulated backlog which can 
be depleted by a temporary surge in supply.  Seasonal variations in supply would also result in a 
sharp rise in waiting list length. 
Moore (1977) reduces customer waiting time for birth and death certificates at the Dallas 
bureau of vital statistics by decreasing the time required to serve each customer.  This research 
first uses queuing theory to calculate the service rate required to achieve a target waiting time of 
15 minutes.  This service rate is converted to the time required to serve one customer.  The 
reduced time required to serve each customer is attained through the use of new equipment and 
more efficient processes. 
Agnihothri and Taylor (1991) seek the optimal staffing at a hospital scheduling 
department that handles phone calls whose intensity varies throughout the day.  There are known 
peak and non-peak periods of the day.  The paper groups periods that receive similar call 
intensity and determines the necessary staffing for each such intensity, so that staffing varies 
dynamically with call intensity.  As a result of redistributing server capacity over time, customer 
complaints immediately reduced without an addition of staff.  Green (2006b) uses the same 
approach and names it Stationary Independent Period by Period (SIPP) to adjust staffing in order 
to reduce the percentage of patients that renege.  However, arguing that congestion starts some 
time after the arrival peak, the staffing levels should lag behind the service demand levels (lag 
SIPP). 
Blocking 
In systems with blocking, congestion not only increases patient waiting time but also 
reduces the throughput of the system.  Bruin et al. (2005) determine the number of beds required 
 9
to achieve a maximum turn away rate of 5% at the emergency cardiac department of the 
university medical centre of Amsterdam which implements the pure loss model.  Cooper and 
Corcoran (1974) deal with the same problem extended to a sequence of two stations each of 
which should have a maximum turn-away rate of 5%.  Milliken et al. (1972) seek a 1% turn-
away rate in an obstetrics department in which vaginal births have priority over scheduled 
caesarian sections.  They point out the benefits of economies of scale so that larger facilities 
incur lower bed investment per additional birth. 
Given a desired maximum turn-away rate, Bruin et al. (2007) determine the optimal 
number of beds in a cardiology department.  The cardiology department is modeled as a network 
of 3 sub-departments.  The research finds that too few beds downstream is the primary cause of 
refused admissions upstream and that congestion effects can add 20-30% to patient length of stay 
in the department.  They characterize having a fixed target utilization rate as unrealistic and 
conclude that a downstream utilization of 55% is necessary to attain a 2% turn-away rate.  As an 
alternative, departments could be merged to gain the benefits of economies of scale thereby 
meeting the goal at higher occupancy rates. 
Blair and Lawrence (1981) seek to design the capacity of horizontally integrated burn 
care facilities throughout the state of New York, so that no more than 5% of patients are turned 
away from the system.  If a patient goes to a facility which is fully occupied, that facility would 
refer to the patient to another which is not filled.  If all facilities are fully occupied, the patient is 
lost to the system.  First, they use queuing theory to determine the capacity of the entire system 
as if it were one queuing system.  This capacity is then allocated to facilities in a manner that 
best attains their individual goals.  They find such a system-planned approach ideal for a system 
with low demand and high infrastructural costs. 
 10
Tucker et al. (1999) consider activating a second operating room (OR) team during the 
night shift.  Using queuing theory, they find that the probability of two patients needing the OR 
services is negligible. 
Minimize Costs 
Determining server capacity by minimizing the costs in a healthcare queuing system is a 
special case of system design.  Most of the research assigns costs to patient waiting time and to 
each server.  After modeling the system using queuing theory, minimizing costs reduces to an 
exercise of finding the resource allocation that costs the least or generates the most profit.   
Keller and Laughhunn (1973) set out to determine the capacity with minimal costs 
required to serve patients at the Duke University Medical center.  They find that the current 
capacity is good but needs to be redistributed in time to accommodate patient arrival patterns. 
Young (1962a, b) proposes an incremental analysis approach in which the cost of an 
additional bed is compared with the benefits it generates.  Beds are added until the increased cost 
equals the benefits. 
Shimshak et al. (1981) consider a pharmacy queuing system with preemptive service 
priority discipline where the arrival of a prescription order suspends the processing of lower 
priority prescriptions.  Different costs are assigned to wait-times for prescriptions of different 
priorities. 
Gupta et al. (1971) choose the number of messengers required to transport patients or 
specimens in a hospital by assigning costs to the messenger and to the time during which a 
request is in queue.  In this problem, non-routine requests are superimposed on top of routine, 
scheduled requests.  The authors also calculate the number of servers required so that a given 
 11
percentage of requests does not exceed a given wait time and the average number of patients in 
the queue does not exceed a given threshold. 
Assuming a phase-type service distribution, Gorunescu et al. (2002a) assign costs based 
on a base stock inventory policy.  In this pure loss model, there is a holding cost associated with 
an empty bed, a penalty cost associated with each patient turned away, and a profit assigned to 
each day a bed is occupied. 
Khan and Callahan (1993) incorporate advertising into their model to control the demand 
for laboratory services.  For each staffing level, they determine the number of clients that would 
maximize profits.  They then choose the staffing level with maximum profits and apply the 
necessary amount of advertising that would attract the desired number of clients.  The model 
assumes that clients would leave without service if they wait above a certain amount of time. 
Rosenquist (1987) chooses staffing capacity in an outpatient radiology service with a 
limited waiting area by minimizing cost.  He suggests scheduling patients when possible and 
segregating patients based on expected examination duration.  Such measures would reduce 
variability and decrease expected waiting times. 
Gorunescu et al. (2002b) use backup beds (only staffed during peak demand) to reduce 
the probability of patient turn-away at a marginal cost.  The model assumes a phase-type service 
distribution. 
Appointment Systems 
Compared to systems without appointments, systems with appointments reduce the 
arrival variability and waiting times at the facility.  However, it is important to note that systems 
with appointments require patients to wait outside the facility.  Of course, because it is not at the 
facility, this waiting can be productive time and therefore has lower cost to the patient.  (Plus, 
 12
they do not occupy space in the facility?s waiting rooms.)  A key issue has been to reduce patient 
waiting times without causing a significant increase in doctor idle time, a significant cost for the 
healthcare facility. 
Bailey (1952, 1954) proposes (a) appointment interval and (b) consultant arrival time as 
two variables that determine the efficiency of an appointment system.  In order to find a balance 
between patient wait time and consultant idle time, first determine the relative values of patient 
time and consultant time.  The ratio of the total time wasted by all patients to the consultant?s 
idle time should equal the value of the consultant?s time relative to the patients?.  He chooses to 
assign individual appointment times at intervals equal to the average patient processing time and 
finds that the consultant should arrive at the same time as the second patient.   
Brahimi and Worthington (1991) design an appointment system to reduce the number of 
patients in the queue at any time, and reduce patient waiting time without significantly increasing 
doctor idle time.  They explore the effect of patients who do not show up for their appointments.  
The clinic starts out with a certain number of patients waiting and a maximum number of 
patients allowed at any time. 
In Vasanawala and Desser (2005) a radiology department has some time slots scheduled 
for routine radiology analysis.  Emergency requests may require rescheduling of scheduled 
requests.  Given a 1% or 5% probability of rescheduling, the authors use queuing theory to 
determine how many scheduled slots to leave empty during routine scheduling. 
Many outpatient appointments allow booking appointments months in advance.  
DeLaurentis et al. (2006) point out that patient no-shows without cancelling appointments could 
lead to waste of resources.  They propose implementing short-notice appointment systems based 
on a queuing network analysis tailored to the realities of any particular outpatient clinic.  Their 
 13
approach assumes the availability of a certain number of staff who can be distributed amongst 
the different stations of the queuing network in several combinations.  A combination is chosen 
based on its resulting utilization per station and expected patient length of stay in clinic.  The 
implementations of these ideas did not improve the appointment system, a failure which they 
attribute to the clinic using many visiting doctors and the patients being unable to schedule visits 
with their primary care physician at short notice. 
Bottlenecks 
In a queuing network, there are several nodes at which services are dispensed.  A patient 
may have to go through several nodes, and thus several queues in order to obtain the desired 
service.  In the context of appointment systems, we can expect nodes where the ratio of demand 
to available service capacity is relatively high to become bottlenecks.  Such bottlenecks would 
have high utilization and increase overall patient waiting times even though other nodes may 
have low utilization. 
Albin et al. (1990) find the bottlenecks at the Hurtado Health Center appointment clinic 
by collecting data and analyzing it using QNA, the queuing network analysis software program.  
Though their model deviates appreciably from assumptions, they are able to find the bottle necks 
by identifying the nodes where wait times are longest.  They then reduce overall waiting time by 
offering common-sense recommendations on a node-by-node basis. 
System Size 
As mentioned in the introduction, the size of healthcare organizations varies greatly.  
Following Hall et al. (2006), we can distinguish between three different scales.  The smallest 
scale is the department, ?a unit within a larger center oriented toward performing a single 
 14
function, or a group of closely related functions.?  The next larger scale organization is the health 
care center, which is a group of proximate, coordinated departments amongst which patients can 
flow.  The largest scale that we consider is the regional health system, a hierarchy of facilities 
with the most routine services provided by local clinics and the most specialized, resource-
intensive services provided at a few regional facilities.   
Hall et al. (2006) also present a macro system scale that considers the life cycle of an 
individual?s state of wellness and his interactions with the healthcare system throughout a 
lifetime.  We have found no research applying queuing theory to this type of system.   
Most of the research reviewed above has been done at the department scale.  Here we 
will highlight some work at the two larger scales.   
Kao and Tung (1981) investigate the redistribution of hospital beds amongst the inpatient 
departments of a hospital.  First, a baseline patient capacity is chosen for each department.  
Additional beds are then allocated to departments in a manner to minimize patient overflows 
from one department to another.  Forecasts are used to determine both the baseline bed allocation 
and the anticipated patient demand in order to minimize overflow. 
Blair and Lawrence (1981) investigate a regional hierarchy of burn care facilities where 
excess demand at one facility is absorbed by other facilities in the same region and overflows at 
one region are absorbed by other regions.  Worthington (1991) considers the coordination of 
patient flow from outpatient clinics to inpatient clinics.  Koizumi et al. (2005) model the mental 
healthcare system as a chain of facilities including acute hospitals, extended acute hospitals, 
residential facilities and supported housing. 
 15
Conclusions 
This paper has surveyed the use of queuing theory for the analysis of different types of 
healthcare processes.  Models for estimating waiting time and utilization, models for system 
design, and models for evaluating appointment systems have been presented.  The survey has 
reviewed models for departments (or units), facilities, and systems. 
We can draw some conclusions from the work surveyed above.  The variability in 
demand for healthcare services and service times mean that simplistic rules like mandating 
specific utilization levels or fixing patient to resource ratios would lead only to congestion and 
poor quality of service and are unlikely to be successful approaches to contain or reduce 
healthcare costs.  Larger organizations with more patients are able to attain the same quality of 
service at higher utilizations than smaller organizations.  Although appointment systems are 
often designed to avoid doctor idle time (without considering patient waiting time), it is possible 
to reduce patient wait time without significantly increasing doctor idle time. 
As long as increasing the productivity of healthcare organizations remains important, 
analysts will seek to apply relevant models to improve the performance of healthcare processes.  
This paper shows that many models are available today.  However, analysts will increasingly 
need to consider the ways in which distinct queuing systems within an organization interact.  
Consider, for instance, the following analogy from manufacturing, where a traditional factory 
(with a functional layout) has been transformed into manufacturing cells whose production is 
closely linked to the final assembly line through simple signals (such as kanbans).  While 
healthcare organizations don?t resemble factories, they do have links between subsystems (such 
as operating rooms and the post-operative recovery unit), and these interfaces need 
 16
improvements.  Developing appropriate models of the links (or interfaces) between the distinct 
queuing systems is an important direction for future research. 
Acknowledgements 
Cooperative Agreement Number U50/CCU302718 from the CDC to NACCHO 
supported this publication. Its contents are solely the responsibility of the University of Maryland 
and the Advanced Practice Center for Public Health Emergency Preparedness and Response of 
Montgomery County, Maryland, and do not necessarily represent the official views of CDC or 
NACCHO. 
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Biographical Sketches 
Samuel Fomundam is a graduate research assistant at the University of Maryland.  He 
earned his B.S. in computer engineering from the United States Military Academy at West Point 
in 2000.  
Jeffrey W. Herrmann is an associate professor at the University of Maryland, where he 
holds a joint appointment with the Department of Mechanical Engineering and the Institute for 
Systems Research. He is the director of the Computer Integrated Manufacturing Laboratory and 
Associate Director for the University of Maryland Quality Enhancement Systems and Teams 
(QUEST) Honors Fellows Program. He is a member of INFORMS, ASME, IIE, SME, and 
ASEE. 
Dr. Herrmann earned his B.S. in applied mathematics from Georgia Institute of 
Technology. As a National Science Foundation Graduate Research Fellow from 1990 to 1993, he 
received his Ph.D. in industrial and systems engineering from the University of Florida. His 
dissertation investigated production scheduling problems motivated by semiconductor 
manufacturing.  
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