The interface between an emergency department and internal wards is often a hospital s bottleneck. Motivated

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1 MANAGEMENT SCIENCE Vol. 58, No. 7, July 2012, pp ISSN (prnt) ISSN (onlne) INFORMS On Far Routng from Emergency Departments to Hosptal Wards: QED Queues wth Heterogeneous Servers Avsha Mandelbaum Faculty of Industral Engneerng and Management, Technon Israel Insttute of Technology, Hafa 32000, Israel, Petar Momčlovć Department of Industral and Systems Engneerng, Unversty of Florda, Ganesvlle, Florda 32611, Yula Tseytln Faculty of Industral Engneerng and Management, Technon Israel Insttute of Technology, Hafa 32000, Israel; and IBM Research Lab, Hafa Unversty, Hafa 31905, Israel, The nterface between an emergency department and nternal wards s often a hosptal s bottleneck. Motvated by ths nteracton n an anonymous hosptal, we analyze queueng systems wth heterogeneous server pools, where the pools represent the wards, and the servers are beds. Our queueng system, wth a sngle centralzed queue and several server pools, forms an nverted-v model. We ntroduce the randomzed most-dle (RMI) routng polcy and analyze t n the qualty- and effcency-drven regme, whch s natural n our settng. The RMI polcy results n the same server farness (measured by dleness ratos) as the longest-dle-server-frst (LISF) polcy, whch s commonly used n call centers and consdered far. However, the RMI polcy utlzes only the nformaton on the number of dle servers n dfferent pools, whereas the LISF polcy requres nformaton that s unavalable n hosptals on a real-tme bass. Key words: queueng systems; heterogeneous servers; healthcare; hosptal routng polces; farness; qualtyand effcency-drven regme; asymptotc analyss Hstory: Receved September 25, 2010; accepted October 1, 2011, by Assaf Zeev, stochastc models and smulaton. Publshed onlne n Artcles n Advance March 9, Introducton Operatons research methodologes, and queueng theory n partcular, have generated valuable nsghts nto operatonal strateges and practces, thus leadng to solutons of sgnfcant problems n healthcare systems. In concert wth ths state of affars, we analyze patent flow n hosptals: specfcally, we focus on the emergency department (ED) and ts nterface wth four nternal wards (IWs) n a large Israel hosptal; we refer to t as Anonymous Hosptal. Two operatonal problems could arse n ths process: patents watng tmes n the ED for a transfer to the IWs could be long, and patent routng to the wards need not be far, as far as workload allocaton among the wards s concerned. In contrast to the majorty of studes that address the problem of long watng tmes n EDs, we explore the process of patent allocaton to wards from the farness perspectve. In search of an allocaton protocol that s farness senstve, we model the EDto-IW process as a queueng system wth heterogeneous server pools: the pools represent the wards, and servers are beds. Wthn ths modelng framework, we compare several routng strateges, thus dentfyng the one most approprate for a hosptal settng Motvaton Two conclusons can be drawn from Anonymous Hosptal data (see Table 1 n 2.3). Frst, the fastest and smallest ward (ward B) s subject to the hghest load: t experences the hghest number of patents per bed per month, wth bed occupancy that s comparable to the other wards. The reasons behnd the hgh turnover rate of ward B are superor manageral and staff practces, as well as (medcally justfed) varyng polces for patents release, whch results n sgnfcantly shorter average length of stay (ALOS). Importantly, a shorter ALOS does not come at the cost of nferor medcal care. Indeed, the level of return rates (wthn three months) s comparable across wards. Hence, one could argue that the allocaton process of patents from the ED to the IWs s not far from the pont of vew of medcal staff, a problem that has been observed n other Israel hosptals as well. 1273

2 Mandelbaum, Momčlovć, and Tseytln: On Far Routng from EDs to Hosptal Wards 1274 Management Scence 58(7), pp , 2012 INFORMS The second emprcal concluson s that some key operatonal characterstcs of the ED-to-IW process concde wth the features that characterze moderateto large-scale queueng systems n the qualtyand effcency-drven (QED) regme (Erlang 1948, Halfn and Whtt 1981, Gans et al. 2003). Such a regme acheves, smultaneously, hgh levels of operatonal servce qualty (ED watng tmes sgnfcantly shorter than servce duratons, specfcally ALOS) and resource effcency (hgh bed occupancy) Contrbutons Our paper focuses on modelng the ED-to-IW process, whch s a key phase of patent flow n hosptals. More broadly, n 2004, the Jont Commsson on the Accredtaton of Healthcare Organzatons (JCAHO 2004) set a standard (LD ) for patent flow leadershp. The standard requres that healthcare leaders develop and mplement plans to dentfy and mtgate mpedments to effcent patent flow throughout the hosptal. It amplfes the need to dentfy the crtcal factors that mpact patent flow, wth the ultmate goal of desgnng and mplementng polces, processes, and procedures that track, montor, and mprove patent flow throughout hosptals. Although we concentrate on the mpact of patent routng on staff farness, our work should be vewed wthn the broader context of mprovng staff effcency by creatng an approprate ncentves structure. Operatonal polces that are not perceved to be far could nternalze neffcences,.e., create stuatons where good work leads to more work. On the other hand, far polces not only reward staff members wth the best practces, but also promote the adopton of such practces. Thus, although far polces come at a certan cost n the short run, n the long run they are lkely to mprove overall system effcency. Our contrbutons are as follows: Based on emprcal data from Anonymous Hosptal, we argue that an nverted-v queueng model n the QED regme s approprate for descrbng the ED-to-IW process, whch exhbts multscale behavor (e.g., IW lengths of stay beng much longer than ED watng tmes). We quantfy operatonal farness toward medcal staff by means of ratos that take nto account bed occupancy levels and bed turnover rates (the average number of admtted patents per bed per unt of tme) two man metrcs as far as workload of medcal staff s concerned. These metrcs serve as operatonal proxes for farness, whch s an ntrcate concept. (Operatonal proxes relate to operatons, are easy to measure, and they approxmate notons that are hard to quantfy.) Although routng algorthms that take nto account farness have been consdered n the lterature, we propose a practcal routng algorthm (randomzed most dle, or RMI) that s sutable for hosptals where only lmted and partal nformaton s avalable for patent routng; that s, we consder the role of nformaton n the performance of routng polces. The proposed algorthm, RMI, s analyzed and compared to known algorthms. Our results demonstrate that RMI acheves a (long-run) level of farness toward medcal staff that s the same as that of algorthms that requre more nformaton to operate. Moreover, wthn our narrow noton of farness, based on occupancy and turnover rates, an extenson of our algorthm, weghted RMI, s flexble enough to acheve any desred farness. Furthermore, our results yeld mportant nsghts on dfferences between varous routng algorthms, dfferences that arse at the subdffuson (room-level) scale. In partcular, these nsghts reveal that nstantaneous mbalance of workload across the IWs s lmted to just a few rooms of patents. Moreover, n the course of just a few days ths mbalance (f present) dsappears due to tme averagng;.e., emprcal workload tme averages converge to the desred longrun averages Bref Lterature Revew There exsts a vast amount of research on healthcare systems n numerous scentfc felds ncludng operatons research. We menton here the most relevant to the present work; addtonal related references can be found n Tseytln (2009), Bekker and de Brun (2010), de Brun et al. (2010), Kc and Terwesch (2009), González and Herrero (2004), de Vércourt and Jennngs (2011), and McManus et al. (2004), as well as n each of the papers mentoned below. Readers are also referred to Green (2004, 2008), who descrbes the general background and ssues nvolved n hosptal capacty plannng. Research papers (e.g., Ltvak et al. 2001) and popular artcles (e.g., New York Tmes 2002) both recognze the mportance of ED proper functonng and the consequences of ts overcrowdng. Patent flow from the ED to other medcal unts n the hosptal (not just the IWs) has receved specal attenton (even becomng the subject of a novel; Wrght and Kng 2006). In fact, t has been acknowledged as a major trgger of ambulance dverson (Ltvak et al. 2001), whch s of great concern. Ramakrshnan et al. (2005) construct a two-tme-scale model for a hosptal system, where the wards operate on a tme scale of days and are modeled by a dscrete tme Markov chan, and the ED operates on a much faster tme scale and s modeled by a contnuous tme Markov chan. Wth the help

3 Mandelbaum, Momčlovć, and Tseytln: On Far Routng from EDs to Hosptal Wards Management Scence 58(7), pp , 2012 INFORMS 1275 of ths model, Ramakrshnan et al. (2005) estmate the expected occupancy of the wards and the probablty of each ward reachng ts capacty. The setup n Ramakrshnan et al. (2005) corresponds to ours n that the ED-to-IW process also operates n several scales. These scales arse naturally from our asymptotc analyss (QED regme). Specfcally, length of say n wards, whch correspond to our servce tmes, are naturally measures n days, whereas watng n the ED for transfer to the wards s naturally measured n hours, whch corresponds to our watng tmes. Relevant analyses of queueng systems wth heterogeneous servers date back to the slow server problem (Rubnovtch 1985a). Intally, the focus was on fndng the best operatng polcy to mnmze the steady-state mean sojourn tme of the customers n the system, whch s equvalent to mnmzng the long-run average number of customers n the system, due to Lttle s law. Under such crtera, t s preferable to use the faster servers more than the slower servers. In fact, under some crcumstances, t s even advantageous to remove the very slow servers and thus reduce sojourn tme. Ths slow server phenomenon s addressed, for the case of two heterogeneous servers and a sngle queue, by Larsen and Agrawala (1983), Ln and Kumar (1984), Rubnovtch (1985a, b), and Stockbrdge (1991); and the general multservers case s addressed by Cabral (2005) (under the random assgnment (RA) polcy, whch routes a customer to one of the dle servers at random). Later, Cabral proved n Cabral (2007) that, for any two servers, the fracton of tme that the faster among them s busy s smaller than that of the slower one, and the effectve servce rate of the faster server s hgher than that of the slower one. Except for Cabral (2007), none of the studes mentoned above touches on the ssue of farness toward servers. In the context of large-scale systems, Armony (2005) analyzed the fastest-servers-frst (FSF) routng polcy that assgns customers to the fastest avalable pool. She shows that FSF s asymptotcally optmal (wthn the set of all nonpreemptve, nonantcpatng frstcome, frst-served (FCFS) polces), n the sense that t stochastcally mnmzes the statonary queue length and watng tme as the arrval rate and number of servers grow large; Armony and Mandelbaum (2012) extended ths result to accommodate abandonments. Yet, under the FSF polcy, asymptotcally only the slowest servers have any dle tme. Ths s obvously unfar toward the fast servers (whch get punshed for beng fast by workng more), and t gves them an ncentve to slow down an undesrable result for the system as a whole. Thus, there exsts a trade-off between operatonal optmalty for the system and farness toward servers (Armony and Ward 2010). There s ample research amed at achevng farness to customers; see, for example, references n Tseytln (2009). However, to our best knowledge, only recently, Atar (2008) was the frst to deal wth the operatonal farness-toward-servers ssue. He studed a sngle-server pools model n the QED regme, where the number of servers and ther servce rates are ndependent and dentcally dstrbuted (..d.) random varables, under a polcy that routes an arrvng customer to the server that has been dle for the longest tme among all dle servers (n a determnstc envronment, ths polcy s called longestdle server frst (LISF) by Armony 2005). Armony and Ward (2010) extended LISF routng to longestweghted-dle-server-frst (LWISF) routng, and proposed a threshold polcy that asymptotcally (n the QED regme) outperforms LWISF whle achevng the same target dleness ratos (IRs). Atar et al. (2011) proposed the longest-dle-pool-frst polcy, that routes a customer to the pool wth the longest cumulatve dleness among the avalable pools; n the QED regme, ths polcy s shown to balance cumulatve dleness among the pools. Most of these papers examne transent system behavor by establshng weak convergence process-level results. In contrast, our work deals wth steady-state behavor. Farness (or justce, or equty) s a well-researched area n the behavoral scences (Colqutt et al. 2001). We shall menton some relevant work later n Organzaton The paper s organzed as follows. In the next secton we descrbe the process for routng patents from an ED to IWs, whch reveals a farness problem n ths process. Our queueng model, as well as the QED regme, are ntroduced n 3. Far routng algorthms are descrbed n 4. Secton 5 contans our theoretcal results and a related dscusson, ncludng manageral mplcatons. Concludng remarks appear n 6. Readers are referred to the onlne appendces (Mandelbaum et al. 2011) for a descrpton of the routng algorthm currently mplemented at Anonymous Hosptal, the state of affars n fve other hosptals, techncal proofs, a table of notaton, and a lst of acronyms. 2. Patent Routng We study patent flow from the ED to the IWs n hosptals. Our research ste s Anonymous Hosptal, whch s a large Israel hosptal wth approxmately 1,000 beds, 45 medcal unts, and approxmately 75,000 patents hosptalzed yearly. Among ts varety of unts, t has a large ED wth an average arrval rate of patents daly, who occupy up to 50 beds, and fve IWs, whch we denote from A to E. The ED s dvded nto two major subunts: nternal and trauma

4 Mandelbaum, Momčlovć, and Tseytln: On Far Routng from EDs to Hosptal Wards 1276 Management Scence 58(7), pp , 2012 INFORMS (the latter beng surgcal and orthopedc patents). An nternal patent, whom the ED decdes to hosptalze, s drected to one of the fve IWs accordng to a certan routng polcy ths routng process s the focus of our research. Departments of nternal medcne are responsble for caterng to a wde range of nternal dsorders, provdng npatent medcal care to thousands of patents each year. Wards A D are more or less smlar n ther medcal capabltes each can treat multple types of patents. Ward E, on the other hand, treats only walkng patents, and routng to t dffers from routng to the other wards. In our study we thus concentrate on the routng process to wards A D only. The exstence of multple wards wth smlar medcal capabltes s common n Israel hosptals and can be attrbuted to varous factors, for example, () there exst constrants n terms of physcal space, e.g., wards can be located on dfferent floors of a buldng; () the exstence of multple wards mples the exstence of multple postons of ward managers (these postons are used by the hosptal management to attract top-performng doctors); and () nformal research n Anonymous Hosptal suggests that healthcare operatons could exhbt dseconomes of scale (ths would, of course, dscourage the formaton of a sngle superward) The Routng Process The decson of routng a to-be-hosptalzed patent to one of wards A D s supported by a computer program, referred to as the Justce Table at the hosptal. As ts name suggests, the algorthm s goal s to make the patent allocaton to the wards far, by balancng the load among the wards. Pror to routng, patents are classfed nto three categores, accordng to the complexty of treatment: ventlated, specal care, and regular. The program accepts a patent s category as an nput parameter and returns a ward for the patent (A D) as an output. For each patent category, there s round-robn order among the wards, whle accountng for the sze of each ward by allocatng fewer patents to a smaller ward: for example, two patents to B per three patents to A, reflectng a 30:45 bed rato. The Justce Table does not take nto account the actual number of occuped beds and the patent dscharge rate. In Mandelbaum et al. (2011), we provde a bref hstory of the Justce Table and a more detaled descrpton of the ED-to-IW process. We recognze a problem n the process of patent routng from the ID to the IWs: patent allocaton to the wards does not appear to be far. In what follows, we frst examne the noton of farness and then dscuss the specfcs present at Anonymous Hosptal. Remark 1. In addton to analyzng the ED-to-IW process n one specfc hosptal, we examned how ths process s managed n fve other Israel hosptals (see Mandelbaum et al. 2011). In partcular, we learned about the routng polces that are beng used, and how successful they are n terms of delays and far allocaton. To ths end, we used a questonnare that ncluded both qualtatve (detaled descrpton of the process, farness consderatons) and quanttatve (operatonal measures of the ED and IWs: capacty, ALOS, watng tmes) questons. Our study revealed that unbalanced loads on the wards due to heterogenety of ALOS are common n all our surveyed hosptals Farness One can analyze farness toward patents (customers) and farness toward wards the latter coverng medcal and nursng staff (servers). There s ample lterature on measurng farness n queues from the customer pont of vew (e.g., see Av-Itzhak and Levy 2004, Larson 1987, Rafael et al. 2002). Varous aspects are nvestgated (for example, sngle queue versus multple queues, or FCFS versus other queueng dscplnes), but all agree that the FCFS polcy s typcally essental for justce percepton. Consequently, customer satsfacton n a sngle queue s hgher than n multple queues (Larson 1987), and watng n a multqueue system produces a sense of lack of justce even when no objectve dscrmnaton exsts (Rafael et al. 2002). The stuaton could be dfferent n nvsble queues (e.g., call centers) and healthcare queues (e.g., EDs). In the latter, clncal prorty naturally domnates FCFS justce. Remark 2. We note that patent servce n a ward actually starts pror to the physcal arrval of the patent to the ward. Indeed, we observed that the ward, once nformed about a to-be-admtted patent, starts preparng for ths specfc patent: dfferent patents, even f they fall under the same classfcaton, mght requre dfferent preparatons. Ths sometmes leads to the followng scenaro. Suppose that a decson for hosptalzaton of patent X was made pror to a decson of hosptalzaton of patent Y (assumng both of them are clncally smlar); say patent X s drected to ward A and patent Y to ward B. Now, suppose that ward B becomes ready to physcally admt the patent earler than ward A hence patent Y jons a ward before patent X, although Y arrved later (Elkn and Rozenberg 2007). In addton to ths need for a ward to prepare for the patent, the hosptal staff refrans from modfyng patent ward assgnments also for a psychologcal reason: a patent awatng hosptalzaton (as well as accompanyng ndvduals) experences hgh levels of stress as s one thus does not wsh to aggravate stress by changng orgnal ward assgnments (Elkn and Rozenberg 2007).

5 Mandelbaum, Momčlovć, and Tseytln: On Far Routng from EDs to Hosptal Wards Management Scence 58(7), pp , 2012 INFORMS 1277 In our work, we focus on the process of patent assgnment to wards, as opposed to the physcal process of patent transfer from the ED to the IWs. We refer to the former as the ED-to-IW process. (In ths process, devatons from FCFS do not rase farness problems among patents snce the assgnment queue s typcally nvsble to them.) The lterature on justce from the server pont of vew s concerned wth Equty Theory (Huseman et al. 1987), accordng to whch workers perceve ther justce by comparng ratos of outputs from the job to nputs to the job. Specfcally, f the output/nput rato of an ndvdual s perceved to be unequal to others, then nequty exsts. The larger the nequty the ndvduals perceve, the more uncomfortable they feel and the harder they work to restore equty (Huseman et al. 1987). Ben-Zrhen et al. (2007) showed that n customer servce centers, servers equty percepton has a postve nfluence on ther performance and job satsfacton. References to addtonal studes on the mportance of perceved justce among employees can be found n Armony and Ward (2010). Our anchor pont s the survey reported n Elkn and Rozenberg (2007), n whch the staff (nurses, doctors, and admnstraton) were asked to grade the extent of farness n dfferent routng polces. When dscussng farness wth ward staff, the consensus was that each nurse/doctor should have the same workload as others. Seemngly, ths s the same as sayng that each nurse/doctor should take care, at any gven tme, over an equal number of patents (assumng homogeneous customers, for smplcty). Because the number of nurses and doctors s usually proportonal to standard capacty, ths crteron s equvalent to keepng occupancy levels of beds equal among the wards. Note that, by Lttle s law, = ALOS, where s the average occupancy level, and s the bed turnover rate. Thus, f one mantans ward occupances equal then wards wth shorter ALOS wll have a hgher turnover rate admt more patents per bed whch gves rse to addtonal farness concerns. Indeed, the load on staff s not spread unformly over a patent s stay, because treatment durng the frst days of hosptalzaton requres much more tme and effort from the staff than n the followng days (Elkn and Rozenberg 2007); moreover, patent admssons and dscharges sgnfcantly consume doctors and nurses tme and effort as well. Thus, even f occupancy among wards s kept equal, the ward admttng more patents per bed ends up havng a hgher load on ts staff. For these reasons, a natural alternatve farness crteron s balancng the turnover rate, or the flux namely, the number of admtted patents per bed per unt of tme (for example, per month) among the wards. One can also combne utlzaton and flux to produce a sngle workload measure; far routng would mantan ths measure equal across wards. For example, load, experenced by medcal staff n a ward, can be roughly dvded nto two parts: load assocated wth treatng hosptalzed patents (quantfed by utlzaton) and load due to patent admssons/ dscharges (quantfed by flux). For example, a sngle (objectve) workload measure for a nurse, based on a lnear combnaton of utlzaton and flux, was proposed by Tseytln (2009, 7.2): N T + N T (1) n where s the flux, s the occupancy level, N s the number of beds n the ward, T s the average amount of tme requred from a nurse to complete one admsson plus dscharge, T s the average tme of treatment requred by a hosptalzed patent per unt of tme, and n s the number of nurses n the ward; all quanttes refer to a gven ward The Settng of Anonymous Hosptal Although wards A D n Anonymous Hosptal provde smlar medcal servces, they do dffer n ther operatonal characterstcs (see Table 1), whch we now elaborate on. Frst of all, each medcal unt s characterzed by ts capacty. A ward s capacty s measured by ts number of beds (standard statc capacty) and number of servce provders doctors, nurses, admnstratve staff, and support staff (dynamc capacty). The maxmal statc capacty stands for the standard statc capacty plus extra beds, whch can be placed n corrdors durng overloaded perods. It s convenent to ntroduce notons of a subward and a patent room; these wll play a role n a dscusson of our results. Wards consst of subwards, whch, n turn, are made up of physcally colocated rooms. There a few (two to three) subwards n Table 1 Internal Wards Operatonal Measures Ward A Ward B Ward C Ward D ALOS a (days) 6.5 (±0.19) 4.5 (±0.15) 5.4 (±0.22) 5.7 (±0.18) Mean occupancy level (%) Mean # patents per month Standard (maxmal) 45 (52) 30 (35) 44 (46) 42 (44) capacty (# beds) Mean # patents per bed per month Return rate (wthn months) (%) Notes. Data refer to the perod May 1, 2006 October 30, 2008 (excludng January March 2007, when ward B was n charge of an addtonal subward). The data cover 16,947 admssons n total. a The level of confdence for average length of stay (ALOS) s 95%.

6 Mandelbaum, Momčlovć, and Tseytln: On Far Routng from EDs to Hosptal Wards 1278 Management Scence 58(7), pp , 2012 INFORMS Table 2 Numbers of Admtted Patents to Wards A D for Each Patent Category IW\Patent type Regular Specal care Ventlated Total Ward A 2,316 (50.3%) 2,206 (47.9%) 83 (1.8%) 4,605 (25.2%) Ward B 1,676 (43.0%) 2,135 (54.7%) 90 (2.3%) 3,901 (21.4%) Ward C 2,310 (49.9%) 2,232 (48.2%) 88 (1.9%) 4,630 (25.4%) Ward D 2,737 (53.5%) 2,291 (44.8%) 89 (1.7%) 5,117 (28.0%) Total 9,039 (49.5%) 8,864 (48.6%) 350 (1.9%) 18,253 Note. Data refer to the perod May 1, 2006 September 1, 2008 (excludng the months January March 2007, when ward B was n charge of an addtonal subward). a ward, and a subward conssts of several (three to four) rooms. In our hosptal IWs, the dynamc capacty durng a partcular shft s determned proportonally to the statc capacty (see, however, dscussons on the approprateness of such proportonal staffng n de Vércourt and Jennngs 2011, Green 2008, Yom- Tov 2007). In partcular, an IW beds-to-nurses rato n mornng shfts s 5:1 or 6:1 (dependng on the number of ntensve care beds n a ward); durng nght shfts, the rato s 8:1 or 9:1. Note that the number of nurses s determned by the number of beds n a ward rather than the number of occuped beds (patents). Hence a unt s operatonal capacty can be characterzed by ts number of beds only denoted as ts standard capacty Farness at Anonymous Hosptal Medcal unts are further characterzed by varous performance measures: operatonal (average bed occupancy level, ALOS, watng tmes for varous resources, number of patents admtted or released per bed per tme unt (flux)) and qualty (patents return rate, patents satsfacton, mortalty rate, etc.). Note that occupancy levels and flux are calculated relatvely to wards standard capactes. (Thus, occupancy can exceed 100%.) Comparng the two basc measures, ward capacty and ALOS, we observe n Table 1 that the wards dffer n both. Indeed, ward B s sgnfcantly the smallest and the fastest (shortest ALOS) among wards A D. We observe that the mean occupancy rate n ths ward s hgh (comparable to that n wards A and D; hgher than n ward C). In addton, the number of patents hosptalzed per month n ths ward s about 90% of those hosptalzed per month n the other wards, although ts sze s merely about two thrds of the others. As a consequence, the flux n ward B (6.38 patents per bed per month) s sgnfcantly the hghest. Because nursers and doctors n Anonymous Hosptal are assgned to partcular wards and are salared workers (as opposed to hourly workers), the load on the ward B staff s hence the hghest. Because the staff-to-beds rato s fxed, f ALOS s kept constant, the occupancy rate n a ward serves a measure of the load on the staff n that ward. Short ALOS could be caused by multple reasons. For example, t can result from a superor effcent clncal treatment or a lberal (versus conservatve) release polcy; a clncally too-early dscharge of patents s clearly undesrable. One possble (and accessble) qualty measure of clncal care s patents return rate (proporton of patents who are rehosptalzed n the IWs wthn a certan perod of tme n our case, three months). In Table 1 we observe that the return rate n ward B does not dffer sgnfcantly from the other wards. Also, patents satsfacton level n surveys another measure of care qualty does not dffer n ward B (Elkn and Rozenberg 2007). Furthermore, the dfference n ALOS across wards could be due to dfferent wards treatng dfferent types of patents and/or performng dfferent types of procedures. However, our data do not support ths hypothess. In fact, accordng to Table 2, ward B handles a dsproportonate share of specal-case and ventlated patents, patents that requre longer ALOS on average. Rather, a shorter ALOS n ward B can be attrbuted to superor staff practces. We conclude that the most effcent ward, nstead of beng rewarded, s exposed to the hghest load; hence, patent allocaton appears unfar, as far as the wards are concerned. Increasng farness n the routng process s expected to ncrease staff satsfacton, as well as provde ncentves for mproved care and cooperaton. 3. Model Formulaton We model the ED-to-IW process as a queueng system wth heterogeneous pools of..d. servers. Arrvals to the system are patents to be hosptalzed n the IWs; pools represent the IWs, whch ndeed have dfferent servce rates (1/ALOS); and the number of servers n each pool corresponds to the number of beds n each ward. To create a tractable system, we assume that arrvals to the wards occur accordng to a Posson process, and LOSs n wards are exponentally dstrbuted. (Both assumptons are mportant for analytcal tractablty, but they are naccurate realty-wse; see Armony et al. (2011) for emprcal fndngs on

7 Mandelbaum, Momčlovć, and Tseytln: On Far Routng from EDs to Hosptal Wards Management Scence 58(7), pp , 2012 INFORMS 1279 Fgure 1 Inverted-V System Hosptal (see 2.3 and 2.4), there exsts a drect connecton between the number of patents (occuped beds) and staff workload n a ward, a key feature that s utlzed n our model. It s possble to conceve of an alternatve, more complex model that focuses on the nursng staff drectly; nevertheless, as explaned n 6.2, our smpler model stll captures the essentals of far routng, f looked at through the approprate lenses. N 1 N 2 N K 1 2 K arrvals and LOS). Although, as remarked n 2.1, patents to be hosptalzed n the IWs are classfed nto several categores, we analyze here a sngle customer class model. Ths, as well as our dstrbutonal assumptons, certanly could present a lmtaton for applcaton of our theoretcal results, but we are stll able to draw useful nsghts about far routng n hosptals The Inverted-V Model Consder the queueng system shown n Fgure 1. Ths -model, or nverted-v model (n the termnology of Armony 2005), conssts of K server pools: pool has N..d. servers, each wth exponental servce tmes of rate (namely, servce rates are equal wthn each pool but vary among the pools). The total number of servers n the system s N = K =1 N. Upon arrval, a customer s routed to one of the avalable pools (f t has one or more dle servers) or jons a centralzed queue of nfnte capacty f all the servers at all pools are busy. Homogeneous customers arrve accordng to a Posson process wth rate > 0. Each customer requres a servce that can be provded by any of the servers, and each server can serve only one customer at a tme. The queueng dscplne s FCFS, nonpreemptve (the servce of a customer can not be nterrupted once started), and work conservng (there are no dle servers whenever there are customers awatng servce n the queue). In addton, we assume that all nterarrval and servce tmes are statstcally ndependent. Remark 3. Our model s centered around beds rather than personnel. In the setup of Anonymous 3.2. The QED Asymptotc Regme In ths secton, we formally ntroduce the QED regme n the context of the nverted-v model. We provde some justfcaton for ts applcablty n modelng the ED-to-IW process as well. The QED regme was frst dscovered by Erlang (1948); t was mathematcally formalzed by Halfn and Whtt (1981), hence t s often referred to as the Halfn-Whtt regme. The regme can be nformally characterzed n terms of any one of the followng condtons: a (steady-state) system wth a large volume of arrvals (demand) and many servers (supply or capacty) s operatng n the QED regme f () the delay probablty s nether near 1 nor near 0, () ts watng tme s one order of magntude shorter than servce tme (e.g., seconds versus mnutes n call centers, hours versus days n our case), or () ts total servce capacty s equal to ts demand up to a safety capacty, whch s of the same order of magntude as the square root of the demand. Characterzatons () and () relate to the qualty aspect, and characterzaton () ponts at hgh server effcences; thus, the QED regme acheves hgh levels of both servce qualty and system effcency by carefully balancng between the two (Armony 2005, Gans et al. 2003). The sutablty of the QED regme to the ED-to-IW process was studed n detal n Armony et al. (2011). Here we emphasze the followng relevant emprcal facts: The number of servers (beds) n each pool (ward) s approxmately (Table 1) the system s large enough for QED approxmatons to apply (Borst et al. 2004, Zhang et al. 2012, Yom-Tov 2010). Servers utlzaton (bed occupancy) s above 85% (Table 1). Watng tmes are ndeed an order of magntude shorter than servce tmes: hours versus days. Observe that a watng tme n our nverted-v model corresponds to a patent watng n the ED due to lack of avalable beds n IWs only,.e., bed allocaton tme the amount of tme from the decson to hosptalze the patent untl a bed becomes avalable n one of the IWs. In partcular, f a bed s avalable at the tme of the hosptalzaton decson, the correspondng bed allocaton tme s zero. However, estmatng bed allocaton tmes s dffcult because they are not tracked by Anonymous Hosptal s nformaton systems. Hence, we estmated the tme between

8 Mandelbaum, Momčlovć, and Tseytln: On Far Routng from EDs to Hosptal Wards 1280 Management Scence 58(7), pp , 2012 INFORMS the hosptalzaton decson and the tme of the frst procedure n the IW (see Mandelbaum et al. 2011). Ths tme serves as an upper bound on bed allocaton tme. Because the latter s a major component of the total watng tme n the ED, we conclude that both quanttes are n the order of hours. A servce tme n the nverted-v model corresponds to the nterval from the tme when a patent occupes a bed untl the tme the patent releases the bed (or, more precsely, untl the bed becomes avalable next). The probablty of encounterng an avalable bed n the desgnated ward, upon hosptalzaton decson (relatvely to the wards standard capactes), s estmatedto be 43%, 48%, 76%, and 55%, for wards A D, respectvely. The probablty of encounterng an avalable bed n any of the wards s approxmately 84%; ths estmate s based on a long perod of tme, whch ncludes lghtly loaded perods (the arrval process s nonstatonary). If one consders only hghly loaded wnter months (November to Aprl), the probabltes of encounterng an avalable bed n wards A D are 29%, 35%, 64%, and 45%, respectvely; the probablty of encounterng an avalable bed n any of the wards s 75% (ths probablty s the lowest n January 59%). Remark 4. The probablty of beng admtted to a ward mmedately (or wthn a short tme) after the hosptalzaton decson s much smaller than the probablty of encounterng an avalable bed (see Fgure 7 n Mandelbaum et al. 2011). Indeed, durng the perod May 1, 2006, to October 30, 2008 (excludng the months January March of 2007), only 2.7% of the patents were admtted to an IW wthn 15 mnutes from ther assgnment to a ward. Ths fact s consstent wth the effcency-drven regme. We further note that, n evenng shfts (when most patents are admtted; Armony et al. 2011), there usually are one or two doctors n each ward; hence, one expects that they operate under hgh loads. In that case, the probablty of encounterng an avalable doctor, whch s a prerequste for beng admtted to a ward, should ndeed be at ED levels (but we do not have any data on staff avalablty). We thus have two parallel queueng systems: beds, whch are QED, and medcal staff, who are ED. As already ndcated, we focus on the former. We use the followng scalng, sutable for the nverted-v model. Consder a sequence of systems, ndexed by (to appear as a superscrpt), wth ncreasng arrval rates, and ncreasng total number of servers N, but wth fxed servce rates 1 K. Then, the servce capacty of pool s c = N, the total servce capacty s c = K =1 c, and the total traffc ntensty s = /c. Both and N tend to nfnty smultaneously, n a way that two lmtng relatons are satsfed. Frst, for large, each pool has a nonneglgble fracton of the total capacty: c /c a as (C1) where a > 0 = 1 2 K), and K =1 a = 1. The scalar a s the lmtng proporton of the servce capacty of pool ( = 1 2 K) out of the total capacty. Second, t s convenent to defne a scalng parameter, = / ˆ where ˆ s the arthmetc-mean servce rate, K ˆ = a =1 t s approprate to thnk of as an effectve system sze. Then, the followng condton s assumed to hold: 1 > 0 as (C2) Ths lmt mples 1 (hgh utlzaton), as, and s (asymptotcally) equvalent to the classcal square-root safety staffng rule: the total servce capacty, c = + ˆ + o, s equal to the arrval rate,, plus a square-root safety capacty, ˆ, where s some qualty-of-servce parameter (the larger the value of, the hgher the servce qualty); here, s a untless quantty, whereas c,, and ˆ are measured n the same unts (e.g., patents/week). Note that fluctuatons of the arrval process from ts mean () are proportonal to. Wth beng the harmonc-mean servce rate, 1 = (C1) and (C2) then mply K =1 a 1 N = K =1 c / as. In vew of ths, (C2) can be rewrtten as N 1 = ˆ/ as. Fnally, we defne q to be the lmtng fracton of pool servers, out of the total number of servers: N N a = q (2) as ; the lmt s due to (C1) and (C2). Because a > 0, for all, we also have that q > 0, for all,.e., the pools are of comparable szes. Clearly, K =1 q = 1, and K =1 q =. Therefore, one can nterpret as the (lmtng) average servce rate of a server n the system, whereas ˆ s the (lmtng) average servce rate at whch customers receve servce. The two quanttes dffer because faster servers serve more customers. (Note that ˆ, wth equalty f and only f all s are equal to each other, n whch case = and = N.)

9 Mandelbaum, Momčlovć, and Tseytln: On Far Routng from EDs to Hosptal Wards Management Scence 58(7), pp , 2012 INFORMS Far Routng Before ntroducng formally two crtera of farness wthn the context of the nverted-v model, we need some notaton. Denote by I the long-run (steadystate) number of dle servers at pool ( = 1 2 K); each I s a random varable that attans values n 0 1 N ( = 1 2 K). Let = 1 ƐI /N be the mean long-run (steady-state) occupancy rate n pool. As the servers wthn each pool are symmetrc, also stands for the utlzaton of servers n pool the fracton of tme that each server s busy n the long run (steady state). Fnally, we denote by the average flux through pool (average number of servce completons per pool server per tme unt): =, by Lttle s law. Clearly, stands also for the average effectve servce rate of a server n pool. When analyzng farness toward servers, we consder the followng two crtera: occupancy balancng and flux balancng. The server s utlzaton, or, equvalently, the pool s occupancy rate, s one of the prevalent measures of the server s workload. As the occupancy rates at all pools tend to one n the QED regme (see (C2)); we thus compare the rato between the proportons of dle servers n the pools 1 /1 j, referred to as the dleness ratos. The closer these ratos are to unty, the more balanced the routng s, accordng to the occupancy-balancng crteron. The second crteron takes nto account the addtonal measure of workload, the average flux through the pools, namely, the number of customers served by a server per tme unt. Hence, our second crteron s based on the flux ratos / j. The closer ths rato s to unty, the more balanced the routng s, accordng to the flux-balancng crteron. Note that, n the QED regme, / j / j as for any work-conservng routng algorthm; that s, pools wth hgher servce rates experence hgher flux. Hence, based on the dscusson n 2.2, t s approprate to have lower utlzaton for pools wth hgher servce rates. Equvalently, f the flux rato for two pools s greater than one (because of the dfference n the servce rates), then the dleness rato should be greater than one as well. The algorthms we dscuss n the next subsecton all acheve ths goal Routng Algorthms In ths subsecton, we descrbe three routng algorthms. All three are work conservng, a choce that s dctated by our goal to reduce queue length (or, equvalently, watng tme by Lttle s law) rather than the number-n-system (sojourn tme). In the EDto-IW settng, the objectve s to mnmze the queue for transfer to the wards, thus reducng the overload on the ED. Work conservaton s dscussed further n the remark at the end of the present subsecton. Next, we descrbe the three routng algorthms and ther mplcatons on server farness. Let I t 0 N denote the number of dle servers n pool ( = 1 2 K) at tme t. When there are no customers awatng servce at tme t, the vector I1 t I K t specfes the state of the system. However, when the watng queue s not empty, an addtonal varable s needed to specfy the number of customers awatng servce; let Q t 0 1 be the number of customers awatng servce at tme t. It s convenent to defne I t N 1 N as (jontly) the total number of dle servers awatng customers/number of customers awatng servers, at tme t: K I t = I t Q t (3) =1 Note that I t can take negatve values: I t =, 1, ndcates that there are customers awatng servce at tme t. Because of work conservaton, one has K =1 I t = I t + and Q t = I t, where x + and x denote the postve and negatve parts of x, respectvely LISF Routng. The LISF polcy routes a customer to a server that has been dle for the longest tme among all dle servers. Ths polcy s commonly used n call centers and consdered to be far (Armony and Ward 2010). It was frst analyzed (n the QED regme) by Atar (2008) n the context of a snglepool system n a random envronment (servce rates were taken to be..d. random varables). Armony and Ward (2010) analyzed LISF routng n the nverted-v model wth two (K = 2) pools. Informally, they show that, for large > 0, the LISF polcy mantans fxed ratos between the number of dle servers n dfferent pools, whenever dle servers exsts; that s, f I t > 0 at tme t, then I t Ij t a I t+ a j I t = a + a j Hence, up to some techncal condtons (see the dscusson pror to Corollary 4.2 n Gurvch and Whtt 2009), under LISF routng one expects 1 1 j = ƐI N / ƐI j a q j = Nj a j q j and / j / j, as ;.e., both the dleness and flux ratos tend to the ratos of server rates. The algorthm leads to a desrable outcome: fast servers work less (have lower utlzaton) but produce more (have hgher flux). Note that even though LISF s a blnd polcy (a polcy that requres, at the tme of routng decson, none or mnmal nformaton on the parameters

10 Mandelbaum, Momčlovć, and Tseytln: On Far Routng from EDs to Hosptal Wards 1282 Management Scence 58(7), pp , 2012 INFORMS of the system or the system state; Atar et al. 2011), mplementng the LISF polcy n the hosptal settng s not straghtforward; namely, one must keep track not only on the number of dle servers (beds) n each pool (ward), but also the relatve orderng of dle servers n terms of ther dle tmes. The latter nformaton s not currently avalable n hosptals. Ths fact motvates us to consder alternatve routng polces that acheve the same (asymptotc) farness toward servers but utlze less nformaton for customer routng IR Routng. The IR routng polcy s a way to acheve the same dleness rato as the LISF polcy, but wthout the nformaton on dleness tmes. Ths polcy s a specal case of queues-and-dlenessrato (QIR) polces, whch were proposed and analyzed by Gurvch and Whtt (2009). They consder a generalzaton of the nverted-v model, a parallelserver system a servce system wth multple server pools and multple customer classes. The problem of dynamc control of such systems s often referred to as skll-based routng (borrowng the termnology from the world of call centers). Adoptng the QIR polcy to the nverted-v model, IR routes a customer to the server pool wth the hghest dleness mbalance. The basc dea s to route each customer n such a way that the vector I t I K t s as close to w 1 I t + w K I t + as possble, where the set of postve constants w s a pror fxed, wth K =1 w = 1. For example, f K = 2 and w 1 = w 2 = 1/2, then the IR polcy routes a customer to the pool wth the hgher number of dle servers. Specfcally, at tme t, a customer s routed to the pool wth the ndex { arg max I t w I t +} Tes are broken n an arbtrary but consstent fashon; the te-breakng rule does not mpact the results at the dffuson ( ) scale. Gurvch and Whtt (2009) showed that (Q)IR controls drve the dleness process to the predetermned proportons w, n the QED regme. Followng the same argument as n the LISF case, one expects 1 1 j = ƐI N / ƐI j w q j Nj w j q and / j / j, as. Therefore, wth the IR algorthm, one can acheve the same rato of the number of dle server (and dleness ratos) as under the LISF algorthm by smply settng w = a, because w q j /w j q = / j (see (2)). Gven ts weghts, the IR polcy s a blnd polcy as only the nformaton on the number of dle servers n each pool s needed. However, determnng the values of a s far from straghtforward n the hosptal settng. In partcular, note that a represents the lmtng rato between the pool servce capacty (c ) and the total servce capacty (c ). Therefore, because one must estmate the servce rates to evaluate the approprate weghts, the consdered verson of the polcy s, effectvely, not blnd. Recall from 2 that the capacty (number of beds) of each ward can vary wth tme; e.g., durng the frst three months of 2007, ward B was n charge of an addtonal subward. These facts serve as a motvaton for consderng yet a thrd routng algorthm, based on further reduced nformaton RMI Routng. We now ntroduce the RMI routng polcy. As our results n the next secton ndcate, the RMI polcy acheves the same ratos of dleness between server pools as the LISF and IR (wth w = a ) polces on the dffuson scale, and yet t requres nformaton on nether dleness tmes nor on pool capactes. Under the RMI polcy, a customer s assgned to one of the avalable pools, wth probablty that equals the fracton of dle servers n that pool out of the overall number of dle servers n the system (hence, the name of the polcy); that s, a customer to be routed at tme t s assgned to pool wth probablty I t /I t +. Tseytln (2009) observed that the RMI polcy, n the nverted-v model, s equvalent to the RA polcy n the sngle-server pool model wth N servers, where N servers have rate ( = 1 K). To llustrate ths, consder the followng example. Assume that a customer arrves to the system wth two avalable pools: pool has two avalable servers, and pool j has three avalable servers. Thus, the customer s routed to pool wth probablty 2/5 and to pool j wth probablty 3/5. As n pool, there are two avalable..d. servers, and each one of them wll serve ths customer wth probablty 1/5; smlarly, each server n pool j wll serve the customer wth probablty 1/5. As a consequence, the customer s assgned to any one of the fve avalable servers wth equal probabltes. An analytcally appealng feature of the RMI polcy s that, when modeled as a Markov chan n contnuous tme, the system s reversble (Kelly 1979) (Tseytln 2009 conjectured that ths s the only routng polcy under whch the nverted-v system nduces a reversble Markov chan). Therefore, one s able to derve ts steady-state probabltes n a straghtforward manner and provde an exact analyss n steady state. (Because of ther complextes, LISF and IR polces were analyzed only asymptotcally.) Fnally, we note that, even though RMI s a randomzed polcy, t can be easly mplemented n a hosptal settng. For example, patent ID numbers can be utlzed as sources of randomness (as stated by Mandelbaum et al. 2011, at least one hosptal n our survey has mplemented some randomzed polcy, based on patent ID numbers).

11 Mandelbaum, Momčlovć, and Tseytln: On Far Routng from EDs to Hosptal Wards Management Scence 58(7), pp , 2012 INFORMS 1283 Remark 5. (Why Work Conservng?). The three consdered algorthms are work conservng. Ths assumpton s not restrctve from the pont of vew of both hosptal and patents. The goal s to reduce the watng tmes n the ED so that the number of patents awatng transfer from the ED to the IWs s mnmzed. Therefore, mplementng a work-conservng polcy s desrable because ntentonal server dlng only ncreases the number of patents n the ED. Tseytln (2009) showed that, under the watng tme crteron and RMI routng, t s not benefcal to dscard slow servers,.e., t s not desrable to ntentonally dle servers. On the other hand, when the objectve s to mnmze the sojourn tme (watng plus servce tmes), cases arse when t s benefcal to dscard slow servers n a system wth heterogeneous servers. Wthn the context of the well-known slow-server problem, ths was shown for two servers (Rubnovtch 1985a) and many servers (Cabral 2005); namely, f some servers are slow enough, roughly speakng, t could turn out preferable for a watng customer to wat for a fast server to become avalable rather than start servce at a slow server. Thus, as far as sojourn tme s concerned (but not watng tme), a non-workconservng polcy could turn out preferable. 5. Theoretcal Results The followng theorem characterzes RMI performance, n the nonasymptotc regme (fnte arrval rate ). The theorem states that when comparng two pools, servers utlzaton n the faster pool s lower than that n the slower pool, but the flux n the faster s hgher than n the slower. Because of the symmetry of servers wthn each pool, ths mples that, when consderng any two servers, the faster between the two wll work less tme than the slower one, but, at the same tme, the faster server wll serve more customers than the slower. The result suggests, frst of all, some form of farness: faster servers are rewarded by workng less tme. In addton, operatonal preferences of the system are accommodated as well: more customers are served by faster servers than by slower servers. We note that Cabral (2007) proved ths result, for the sngle-server-pool system under the RA polcy, ndependently. Theorem 1. In the nverted-v model under the RMI polcy, for any two pools and j, f > j, then < j and > j. Proof. See Mandelbaum et al. (2011). The theorem provdes an upper and lower bound on the rato of server utlzatons (assumng > j j / < / j < 1. Ths suggests that the dfference n utlzatons of any two servers s more sgnfcant the more ther servce rates dffer: for j, one has j, but as j grows smaller than, the server-utlzaton rato decreases. The bounds on the flux ratos are as follows (assumng > j 1 < / j < / j, where the second nequalty follows from / j < 1. The latter upper bound s mportant although the fact that faster servers serve more customers contrbutes to system performance, one should not forget that, n certan cases, hgher flux actually mples hgher workload. In partcular, ths s the case n our ED-to-IW process, because servce admssons and releases mpose workload that s plausbly proportonal to flux. For the RMI polcy, the servers flux rato / j s bounded by the rato of ther servce rates: / j = / j j < / j when > j ; thus, f server rates are comparable, a faster server s flux can not be much hgher than that of a slower one. Remark 6. The fact that utlzaton decreases the faster the server gets, provdes an ncentve for servers to work faster, whch s postve on one sde but, on the other, mght harm servce qualty, f one starts servng customers too fast. For example, see the study by Gans et al. (2003), who descrbe telephone agents that ntentonally hang up on customers to mantan low average servce tmes. Another ssue s that f the slow server s not responsble for beng slow (for example, a new server versus a veteran), punshng the server for beng slow appears qute unfar. However, as noted earler, hgher flux may be consdered n certan cases a punshment as well. Hence, to decde whch servers perceve themselves as better off (the faster or the slower) or, alternatvely, what servers ncentves are (to ncrease or decrease hs/her servce rate), one must account for the servers utlty functons (the ones they strve to optmze), whch combne both crtera (utlzaton and flux); for an example of such a utlty functon, recall (1). The next theorem characterzes the nverted-v model under RMI routng n the QED regme. It can serve as a means for evaluatng performance measures (probablty of wat, expected watng tme, etc.) of the nverted-v model n the QED regme. A summary of performance measures, both for fnte and n the lmt, as, can be found n Mandelbaum et al. (2011). Recallng (3), let I be the statonary total number of dle servers/customers awatng servce n the system wth arrval rate ; the random varable I takes values n N 1 N ; by defnton, I + = K =1 I ; and I = for negatve ndcates that there are customers awatng servce. As the system sze ncreases ( ), the varablty of demand ncreases as well, mplyng that the magntude of I gets larger and larger. Hence, to gan understandng of the behavor of I, we consder ts scaled verson: Î = I /

12 Mandelbaum, Momčlovć, and Tseytln: On Far Routng from EDs to Hosptal Wards 1284 Management Scence 58(7), pp , 2012 INFORMS such a scalng s typcal for the QED regme and s referred to as a dffuson scalng. Informally, the theorem states that, n statonarty, there exsts a dmensonalty reducton n the sense that the statonary number of dle servers n pool satsfes I a I +, for large ;.e., dle servers are dstrbuted across the pools proportonally, accordng to the relatve pool capactes. Moreover, the theorem turns out to provde an explct estmate of how close I s to a I +. In partcular, fluctuatons of I around a I + grow wth at a rate 4, and thus we consder Î = 1 ( I I c c I ) = 1 K We refer to the 4 scale as a subdffuson scale because 4 for large. Characterzaton of RMI behavor on the subdffuson scale provdes addtonal nsghts nto the operaton of the algorthm; we dscuss ths further n the next subsecton. Denote by and the standard normal densty and dstrbuton functons, respectvely. Let denote convergence n dstrbuton. Theorem 2. Consder the nverted-v model n steady state under the RMI routng algorthm n the QED regme (C1 C2). Then, as, (Î Î 1 Î K 1 ) (Î ) Î >0 Î1 Î K 1 Î>0 (4) where Î and Î 1 Î K are ndependent; ( Î 0 = 1 + ) 1 Î > x Î > 0 = x/ x 0; Î x Î 0 = e x, x 0; and Î 1 Î K s zero-mean multvarate normal, wth ƐÎ Î j = a 1 =j a a j. Proof. See Mandelbaum et al. (2011). The contrbuton of the theorem to understandng system behavor under RMI s twofold. Frst, RMI acheves the same server farness as LISF and IR (wth approprate weghts) n the sense that dle servers are dstrbuted across the pools accordng to the pools relatve capactes (a s). Ths s of sgnfcance because RMI requres less nformaton for ts operaton than LISF and, unlke IR, RMI does not utlze nformaton on pool capactes. Second, the qualty of allocaton of dle servers across the pools under RMI s revealed. Specfcally, the number of dle servers n a pool devates from c /c I (a number determned by pools relatve capactes) by a random quantty (normally dstrbuted) of the order 4. In vew of the fact that the number of dle servers n a pool s proportonal to, fluctuatons of order 4 are neglgble when s not small. Remark 7 (Equal Servce Rates). When the servce rates are equal across the server pools,.e., 1 = 2 = = K, then = ˆ = 1, =, and N / 1, as, and one recovers the well-known Erlang-C QED approxmaton (Halfn and Whtt 1981). Remark 8. Note that K =1 Î 1 Î >0 = 0 by defnton. The lmt (4) s consstent wth ths condton: K =1 Î = 0 = 1, because Ɛ K =1 Î 2 = 0. Remark 9. The dmensonalty reducton (on the dffuson scale) can be deduced from the hydrodynamcal equatons n Da and Tezcan (2011). For example, consder the case when there are K = 2 pools. If the fracton of dle servers that are n the frst pool exceeds c 1 /c, then a dsproportonate number of customers wll be routed to the frst pool, resultng n a lower number of dle servers n the frst pool. As smlar stuaton occurs when the fracton of dle servers that are n the frst pool drops below c 1 /c. Hence, the rato of the number of dle servers n dfferent pools should be equal to the rato of the pool capactes. Example 1 (Probablty of Delay). Consder an nverted-v system wth the followng parameters: K = 2, q 2 = 2q 1 = 2/3, and 1 = 2 2 = 2 (e.g., patents/ week). The total number of servers (e.g., beds) s taken to be + 05 N = q q 2 2 whereas N1 = roundq 1N and N2 = N N1. We vary the arrval rate from 10 to 500 ths corresponds to varyng N1 and N 2 from 3 to 128 and 6 to 256, respectvely. In Fgure 2, we plot the exact probablty of delay along ts QED approxmaton. Note that because of the PASTA property, the probablty of delay s equal to I 0, and, hence, Theorem 2 renders ( delay 1 + ) 1 (5) as. The precedng lmt serves as the bass for calculatng QED approxmaton for the fnte system. To ths end, the requred QED parameter changes slghtly wth because the number of servers n each pool s a natural number. Thus, when evaluatng the QED approxmaton, we compute for each value of : ( ) / = 1 N1 1 + N2 ˆ 2 where ˆ = N N N1 1 + N2 2 A QED approxmaton for the probablty of delay s obtaned by evaluatng the rght-hand sde of (5), wth

13 Mandelbaum, Momčlovć, and Tseytln: On Far Routng from EDs to Hosptal Wards Management Scence 58(7), pp , 2012 INFORMS 1285 Fgure Exact Values and QED Approxmatons of the Probablty of Delay for the Sequence of Systems Descrbed n Example 1 are more than arrvals/departures n the tme nterval 0 900). One can observe that the processes I t t 0 and I 1 t a 1I t + t 0 evolve on two dfferent countng scales the frst one on the -scale ( 187), and the second one on the 4 -scale ( 4 43). Probablty of delay From the followng corollary, t s mmedate that, under RMI, the dleness ratos satsfy 1 /1 j / j as ; that s, the RMI polcy acheves the same dleness ratos as the LISF polcy Exact value QED approxmaton Arrval rate =. As can be seen n Fgure 2, the QED approxmaton has a reasonable (useful) accuracy when system szes and servce rates are smlar to those n Anonymous Hosptal (see 2). Example 2 (Dmensonalty Reducton). To llustrate the dmensonalty reducton that arses n Theorem 2, we smulated an nverted-v system under the RMI polcy, wth parameters that adhere to the QED scalng (C1) and (C2): K = 2, = 3 950, 1 = 15, 2 = 75, N 1 = 138, and N 2 = 276 (a 1 = a 2 = 1/2, ˆ = 1125, 086). In Fgure 3, we plot typcal realzatons of the total number of dle servers, I t t 0, and the centered number of dle server n the frst pool, I 1 t a 1I t + t 0. Intally, at tme t = 0, there are no dle servers and no customers awat servce,.e., I 0 = 0. By tme t = 900, the system s close to ts statonary regme (there Corollary 1. Consder the nverted-v model n steady state under the RMI routng algorthm n the QED regme (C1 C2). Then, as, ƐÎ ƐÎ, ƐÎ + ƐÎ +, and ƐI / a ƐÎ + for = 1 K, where Î s as n Theorem 2. Proof. See Mandelbaum et al. (2011). Remark 10 (Loss of Performance). As stated n the ntroducton, FSF routng s asymptotcally optmal (wthn the set of all nonpreemptve, nonantcpatng FCFS polces) n the sense that t stochastcally mnmzes the statonary queue length and watng tme as the arrval rate and number of servers grow large n the consdered nverted-v model. When analyzng the trade-off between farness and performance (.e., RMI versus FSF), one must consder the followng two aspects. On one hand, wthn our mathematcal model, farness comes at the cost of a decrease n system performance (e.g., an ncrease of the probablty of delay). Theorem 2 (and Corollary 1), together wth results Fgure 3 Illustraton for Example I (t) I 1 (t) a1 (I (t)) Tme Tme

14 Mandelbaum, Momčlovć, and Tseytln: On Far Routng from EDs to Hosptal Wards 1286 Management Scence 58(7), pp , 2012 INFORMS on FSF routng n Armony (2005), provdes means for quantfyng the loss of performance under the RMI polcy (relatve to FSF). In partcular, results n Armony (2005) ndcate that, under the FSF polcy, ( ) Î 0 = Î > x Î > 0 = x / ˆ/, x 0; Î Î 0 = e x, x 0; = mn ; and = ˆ/. Thus, the probablty of delay under the RMI polcy s hgher (because ) than the probablty of delay under the FSF polcy by a factor of 1 + / 1 + / (6) whch depends on the spare capacty parameter and the rato of the arthmetc-mean servce rate ˆ to the mnmum servce rate ; recall that n heavy traffc, only servers wth the smallest servce rate are dled under the FSF polcy. The ncrease n the expected watng tme s gven by the same rato. Ths follows from the fact that the condtonal expected wat, gven that t s postve, s the same under the two routng polces. The decrease n performance should be carefully revewed to determne f ncreased delays are clncally acceptable. For example, n Anonymous Hosptal (see Table 1), ˆ 118 patents/week, 108 patents/week, and 087, leadng to the rato n (6) beng equal to 1.07,.e., the probablty of delay ncreases by 7% when employng the RMI nstead of the FSF polcy. On the other hand, as stated n 1, the ssue of farness needs to be examned beyond the mathematcal model wth fxed servce rates. In partcular, ensurng farness toward medcal staff should be vewed wthn the context of provdng a rght set of ncentves for staff. Although the watng tme of customers (patents) s stochastcally mnmzed under the FSF polcy, medcal staff workng n wards wth nonmnmal servce rates have no ncentve to further reduce LOS any mprovement would lead to a hgher load. Thus, although the FSF polcy nomnally results n shorter watng tmes, the RMI polcy can be more benefcal for customers n the long run because the mplementaton of the RMI polcy can eventually lead to shorter LOSs and, as a result, shorter watng tmes. Weghted RMI. Fnally, we note that the RMI algorthm can be generalzed to a weghted RMI (WRMI) algorthm as follows. Gven a set of weghts w > 0 ( = 1 K) such that K =1 w = 1, the WRMI algorthm routes a customer to pool, at tme t, wth probablty I w t / K j=1 I w j t, where I w t = w I t. The weghts can be used to adjust dleness ratos to a desred target (relatve to the rato of servce rates) n the QED regme. Unless all weghts are equal, the resultng system s not reversble, and thus our analyss of RMI can not be extended to WRMI. However, nsghts ganed from RMI can be used to heurstcally analyze WRMI as follows. Consder a tme nstance t such that I t/ > 0. Then, the departure rate of customers from pool s c I t (where c I t); on the other hand, customers enter servce at pool wth rate I w t/ K j t. Therefore, for large, j=1 I w a c I t a j cj j Ij t I w t I w j t = w I t w j Ij t In vew of the precedng, one expects that the dleness ratos satsfy, as, 1 1 j w j w j 5.1. Comments on the Subdffuson Scale Relevance and Implcatons. Based on emprcal data from our Anonymous Hosptal, we estmated that 1897 patents/week, and ˆ 118 patents/week (see Table 1); thus, = / ˆ These numbers and our theoretcal analyss reveal that there exst three relevant countng scales (bed, room, subward) and three correspondng tme scales (hour, day, week). To descrbe a stochastc process of nterest (e.g., the number of avalable beds), one needs to defne not only an approprate countng scale, but also approprate tme ntervals (scale) over whch relevant changes n the process occur. On tme ntervals that are too short, no changes n the value of the process can be observed on the countng scale; on the other hand, on tme ntervals that are too long, the process covers the whole countng scale, and tme varablty can not be studed. The fnest countng scale s at the level of an ndvdual bed/patent (order-1 scale), whereas the correspondng tme scale s at the level of an hour (order-1/). Indeed, when consderng ndvdual patent arrvals, the relevant tme scale s based on hours because 1/ 086 hours. The coarsest countng scale (order- ; dffuson scale) s used to descrbe the total number of avalable beds and patents awatng hosptalzaton. For a large hosptal, ths scale s defned by a subward because 127 (beds or patents) s approxmately the sze of one-quarter to one-thrd of a ward, and the number of avalable beds/patents watng s proportonal to (see Theorem 2). The correspondng tme scale s defned by a typcal LOS a week n our case, because patents stay n a ward a bt less than a week on average (or 1/ ˆ 085 weeks; see Table 1).

15 Mandelbaum, Momčlovć, and Tseytln: On Far Routng from EDs to Hosptal Wards Management Scence 58(7), pp , 2012 INFORMS 1287 The precedng two pars of scales are standard for systems operatng n the QED regme. The thrd par of scales s ntermedate and due to RMI. The countng scale (subdffuson scale) s defned by a room (rooms n Anonymous Hosptal have four beds) and s an order- 4 scale ( 4 36 (beds or patents)). Ths scale s relevant n descrbng the number of patents that need to be moved between wards to make the nstantaneous (at a specfc moment of tme) dleness ratos equal to the long-run (average) dleness ratos ntroduced earler. The latter ratos do not provde nformaton on the system behavor at a specfc moment of tme. For example, consder the followng two scenaros, assumng two symmetrc wards: () the number of avalable beds n the two wards s the same at all tmes, and () the number of avalable beds n the frst ward s hgher than n the second one for a long perod of tme, and then the stuaton s reversed for the same amount of tme. Under both scenaros, the dleness rato s unty. The room-level ( 4 ) countng scale provdes nformaton on devatons of the numbers of avalable beds (n steady state) from the numbers that ensure that the actual ratos of dle servers at a gven moment of tme are equal to the dleness ratos (average quanttes). In partcular, our results mply that fluctuatons of nstantaneous dleness ratos around ther long-run averages are of the order 1/ 4. The assocated tme scale s based on a day (order-1/ ˆ; 1/ ˆ 087 days) and descrbes relevant tme ntervals over whch these relatvely mnor fluctuatons of dleness ratos average out. Recall the two scenaros mentoned above. In the second one, several cycles are needed for dleness ratos to converge to unty. Indeed, f one observes the system for a short perod of tme only, then the system appears unfar. Yet, over long perods of tme the system s far. Fnally, we note that the ntermedate countng and tme scales are related, n the sense that the larger the fluctuatons of the number of avalable beds, the longer tme ntervals one requres for convergence of the dleness ratos. Informally, our results ndcate that, under the RMI polcy, fluctuatons of the numbers of dle beds n dfferent wards, around values that ensure / j nstantaneous dleness ratos, are of the order 36 4 (beds or patents);.e., the room-based countng scale s relevant. Indeed, Theorem 2 mples I c c I + Î I and hence the dleness ratons obey I /N Ij /N j ( ( )) Î j I /a I j /a j Moreover, when avalable beds exst, even the mnor dfferences (order-1/ I or, equvalently, order- 1/ 4 ) between nstantaneous dleness ratos and the correspondng long-run dleness ratos ( / j ) are averaged out on tme ntervals of order 1/ ˆ (tme ntervals that contan -order arrvals/departures are of nterest here, because central lmt theorem devatons are of the order 4 n that case; consequently, ths corresponds to tme ntervals of length / = 1/ ˆ). In Anonymous Hosptal, ths corresponds roughly to days (1/ ˆ 087 days), and consequently one expects desred dleness ratos (wth very hgh accuracy) on a weekly bass. Interestngly, the same behavor of ntermedate scales occurs when the system operates under the LISF rule (see below) Techncal Dscusson. Even though the three consdered algorthms operate n dfferent ways, they result n the same behavor on the dffuson scale. However, dfferences arse on the subdffuson scale. Informally, Theorem 2 states that, for large, RMI devatons of I around c /c I + are on the order 4 of ; note that both I and I + are -order random varables. On the other hand, under IR polcy, I t a I + s an order-1 random varable, as. Consequently, although mplementng RMI n a hosptal settng wll not ensure that an nstantaneous dleness rato s equal exactly to the desred value / j at all tmes, the number of avalable beds n a ward wll only dffer from the desred one by a 4 -quantty, whch s neglgble n comparson wth the number of avalable beds n a ward. Furthermore, t should be noted that, under both the IR and RMI polces, there exsts a separaton of tme scales; namely, the processes I t/ t 0 and I t/ t 0 evolve on the order-1 tme scale under both polces. Ths s typcal n the QED regme no tme speedup s needed, n contrast to the case of conventonal heavy traffc. However, the processes I t a I t + t 0 and I t c /c I t + / 4 t 0 evolve on the 1 - and ˆ 1/2 -tme scales, as, under IR and RMI routng, respectvely. Hence, there s a separaton of tme scales, because the latter processes evolve on much faster tme scales (order- 1 and order- ˆ 1/2 ) than the former processes (order-1 tme scale). In Fgure 4, we plot typcal sample paths of I1 a 1 I t + t 0 under RMI and IR routng, for the system descrbed n Example 2 the dfference n countng and tme scales s evdent. Therefore, n the context of Anonymous Hosptal, the desred dleness ratos are mantaned not only n the long run, but also on shorter tme ntervals. In partcular, provded that dle servers exst (I 0 > for some > 0), ntervals of order 1 and ˆ 1/2 are requred under IR and RMI, respectvely, for convergence of

16 Mandelbaum, Momčlovć, and Tseytln: On Far Routng from EDs to Hosptal Wards 1288 Management Scence 58(7), pp , 2012 INFORMS Fgure 4 Typcal Sample Paths of I 1 t a 1I t + t 0 n an Inverted-V System Under the RMI (left) and IR (Rght) Polces I 1 (t) a1 (I (t)) + 0 I 1 (t) a1 (I (t)) Tme Note. Parameters of the system are the same as n Example 2. Tme the emprcal dleness ratos to the long-run averages 1 /1 j, n the sense that (for large ) ( 1/N ) t/ I 0 u du 1 and u du 1 j 1/N j t/ ( 4 1/N t/ 1/Nj t/ I 0 j ˆ 0 I ˆ 0 I ) u du 1 j u du 1 j converge to 0 as t ncreases (t s an order-1 quantty here) for IR and RMI, respectvely. In certan cases, the tme scale separaton can be used to explctly evaluate the subdffuson behavor of the system under IR routng n the QED regme. As seen n the followng example, the dea s to explot the fact that the subdffuson process evolves on a faster tme scale than dffuson processes. Recall that the subdffuson behavor under RMI s characterzed n Theorem 2. Example 3 (Subdffuson Scale Under IR). Consder the nverted-v model n the QED regme, under IR routng, wth K = 2, w 1 = w 2 = a 1 = a 2 = 1/2, and note that I1 t I t/2 I2 t I t/2 = I1 t I2 t/2 I 2 t I 1 t/2. The heavy-traffc averagng prncple (e.g., see Coffman et al. 1995) states that, when consderng the dstrbuton of I1 t I 2 t, as, one can act as f the total number of (scaled) dle servers s fxed (Whtt 2002, p. 70). In partcular, on the event I t/ >, > 0, the dstrbuton of I1 t I 2 t converges, as, to the statonary dstrbuton of the brth death contnuoustme Markov chan, wth transton rates r +1 = 1/2 + 1 < =0 and r 1 = 1/2 + 1 >0 + 1 =0 (here we assume that, f the two pools have the same number of dle servers, then a customer s routed to the frst pool wth probablty 0 1). Indeed, f I1 t I 2 t s postve, then departures from pool 2 and new arrvals contrbute to a decrease of ths quantty; on the other hand, departures from pool 1 ncrease I1 t I 2 t. Ths leads, for any > 0, to [ I 1 t I 2 t = I t/ > ] as. ( > <0 ) 3 1 We conjecture that the subdffuson behavor of the system under the LISF algorthm s the same as the one under RMI. The conjecture s based on the followng heurstc reasonng. A way to mplement the LISF polcy s to have servers completng servce jon a queue of dle servers. Ths queue operates n an FCFS fashon. Whenever a customer needs to be assgned to a server, t s routed to the server at the head of the queue. Observe that, under the descrbed scheme, the server that has been dle for the longest tme s assgned a customer before any other server. The state of the queue of dle servers at tme t s an ordered lst that conssts of pool labels (1 2 K). Now, consder an nverted-v model n the QED regme ( ) at a tme nstance t such that I t/ > 0. Then, all the servers n the dle queue joned the queue wthn a tme nterval that s on the order of 1/ ˆ;.e., I t/ remans approxmately constant durng ths nterval of tme. The server pool labels n the dle queue are approxmately ndependent, wth a label beng equal to wth probablty c I t c c a K =1 I t = a I t c ˆI t + 1/ for large ; the standard asymptotc notaton 1/ ndcates that the second term s of the

17 Mandelbaum, Momčlovć, and Tseytln: On Far Routng from EDs to Hosptal Wards Management Scence 58(7), pp , 2012 INFORMS 1289 order 1/. The precedng equaton s due to the fact that a gven label s of type f a server n pool completes servce before any other server n the other pools. As a consequence, the random varable I 1 t a 1I t s of the order 4, because of the central lmt theorem and the fact that the total number of labels n the queue s I t, a quantty proportonal to. 6. Concludng Remarks We consdered routng algorthms that are applcable to routng hosptal patents from the emergency department to nternal wards. Gven the heterogenety of the wards, the objectve s to acheve farness from the pont of vew of hosptal staff, whle not hurtng effcency too severely. Wards are modeled as server pools, and two types of quanttes are used to quantfy farness: dleness ratos and flux ratos. Under the LISF polcy, whch s consdered to be far and s commonly used n call centers, both ratos tend to the ratos of servce rates n the respectve server pools, when the system s n the QED regme; the approprateness of the QED regme n modelng the ED-to- IW process s supported by emprcal data, collected n Anonymous Hosptal. In other words, LISF routng leads to a desrable outcome: faster servers work less, yet they produce more (serve more customers). However, the applcablty of LISF routng n hosptals s lmted because the algorthm requres nformaton unavalable n hosptals on a real-tme bass. The same dleness and flux ratos can be acheved by IR routng; yet, n that case, one must estmate (tme-varyng) ward (server pool) servce capactes. We thus propose a randomzed routng polcy, RMI, that attans the same desred performance ratos as LISF, but requres only the number of dle servers n server pools for makng decsons. The polcy can be mplemented n a hosptal by usng, for example, patent ID numbers as sources of randomness. A generalzed verson of RMI, WRMI, can be used to fne-tune the desred outcome. The three algorthms (LISF, IR, and RMI) share one feature n common; namely, these algorthms acheve the desred dleness ratos (equal to servce-rate ratos) by attemptng to mantan the ratos of the numbers of dle servers n dfferent pools equal to these dleness ratos at all tmes. However, because dleness ratos represent an average performance measure, one can vary the nstantaneous ratos of dle servers dependng on the total number of dle servers and stll acheve the target (average) dleness ratos. Ths approach was proposed by Armony and Ward (2010). The advantage of such an algorthm s that t delvers a lower average watng tme compared to LISF, whle mantanng the same long-run dleness ratos as LISF. However, the gan n performance does not come for free. In partcular, one must determne the optmal number of dle servers ratos as a functon of the total number of servers and, therefore, the parameters of the system must be known (arrval rate, pool servce capactes),.e., the polcy s effectvely not blnd Partal Informaton Routng Smulaton Analyss We mentoned above that avalablty of nformaton s crtcal for determnng an approprate routng polcy n hosptals. Our proposed routng, RMI, requres the nformaton on the number of avalable beds at each ward at the moment of routng (nformaton that s qute mnmal compared to the other routng polces consdered n ths paper). However, the occupancy status n the IWs s not avalable on a real-tme bass n Anonymous Hosptal; nstead, the ED reles on one bed census update per day (n the mornng). Thus, to mplement RMI routng, t s necessary to estmate the system state at decson tmes, based on the system state at the last update tme pont. An example of such partal-nformaton routng can be found n Tseytln and Zvran (2009), where the authors created a computer smulaton model of the ED-to-IW process n Anonymous Hosptal. They used t to examne varous routng polces, accordng to some farness and performance crtera, whle accountng for the scarcty of nformatonn the system. Smulatng the ED-to-IW process helped acheve addtonal practcal nsghts, by accommodatng some analytcally ntractable features (such as tme-nhomogeneous Posson arrvals), and allowed analyss of more complex routng algorthms. The best-performng algorthm (n terms of both staff farness and operatonal performance) proposed by Tseytln and Zvran (2009) was an algorthm that mnmzed at each decson pont a convex combnaton of the two conflctng demands: balanced occupancy rates and balanced flux. The mplementaton under partal nformaton resulted n almost no deteroraton of performance. To conclude, we now explan brefly the way that the lackng nformaton s predcted. Denote by M j the number of occuped beds (busy servers) n ward (pool) j. Ths counter s updated at some tme pont T and s estmated at other decson tme ponts accordng to the patent routng and the ward servce rate; namely, we estmate the number of occuped beds n ward j at tme t > T to be equal to M j M j j t T +, j = After a routng decson s made, we update M k = M k + 1 (k denotes the ward chosen to admt the next patent) Routng at the Level of Indvdual Provders Instead of examnng farness va our bed-based model, one can study farness be means of a more

18 Mandelbaum, Momčlovć, and Tseytln: On Far Routng from EDs to Hosptal Wards 1290 Management Scence 58(7), pp , 2012 INFORMS complex staff-based model. Such a more detaled model could potentally be used to study farness at the level of ndvdual care provdes rather than wards. (It would thus be more relevant to the U.S. healthcare system, where typcally nurses do not have fxed ward assgnments and are pad on an hourly bass.) In that case, one stll needs to keep track of the number of patents n wards. However, two addtonal aspects must be modeled explctly: () patent servce requests,.e., when certan tasks need to be performed by nurses/doctors (for example, n Belgum, nurses work s broken down nto 23 representatve tasks for plannng purposes; Wllams 2000); and () the process of assgnng of these tasks to ndvdual staff members. For example, a model can be constructed from the followng components: Parallel-server systems wth heterogeneous serves (Atar 2008): A system represents a ward, and servers represent medcal staff; tasks n a ward are assgned to ndvdual doctors/nurses accordng to a (ward) schedulng polcy. Erlang-R ( R for reentrant ) model wth a bounded number of customers (Yom-Tov and Mandelbaum 2011): The reentrant aspect of the model captures the fact that customers (patents) requre servce (tasks) multple tmes durng ther sojourn tmes (LOS) n the system; the bounded number of customers s due to the fnte number of beds n a ward. A (hosptal) routng polcy that assgns customers (patents) to one of parallel Erlang-R systems (wards) wth heterogeneous servers. Overall farness can be acheved by enforcng farness both at the nter- and ntraward levels. There exst multple tme scales n ths model: a task scale (mnutes/tens of mnutes), a content scale (tens of mnutes/hours; tme between tasks for a sngle patent), a shft scale (hours), and fnally a length-of-stay (a week) scale. These scales can potentally be used to smplfy the analyss of the system. Indeed, durng a patent s stay n the hosptal, not only many tasks are performed, but also many shfts are rotated. Hence, one expects that a patent experences an equvalent servce (task) rate on the LOS scale. Ths equvalent rate s a functon of staff n the ward as well as the task assgnment polcy. As a consequence, when consderng routng at the hosptal level, one can plausbly replace the heterogeneous Erlang-R model wth a parallel-server system wth an equvalent servce rate,.e., one obtans our nverted-v model Future Research We propose a number of research drectons motvated by our work. Frst, robustness of our nsghts aganst dstrbutonal assumptons on servce tmes remans to be nvestgated. Second, the nverted-v model takes nto account prmarly the number of beds n each ward, whereas hosptal staff (nurses and doctors) affect the model ndrectly through server servce rates. The model can be mproved by explctly modelng staff as well. In that case, two-scale (doctors and beds) models arse. Thrd, patents to be hosptalzed n the IWs are classfed nto several categores. When arrvng to the ED, patents are classfed as walkng or lyng ; n addton, pror to runnng the Justce Table, they are classfed as regular, specal care, or ventlated. The load, mposed on the hosptal by patents, vares sgnfcantly among dfferent categores, n LOS, complexty of treatment, and watng tmes. Thus, t s of prme nterest to extend our model to accommodate multple customer classes. Last, n the present work, servce rates are taken to be exogenous quanttes,.e., there s no attempt to capture possble dependency between the routng algorthm and servce rates of doctors and nurses. However, such dependency does exst because the hosptal staff adapts to routng polces by ncreasng/ decreasng ther servce rates and/or qualty of care. Tools from game theory can be applcable n modelng such effects. Acknowledgments Avsha Mandelbaum s research was supported n part by the Bnatonal Scence Foundaton [Grants and ]; the Israel Scence Foundaton [Grant 1357/08]; and by the Technon funds for the promoton of research and sponsored research. Petar Momčlovć s research was supported n part by the Natonal Scence Foundaton [Grant CNS ]. References Armony, M Dynamc routng n large-scale servce systems wth heterogeneous servers. Queueng Syst. Theory Appl. 51(3 4) Armony, M., A. Mandelbaum Routng and staffng n largescale servce systems: The case of homogeneous mpatent customers and heterogeneous servers. Oper. Res. 59(1) Armony, M., A. Ward Far dynamc routng polces n large-scale systems wth heterogeneous servers. Oper. Res. 58(3) Armony, M., S. Israelt, A. Mandelbaum, Y. Marmor, Y. 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