title: A spoonful of math helps the medicine go down: an illustration of how healthcare can benefit from mathematical modeling and analysis.
abstract: OBJECTIVE
  A recent joint report from the Institute of Medicine and the National Academy of Engineering, highlights the benefits of--indeed, the need for--mathematical analysis of healthcare delivery. Tools for such analysis have been developed over decades by researchers in Operations Research (OR). An OR perspective typically frames a complex problem in terms of its essential mathematical structure. This article illustrates the use and value of the tools of operations research in healthcare. It reviews one OR tool, queueing theory, and provides an illustration involving a hypothetical drug treatment facility.
  
  
  METHODS
  Queueing Theory (QT) is the study of waiting lines. The theory is useful in that it provides solutions to problems of waiting and its relationship to key characteristics of healthcare systems. More generally, it illustrates the strengths of modeling in healthcare and service delivery.Queueing theory offers insights that initially may be hidden. For example, a queueing model allows one to incorporate randomness, which is inherent in the actual system, into the mathematical analysis. As a result of this randomness, these systems often perform much worse than one might have guessed based on deterministic conditions. Poor performance is reflected in longer lines, longer waits, and lower levels of server utilization.As an illustration, we specify a queueing model of a representative drug treatment facility. The analysis of this model provides mathematical expressions for some of the key performance measures, such as average waiting time for admission.
  
  
  RESULTS
  We calculate average occupancy in the facility and its relationship to system characteristics. For example, when the facility has 28 beds, the average wait for admission is 4 days. We also explore the relationship between arrival rate at the facility, the capacity of the facility, and waiting times.
  
  
  CONCLUSIONS
  One key aspect of the healthcare system is its complexity, and policy makers want to design and reform the system in a way that affects competing goals. OR methodologies, particularly queueing theory, can be very useful in gaining deeper understanding of this complexity and exploring the potential effects of proposed changes on the system without making any actual changes.
authors.affiliation: Department of Maternal and Child Health, UNC Gillings School of Global Public Health, 407A Rosenau Hall, CB# 7445, Chapel Hill, NC 27599 USA. emfoster@unc.edu
authors.forename: E Michael
authors.initials: EM
authors.lastname: Foster
authors.forename: Michael R
authors.initials: MR
authors.lastname: Hosking
authors.forename: Serhan
authors.initials: S
authors.lastname: Ziya
mesh_terms: D003661:Decision Support Techniques; D003695:Delivery of Health Care / Q000458:organization & administration*; D017598:Efficiency, Organizational; D006276:Health Facility Size; D006801:Humans; D008962:Models, Theoretical; D009874:Operations Research; D016320:Substance Abuse Treatment Centers / Q000458:organization & administration* / Q000706:statistics & numerical data; D019966:Substance-Related Disorders / Q000534:rehabilitation; D013597:Systems Analysis; D014600:Utilization Review; D014850:Waiting Lists