Modelling activities at a neurological rehabilitation unit

A queuing model of a specialist neurological rehabilitation unit is studied. The application is to the Neurological Rehabilitation Centre at Rookwood Hospital (Cardiff, UK), the national rehabilitation unit for Wales. Due to high demand this 21-bed inpatient facility is nearly always at maximum occupancy, and with a significant bed-cost per day this makes it a prime candidate for mathematical modelling. Central to this study is the concept that treatment intensity has an effect on patient length of stay. The model is constructed in four stages. First, appropriate patient groups are determined based on a number of patient-related attributes. Second, a purpose-built scheduling program is used to deduce typical levels of treatment to patients of each group. These are then used to estimate the mean length of stay for each patient group. Finally, the queuing model is constructed. This consists of a number of disconnected homogeneous server queuing systems; one for each patient group. A Coxian phase-type distribution is fitted to the length of time from admission until discharge readiness and an exponential distribution models the remainder of time until discharge. Some hypothetical scenarios suggested by senior management are then considered and compared on the grounds of a number of performance measures and cost implications.

[1]  M. R. Osborne.  Fisher's Method of Scoring , 1992 .

[2]  Paul R. Harper,et al.  TreeWorks: Advances in Scalable Decision Trees , 2008, Int. J. Heal. Inf. Syst. Informatics.

[3]  Lynne Turner-Stokes,et al.  Cost-efficiency of longer-stay rehabilitation programmes: Can they provide value for money? , 2007, Brain injury.

[4]  Philip J. Kiviat,et al.  GASP--A General Activity Simulation Program. , 1964 .

[5]  Pierre Hansen,et al.  First vs. best improvement: An empirical study , 1999, Discret. Appl. Math..

[6]  Ren Asmussen,et al.  Fitting Phase-type Distributions via the EM Algorithm , 1996 .

[7]  Lester Lipsky,et al.  The Importance of Power-Tail Distributions for Modeling Queueing Systems , 1999, Oper. Res..

[8]  D Lee,et al.  An analysis of short-stay hospital records and measurement of the probability discharged as cured from the Severance Hospital, 1967-1969. , 1970, Yonsei medical journal.

[9]  P H Millard,et al.  Balancing acute and long-term care: the mathematics of throughput in departments of geriatric medicine. , 1991, Methods of information in medicine.

[10]  B. Jennett,et al.  ASSESSMENT OF OUTCOME AFTER SEVERE BRAIN DAMAGE A Practical Scale , 1975, The Lancet.

[11]  B L Hanson,et al.  A statistical model for length of stay in a mental hospital. , 1973, Health services research.

[12]  Philip Mandy,et al.  Interdisciplinary rather than multidisciplinary or generic practice , 1996 .

[13]  D. Cope,et al.  Head injury rehabilitation: benefit of early intervention. , 1982, Archives of physical medicine and rehabilitation.

[14]  Alexander Kolker,et al.  Process Modeling of Emergency Department Patient Flow: Effect of Patient Length of Stay on ED Diversion , 2008, Journal of Medical Systems.

[15]  L Turner-Stokes,et al.  Efficiency of specialist rehabilitation in reducing dependency and costs of continuing care for adults with complex acquired brain injuries , 2006, Journal of Neurology, Neurosurgery & Psychiatry.

[16]  Herbert A. Sturges,et al.  The Choice of a Class Interval , 1926 .

[17]  M. A. Johnson,et al.  Matching moments to phase distributions: nonlinear programming approaches , 1990 .

[18]  D. Gross,et al.  Using the Transform Approximation Method to Analyze Queues with Heavy-Tailed Service , 2003 .

[19]  A Marazzi,et al.  Fitting the distributions of length of stay by parametric models. , 1998, Medical care.

[20]  J M Linacre,et al.  FUNCTIONAL STATUS AND THERAPEUTIC INTENSITY DURING INPATIENT REHABILITATION , 1995, American journal of physical medicine & rehabilitation.

[21]  Cynthia Barnhart,et al.  Planning for Robust Airline Operations: Optimizing Aircraft Routings and Flight Departure Times to Minimize Passenger Disruptions , 2006, Transp. Sci..

[22]  M. A. Johnson,et al.  An investigation of phase-distribution moment-matching algorithms for use in queueing models , 1991, Queueing Syst. Theory Appl..

[23]  Gregory K. Miller.  Maximum likelihood estimation for the Erlang integer parameter , 1999 .

[24]  D. Mclellan,et al.  The effects of increased rehabilitation therapy after brain injury: results of a prospective controlled trial , 2001, Clinical rehabilitation.

[25]  Mark Speechley,et al.  A systematic review of the rehabilitation of moderate to severe acquired brain injuries , 2007, Brain injury.

[26]  Jeffrey Deacon Griffiths,et al.  Scheduling physiotherapy treatment in an inpatient setting , 2012 .

[27]  M Rappaport,et al.  Disability rating scale for severe head trauma: coma to community. , 1982, Archives of physical medicine and rehabilitation.

[28]  L Mayhew,et al.  Using queuing theory to analyse the Government’s 4-h completion time target in Accident and Emergency departments , 2008, Health care management science.

[29]  S. A. Nozaki,et al.  Approximations in finite-capacity multi-server queues by Poisson arrivals , 1978, Journal of Applied Probability.

[30]  Maria-Pia Victoria-Feser,et al.  A Robust Test for Non‐nested Hypotheses , 1997 .

[31]  U. Narayan Bhat,et al.  Estimation for renewal processes with unobservable gamma or Erlang interarrival times , 1997 .

[32]  Mohamed Shoukri,et al.  Sampling properties of estimators of the log‐logistic distribution with application to Canadian precipitation data , 1988 .

[33]  Fred Glover,et al.  Tabu Search: A Tutorial , 1990 .

[34]  Sanja Petrovic,et al.  Comparison of Algorithms for Nurse Rostering Problems , 2008 .

[35]  Charles Knessl,et al.  An Integral Equation Approach to the M/G/2 Queue , 1990, Oper. Res..

[36]  Jeffrey Jutai,et al.  The efficacy of acquired brain injury rehabilitation , 2007, Brain injury.

[37]  Jeffrey W. Herrmann,et al.  A Survey of Queuing Theory Applications in Healthcare , 2007 .

[38]  W. F. Blackerby,et al.  Intensity of rehabilitation and length of stay. , 1990, Brain injury.

[39]  Graham Kendall,et al.  Scheduling English football fixtures over holiday periods , 2008, J. Oper. Res. Soc..

[40]  Wayne A. Gordon,et al.  Factors that influence costs and length of stay of persons with traumatic brain injury in acute care and inpatient rehabilitation , 1993 .

[41]  Richard E. Nance,et al.  A history of discrete event simulation programming languages , 1993, HOPL-II.

[42]  E. Mackenzie,et al.  HEAD INJURIES: COSTS AND CONSEQUENCES , 1991 .

[43]  Peter H. Millard,et al.  Where to treat the older patient? Can Markov models help us better understand the relationship between hospital and community care? , 2007, J. Oper. Res. Soc..

[44]  Gustave J. Rath,et al.  Nurse Scheduling Using Mathematical Programming , 1976, Oper. Res..

[45]  E. Maaløe.  Approximation Formulae for Estimation of Waiting-Time in Multiple-Channel Queueing System , 1973 .

[46]  J. O. Mayhugh,et al.  Steady State Solution of the Queue M/E k /r , 1968 .

[47]  Maria Sandhaug,et al.  Functional level during sub-acute rehabilitation after traumatic brain injury: Course and predictors of outcome , 2010, Brain injury.

[48]  Preeti Wanti Srivastava,et al.  A Log-Logistic Step-Stress Model , 2008, IEEE Transactions on Reliability.

[49]  A. Tennant,et al.  A randomised controlled trial to determine the effect of intensity of therapy upon length of stay in a neurological rehabilitation setting. , 2002, Journal of rehabilitation medicine.

[50]  Millard Ph,et al.  Geriatric medicine : a new method of measuring bed usage and a theory for planning. , 1988 .

[51]  Lynne Turner-Stokes,et al.  The Northwick Park Care Needs Assessment (NPCNA): a directly costable outcome measure in rehabilitation , 1999, Clinical rehabilitation.

[52]  Ian Roberts,et al.  Systematic review of prognostic models in traumatic brain injury , 2006, BMC Medical Informatics Decis. Mak..

[53]  Sally McClean,et al.  Using a continuous-time Markov model with Poisson arrivals to describe the movements of geriatric patients , 1998 .

[54]  M. Donaghy,et al.  Principles of neurological rehabilitation , 2003, Journal of neurology, neurosurgery, and psychiatry.

[55]  Ann Turner,et al.  Occupational therapy and physical dysfunction : principles, skills and practice , 1992 .

[56]  T W Meade,et al.  Remedial therapy after stroke: a randomised controlled trial. , 1981, British medical journal.

[57]  F. Ballester,et al.  Length of stay in a hospital emergency room due to asthma and chronic obstructive pulmonary disease: Implications for air pollution studies , 2000, European Journal of Epidemiology.

[58]  S McClean,et al.  Stochastic models for geriatric in-patient behaviour. , 1994, IMA journal of mathematics applied in medicine and biology.

[59]  F. Mahoney,et al.  FUNCTIONAL EVALUATION: THE BARTHEL INDEX. , 2018, Maryland state medical journal.

[60]  D. W. Scott.  On optimal and data based histograms , 1979 .

[61]  D. Ruppert.  Statistics and Data Analysis for Financial Engineering , 2010 .

[62]  Hassan S. Bakouch,et al.  Probability, Markov chains, queues, and simulation , 2011 .

[63]  W. J. Studden,et al.  Tchebycheff Systems: With Applications in Analysis and Statistics. , 1967 .

[64]  Gilbert Laporte,et al.  Metaheuristics: A bibliography , 1996, Ann. Oper. Res..

[65]  A S Morgan,et al.  Early intervention in severe head injury: long-term benefits of a formalized program. , 1992, Archives of physical medicine and rehabilitation.

[66]  Saul Shapiro.  The M-Server Queue with Poisson Input and Gamma-Distributed Service of Order Two , 1966, Oper. Res..

[67]  Daniel J Vreeman,et al.  Evidence for electronic health record systems in physical therapy. , 2006, Physical therapy.

[68]  P. Harper.  A Framework for Operational Modelling of Hospital Resources , 2002, Health care management science.

[69]  Pia Veldt Larsen,et al.  In All Likelihood: Statistical Modelling and Inference Using Likelihood , 2003 .

[70]  Miklós Telek,et al.  Matching More Than Three Moments with Acyclic Phase Type Distributions , 2007 .

[71]  G. Schwarz.  Estimating the Dimension of a Model , 1978 .

[72]  J. Salonen,et al.  The significance of intensity of rehabilitation of stroke--a controlled trial. , 1985, Stroke.

[73]  S I McClean,et al.  Markov Chain Modelling for Geriatric Patient Care , 2005, Methods of Information in Medicine.

[74]  L Nyström,et al.  Statistical Analysis , 2008, Encyclopedia of Social Network Analysis and Mining.

[75]  Michael A. Johnson,et al.  Matching moments to phase distri-butions: mixtures of Erlang distribution of common order , 1989 .

[76]  Jeffrey C. Lagarias,et al.  Convergence Properties of the Nelder-Mead Simplex Method in Low Dimensions , 1998, SIAM J. Optim..

[77]  Dieter Fiems,et al.  Computationally efficient evaluation of appointment schedules in health care , 2014, Eur. J. Oper. Res..

[78]  P H Millard,et al.  Developing a Bayesian belief network for the management of geriatric hospital care , 2001, Health care management science.

[79]  Miklós Telek,et al.  PhFit: A General Phase-Type Fitting Tool , 2002, Computer Performance Evaluation / TOOLS.

[80]  Y. Barlas,et al.  Discrete vs. Continuous Simulation: When Does It Matter? 1 , 2009 .

[81]  Walter M. High,et al.  Functional Measures After Traumatic Brain Injury: Ceiling Effects of FIM, FIM+FAM, DRS, and CIQ , 1996 .

[82]  Lynne Turner-Stokes,et al.  Rehabilitation following aquired brain injury: national clinical guidelines , 2003 .

[83]  C. O'Cinneide.  On non-uniqueness of representations of phase-type distributions , 1989 .

[84]  Seyfettin Noyan Ogulata,et al.  Personnel and Patient Scheduling in the High Demanded Hospital Services: A Case Study in the Physiotherapy Service , 2008, Journal of Medical Systems.

[85]  Virginia Mary Lo,et al.  Heuristic Algorithms for Task Assignment in Distributed Systems , 1988, IEEE Trans. Computers.

[86]  R. J. Lawrence.  The Lognormal Distribution of the Duration of Strikes , 1984 .

[87]  H J Steadman,et al.  Assessing the New York City involuntary outpatient commitment pilot program. , 2001, Psychiatric services.

[88]  Maliha S. Nash.  Data Preparation for Analytics Using SAS , 2008, Technometrics.

[89]  C. Granger,et al.  The functional independence measure: a new tool for rehabilitation. , 1987, Advances in clinical rehabilitation.

[90]  A. H. Marshalla,et al.  Conditional phase-type distributions for modelling patient length of stay in hospital , 2003 .

[91]  A J Thompson,et al.  Measuring change in disability after inpatient rehabilitation: comparison of the responsiveness of the Barthel Index and the Functional Independence Measure , 1999, Journal of neurology, neurosurgery, and psychiatry.

[92]  Kryss McKenna.  OCCUPATIONAL THERAPY AND PHYSICAL DYSFUNCTION: PRINCIPLES, SKILLS AND PRACTICE (5TH ED.) , 2004 .

[93]  R. Hanlon.  Motor learning following unilateral stroke. , 1996, Archives of physical medicine and rehabilitation.

[94]  Mukundja Hilde,et al.  HIV/AIDS epidemic , 2003 .

[95]  Mor Harchol-Balter,et al.  A Closed-Form Solution for Mapping General Distributions to Minimal PH Distributions , 2003, Computer Performance Evaluation / TOOLS.

[96]  Chetwyn C. H. Chan,et al.  Does intensive rehabilitation improve the functional outcome of patients with traumatic brain injury (TBI)? A randomized controlled trial , 2007, Brain injury.

[97]  D. Kendall.  Stochastic Processes Occurring in the Theory of Queues and their Analysis by the Method of the Imbedded Markov Chain , 1953 .

[98]  Gwo-Hshiung Tzeng,et al.  Multi-criteria task assignment in workflow management systems , 2003, 36th Annual Hawaii International Conference on System Sciences, 2003. Proceedings of the.

[99]  S E Ross,et al.  Effects of intensity of treatment and length of stay on rehabilitation outcomes. , 1992, Brain injury.

[100]  G. Spivack,et al.  Admission examination factors predicting cognitive improvement during acute brain injury rehabilitation. , 1997, Brain injury.

[101]  Paola Festa,et al.  GREEDY RANDOMIZED ADAPTIVE SEARCH PROCEDURES , 2003 .

[102]  E W Lee,et al.  The analysis of correlated panel data using a continuous-time Markov model. , 1998, Biometrics.

[103]  S. Kolakowsky-Hayner,et al.  The relationship between therapy intensity and rehabilitative outcomes after traumatic brain injury: a multicenter analysis. , 2003, Archives of physical medicine and rehabilitation.

[104]  Adele H. Marshall,et al.  RECENT DEVELOPMENTS IN FITTING COXIAN PHASE- TYPE DISTRIBUTIONS IN HEALTHCARE , 2009 .

[105]  J D Littler,et al.  A PROOF OF THE QUEUING FORMULA , 1961 .

[106]  Adele H. Marshall,et al.  Experimenting with the Coxian Phase-Type Distribution to Uncover Suitable Fits , 2012 .

[107]  S. McClean,et al.  Patterns of Length of Stay after Admission in Geriatric Medicine: An Event History Approach , 1993 .

[108]  Vijay P. Singh,et al.  Parameter estimation for 2-parameter log-logistic distribution (LLD2) by maximum entropy , 1995 .

[109]  H. T. Papadopoulos.  Analysis of production lines with Coxian service times and no intermediate buffers , 1998 .

[110]  Vicent Pla,et al.  ON A MULTISERVER FINITE BUFFER QUEUE WITH IMPATIENT CUSTOMERS ∗ , 2004 .