IOSR Joural of Mathematis (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X. Volume 10, Issue 4 Ver. VI (Jul-Aug. 2014), PP 47-53 Queueig Aalysis of Patiet Flow i Hospital Olorusola S. A, Adeleke R. A ad Ogulade T. O Departmet of Mathematial Siees, Ekiti State Uiversity of Ado-Ekiti, Ekiti State, Nigeria. Abstrat: Waitig o a queue is ot usually iterestig, but redutio i this waitig time usually requires plaig ad extra ivestmets. The ireasig populatio ad health-eed due to adverse evirometal oditios have led to esalatig waitig times ad ogestio i hospitals espeially i the Emergey ad Aidet Departmets (EAD). It is uiversally akowledged that a hospital should treat its patiets, espeially those i eed of ritial are, i timely maer. Iidetally, this is ot ahieved i pratie partiularly i govermet owed health istitutios beause of high demad ad limited resoures i these hospitals. To ehae the level of admittae to are, optimal beds required i hospital is eeded ad this a be ahieved by adequate kowledge of patiet flow. I this paper, we show that queue theory a aurately model the flow of i-patiet i hospital; we determie the optimal bed out ad its performae measure. Keywords: M/M/C queue, Poisso arrival, Expoetial distributio, I-patiet. I. Itrodutio Oe of the major elemets i improvig effiiey i the delivery of health are servies is i-patiet flow. From a liial perspetive, i-patiet flow represets the progressio of a patiet s health status. As suh, a uderstadig of patiet flow a offer eduatio ad isight to health are providers, admiistrators, ad patiets about the health are eeds assoiated with medial oers like disease progressio or reovery status. Equally importat, a uderstadig of patiet flow is also eeded to support a health are faility s operatioal ativities. From a operatioal perspetive, patiet flow a be thought of as the movemet of patiets through a set of loatios i a health are faility. The, effetive resoure alloatio ad apaity plaig are otiget upo patiet flow beause patiet flow, i the aggregate, is equivalet to the demad for health are servies (M. J Côté, 2000). The risig populatio ad health-eed due to adverse evirometal oditios have led to esalatig waitig times ad ogestio i hospital Emergey Departmets (ED) Derlet R. W et al (2001). It is uiversally akowledged that a hospital should treat its patiets, espeially those i eed of ritial are, i a timely maer. Iidetally, this is ot ahieved i pratie partiularly i govermet owed health istitutios beause of high demad ad limited resoures i these hospitals. Queueig theory is used widely i egieerig ad idustry for aalysis ad modelig of proesses that ivolve waitig lies. I appropriate systems, it eables maagers to alulate the optimal supply of fixed resoures eessary to meet a variable demad. I the past, attempts have bee made to apply queuig aalysis to a variety of hospital ativities, iludig ardia are uits, obstetri servies, operatig rooms ad emergey departmets, as a meas of diretig the alloatio of ireasigly sare resoures. More reetly, health poliy ivestigators have also sought to apply these tehiques more widely aross etire healthare systems. Ufortuately, most proposed queuig models lak real-world validatio ad perhaps for this reaso, have yet to be embraed by physiias ad hospital admiistrators. Therefore, to explore the utility ad impliatios of queuig theory as it relates to the supply ad demad for ritial are servies, we sought to validate a simple queuig model i a busy hospital. Queueig theory is a mathematial approah i Operatios Researh applied to the aalysis of waitig lies. A.K. Erlag first aalyzed queues i 1913 i the otext of telephoe failities. The body of kowledge that developed thereafter via further researh ad aalysis ame to be kow as Queueig Theory, ad is extesively applied i idustrial settigs ad retail setors. Waitig o a queue is ot usually iterestig, but redutio i this waitig time usually requires plaig ad extra ivestmets. Queuig theory ivolves the mathematial study of waitig lies. Queuig systems is a system osistig of flow of ustomers requirig servie where there is some restritio i the servie that a be provided. Three mai elemets are ommoly idetified i ay servie etre amely; a populatio of ustomers, the servie faility ad the waitig lie. We usually ivestigate queues i order to aswer questios like, the mea waitig time i the queue, the mea respose time i the system, utilizatio of servie failities, distributio of umber of ustomers i the queue, et. Deisios regardig the amout of apaity to provide a servie must be made frequetly by ay servie provider for optimality. The study of queueig theory requires some bakgroud study i probability theory ad stohasti aalysis. 47 Page
Queueig Aalysis of Patiet Flow i Hospital II. Literature Review Queueig theory has effetively bee applied to various field of edeavour like traffi maagemet, supermarket ad health are et. Weiss ad MCliam (1986) used the M/G/ system to model the queue of patiets eedig alterative levels of are i a aute are failities whose treatmet is ompleted ad are waitig to be trasferred to a exteded are faility. Adeleke R. A et al (2009) osidered appliatio of queuig theory to the waitig time of out-patiets i a hospital. The average umber of patiets ad the time eah patiet waits i the hospital were determied. Likewise i his paper Worthigto (1987) used queuig theory to model hospital waitig list. He used a M/G/C queue with state depedet arrival rate to address the log- wait list problem. He experimeted with various maagemet atios suh as ireasig the umber of beds or dereasig the mea servie time through appropriate meas. DeBrui et al (2006) ivestigated the emergey i-patiet flow of ardia patiet i a hospital i order to determie the optioal bed alloatio so as to keep the fratio of those refused admissio uder a target hit. The authors fid a relatio betwee the size of a hospital uit, oupay rate ad target admissio rates. After aalytially estimatig the required umber of beds i first ardia Aid (FCA) uit, they also used umerial method to estimate the umber of beds i the Critial Care Uit (CCU) ad Normal are liial ward (NC). Joatha E. H et al (2009) haraterize a optimal admissio threshold poliy usig otrol o the sheduled ad expedited gate way for a ew Markov Deisio proess model. I their work, they preseted a pratial poliy base o isight from the aalytial model that yield redued emergey blokages, aelatios ad off-uits through simulatio based o historial hospital data. Appliatio of queueig theory to model health are is growig more popular as hospital maagemet teams are beomig aware of the advatages of these tehiques. I this researh we will use both aalytial tehiques ad simulatio to study a simple queuig etwork omposed of oly two servie statios plaed i tadem. I this paper, we studied all admissios ito the Emergey ad Aidet Departmet (EAD) of a tertiary hospital. We will show that admissios ito this system has a Poisso distributio, hee it has expoetial iter-arrival rate. We also examie the average legth of stay, the oupay rate ad we determie the optimal bed out i the Itesive Care Wards (ICW) ad the Medial ad Surgial Wards (MSW). Sie the ICW ad MSW have multiple beds we will osider the M/M/ queue. III. The M/M/ Model For this queueig system, it is assumed that the arrivals follow a Poisso probability distributio at a average of ustomers (patiets) per uit of time. It is also assumed that they are served o a first ome, firstserved basis by ay of the dotors. The servie times are distributed expoetially, with a average of µ ustomers (patiets) per uit of time ad umber of servers. Cosider the M/M/ queue where the arrival ad servie rates are ad µ, respetively. Assumig that steady state exists, let p be the steady state distributio of the umber of uits i the system. By the rate-equality priiple p µp p 1 1 µp 1 3.1 Similarly, for the ase of, we get p µp p 1 µp 1 3.2 p 1 µp 1 p 1 µp p 2 ( 1)µp 1 µp 1 0 By rearragig terms ad iteratig we obtai that for 0 <. p 2 () 2 p p 1 1 p 2 p! 0 3.3 I a similar fashio, we get that for > (i. e, 1, 2 ). to fators p 2 to fators!! p 1 ρ p 3.4 Now for /(µ) < 1, the ormalizatio oditio p 1 gives 48 Page
!! (1 ) 1 Queueig Aalysis of Patiet Flow i Hospital Note that the probability a arrival uit has to wait o arrival is give by the probability P N p ( )! (1 ρ) p 1 ρ This is kow as Erlag s C formula or Erlag delay probability. We ow proeed to ompute some performae measures. We ow proeed to ompute some performae measures. The expeted queue legth L a be omputed as 3.5 3.6 3.1 Expeted Nuber Of Busy Ad Idle Servers The expeted umber of busy servers E(B) is give by E B p p 0 ( ) 1 ( 1! ) 1! 1 ρ 3.7 2 m0 ( )m m! { 1 ρ ρ}( ) 1! 1 ρ m 0 ( )m ( ) m!! 1 ρ 1 ρ 3.8 Hee the expeted umber of idle servers E(I) is give by E I E B E E B ρ 1 ρ 3.9 The expeted queue legth L a be omputed as L p ( )! ρp (1 ρ) 2 ρ P N 3.10 1 ρ A speial ase where 1, L ρ 1 ρ Where ρ /µ < 1 is referred to as the server utilizatio. Applyig Little s formula, we also obtai the expeted waitig time i the queue W L ρ (1 ρ) 2 3.11 We a fid the steady state waitig-time distributio for the M/M/ queue. Let w q x ad w(x) be the PDFs of the waitig time W q ad W s i the queue ad i the system, respetively. We obtai w (s) by oditioig. If a patiet fids o arrival <, he does ot have to wait, ad his waitig time i the system equals to his servie time, that is, w s for < 3.12 s If he fids patiets i the hospital, he has to wait i the hospital util the ompletio of servie of (-1). All the beds beig oupied, the the servie rate is. Takig ito osideratio his ow servie time, he has to wait i the system for the ompletio of (-1) servies at the rate ad his ow servie at the rate. That is, 49 Page
w s s Usig PASTA property, a p. Thus 1 s Queueig Aalysis of Patiet Flow i Hospital for 3.13 s s 0 0! w s w s p w s p p p s!! Ivertig the trasform, we get w t p! 0 e t 0 A speial ase for 1 s!! 1 s 2 1 3.14 1 s 1 s 2 1 [e t e 1 ρ t ] w t (1 ρ)e (1 ρ)t 3.15 Sie w s w q s [ (s )], we get from 3.14 Ivertig the trasform gives Simplifyig, we get! w q s w q t p! p s p p s δ t 1 p (t) e t ( )! w q t 1 1 ρ δ t p e (1 ρ)t, t > 0 3.17 Where δ(t) is the Dira delta (or impulse futio). Note that the oeffiiet of δ(t) is the probability of zero wait, or the probability that there is a free server upo arrival. IV. Legth Of Stay Distributio The umber of days i hospital for a patiet is desribed by the term legth of stay (LOS). LOS is defied as the time of disharge mius time of admissio. Followig, the average legth of stay is abbreviated as ALOS. The average legth of stay i hospitals is a statistial alulatio ofte used for health plaig Total disharge days purposes. Average Legth of stay i days Or Average Legth of stay i days Total ipatiet days of are Total disharges Total admissios Below are the defiitios for eah of the four data items iluded i the above alulatios: TOTAL DISCHARGE DAYS - The sum of the umber of days spet i the hospital for eah ipatiet who was disharged durig the time period examied regardless of whe the patiet was admitted. TOTAL DISCHARGES - The umber of ipatiets released from the hospital durig the time period examied. This figure iludes deaths. Births are exluded uless the ifat was trasferred to the hospital's eoatal itesive are uit prior to disharge. TOTAL INPATIENT DAYS OF CARE - Sum of eah daily ipatiet esus for the time period examied. TOTAL ADMISSIONS - The total umber of idividuals formally aepted ito ipatiet uits of the hospital durig the time period examied. Births are exluded from this figure uless the ifat was admitted to the hospital's eoatal itesive are uit. 3.16 50 Page
Queueig Aalysis of Patiet Flow i Hospital V. Bed Oupay It is ommo pratie i health servies to estimate the required umber of beds as the average umber of daily admissios times average legth of stay i days ad divided by average bed oupay rate (average umber of oupied beds durig a day) Huag X (1995) average o. of daily admissios bed requiremet average legth of stay 5.1 average bed oupay rate Hospital bed apaity deisios have bee made based o Target Oupay Rate (TOR) the average peretage of oupied beds ad the most ommoly used oupay target has bee 85% Lida V. Gree (2002). Aother metri ofte ited i the literature is the Target Aess Rate (TAR), whih measures the peretage of the time that a esus out will show that the hospital otais at least oe empty bed, de Brui et al (2007), Kumar ad Joh (2010). VI. Numerial Solutio We osider a tertiary hospital i the south-wester Nigeria ad studied the admissios through its Aidet ad Emergey departmet to the ICW ad MSW. The queueig model seleted assumes that daily admissios rate follow a Poisso distributio ad this behavior was ofirmed here by goodess of fit test as illustrated i the fig1. Gree L.V(2002) ad Mile ad Whitty (1995) have show that the arrival rate ito Itesive Care Uit follows a Poisso distributio. Fig2 Distributio of arrivals ito the System. The oupay rate (ρ) is related to the real demad () ad LOS () ad a be defied as follows, Average umber of beds oupied ρ (1 P ) 7.1 umber of beds available The term (1 P ) a be etitled as the effetive demad as the refused admissios are subtrated from the real demad. Furthermore, the produt whih is the expeted umber of patiets i the system is also kow as the workload of the system. May hospitals use the same target oupay rate for all hospital uits, o matter the size of the uit. The target oupay rate is typially set at 85% ad has developed ito a golde stadard [Gree (2002)]. The olusio is lear ad importat. Larger hospital uits a operate at higher oupay rates tha smaller oes while attaiig the same peretage of refused admissios. Therefore, oe target oupay rate for all hospital uits is ot realisti [De Brui (2007)]. 51 Page
Queueig Aalysis of Patiet Flow i Hospital Total admissio ito ICW634, ALOS i ICW4.44days, peretage of patiets reeged, k3.4%. We also have the followig set of data for the MSW: Total admissio ito MSW5073, ALOS7.24 days. From the parameter values speified, we estimate the arrival rate to eah statio as N ICW ICW 365days 1.74days 1 MSW N MSW 365days 13.9days 1 But the queue leadig to the MSW is omposed of ew arrivals ad bloked patiets from the ICW. Also we have oly a fratio 1 k 96.6% of patiets arrived ito ICW without reegig durig servie. So that the effetive arrival ito the ICW is e ICW 1 k 1.681days 1 Ad e MSW ICW e ICW ICW 15.581days 1 Table1: Performae Measure for ICW 1 % server utilizatio Probability of delay 10 74.64 0.299850 12 62.20 0.093022 14 53.31 0.023395 16 46.65 0.004763 18 41.46 0.000791 20 37.32 0.000108 22 33.93 12 24 31.10 001 26 28.71 00 28 26.66 00 30 24.88 00 2 % server utilizatio 114 98.95 116 97.25 118 95.60 120 94.01 122 92.46 124 90.97 126 89.53 128 88.13 130 86.77 132 85.46 134 84.18 136 82.95 Mea waitig time 4.964899 4.531046 4.455892 4.442477 4.440333 4.440038 4.440004 Table2: Performae Measure for ICW Probability delay 0.869420 0.679725 0.523275 0.396277 0.294935 0.215531 0.154511 0.108572 0.074723 0.050335 0.033167 0.021366 of Mea waitig time 12.513803 8.780979 7.969463 7.638836 7.472264 7.379405 7.324789 7.291737 7.271465 7.258987 7.251330 7.246670 Mea waitig time i queue (hrs) 12.600000 2.1900 0.5600 0.0590 0.0080 0.0010 0.0001 Mea waitig time i queue (hours) 126.570000 36.980000 17.507112 9.572064 5.574336 3.345720 2.034936 1.241688 0.755160 0.455688 0.267192 0.160080 Tables1 ad 2 show result for various values of 1 ad 2. From the tables we a see that 1 24 guaratees that there is o waitig at the EAW, sie urget patiet eedig urget are are brought i through it. I the MSW, 1 132, will guaratee a approximate of 85.46% server utilizatio ad a miimum waitig time i queue. 52 Page
Queueig Aalysis of Patiet Flow i Hospital VII. Summary A admissio guaratee should be oe of the mai goals of ay hospital for patiets eterig through its Emergey ad Aidet Departmet. I this work, we aalyzed a queueig etwork model with reegig to study how waitig time i the EAD of a hospital is iflueed by the umber of beds i the ICW ad MSW. The system was deomposed ito two idepedet multi-server queues so as to obtai estimates for the required umber of beds i the wards. We foud that the required umber of beds to esure that emerget patiets are promptly atteded ad there is easy flow is approximately 24 i the ICW ad 132 i the MSW for the test hospital uder osideratio. Referees [1]. Adeleke. R. A, Oguwale O. D, Halid O. Y (2009) Appliatio of Queueig Theory to Waitig Time of Out-patiets i Hospitals. Paifi Joural of Siee ad Tehology Vol. 10(2) 270-274 [2]. Aru Kumar ad Joh Mo (2010) Models for Bed Oupay Maagemet of a Hospital i Sigapore. Proeedigs of the 2010 Iteratioal Coferee o Idustrial Egieerig ad Operatios Maagemet Dhaka, Bagladesh. [3]. DeBrui A.M, A.C va Rossu MC Visser, GM Koole(2006) Modelig the emergey ardia i-patiet flow: a appliatio of queuig theory. Health Care Maagemet Siee Vol. 10, No 2, pp 125-137 [4]. Gree L. V (2002) How may Hospital Beds? Deisio, Risk ad Operatio. Workig Paper Series [5]. Gree LV, Soares J, Giglio JF, Gree RA (2006) Usig queueig theory to irease the effetiveess of emergey departmet provider staffig. Aademi Emergey Mediie 13(1):61-68 [6]. Joatha. E H, shervi Ahmad Beygi, Mark P.Va Oye (2009), Desig ad Aalysis of Hospital Admissio Cotrol for operatioal Effetiveess. Tehial Report 09-05. Uiversity of Mihiga. Mihiga [7]. Murray J. Cote (2000) Uderstadig Patiet Flow. Produtio/Operatios Maagemet Deisio lie pp 8-10 [8]. Vaberke P. T Bouherier R. J Has E. W Huri J. L, Litvak N (2010). A Survey of Health Care Models that Eompasses Multiple Departmet. Itl Joural of Health Maagemet Iformatio 1(1):37-69 [9]. Weiss E.N, J.O Mlai (1986) Admiistrative days i aute are failities: a queig aalyti approah operatios Researh Vol. 35 No 1 pp 35-44 [10]. Worthigto D.J (1987) queueig models for hospital waitig list. The joural of the operatioal researh Soiety Vol. 38, No 5, pp 413-422 53 Page