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Modelling emergency medical services with phase-type distributions

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Health Systems

Abstract

Effective and efficient emergency medical services (EMS) are a critical part of a national healthcare system. This paper describes research to model EMS by better capturing ambulance service times using Coxian phase-type (PH) distributions. Distributions are fitted to both the overall cycle time for different classes of patient priorities, as well as to sub-cycles. Sub-cycles are the distinct identifiable parts of the ambulance cycle time, such as travel times, time on scene and turnaround time at the hospital. The Coxian PH fits have then been used within a priority simulation model to provide guidance on the number of ambulances required to meet response time targets. Results from using various numbers of phases from the fitted Coxian distributions are compared. The proposed benefit of using sub-cycle fits is that it more readily permits scenario modelling within the simulation, such as evaluating the impact of reducing the turnaround time on the overall response times in the ambulance system.

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References

  • Asmussen S, Nerman O and Olsson M (1998) EMpht. [WWW document] http://home.imf.au.dk/asmus/pspapers.html (accessed 2 February 2012).

  • Asmussen S, Nerman O and Olsson M (2008) Fitting phase-type distributions via the EM algorithm. Scandinavian Journal of Statistics 23 (4), 419–441.

    Google Scholar 

  • BBC (2011) Patient handovers at hospitals ‘delaying ambulances’. [WWW document] http://www.bbc.co.uk/news/uk-wales-politics-12190751 (accessed 12 August 2011).

  • Brotcorne L, Laporte G and Semet F (2003) Ambulance location and relocation models. European Journal of Operational Research 147 (3), 451–463.

    Article  Google Scholar 

  • Burt CW, McCaig LF and Valverde RH (2006) Analysis of ambulance transports and diversions among US emergency departments. Annals of Emergency Medicine 47 (4), 317–326.

    Article  Google Scholar 

  • Cairns KJ and Marshall AH (2011) Assessment of the benefits of discrete conditional survival models in modelling ambulance response times. International Journal of Health Management and Information 2 (l), 1–23.

    Google Scholar 

  • Chakravarthy S and Alfa AS (1996) Matrix-Analytic Methods in Stochastic Models (Lecture Notes in Pure and Applied Mathematics). CRC Press, New York, USA.

    Google Scholar 

  • Cox DR (1955) A use of complex probabilities in the theory of stochastic processes. In Mathematical Proceedings of the Cambridge Philosophical Society (BJ Green, Ed), Vol. 51, pp 313–319, Cambridge University Press, University of Cambridge, UK.

    Google Scholar 

  • Etzioni RD, Feuer EJ, Sullivan SD, Lin D, Hu C and Ramsey SD (1999) On the use of survival analysis techniques to estimate medical care costs. Journal of Health Economics 18 (3), 367–382.

    Article  Google Scholar 

  • Fackrell M (2009) Modelling healthcare systems with phase-type distributions. Health Care Management Science 12 (l), 11–26.

    Article  Google Scholar 

  • Faddy MJ and McClean SI (1999) Analysing data on lengths of stay of hospital patients using phase-type distributions. Applied Stochastic Models in Business and Industry 15 (4), 311–317.

    Article  Google Scholar 

  • Goldberg JB (2004) Operations research models for the deployment of emergency services vehicles. EMS Management Journal 1 (2039), 20–39.

    Google Scholar 

  • Green LV and Kolesar PJ (2004) Improving emergency responsiveness with management science. Management Science 50 (8), 1001–1014.

    Article  Google Scholar 

  • Harper PR (2002) A framework for operational modelling of hospital resources. Health Care Management Science 5 (3), 165–173.

    Article  Google Scholar 

  • Harper PR, Knight VA and Marshall AH (2009) Discrete conditional phase-type models utilising classification trees: application to modelling health service capacities. To Appear in European Journal of Operational Research 219 (3), 522–530.

    Article  Google Scholar 

  • Harper PR and Shahani AK (2002) Modelling for the planning and management of bed capacities in hospitals. Journal of the Operational Research Society 53 (1), 11–18.

    Article  Google Scholar 

  • Ingolfsson A, Akhmetshina E, Budge S, Li Y and Wu X (2007) A survey and experimental comparison of service-level-approximation methods for nonstationary M(t)/M/s(t) queueing systems with exhaustive discipline. INFORMS Journal on Computing 19 (2), 201–214.

    Article  Google Scholar 

  • Knight VA (2011) Personal webpage. http://www.vincent-knight.com (accessed 12 August 2011).

  • Knight VA, Harper PR and Smith L (2012) Ambulance allocation for maximal survival with heterogeneous outcome measures. Omega – The international Journal of Management Science (In Press) 10.1016/j.omega.2012.02.003.

  • Law AM and Kelton WD (2000) Simulation Modelling and Analysis. McGraw Hill Higher Education, India.

    Google Scholar 

  • Lightfoot Solutions (2007) Trasforming ambulance services and NHS direct wales. [WWW document] http://www.ambulance.wales.nhs.uk/assets/documents/c4cc0416-9fab-4dea-8753-247a9431c4c7633446359123733750.pdf (accessed 2 February 2012).

  • Lowthian JA, Jolley DJ, Curtis AJ, Currell A, Cameron PA, Stoelwinder JU and McNeil JJ (2011) The challenges of population ageing: accelerating demand for emergency ambulance services by older patients, 1995–2015. The Medical Journal of Australia 194 (11), 574–578.

    Google Scholar 

  • Marshall AH, Burns L and Shaw B (2007) Patient activity in hospital using discrete conditional phase-type (DC–Ph) models. Recent Advances in Stochastic Modeling and Data Analysis 1 (1), 154–161.

    Article  Google Scholar 

  • Marshall AH and McClean SI (2004) Using Coxian phase-type distributions to identify patient characteristics for duration of stay in hospital. Health Care Management Science 7 (4), 285–289.

    Article  Google Scholar 

  • McLachlan GJ and Krishnan T (1996) The EM Algorithm and Extensions (Wiley Series in Probability and Statistics). Wiley-Blackwell, New York, USA.

    Google Scholar 

  • Neuts MF (1995) Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. Dover Publications Inc, Toronto, Ontario, Canada.

    Google Scholar 

  • Olsson M (1996) Estimation of phase-type distributions from censored data. Scandinavian Journal of Statistics 23 (4), 443–460.

    Google Scholar 

  • Peacock PJ, Peacock JL, Victor CR and Chazot C (2005) Changes in the emergency workload of the London ambulance service between 1989 and 1999. Emergency Medicine Journal: EMJ 22 (1), 56–59.

    Article  Google Scholar 

  • Pryor S (2009) Welsh ambulance services NHS trust. Trust board operational performance: July. Technical report.

  • Schwarz G (1978) Estimating the dimension of a model. The Annals of Statistics 6 (2), 461–464.

    Article  Google Scholar 

  • Shaw B and Marshall AH (2011) Modelling the flow of congestive heart failure patients through a hospital system. Journal of the Operational Research Society 58 (2), 212–218.

    Article  Google Scholar 

  • Snyder DE, White RD and Jorgenson DB (2007) Outcome prediction for guidance of initial resuscitation protocol: shock first or CPR first. Resuscitation 72 (1), 45–51.

    Article  Google Scholar 

  • Svenson JE (2000) Patterns of use of emergency medical transport: a population-based study. The American Journal of Emergency Medicine 18 (2), 130–134.

    Article  Google Scholar 

  • Taylor GJ, McClean SI and Millard PH (2000) Stochastic models of geriatric patient bed occupancy behaviour. Journal of the Royal Statistical Society: Series A (Statistics in Society) 163 (1), 39–48.

    Article  Google Scholar 

  • Vandeventer S, Studnek JR, Garrett JS, Ward SR, Staley K and Blackwell T (2011) The association between ambulance hospital turnaround times and patient acuity, destination hospital, and time of day. Prehospital Emergency Care: Official Journal of the National Association of EMS Physicians and the National Association of State EMS Directors 15 (3), 366–370.

    Article  Google Scholar 

  • Vile JL, Gillard J, Harper PR and Knight VA (2012) Predicting ambulance demand using singular spectrum analysis. Journal of the Operational Research Society (In Press) 10.1057/jors.2011.160.

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Authors and Affiliations

  1. School of Mathematics, Cardiff University, Cardiff, U.K.

    Vincent A Knight & Paul R Harper

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  1. Vincent A Knight
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  2. Paul R Harper
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Correspondence to Vincent A Knight.

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Knight, V., Harper, P. Modelling emergency medical services with phase-type distributions. Health Syst 1, 58–68 (2012). https://doi.org/10.1057/hs.2012.1

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  • DOI: https://doi.org/10.1057/hs.2012.1

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