In this paper, we present a modelling framework for patient flow in a healthcare system using semi-open queueing network models, which introduces a total bed constraint, above which new patients will be refused admission. Hence this model provides a realistic representation of a real system. This approach enables us to have access to a range of established methods that deals with queueing network models. We demonstrate the usefulness of the model in the context of a geriatric department and show that hospital managers can use this model to gain better understanding of the dynamics of patient flow and to study potential long-term impacts of policy changes

Chaussalet, T.J.

Rees, M.

Xie, H.

WestminsterResearch

   WestminsterResearch http://www.wmin.ac.uk/westminsterresearch   A semi-open queueing network approach to the analysis of patient flow in healthcare systems.  Haifeng Xie Thierry Chaussalet Mike Rees  School of Informatics   Copyright © [2007] IEEE. Reprinted from the proceedings of the 20th IEEE International Symposium on Computer-Based Medical Systems. IEEE CBMS 2007, Maribor, Slovenia, 20-22 June 2007. IEEE, Los Alamitos, USA, pp. 719-724. ISBN 0769529054.  This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of the University of Westminster's products or services.  Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.   The WestminsterResearch online digital archive at the University of Westminster aims to make the research output of the University available to a wider audience.  Copyright and Moral Rights remain with the authors and/or copyright owners. Users are permitted to download and/or print one copy for non-commercial private study or research.  Further distribution and any use of material from within this archive for profit-making enterprises or for commercial gain is strictly forbidden.      Whilst further distribution of specific materials from within this archive is forbidden, you may freely distribute the URL of the University of Westminster Eprints (http://www.wmin.ac.uk/westminsterresearch).  In case of abuse or copyright appearing without permission e-mail wattsn@wmin.ac.uk. A Semi-open Queueing Network Approach to the Analysis of PatientFlow in Healthcare SystemsHaifeng Xie, Thierry Chaussalet, Mike ReesHealth and Social Care Modelling GroupSchool of InformaticsUniversity of Westminster, London, UK{xieh, chausst, reesm}@wmin.ac.ukAbstractIn this paper, we present a modelling framework for patient flow in a healthcare system usingsemi-open queueing network models, which introduces a total bed constraint, above which newpatients will be refused admission. Hence this model provides a realistic representation of a realsystem. This approach enables us to have access to a range of established methods that deals withqueueing network models. We demonstrate the usefulness of the model in the context of a geriatricdepartment and show that hospital managers can use this model to gain better understanding of thedynamics of patient flow and to study potential long-term impacts of policy changes.1 IntroductionPatient flow has been recognised as a major factor in improving the efficiency of a healthcaresystem [1]. In a healthcare system, patients can be regarded as progressing through a set of logicalstages that resemble the process of care, such as assessment, diagnosis, rehabilitation and long-staycare, etc [2, 3]. Adopting this view, compartmental models have been developed to explain patientflow through a geriatric department [2]. These models show different phases of care interact toinfluence admission and discharge of patients in a department of geriatric medicine. Most studiesfocus on deriving steady-state solutions for systems without bed capacity constraints. That is asystem assuming to have as many beds available as required at any time. Clearly this assumptionis generally untrue in reality. Therefore, an introduction of bed capacity constraint in such modelswill provide a more realistic representation of the real world situation.In a previous study [5], we introduced a modelling framework for patient flow in a healthcaresystem using a closed queueing network model [6]. This approach enabled us to have access to arange of established methods that deals with queueing network models. It also provides a naturalplatform for incorporating a capacity constraint on the system, e.g. total number of beds availablein a hospital department. In this paper, we present an extension to this approach, which relaxesthe assumption in our previous study [5] that the system is always full. This approach, which isbased on semi-open queueing network models, introduces a total bed constraint (above which newpatients will be refused admission), which relaxes the assumption that the system is always full(i.e. a closed network) and hence provides a more realistic representation of a real system. Weshow that our results are applicable to both the closed network model and open network model (asdiscussed by Irvine et al [4]) since they are simply special cases of the semi-open network model.Twentieth IEEE International Symposium on Computer-Based Medical Systems (CBMS'07)0-7695-2905-4/07 $20.00  © 2007Authorized licensed use limited to: University of Westminster. Downloaded on June 12, 2009 at 05:26 from IEEE Xplore.  Restrictions apply.Figure 1. Health care system modelled with M phases. pi,i+1 (i = 1, . . . , M −1) represent the transfer probabilities and pi,0 (i = 1, . . . , M) the dischargeprobabilities. Subscript 0 represents the state of being outside the healthcaresystem.In the following section, we introduce the queueing network approach to the analysis of thehealthcare system. In Section 3, we derive the corresponding performance measures, such as lossprobability of the system and expected occupancy level. In Section 4, a case study is presented toillustrate the application of the proposed model to a hospital geriatric department. Conclusions anddirections for further research are given in Section 5.2 Queueing network models for healthcare systemsThe progression of patients in a healthcare system is modelled as consisting of a set of conceptualphases. On admission, patients enter the 1st phase, which could correspond to assessment,diagnosis, etc. After staying a period of time, they are either discharged (including by death), ortransferred into the 2nd phase, which could be some form of rehabilitation. The process continuesuntil at the final phase all patients are eventually discharged. A diagrammatic representation ofthe system with M phases is shown in Fig. 1. The transfer probability pi,i+1 (i = 1, . . . ,M − 1)represents the probability that a patient will move to phase i + 1 upon completion of phase i; andpi,0 (i = 1, . . . ,M) is the discharge probability upon leaving phase i. We assume new patientsarrive at a constant rate λ per unit time (taken to be days without loss of generality). Further, weassume that there are a total of K beds in the system. That is there are at most K patients in thesystem at any one time. Under this setting, the system will admit patients for as long as there isavailable beds, and patients will be turned away when there is no empty beds left in the system. Aservice node of the queueing network is one of the phases. The servers at a service node are allhospital beds providing the corresponding phase of care at the time. The number of servers at eachservice node varies with time depending on the dynamic use of each bed, as patients move throughcare phases.More specifically, the number of servers at node i, denoted by random variable Li (i =1, . . . ,M), varies between 0 and K and satisfies the condition∑Mi=1 Li ≤ K. We further assumethat, for a server of service node i, its service time follows an exponential distribution with mean1/µi, where µi is the constant service rate per bed, i.e. the average number of patients treated perday per bed.We solve the homogeneous linear system ei =∑Mj=0 ejpj,i (i = 0, 1, . . . ,M), for the relativearrival rates ei (i = 1, . . . ,M) to within a multiplicative constant [6]. The relative arrival rates arethe effective rates of patients entering each phase per day. Assuming e0 = 1, the relative arrivalTwentieth IEEE International Symposium on Computer-Based Medical Systems (CBMS'07)0-7695-2905-4/07 $20.00  © 2007Authorized licensed use limited to: University of Westminster. Downloaded on June 12, 2009 at 05:26 from IEEE Xplore.  Restrictions apply.rate ei becomesei =i−1∏j=1pj,j+1, (i = 2, . . . ,M) (1)which is the proportion of admitted patients who will enter phase i.It can be shown that the joint probability distribution of the number of patients in each phase isgiven by [6]pi(l1, l2, . . . , lM ) =1H(K)M∏i=1ρliili!, forM∑i=1li ≤ K, (2)where li is the number of patients in the ith phase; and ρi = λei/µi (i = 1, . . . ,M), in which eiis the relative arrival rate to phase i, µi is the constant service rate per bed of phase i, and H(K) isthe normalisation constant. By the use of the multinomial theorem, H(K) can be computed in thefollowing closed-formH(K) =K∑k=01k!(M∑i=1ρi)k. (3)2.1 Special case: closed networkA closed network is a special case of the semi-open network model where the system is alwaysfull. In other words, a discharge from the system is immediately replaced by an admission. Suchcondition is reasonable for hospital departments that usually run at full occupancy level. Underthis condition, the normalisation constant is given as H(K) = (∑Mi=1 ρi)K/K!, and therefore (2)becomespi(l1, l2, . . . , lM ) =K!l1! · · · lM !M∏i=1ri, where ri =ei/µi∑Mj=1 ej/µj. (4)This suggests that the number of patients in each phase jointly follow a multinomial distributionwith parameters K and ri (i = 1, . . . ,M ).2.2 Special case: open networkAn open network is another special case of the semi-open network model where the system isassumed to have infinite capacity to admit patients. Under this condition, we can show that thenormalisation constant reduces to H(K) = exp(−∑Mi=1 ρi), and (2) becomespi(l1, l2, . . . , lM ) =M∏i=1ρlii eρili!, (5)which implies that the number of patients in phase i (i = 1, . . . ,M ) are independent Poissondistributed random variables with rate ρi.Although it is rarely true in reality that a hospital department will have almost infinite bedcapacity, this special case can still be useful in representing an ideal scenario for comparisonpurpose.Twentieth IEEE International Symposium on Computer-Based Medical Systems (CBMS'07)0-7695-2905-4/07 $20.00  © 2007Authorized licensed use limited to: University of Westminster. Downloaded on June 12, 2009 at 05:26 from IEEE Xplore.  Restrictions apply.Figure 2. Parameters reported in Parry [7] for a geriatric department.3 Performance measuresMeasures of system performance, such as the loss probability of the system (i.e. the probabilitythat a patient has to be turned away because the system is full), expected occupancy level, effectiveadmission rate, etc, can be easily computed.Since the probability that there are l (l ≤ K) patients in the system is given bypil =∑∑Mi=1li=lpi(l1, l2, . . . , lM ) =1H(K)(∑Mi=1 ρi)ll!, (6)therefore, the loss probability of the system, i.e. the probability that a patient has to be turned awaybecause the system is full, is the probability that there are exactly K patients in the system, whichis given byPloss = piK =1H(K)(∑Mi=1 ρi)KK!. (7)The expected number of patients in the system is given byEL =K∑k=1kpik = (1 − Ploss)M∑i=1ρi, (8)with the corresponding variance asVarL =(piK − piK−1 − pi2K)( M∑i=1ρi)2+ (1 − piK)M∑i=1ρi. (9)Similarly, the expected number of patients at the ith phase (i = 1, . . . ,M ) of the system is given asELi = (1− Ploss) ρi, (10)with the corresponding variance asVarLi =(piK − piK−1 − pi2K)ρ2i + (1 − piK) ρi. (11)Twentieth IEEE International Symposium on Computer-Based Medical Systems (CBMS'07)0-7695-2905-4/07 $20.00  © 2007Authorized licensed use limited to: University of Westminster. Downloaded on June 12, 2009 at 05:26 from IEEE Xplore.  Restrictions apply.bed capacityarrival rateexpected occupancy levelbed capacityarrival rate300 400 500 60016182022242628Figure 3. Expected occupancy level as a function of bed capacity and arrival rate.The graph on the right is the contour plot of the 3D version on the left. Currentvalues of bed capacity and arrival rate are indicated by the gray lines.4 Application of the modelWe demonstrate the use of the model using parameters reported by Parry [7], who applied themethodology proposed in [2] to analyse the data of a geriatric department, which had 473 beds.Three phases were suggested - acute care (1st phase), rehabilitation (2nd phase) and long-stay care(3rd phase) with average length of stay being 9 days, 67 days and 863 days, respectively. Uponleaving the 1st phase, 7% of the patients move to the 2nd phase, of which 17% move to the 3rdphase (see Fig. 2). New patients arrived at an estimated rate of 19 patients per day.Using equations derived in the previous sections, we compute that the number of patients in thedepartment is expected to be 447.9 (s.d = 16.2), and the expected number of patients in each phaseis 168.2 (s.d = 11.9), 87.7 (s.d = 9.0), 192.0 (s.d = 12.6), respectively. This suggests that morethan 40% of the beds are expected to be occupied by long-stay patients. The expected occupancylevel is 94.7%, and the probability that a patient will be refused admission because the departmentis full is expected to be 0.016.In addition, we can use the model to study how the performance measures change in relationsto changes in system parameters, such as bed capacity, arrival rate, etc. Fig. 3 shows how changesin bed capacity and patient arrival can have a large impact on expected occupancy level in thedepartment. Hospital managers can use the contour plot to estimate the number of extra bedsrequired to cope with increased demand if the department is simply to maintain the same level ofoccupancy level. Generally, the number of beds in a department represents a trade-off between thedesire to lower the risk of turning patient away because the department is full, and the preference ofhaving high occupancy level. Fig. 4 shows the impact of changing bed capacity on loss probabilityand expected occupancy level. This plot allows hospital managers to visualise and balance theseconflicting goals when deciding on the number of beds.Twentieth IEEE International Symposium on Computer-Based Medical Systems (CBMS'07)0-7695-2905-4/07 $20.00  © 2007Authorized licensed use limited to: University of Westminster. Downloaded on June 12, 2009 at 05:26 from IEEE Xplore.  Restrictions apply.300 400 500 6000.00.10.20.30.4bed capacityloss probability0.650.70.750.80.850.90.951expected occupancy levelloss prob.exp. occ. levelFigure 4. The effect on loss probability and expected occupancy level due tochanges in bed capacity. Current bed capacity is indicated by the gray line.5 ConclusionIn this paper, we introduce a novel approach to the development of a realistic and generic modelof patient flow in a health care system. By incorporating a total bed capacity constraint, above whichnew patients will be refused admission, this model provides a realistic representation of a healthcaresystem. Health care policy makers will be able to use this model to gain better understanding ofthe dynamics of patient flow and to study potential long-term impacts of policy changes, which willcontribute to better care for patients.References[1] M. J. Coˆte, “Understanding patient flow”, Decis Line, 2000, 31:8–10.[2] G. W. Harrison, and P. H. Millard, “Balancing acute and long-term care: The mathematics ofthroughput in departments of geriatric medicine”, Methods Inf Med, 1991, 30:221–228.[3] M. J. Faddy, and S. I. McClean, “Markov chain modelling for geriatric patient care’, MethodsInf Med, 2005, 44:369–373.[4] V. Irvine, S. I. McClean, and P. H. Millard, “Stochastic models for geriatric in-patientbehaviour”, IMA J Math Appl Med Biol, 1994, 11:207–216.[5] T. J. Chaussalet, H. Xie, and P. H. Millard, “A closed queueing network approach to theanalysis of patient flow in health care systems”, Methods Inf Med, 2006, 45:492–497.[6] H. Chen, and D. D. Yao, Fundamentals of Queueing Networks: Performance, Asymptotics,and Optimization, Springer-Verlag, New York, 2001.[7] A. Parry, “An age-related service revisited”, In: P. H. Millard, and S. I. McClean (editors),Go with the Flow: A system approach to healthcare planning, pp. 127-130, Royal Society ofMedicine Press, London, 1996.Twentieth IEEE International Symposium on Computer-Based Medical Systems (CBMS'07)0-7695-2905-4/07 $20.00  © 2007Authorized licensed use limited to: University of Westminster. Downloaded on June 12, 2009 at 05:26 from IEEE Xplore.  Restrictions apply.

A semi-open queueing network approach to the analysis of patient flow in healthcare systems

Crossref

A Semi-open Queueing Network Approach to the Analysis of Patient Flow in Healthcare Systems

https://westminsterresearch.westminster.ac.uk/download/e225cad10479003701b3af7db688e1d2b6ce85eba1fb135a5c3e489a8bab1161/225509/Xie_Chaussalet_Rees_2007_as_published.pdf

A semi-open queueing network approach to the analysis of patient flow in healthcare systems

Abstract

Similar works

Full text

Available Versions

WestminsterResearch

Crossref