Concept Architecture
Concept
Theoretically, Queuing Model is a mathematical model that represents the formation and movement of queues arising when demand for healthcare services exceeds available service capacity. It is founded on queueing theory and stochastic processes, describing the interaction between arrivals, waiting, service and departures. In health economics, queuing models are used to evaluate healthcare resource utilisation, waiting times, service capacity and the economic consequences of alternative delivery strategies.
Mathematically, queuing models are represented using stochastic processes that describe arrival rates, service rates, queue disciplines and the number of servers. The most widely recognised framework is Kendall's notation, such as M/M/1 or M/M/c, where arrival and service processes follow specified probability distributions. Model equations estimate expected queue lengths, waiting times, resource utilisation and system performance measures.
In practice, queuing models are parameterised using observed arrival rates, service times and resource availability obtained from hospital information systems, administrative databases or observational studies. Parameters are estimated statistically and used to compare alternative staffing levels, service configurations or intervention strategies. Model outputs are frequently incorporated into health economic evaluations to quantify waiting costs, resource use and operational efficiency.
Purpose
Used to estimate waiting times, queue lengths, resource utilisation and service capacity, supporting economic evaluation and optimisation of healthcare delivery systems.
Mathematical Formulae
Primary Formula
For an M/M/1 queue:
L = ?/(? ? ?)
where:
- L = expected number of entities in the system
- ? = arrival rate
- ? = service rate
- ? < ?
Supporting Formulae
Utilisation:
? = ?/?
Expected waiting time in the system:
W = 1/(? ? ?)
Little's Law:
L = ?W
Expected waiting time in the queue:
Wq = ?/[?(? ? ?)]
Expected queue length:
Lq = ?Wq
Related Mathematical Methods
- Queueing theory
- Markov processes
- Birth-death processes
- Stochastic processes
- Probability distributions
- Discrete-event simulation
- Monte Carlo simulation
Example
An outpatient clinic receives patients at an average rate of:
? = 8 patients/hour
The clinician treats:
? = 10 patients/hour
Utilisation is:
? = 8/10 = 0.80
Expected number of patients in the system:
L = 8/(10 ? 8) = 4
Expected time spent in the system:
W = 1/(10 ? 8) = 0.5 hours
or 30 minutes. These estimates can be incorporated into an economic evaluation to quantify patient waiting costs and assess the value of increasing clinical capacity.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
/ | =ArrivalRate/ServiceRate | Calculate resource utilisation |
| IF | =IF(ArrivalRate<ServiceRate,ArrivalRate/(ServiceRate-ArrivalRate),""Unstable"") | Estimate expected system size for a stable queue |
| AVERAGE | =AVERAGE(ServiceTimes) | Estimate mean service duration |
| COUNTIFS | =COUNTIFS(Status,""Waiting"") | Measure observed queue length |
| Data Table | What-If Analysis | Evaluate alternative staffing and service capacities |
VBA (Optional)
Automate simulation of patient arrivals, service completion and waiting-time analyses under alternative healthcare capacity scenarios.
Sources
- Kendall DG. Stochastic Processes Occurring in the Theory of Queues and Their Analysis by the Method of the Imbedded Markov Chain. Annals of Mathematical Statistics. 1953.
- Gross D, Shortle JF, Thompson JM, Harris CM. Fundamentals of Queueing Theory. 5th ed. Wiley.
- Green LV. Queueing analysis in healthcare. In: Hall RW, ed. Patient Flow: Reducing Delay in Healthcare Delivery. Springer.
- Karnon J, Stahl J, Brennan A, et al. Modeling using discrete event simulation: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force-4. Medical Decision Making. 2012;32(5):701?711.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press; 2006.
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. 4th ed. Oxford University Press.
Related Concepts (3)
Library
Tools & Resources
1
hesim — Health Economic Simulation Modeling and Decision Analysis (R package) — Devin Incerti & Jeroen P. Jansen, R package ed., 2024 (CRAN)
A modular, computationally efficient R package for building and analysing health economic simulation models — cohort state-transition, partitioned survival, and individual-level continuous-time models — with fast individual-patient simulation and PSA via C++.
Software (R package)View source →
Frequently Asked Questions (6)
What is a queuing model?
A mathematical model describing how entities wait in line for limited servers, used to analyse average waiting time and server utilisation.
Source: Erlang 1909
Who founded queuing theory?
Queuing theory began with the work of the Danish engineer Agner Krarup Erlang in the early twentieth century, who studied congestion in telephone exchanges to work out how many lines were needed to keep waiting acceptable. His mathematics of arrivals, service, and waiting became the foundation for analysing any system where demand meets limited capacity. The same theory is now applied to clinics, emergency departments, and operating theatres. Erlang (1909) laid the groundwork.
Source: Erlang 1909
What are the components of a queuing model?
A queuing model comprises an arrival process, describing how entities arrive, often assumed random at a constant average rate; a service process, describing how long service takes; the number of servers available; the queue discipline, the order of service; and the system capacity. From these, the model derives performance measures such as average waiting time, queue length, and the proportion of time servers are busy. Standard notation describes a model by its arrival and service distributions and number of servers, such as the M/M/1 queue.
Source: Erlang 1909
What does a queuing model calculate?
A queuing model calculates performance measures of a waiting system: the average time entities wait, the average number waiting in the queue and in the system, the utilisation of the servers, and the probability of a given number in the system. These follow from the arrival and service rates and the number of servers. For example, they show how waiting time rises sharply as utilisation approaches capacity. Such measures let the model quantify congestion and assess how it depends on demand and the resources provided.
Source: Erlang 1909
In a queuing model, why does waiting time rise as utilisation increases?
Waiting time rises as utilisation increases because, as the arrival rate approaches the capacity to serve, the system has less spare time to clear the queue, and random variation in arrivals and service times causes queues to build up that take longer to dissipate. Queuing models show that waiting time increases non-linearly, rising sharply as utilisation nears one, so a system run close to full capacity suffers long and volatile waits. This relationship is important for deciding how much capacity is needed to keep waiting acceptable.
Source: Erlang 1909
How are queuing models used in health?
Queuing models are used in health to analyse and plan services where patients wait for limited resources, such as staff, beds, clinics, or equipment, estimating waiting times and the capacity needed to keep them acceptable. They show how waiting depends on demand and capacity, helping decide staffing and bed numbers and understand why running services near full capacity causes long waits. While real health systems may be too complex for simple formulas, queuing models provide insight and inform more detailed simulation of patient flow.
Source: Erlang 1909
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 6 Oct 2025
Content version: 1.0.0
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