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Nurse-To-Patient Ratios in Hospital Staffing:a Queuing Perspective

Francis de VericourtFuqua School of Business, Duke University, Durham, NC 27708 [email protected]

Otis B. JenningsFuqua School of Business, Duke University, Durham, NC 27708, [email protected]

The immediate motivation of this paper is California Bill AB 394, a piece of legislation which mandates

fixed nurse-to-patient staffing ratios as a means to address the current crisis in the quality of health care

delivery. With a predictive queuing analytic approach we seek to determine whether or not ratio policies are

effective and, if not, to provide an alternative staffing paradigm.

We assume a significant correlation between delays in addressing patient needs and averse medical out-

comes and we promote using the frequency of excessive delay as a measure of staffing policy performance.

By applying new many-server asymptotic results, we develop two heuristic staffing policies that perform

very well and are easy to implement. This is the first many-server asymptotic analysis of health care issues.

Among the insights gained from the heuristics is the realization that no ratio policy can provide consistentlygood quality of service across medical units of different sizes. Moreover, the optimal staffing levels for larger

systems display a type of super pooling effect in which the requisite workforce is significantly smaller than

the nominal patient load.

Key words: Queuing System, Health Care, Public Policy, Nursing, Staffing, Many-Server Limit Theorems

1. Introduction

In 1999, California introduced the nations first law mandating nurse-to-patient ratios in hospitals,Bill AB 394. The legislation, implemented in 2004, specifies the minimum number of nurses that

should be staffed for each hospital unit, given the current number of patients therein. The min-

imum number of nurses is a fixed, unit-specific fraction of the number of patients. For instance,

according to AB 394 at least one registered nurse for every four patients should now be present at

any time in any pediatrics unit within California. This legislation has inspired other states to con-

sider establishing similar requirements. Further, two House bills in a recent Congressional session

proposed regulating nurse staffing among Medicare-participating hospitals (see for instance Spetz

2005).

The rationale for implementing these ratios stems from the association between nurse staffing

level and patient safety. Research studies suggest a significant connection between nurse workload

and clinical outcomes. For instance, Aiken et al. (2002) conclude that the addition of one surgical

1


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Article submitted to Management Science; manuscript no. MS- 3

where n is the fixed number patients, who alternate between stable and needy states, and where

s is the number of nurses. Each needy patient is served by a nurse, if available. Otherwise, the

patient must wait in a (virtual) queue until service can be provided.

In our framework, the principal metric of concern is the probability of excessive delay, i.e., the

probability that the delay between the onset of neediness and the provision of care from a nurse

exceeds a given time constraint. To our knowledge, no empirical studies have explored the impact

of this metric on patient care and safety. Nonetheless, we argue that excessive delays are akin to

possible adverse events from the supply side (pressure experienced by the nurses) and from the

demand side (waiting patients), both of which factor into the overall quality of care. Accordingly,

we pose the nurse staffing problem in terms of finding staffing levels that guarantee a bound

on a specified probability of excessive delay. We prove new many-server asymptotic results to

approximate this probability of excessive delay. To our knowledge, this constitutes the first many-

server asymptotic analysis of health care related issues. Based on these findings we derive two

heuristics corresponding to the so-called efficiency-driven (ED) and quality- and efficiency-driven

(QED) staffing regimes. In addition to performing well, the heuristics provide managerial insights.

In particular, we observe a super pooling effect in large medical units where the optimal number

of required nurses is dramatically less than the nominal patient workload.

It is worth noting that we do not try to find what an acceptable probability of excessive delay

should be. Neither do we determine the precise values of the parameters describing our model (i.e.,

average nurse service times and frequency of patient needs). Nevertheless, we argue that, whatever

constitutes an excessive delay and our tolerance thereof, mandating nurse-to-patient ratios cannot

ensure uniform quality of care across all hospitals.

We present our basic model in Section 3. The impact of Bill AB 394 on probabilities of excessive

delay is explored in Section 4. We formulate the optimal nurse staffing problem in Section 5, and

derive and test two heuristics in Section 6. Section 7 presents our main managerial insights, which

make use of the heuristics provided earlier. We conclude by discussing our results and suggesting

future analytical and empirical research. The following section contains a short survey of the

relevant literature.

2. Literature Review

Recent years have witnessed considerable debate in health-care practice, both within government

and throughout academia, over the issue of administrating nurse staffing practices in health-care

facilities. Spetz (2004) provides a comprehensive overview of the history and surveys some prelim-

inary effects of nurse-to-patient ratios in California. Spetz (2005) also describes several alternative


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4 Article submitted to Management Science; manuscript no. MS-

policies, including the development of pay-for-performance systems in hospitals. The concept of

mandatory nurse-to-patient ratios, which should help improve nurse working conditions, is moti-

vated in part by the persistent shortage of registered nurses in the United States (Sochalski 2004,

Kaestner 2005). But the primary justification for these ratios lies in the assumption that nurse

staffing levels have a significant impact on patient safety and outcomes. The bulk of research in this

field focuses on trying to establish (or negate) this basic premise. Data that are usually collected

to explore this link include such outcome measures as mortality rates, adverse incidents, length of

stay, and patient and nurse dissatisfaction (Neisner and Raymond 2002). Lang et al. (2004) provide

a fairly exhaustive and systematic review of articles investigating this link.2

The link between workload and quality of care is often thought of in terms of adverse events.

For instance, in evaluating how hospitals improve quality of care, Tucker and Edmondson (2003)

identify failures that occur in care delivery processes, such as tasks that are unnecessarily or

incompletely performed. In our setting, staffing levels are set to prevent the assistance of patients in

need from being delayed longer than a specified time constraint. We assume that the frequency at

which delays exceed this constraint is significantly correlated with the frequency of adverse events.

This is a reasonable assumption, since delaying certain procedures can endanger patient health.

For instance, the medical guidelines for certain myocardial infarctions recommend the immediate

administration of aspirin (American College of Cardiology 2002). Delays also give rise to unfinished

tasks, either because nurses fail to remember them later or because they abandon them in order

to take care of more urgent procedures. Unfinished care has been identified as a strong factor

impacting the quality of nursing care (Sochalski 2004).

Note also that in business settings, timely service has long been recognized as a main driver ofcustomer satisfaction and is widely used as a service quality indicator (see for instance Gans et al.

2003) for applications in call centers). For businesses, excessive delays can lead to lost business

opportunities.

This paper deals exclusively with the nurse staffing regulatory issue. However, there are several

other factors that go into nursing workforce management. For example, Kao and Tung (1980) use

statistics to forecast patient loads a year into the future. Such forecasts are necessary for under-

standing nursing requirements and for effectively setting budgets to cover nurse salaries. Another

aspect of workforce management is the setting of nurse schedules, which typically is completed in

the days or weeks preceding the actual work shifts. Approaches to solving the scheduling problem

2 The study was commissioned by the CDHS and was included as part of the final statement of evidence justifyingthe implementation of California Bill AB 394 in 2003.


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include linear programming (Jaumard et al. 1998), math programming (Warner and Prawda 1972,

Miller et al. 1976), and genetic algorithms (Aickelin and Downsland 2003, Bailey et al. 1997). In

some hospital settings, patients are matched with nurses; see Punnakitikashem et al. (2006) for a

stochastic programming approach to the nurse assignment problem.

Operations management studies also extend to more general hospital capacity issues beyond

the nurse staffing problem. Green (2004) presents a recent overview of queuing models used in

capacity planning and management of hospitals (see also Smith-Daniels et al. 1988). Typically,

medical units are modelled as multi-server, open-loop, M/M/s queues, where the arrival process

represents the stream of incoming patients and the service time captures the average length of stay.

Studies of these models provide, for instance, the number of required beds to achieve availability

targets in a given unit. On the other hand, our model seeks to capture the workload variability

experienced by the nurses. During a time period when the number of patients is relatively constant,

the dynamics of the system are better captured by a closed queue with a finite population of

patients (an M/M/s//n queue), where each patient can independently require processing by a

nurse. In a previous paper (de Vericourt and Jennings 2006a) we provide preliminary asymptotic

results for this system via many-server limit theorems. Those results, combined with extensions

herein, allow us to derive efficient staffing rules with general targets for the probability of excessive

delay, as well as to evaluate the performance of nurse-to-patient ratio policies relative to this metric.

Many-server asymptotic analysis has proven exceptionally useful for other service-oriented capac-

ity management problems, specifically those focusing on call centers. Many recent call center papers

can be directly linked to the conceptual breakthrough of Halfin and Whitt (1981), who recognized

that, at least for large open queueing systems, one can staff service operations that achieve not only

a high level of utilization but also very fast customer response times. In other words, the staffing

policies studied by Halfin and Whitt are efficiency-driven (ED) from the standpoint of the service

provider and quality-driven (QD) from the perspective of the customer, yielding a hybrid quality-

and efficiency-driven (QED) staffing regime. Garnett et al. (2002) later studied the distribution of

the virtual waiting time of call centers staffed in the QED regime in more detail. They also observed

that by including abandonment in many-server queueing models, one can consider the ED case, for

which the nominal load exceeds the processing capacity. Whitt (2004), Mandelbaum and Zeltyn

(2006) and Baron and Milner (2006) have all extended the study of open many-server systems with

abandonment and staffed in the ED regime. The latter two, both contemporaneous papers, also

consider minimizing service capacity, subject to constraints on the frequency of excessive delay, as

the primary objective of system design.


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3. The Queuing Model

Consider a medical unit of s nurses serving n patients. Over time, each patient repeatedly requires

the assistance of a nurse for various care procedures. When a patient requires the assistance of a

nurse, we refer to the patients state as needy. Otherwise the patient is said to be stable. We assume

that the time for a stable patient to become needy, referred to as the activation time, is random

and independent of all other system dynamics. Upon becoming needy, a patient is immediately

treated by a nurse, if an idle nurse is available. Otherwise, a needy patient waits for an available

nurse; queued patients are assisted in a first-come-first-served fashion. The nurses service time is

also random and independent of other system features. Once treated, a patient becomes stable

again, until she needs other care procedures. We assume that nurses work as a team so that any

one of them can serve any patient. In some medical units however, a nurse is assigned to a specific

set of patients. Our model remains relevant in such settings, provided that nurses whose patients

temporarily require less care will help their busier colleagues, as is often the case in practice.

In general, activation times and nurse service times depend on patient acuity levels and nurses

skill levels. Unfortunately, when setting public policy, the possible combinations of these levels

are too vast to accommodate fully using specific guidelines. Instead, policy makers may want to

set nurse-to-patient ratios assuming either the lowest or the highest possible acuity level for all

patients in the unit.3 For practical modeling purposes, the possible discrepancy in acuity levels

and different skill levels of nurses can be captured by adjusting the probability distribution of the

activation time and the nurse service time accordingly.

To our knowledge, no empirical studies have tried to estimate the probability distributions of

activation and service times. We suspect that these distributions can be quite general in practice.

For analytical tractability, however, we assume that activation and service times are exponentially

distributed with rates and , respectively. In some respects, the exponential assumption is a

reasonable one. The wide range of possible patient needs and nursing tasks suggests a high degree

of variability in the health care process, which is consistent with the high coefficient of variation

of the exponential distribution.

In a system in which each patient has a dedicated nurse, the long run fraction of nursing time

required by a typical patient is equal to r := /( + ) while the long run fraction of time a patient

is stable is given by r := 1 r . The quantity rn then is the nominal patient load (also referred3 In fact, California Bill AB 394 specifically advises that, in order to account for case mix and patient acuity levels,hospitals use the mandated minimum staffing ratio recommendations in conjunction with staffing policies and proce-dures developed locally at the hospitals. We consider the case of homogeneous patients for which such adjustmentsare not required.


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rate intensity. Letting k := (n k) denote the activation rate when k patients are needy (andsuppressing the parameter n except when necessary), we obtain

pn(s, t) =n

k=s

kkni=0 ii

P(V > t|N= k) = estn

k=s

(n k)kni=0(n i)i

ksj=0

(st)j

j!. (2)

Recall that the parameter T 0 delineates between acceptable and excessive delays. The probabil-ity of excessive delay for a system with n patients and s nurses is denoted pn(s, T), or simply pn(T)

when no confusion is possible, and is a point along the function pn(s, ). An interesting property ofthe expression (2) when evaluated for a specific T is that, given the number of nurses and patients

in the unit, the probability of excessive delay is entirely described by the parameters r and the

target T since k in (1) only depends on = r/(1 r). For ease of exposition, we will alwaysspecify the time target T as an integer multiple of the service time (T = i/).

In the next section we study the impact of nurse-to-patient ratios on the probability of excessive

delay.

4. A Queueing Model Perspective on California Bill AB 394

Let us consider California Bill AB 394, which has governed nurse staffing over the past few years,

and pediatric units in the state. The law requires that at all times, regardless of the size of a

hospital, at least one nurse must be staffed for every four patients under care (see Table 2 in the

appendix for the full list of these mandated ratios). In the following, we investigate the theoretical

effect of the mandated ratios, as predicted by our queueing model. We focus our investigation on

pediatric units exclusively, with the understanding that our analysis extends to all units for which

pooling of nurses is a reasonable operating assumption.

The main idea when deriving the collection of staffing ratios was to evaluate, for each hospital

unit, the proportion of nursing time a patient requires during a typical shift (controlling for patients

acuity level and nurses skills). One should note that this proportion of time is precisely the same

as the load factor r in our queuing framework. Because quantitative data are lacking, the interest

groups and research teams who framed Bill AB 394 and ultimately set the ratio values, relied

heavily on expert panels comprised primarily of highly qualified and experienced registered nurses

and nurse administrators (see California Department of Health Services 2003, Institute for Health

& Socio-Economic Policy 2001). The suggested ratio values from these panels differed greatly (see

Table 2), perhaps as a result of the various constituents involved in the process. Nevertheless,

compromises among the various suggestions were made and the results are our best estimates for

the nominal patient load for each hospital unit.


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In this paper we consider only pediatric units, primarily b ecause of the relative consensus among

the expert panels suggested pediatric unit ratio values. The mandated ratio policy sets the number

of nurses sn equal to the nominal patient load rn, when there are n patients in the unit:

sRn = rn, (3)

with r = 1/4 (for pediatric units) and where

x

is equal to the smallest integer larger or equal to

x. We also refer to this staffing rule as the nominal ratio policy.

Given the size n of the pediatric unit, we can compute the required number of nurses sn based

on (3) and deduce the resulting probability of excessive delay using (2) for different time thresholds

T. Figures 13 show the impact of the size of the pediatric unit on the probability of excessive

delay experienced by the needy patients for T = 0 (any delay), 1/ (delay longer than an average

nurse service time), and 2/ (delay longer than two average nurse service times), respectively. The

sharp oscillations in performance can be explained by the step-wise increase in the staffing level

that occurs at four-patient intervals when rn is rounded up to the nearest integer. The magnitudeof these oscillations decreases with n. For T = 0, Figure 1 suggests that when disregarding these

oscillation effects, the excessive delay probability remains roughly stable around 65% for most n.

On the other hand, for T > 0, Figures 2 and 3 indicate that as the size of the unit grows, patients

can expect to see progressively lower probabilities of excessive delay. In other words, patients in

larger pediatric units appear to be better off than patients in smaller units, although all units are

consistent in the implementation of the 1:4 staffing ratio dictated by law.

In order to explore the magnitude of this inconsistency in California we need information on the

distribution of the number of patients in Californian pediatric units, which can significantly vary

over time. Unfortunately such detailed information is not publicly available. As an alternative,

we use the data provided by the California Office of Statewide Health Planning and Development

(OSHPD) as part of the state-wide hospital utilization data it collects annually. 5 In this study

we use the data from 2003 which contains the patient numbers reported by all pediatric units of

Californian hospitals, averaged over the year. These census averages are used as if they represent

the typical sizes of corresponding units at any given time. For our sample, a total of 134 pediatric

units in California with average occupancies between 2 and 154 were identified (see Table 3 in theappendix).

5 The database was and continues to be used by many of the interest groups and research teams that framed Bill AB394 and its implementation.


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Figure 1 Delay Probability under Nominal Ratio Staffing, T = 0

0 10 20 30 40 50 600.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

n

pn(

T)

Figure 2 Excessive Delay Probability under Nominal Ratio Staffing, T =

1

,

0 10 20 30 40 50 600

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

n

pn

(T)

The first two rows of Table 1 contain our chosen partition of the pediatric unit sizes in California.

Interestingly, the average pediatric unit occupancies are considerably skewed, with a majority of

hospitals reporting average occupancies below 20 patients; only 11% of the 134 pediatric units

reported average occupancies greater than 20. The remaining rows of the table provide excessive

delay probabilities, as predicted by our model, for the specified choice of time constraint parameter

T. The data is presented in a fashion that controls for the rounding-up effect. Namely, for each unit


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Figure 3 Excessive Delay Probability under Nominal Ratio Staffing, T = 2

,

0 10 20 30 40 50 600

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

n

pn(

T)

Table 1 Excessive Delay Probabilities in Californian

Pediatric Units

n n 5 5 < n 10 10 < n 20 20 < n

%Units 36% 38% 15% 11%

T = 0 66% 67% 67% 67%

T = 1/ 38% 28% 24% 15%

T = 2/ 20% 10% 6% 2%

size partition / delay threshold pair, the table reports the largest predicted probability of excessive

delay among all unit sizes in the partition. The table summarizes the theoretical performance

of the current law under alternative definitions of acceptable delay. For instance, when T = 0,

the probability of delay hovers around 66% for all n. Although the performance is consistent,

it is consistently poor. For acceptable delays that are strictly positive (T > 0), Table 1 suggests

significant discrepancies in service quality across the different hospitals in California. For instance,

in 36% of pediatric units (for which n 5) the probability that delays exceed a time equivalentto a nurses average service time (T = 1/) can be as high as 38%, whereas in another 26% of

the units (for which n > 10) this probability is less than 24%. When T = 2/, the probability of

excessive delay appears to be reasonable (less than 6%) for 26% of the units, but can still reach

20% in small units.

The nominal ratio policy is but one of many within the spectrum of possible ratio policies, each

having the form sn = n for some real number > 0. Studies have argued that decreasing the


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number of patients per nurse leads to a reduction in adverse events. In Figure 4, we provide the

predicted probability of delay (T = 0) when r = 1/4 and a ratio policy with = 1/3 is employed

(see Figure 10 in the appendix for the case where T = 1/). As expected, performance is improved,

but exhibits a significant inconsistency across unit sizes. On the other hand, when = 1/5, the

ratio policy provides consistent but poor performance for the delay threshold T = 1/, disregarding

again the oscillation effect; see Figure 5 (see also Figure 11 in the appendix for T =0).

Figure 4 Delay Probability under Ratio Staffing, r = 14

, = 13

, T = 0

0 10 20 30 40 50 600.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

n

pn(T

)

In summary, the nurse-to-patient staffing ratios

appear to have different performance implications for smaller units than for units with largerpatient populations, with the trend showing that patients in smaller units will see higher proba-

bilities of excessive delay in relation to units of larger size,

can, in many cases, result in very high probabilities of excessive delays which in turn mightlead to poor levels of quality of care.

In the remainder of this paper, we use a more analytic approach to formalize and generalizethe findings of this section. In particular, we will show that these insights depend neither on the

precise values of the different parameters which describe our system, nor on the choice of the ratios

mandated by law.


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Figure 6 Optimal versus nominal ratio staffing, r = 14

, = 1%, T = 0

0 10 20 30 40 50 600

5

10

15

20

25

n

sn

sn

*

sn

R

system exhibits any level of variability, such a target can only be achieved with a dedicated nurse

for each patient, an unreasonable expense in most units. Determining acceptable values of and

T is a subject for future research. On the other hand, our analytic approach, and in particular

our test for a range of values for and T, allows one to develop insights into the form of the best

possible staffing policy, as well as pointing out the shortcomings of the current legislation, without

assuming the exact levels of the probability and/or defining what constitutes excessive delays.

For each n 1, the optimal staffing level sn satisfying (4) can be obtained using a simpleenumerative search algorithm over all possible values of sn. For each step of the procedure, the

probability of excessive delay pn(sn, T) is calculated using equation (2) and compared to the bound. Of course, the optimum staffing level is an increasing function of n. So sn provides a natural

starting point in the search for sn+1.

Figures 68 compare the staffing of pediatric units (r = 1/4) under the mandated nominal ratio

policy (sRn = rn) with the optimal staffing policy sn, for = 1% and for T = 0, 1/ and 2/,respectively. When T = 0, Figure 6 shows that the nominal ratio policy underestimates the staffing

requirements that ensure that the excessive delay probability is less than 1%. Furthermore, the gap

between the ratio staffing rule and the optimum one seems to be increasing with n. For T = 1/,

the number of nurses scheduled in the nominal ratio policy is still not large enough to guarantee

that at most 1% of needy patients will wait more than a time equivalent to a nurses average service

time, 1/. The deviation from the optimal staffing rule quantity seems however to decrease with

n, at least within the range of unit sizes considered. Figure 8 provides a more intriguing situation


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Figure 7 Optimal versus nominal ratio staffing, r = 14

, = 1%, T = 1

0 10 20 30 40 50 600

2

4

6

8

10

12

14

16

n

sn

sn

*

sn

R

Figure 8 Optimal versus nominal ratio staffing, r =

1

4 , = 1%, T =

2

0 10 20 30 40 50 600

5

10

15

n

sn

sn

*

sn

R

in which the nominal ratio policy underestimates the number of required nurses needed by small

units (n 25), but is conservative relative to the optimal number needed by larger units. In effect,

only larger units achieve the target probability of excessive delay of at most 1%, but the staffing

for these situations is needlessly excessive.

The poor performance of the nominal ratio policy is also confirmed by the significant gap in the

probability of excessive delay for the nominal ratio policy versus the optimum policy. For instance


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Figure 9 Excessive Delay Probability as a Function of n, r = 0.25, = 0.05, T = 1/

0 10 20 30 40 50 600

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

n

pn(

T)

p*

n(T)

pR

n(T)

Figure 9 illustrates the significant gap in the probability of excessive delay for the nominal ratiopolicy versus the optimum policy for r = 1/4, = 5% and T = 1/. As shown by the figure, the

nominal ratio policy meets the service level requirement consistently only for large unit size (n 45)(see de Vericourt and Jennings 2006b, for more details).

These results suggest that nominal ratio policies do not possess the structure of the optimal

staffing rule. In particular, the unit size n seems to affect the slop of sn as a function of n, while

the slop of a ratio policy is independent of n. This will be confirmed in the following section.

6. Heuristics

In the following, we introduce and test alternative staffing rules that perform well and are easy

to implement. The heuristics provide important insights into the structure of appropriate nurse

staffing rules that cannot be directly obtained from the optimization problem in (4).

We derive the heuristics via many-server asymptotics, an approach that investigates properties

of families of staffing rules (i.e., mappings n sn) as the number of patients becomes very large. Itturns out that depending on the limit s =limn sn/n, the system exhibits two distinct behaviors.

When s < r, for large values of n, all nurses are practically always busy. Hence we refer to such

policies as efficiency-driven (ED), or belonging to the ED staffing regime. Another phenomenon in

this setting is that all needy patients must wait a non-trivial amount of time before being assisted.

This suggests that, when staffing in the ED regime, the time delay threshold T must be strictly

positive. On the other hand, when s > r , needy patients experience little or no wait and, hence,


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such policies are referred to as quality-driven. Moreover, the corresponding value of T is small or

zero. In this case there will often be some idle nurses. The so-called QED regime, arising when

s = r, exhibits the defining characteristics of both the ED and QD regimes. (See de Vericourt and

Jennings (2006b) for more details on the many-server framework.)

6.1. QED-Based Heuristics

Our first heuristic is based on the assumption that optimal staffing levels are slight deviations from

those provided by the nominal ratio policy. Under this assumption, the resulting staffing policy

can initially be thought of as being in the QED regime.

Suppose we have a QED staffing policy sn. To capture the microscopic dynamics of the system,

we consider the following process

Nn(t) :=Nn(t) rn

n, t 0, (5)

where the process Nn tracks the number of needy patients and its initial value Nn(0) is assumed

to be approximately equal to the centering constant rn. We refer to the transformation in (5) asdiffusive scaling. In de Vericourt and Jennings (2006a), functional central limit theorems (FCLTs)

are taken for this sequence of processes. The result, summarized below, is a diffusion process which

serves to approximate the stochastic dynamics of Nn.

It was shown in de Vericourt and Jennings (2006a) that the sequence of virtual hitting time

processes {Vn, n 1} converges to zero at the same rate as 1/

n converges. It follows that, in order

to approximate needy patient waiting times, we must zoom in on this process. Define the inflated

virtual hitting time processes{Vn, n 1}, where Vn :=

nVn, for each n 1. As Proposition 1states below, under the QED staffing regime, the diffusion-scaled needy patient process and the

inflated hitting time process converge together to scalar multiples of the same diffusion process.

Consider first the following series {n, n 1} associated with the staffing rule n sn,

n :=sn

n r

n. (6)

Modulo round-off error, the quantity n

n is the deviation of sn from the nominal ratio policy.

The following proposition states that if n converges to a finite value , which also requires that

s = r (confirming that indeed the policy operates in the QED limiting regime), then the inflated

virtual hitting time converges in distribution to a diffusion process.

Proposition 1. Consider a staffing rule n sn such that n where (,) as n . If Nn(0) converges in distribution to a proper random variable N(0) as n , then Vn and


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(Nn )+/(sn/n) converge jointly in distribution to (N )+/(r). Moreover, the steady statedistribution of N has probability density

fQED(x) =

1()rr

xrr

rr

x < ,

()r

rx+rrr

rr

x .

(7)

with

() =

1 + e2

2r2r

rrrr

1

, (8)

and where the function () is the standard normal cumulative distribution function.

Proof. The result is a direct consequence of Theorem 2 and Corollary 2 of de Vericourt and

Jennings (2006a).

To derive our first heuristic, consider a staffing rule of the form sn = rn + n

n and assume

that n converges to some constant . The excessive delay probability can be approximated by the

tail of the virtual hitting time distribution: pn(sn, T) P(Vn >

nT). 6 Using Proposition 1, we

approximate the distribution of Vn with that of (N)+/(sn/n):

P(Vn >

n T) P

(Nn)+sn/n

> T

n

.

We then consider the distribution of N in steady state given by (7) so that

pn(sn, T) (n)r

1

nrr

n+

snTn

rx + rn

r

r

dx

= (n) n

r

r snT

n

n

r

r

. (9)

where () is the standard normal distribution function. Our first heuristic consists of finding nsuch that the approximation in (9) is equal to the target :

Heuristic 1. (H1)

For a given > 0 and T 0,sH1n = rn + n

n (10)

where n satisfies

= (n) n(1 + rT)rr rTnrnrr , (11)and where (n) is given by (8).

6 Notice the implicit asymptotic PASTA assumption.


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Note that when T = 0, (11) simplifies considerably: = (). It follows that, when T = 0, we

have n = 1() for each n. When T = 0, our first heuristic is equivalent to the one developed in

Section 5 of de Vericourt and Jennings (2006a) and performs exceptionally well. In particular, the

heuristic policy never deviates from the optimal one by more than one nurse; i.e., for each n 1,|sn sH1n | 1.

When T > 0, finding sH1n involves solving equation (11) in its general form. The next proposition

guarantees the existence and uniqueness of the solution. It also leads to efficient numerical methodsto determine n.

Proposition 2. The function : + such that

(x) = (x)xrr

(1 + rT)rTnr

xrr

is strictly decreasing in x.

Proof. Note first that () can be written as

(x) = (x)(ax b)

(cx) ,

where a, b and c are positive constants. Denote by h() the hazard rate of the standard normaldistribution, i.e., h(x) = (x)/(x) for every real x. We can restate (8) as

(x) =

1 +r h

x

rr

hxrr

1

.

Because the hazard rate of the normal distribution is increasing, the function () is decreasing. It

remains to show that x (ax b)/(cx) is also decreasing. Taking the first order condition,this is equivalent to showing that,

a(ax b)(cx) > c(ax b)(cx),

or equivalently, because (x) = (x) for the standardized normal distribution, that

h(ax + b) >c

ah(cx).

This final relation follows because h() is increasing, c/a = 1/(1 + rT) < 1, and ax + b > c x for any

real x.

The monotonicity of allows us to use a simple bisection search algorithm in order to solve

equation (11). This procedure is more efficient than solving the optimal problem in (4) but still

requires some computation effort.


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6.2. ED-Based Heuristics

In designing Heuristic 1 we assumed that the optimal staffing policy was a slight deviation from the

nominal ratio policy. This approach is valid for small values of T, a feature that suggests staffing in

the QED regime. However, when T is large we should staff in the ED regime because the nominal

ratio policy is no longer close to the optimal one. Our second heuristic is based on the assumption

that, for every value of T > 0, there exists a unique quantity rT < r such that the optimal policy is

an order n deviation from rTn.Heuristic 2. (H2) For a given > 0 and T > 0,

sH2n = rTn + T

n (12)

where

rT =r

1 + rT(13)

and

T =r 1

() r + rT

(1 + rT)3 . (14)

The following argues that Heuristic 2 is asymptotically optimal as n , thereby justifying itsuse for large values of n. As discussed later, it works well for all values of n.

Proposition 3. For any given T > 0, the probability of excessive delay pn(sn, T) has a nonde-

generate limit (0, 1) if and only ifsn

n rT

n , as n,

for some

(

,

), with

=

r

(1 + rT)

3

r + rT

. (15)

Proof. The probability of excessive delay can be written as

pn(sn, T) =

n

k=0

k(n k)1 n

k=sn

k(n k)esTksnj=0

(sT)j

j!

=DnBn

1 +

AnBn

1,

where

An sn1k=0

nk

(n k)k,

Bnn

k=sn

n!

sn!(n k)ssnn

sn

k


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and

Dn n

k=sn

n!

(n k)!ssnnsn!

sn

kesT

ksnj=0

(snT)j

j!.

The quantity An can be expressed as

An = n(1 + )n1

sn1k=0

n 1

k

rkrnk

= n(1 + )n1P(Xn sn 1),

where for each n, Xn is a binomial random variable with parameters n 1 and r. Similarly, onecan express Bn as

Bn =n!

sn!ssnn

sn

n1esn/

nsn1k=0

sn

k1

k!esn/

=n!

sn!ssnn

sn

n1esn/ P(Yn n sn 1)

and Dn as

Dn = n!sn!

ssnn

sn

n1esn/

nsn1k=0

nsnk1j=0

snk

1k!

(snT)j

j!e(sn/+snT)

=

n!

sn!ssnn

sn

n1esn/ P(Zn n sn 1),

where Yn and Zn are independent Poisson random variables with parameters sn/ and sn/+ snT,

respectively. For each sequence of the random variables {Xn}, {Yn} and {Zn}, we generate a centrallimit theorem. Starting with Xn, we center and rescale to obtain

P(Xn sn 1) = P

Xn (n 1)r(n 1)rr

sn 1 (n 1)r(n 1)rr

, (16)

P(Yn n sn 1) = P

Yn sn/sn/

n 1 sn/rsn/

(17)

and

P(Zn n sn 1) = PZn sn( 1 + T)

sn(1

+ T) n 1 sn(

1r

+ T)sn(

1

+ T)

. (18)

The sequences (Xn (n 1)r)/

(n 1)rr, (Yn sn/)/

sn/ and (Zn sn( 1 +T))/

sn(

1

+ T) each converge in distribution to standard normal random variables (i.e., with

mean zero and variance one). It is given that the staffing level as a function of n is sn = rn/(1 +rT) + o(n). It follows that [sn 1 (n 1)r]/

(n 1)rr as n . By (16) one can

conclude that

limn

P(Xn sn 1 ) = 0. (19)


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Likewise, (n 1 sn/r)/

sn/ as n, so that by (17),

limn

P(Yn n sn 1 ) = 1. (20)

Finally, by the conditions provided for sn,

n 1 sn( 1r + T)

sn( 1 + T)

r

(1 + rT)

3

r + rT

as n, and by (18) we have

limn

P(Zn n sn 1 ) =

r

(1 + rT)

3

r + rT

. (21)

The ratio An/Bn can be written as

AnBn

= CnP(X sn t)

P(Y n sn 1) ,

where

Cn = sn!

n!

sn

n1 (1 + )n1ssnn

esn/.

Following the analysis in the proof of Proposition 1 of de Vericourt and Jennings (2006a), we have

Cn r

1r

+ T

e2/2r2 as n. The relevant feature is that the limit of Cn is finite. It follows

then by (19) and (20) that An/Bn 0. Finally, notice that Dn/Bn = P(Zn n sn 1)/P(Yn n sn 1) so that by (20) and (21),

limn

pn(sn, T) =

r

(1 + rT)

3

r + rT

.

6.3. Numerical Study

We explore the performance of the previously introduced heuristics through numerical examples.

Tables 4-6 report the differences sn sH1n and sn sH2n for different values of T, n and for r =0.1, 0.25, 0.9 and = 1%. (de Vericourt and Jennings (2006b) provides supplementary numerical

results.)

For T = 0, H1 performs extremely well with |sn sH1n | 1 for all tested cases. This is consistent

with the results in de Vericourt and Jennings (2006a). (Note that since H2 is only defined for

positive T, the tables do not report its performance for T = 0.) For T > 0, H1 also performs well

with |snsH1n | 1 for r = 0.1 and 0.25. However the performance of H1 deteriorates as n increasesfor r = 0.9, as shown by Table 6.


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This is not the case for H2, which exhibits stable performance. In all tested cases, we find that

|sn sH2n | 1 except in Table 4, where r = 0.1 and = 1%, for which |sn sH2n | 2. Further, forsmall r (r = 0.1 and r = 0.25), H1 seems to dominate H2 when T = 1/.

Overall both heuristics perform well for reasonable values of r and n. When T and r are small

(T 1/ and r 0.25) H1 outperforms H2 and guarantees that |sn sH1n | 1; H2 outperformsH1 otherwise, with |sn sH2n | 1.

7. Managerial Insights

In the previous section we developed heuristics for nurse staffing and, using numerical examples,

demonstrated that the heuristics do well in approximating actual optimal staffing levels. In this

section we use the heuristics to evaluate the performance of ratio policies. In particular, we find

that only in limited, and arguably impractical, circumstances does the ratio policy mandated by

California law i.e., the nominal ratio policy provide consistent quality of service. We also use the

heuristics to investigate the structure of the optimal policy. Of particular interest is the unusually

strong (super) pooling effects for large units that the optimal staffing policy exhibits.

Recall that when the delay constraint is T = 0, we are forced to use Heuristic 1; Heuristic 2 is

undefined. For strictly positive excessive delay thresholds (T > 0), the previous section suggests

that we use Heuristic 1 for small values of T and Heuristic 2 for large values.

Consider now the probability of delay (T = 0). If we assume that staffing levels and unit sizes

take real values (i.e. we ignore round-up effects), the optimal staffing rule can be approximated

by sn = rn + n

n where n is the solution of (11). One interpretation is that the nominal ratio

policy is close to optimal whenever n = 0; that is, when plugging n = 0 and T = 0 into (11) and

solving for the probability of delay, we obtain

=2

1 +

r(0) =

1

1 +

r. (22)

In this case, the probability of delay is at least as large as 50%. In fact, for pediatric units in

California, this probability of delay under the ratio policy is predicted by our heuristic to be 2 /3,

a quantity consistent with Figure 1: when controlling for the oscillation effects (due to the staffing

level round-up), the ratio policy seems to guarantee a consistent delay probability of roughly 65%.

Now consider another ratio policy sn = rTn with rT = r. If rT < r then for any reasonable targetprobability of delay , this policy will lead to understaffing. Supposing instead that rT > r, then

for only one value of n (using real values for n), will the ratio policy deliver the target probability

of delay:

n =

rT r2

,


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where is chosen so that () as expressed in (8) yields the target delay probability. Of course,

this yields only one unit size for which this particular ratio policy is optimal. This leads to our first

observation.

Observation 1. For any given unit type, only one ratio policy can yield a consistent probability

of delay across all hospitals, the nominal one mandated by California legislation. However, the

nominal ratio policy cannot guarantee a probability of delay smaller than 50%.

Consider the probability of excessive delays with T > 0. Suppose that the nominal patient need

is r, the threshold for delay is T > 0, and a ratio staffing policy is implemented for which (ignoring

round-up effects) sn = rTn, where rT satisfies (13). For a given target probability of excessive delay

, the approximately optimal staffing policy is, by Heuristic 2, rTn + T

n, where T is given by

(14). This implies that for sn to be optimal, T = 0, and hence, by (15), the probability of excessive

delay must be 50%.

Suppose a different ratio policy is chosen perhaps even the nominal policy and can be

expressed as n, for some

= rT. Then, given any target excessive delay probability , this ratio

policy is optimal for at most one value of n. This leads to our second observation.

Observation 2. For any given unit type and specified delay threshold T > 0, only one ratio

policy can yield a consistent probability of excessive delay across all hospitals, and it is not the

nominal ratio policy mandated by California legislation. Moreover, the consistent ratio policy cannot

guarantee a probability of delay smaller than 50%.

Suppose staffing conforms to the nominal ratios dictated by California Bill AB 394. When the

delay threshold is strictly positive and the target probability of excessive delay is less than 50%,

solvingrn = rTn + T

n

for n leads to a unique solution

n =

1()

T

2r + rT

1 + rt

.

If > 50%, then the nominal ratio policy exceeds the optimal policy for all values of n. Otherwise,

the nominal ratio policy understaffs for n < n and staffs more than the optimal staffing policy does

for n > n. This leads to the following observation.

Observation 3. For a strictly positive delay threshold (T > 0) and a reasonable target proba-

bility ( n).


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This observation is consistent with Figure 3 where n is roughly equal to 28. In Figure 2 the crossing

point n is larger than 60.

The next observation summarizes the trends in quality of service and identifies the limitation of

the ratio policy mandated by California law.

Observation 4. For any medical unit, any delay threshold, and a ratio policy sn = n, the

probability of excessive delay increases (resp. decreases) monotonically with the size of the unit andasymptotically approaches 1 (resp. 0) if is less than (resp. greater than) the nominal patient load

of that unit.

Observation 4 is consistent with Figure 11 (in the appendix) which shows an increasing trend of

the probability of excessive delay for = 1/5 < r = 1/4.

In short, according to Observations 1-4 the Californian legislation appears to provide consistent

performances only for poor service levels when the target times are restricted to zero. The previous

discussion also sheds light on pooling effects under the optimal staffing rule. Pooling effects appear

when the number of additional nurses required per new patient is decreasing in the unit size n,

that is when sn n is concave in n. When unit size and the staffing levels are assumed to take realvalues, Heuristic 2 can easily be shown to be concave, which suggests that the optimal staffing level

exhibits this pooling effect7. However, the effects are more dramatic than just concavity. Recall

that rn represents the nominal patient load, that is, the long run cumulative fraction of nursing

time required in the unit provided nurses are always available when patients become needy. For

large system (n > n), Heuristics 1 and 2 actually specify a number of nurses less than the baseline

rn (since n < 0 under Heuristic). In fact the deviation from the baseline is order n.

Observation 5. When the time target is positive (T > 0), the optimal staffing rule shows super

pooling effects for large units (n > n): the staffing level is less than the nominal patient load (sn 1. When n = 1, most reasonable staffing policies

will dictate that one nurse is present.

Table 4 Performance of Heuristics H1 and H2; r = 0.1; = 0.01

T = 0 T = 1/ T = 2/

n sn sn s

H1n s

n s

n s

H1n s

n s

H2n s

n s

n s

H1n s

n s

H2n

2 2 0 2 1 1 2 1 1

3 3 1 2 1 1 2 1 1

4 3 1 2 0 1 2 1 1

5 3 0 2 0 1 2 0 1

6 3 0 3 1 1 2 0 1

7 4 1 3 1 1 2 0 0

8 4 1 3 1 1 2 0 0

9 4 0 3 1 1 3 1 1

10 4 0 3 0 1 3 1 1

15 5 0 4 1 1 3 0 1

20 6 0 5 1 2 4 1 1

25 7 0 5 1 1 4 0 0

50 11 0 8 1 2 7 1 1

100 18 0 13 1 1 11 0 0

200 31 0 23 1 1 20 0 0

500 67 1 51 0 1 46 0 0


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ratios 3te

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