Title: Models for hospital census prediction and allocation
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 Material Information
Title: Models for hospital census prediction and allocation
Physical Description: xiii, 224 leaves : ill. ; 28 cm.
Language: English
Creator: Nguyen, Khanh-Luu Thi, 1947-
Copyright Date: 1977
 Subjects
Subject: Hospitals -- Admission and discharge -- Mathematical models   ( lcsh )
Hospitals -- Planning -- Mathematical models   ( lcsh )
Industrial and Systems Engineering thesis Ph. D
Dissertations, Academic -- Industrial and Systems Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Statement of Responsibility: by Khanh-Luu Thi Nguyen.
Thesis: Thesis--University of Florida.
Bibliography: Bibliography: leaves 218-223.
General Note: Typescript.
General Note: Vita.
 Record Information
Bibliographic ID: UF00099394
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000187485
oclc - 03406998
notis - AAV4085

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MODELS FOR HOSPITAL CENSUS PREDICTION AND
ALLOCATION








By

KHANH-LUU THI NGUYEN














A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY








UNIVERSITY OF FLORIDA

1977
































To Khai















ACKNOWLEDGMENTS


The author would like to express her appreciation to the members

of her doctoral committee for their guidance and support. In

particular, the author expresses gratitude to her chairman, Dr. Kerry

Kilpatrick, and her co-chairman, Dr. Thom Hodgson, for their assist-

ance and encouragement during both the research and writing phases of

the study. The author would also like to thank the other members of

the committee, Dr. Timothy Lowe, Dr. James McClave, and Dr. Ralph Swain

for their comments on various drafts of the manuscript. In addition,

the author thanks Dr. Eginhard Muth and Dr. Lee Schruben for their

helpful suggestions.

It is impossible to mention all of the individuals at Shands

Teaching Hospital, Gainesville Veterans Administration Hospital, North

Florida Regional Hospital, and Alachua General Hospital to whom the

author is indebted.

Financial support for this research was provided by the Health

Services Research and Development Trainee Program, Gainesville Veterans

Administration Hospital, and the Health Systems Research Division,

J. Hillis Miller Health Center.

Ms. Carol Brady deserves many thanks for her editorial assistance,

as does Ms. Beth Beville for her excellent typing of this dissertation.















TABLE OF CONTENTS


PAGE
ACKNOWLEDGMENTS .................................................. iii

LIST OF TABLES .......................... ......................... vi

LIST OF FIGURES ...................................................viii

ABSTRACT ...................................................... xii

CHAPTER

ONE INTRODUCTION ........................................... 1

TWO LITERATURE REVIEW ........................................ 5

2.1 Bed Allocation ..................................... 5
2.2 Census Process ...................................... 13
2.3 Length of Stay Estimation ........................... 24
2.4 Arrival Process ..................................... 27

THREE THE BED ALLOCATION PROBLEM .............................. 28

3.1 Introduction ...................................... 28
3.2 Development of Descriptive Models .................. 31
3.3 Development of Probability Distribution Functions ... 41
3.4 Solution Techniques for the Bed Allocation Problem .. 64
3.5 The Relative Costs in Bed Allocation Models ......... 66

FOUR THE SCHEDULING OF ADMISSIONS ............................ 91

4.1 A General Description of the Admissions Process ..... 92
4.2 A Census Prediction Model ........................... 93
4.3 Methods for Predicting Unscheduled Admissions ....... 101
4.4 Methods for Predicting Patients' Length of Stay ..... 106
4.5 Prediction of Future Census ......................... 125
4.6 Characteristics of the Hospitals in the Study ....... 160
4.7 The Scheduling of Elective Admissions ............... 165

FIVE FURTHER RESEARCH ........................................ 172











APF


ENDIX

A THE M/Ek/sj/s N+ QUEUEING SYSTEM ........................ 176

B BLOCK DIAGRAMS AND PROGRAM LISTING FOR A GPSS
SIMULATION FOR BED ALLOCATION .......................... 183

C SUMMARY OF DATA COLLECTION AND DESCRIPTION .............. 188

D THE EXPECTED VALUE AND VARIANCE OF A RANDOM SUM Z. OF
INDEPENDENT BERNOULLI TRIALS ....................J....... 199

E PROGRAM LISTING FOR THE BOUNDS ON STATE PROBABILITIES
OF THE M/Ek//s/* QUEUEING SYSTEM ..................... 202

F PROGRAM LISTING FOR THE CENSUS PREDICTION MODEL ......... 211


REFERENCES ..... ..................................................

BIOGRAPHICAL SKETCH ..............................................















LIST OF TABLES


TABLE PAGE

2.1 Summary of Admissions Models in Literature ............ 6

3.1 Parameters for a Three-Service Hospital System Used in
the Example for State Probability Bounds .............. 62

3.2 Computational Costs for Analytic Results .............. 63

3.3 Parameters for a Three-Service Hospital System Used in
an Example of Model 3 ................................. 67

3.4 Computational Results for an Example of Model 3 ....... 68

3.5 Relative Costs of Bed Borrowing ....................... 87

3.6 The Costs for the Existing Allocation and Its
Neighboring Allocations ............................... 90

4.1 Results of Testing Hypothesis H : =0 for Service,
Gainesville Veterans Administration Hospital .......... 108

4.2 Results of Testing Hypothesis H : B1=0 ................ 109

4.3 Results of the Chi-Square for Testing Goodness of Fit.. 116

4.4 The Empirical Prediction Errors for Shands Teaching
Hospital Pediatrics Unit .............................. 134

4.5 Mean and Variance of Length of Stay by Service for
Gainesville Veterans Administration Hospital .......... 137

4.6 Mean and Variance of Unscheduled Admissions by Day of
the Week for Gainesville Veterans Administration
Hospital ........................................... 143

4.7 The Empirical Prediction Errors for Gainesville
Veterans Administration Hospital ..................... 144

4.8 Mean and Variance of Emergency Admissions by Day of the
Week for North Florida Regional Hospital .............. 151

4.9 Mean and Variance of Unscheduled Admissions by Day of
the Week for Alachua General Hospital ................. 153









TABLE PAGE

4.10 Summary of Census Prediction Results .......... ..... .... 161

4.11 Effects of Hospital Characteristics on the Census
Prediction ............................................ 166















LIST OF FIGURES


FIGURE PAGE

3.1 The flow of patients in Model 1 ........................... 33

3.2 The flow of patients in Model 2 .......................... 36

3.3 The flow of patients in Model 3 .......................... 38

3.4 The flow of patients in Model 4 .......................... 40

3.5 The bounds on the probability function of the M/Ek/s/N+1 47
queueing system .................................... ... 47

3.6 The lower bounds on the state probability for the
M/Ek/Sj/s+1l queueing system .............................. 55

3.7 The upper bounds on the state probability for the
M/Ek/Sj/s,+1 queueing system .............................. 55

3.8 The state probabilities and bounds for service 1 .......... 60

3.9 The state probabilities and bounds for service 2 .......... 60

3.10 The state probabilities and bounds for service 3 .......... 61

3.11 The probability distribution of the number of beds occupied
for Medicine service, Gainesville Veterans Administration
Hospital ............................................... 72

3.12 The probability distribution of the number of beds occupied
for Neurology service, Gainesville Veterans Administration
Hospital ................................................. 73

3.13 The probability distribution of the number of beds occupied
for Surgery service, Gainesville Veterans Administration
Hospital ................................................. 74

3.14 The decomposition of the V.A. Hospital system.............. 76

3.15 The probability distribution of the number of beds occupied
for Neurology service, Gainesville Veterans Administration
Hospital ................................................. 77

3.16 The probability distribution of the number of beds occupied
for E.N.T. service, Gainesville Veterans Administration
Hospital ............................................... 78









FIGURE PAGE

3.17 The probability distribution of the number of beds
occupied for General Surgery service, Gainesville
Veterans Administration Hospital .......................... 79

3.18 The probability distribution of the number of beds
occupied for Neurosurgery service, Gainesville Veterans
Administration Hospital ................................... 80

3.19 The probability distribution of the number of beds
occupied for Ophthalmology service, Gainesville Veterans
Administration Hospital ................................... 81

3.20 The probability distribution of the number of beds
occupied for Orthopedics service, Gainesville Veterans
Administration Hospital ................................... 82

3.21 The probability distribution of the number of beds
occupied for Plastic Surgery service, Gainesville Veterans
Administration Hospital ................................... 83

3.22 The probability distribution of the number of beds
occupied for Thoracic Surgery service, Gainesville
Veterans Administration Hospital .......................... 84

3.23 The probability distribution of the number of beds
occupied for Urology service, Gainesville Veterans
Administration Hospital ................................... 85

4.1 A general hospital admissions system ...................... 94

4.2 The flow of census in time ................................ 96

4.3 An example of admissions prediction using Winters' time
series model for Gainesville Veterans Administration
Hospital ................................................ 103

4.4 Scatter diagrams for unscheduled and scheduled admissions
and the least square lines, Shands Teaching Hospital
Pediatrics unit .......................................... 105

4.5 Mean values of length of stay by day of the week,
Gainesville Veterans Administration Hospital .............. 111

4.6 Mean residual life function for the gamma, lognormal, and
Weibull distributions with mean 1 and coefficient of
variation /75 ................. .......................... 119

4.7 Mean residual length of stay function for Surgery service,
Shands Teaching Hospital .................................. 120









FIGURE PAGE

4.8 Mean residual length of stay function, Shands
Teaching Hospital Pediatric Medicine service, Thursday
admissions ............................................... 121

4.9 Mean residual length of stay function, Shands Teaching
Hospital Pediatric Surgery service, Monday admissions ..... 122

4.10 Mean residual length of stay function, Shands Teaching
Hospital Pediatric Medicine service, Monday admissions .... 123

4.11 One-day census prediction errors, Shands Teaching Hospital
Pediatrics unit ........................................... 127

4.12 Seven-day census prediction errors, Shands Teaching
Hospital Pediatrics unit .................................. 128

4.13 Frequency of one-day prediction errors, Shands Teaching
Hospital Pediatrics unit .................................. 129

4.14 Frequency of seven-day prediction errors, Shands Teaching
Hospital Pediatrics unit .................................. 130

4.15 Prediction errors for admissions, Shands Teaching Hospital
Pediatrics unit .......................................... 132

4.16 Prediction errors for discharges, Shands Teaching Hospital
Pediatrics unit .......................................... 133

4.17 Fraction errors of one-day census prediction, Shands
Teaching Hospital Pediatrics unit ......................... 136

4.18 One-day census prediction errors, Gainesville Veterans
Administration Hospital ................................... 139

4.19 Seven-day census prediction errors, Gainesville Veterans
Administration Hospital ................................... 140

4.20 Frequency of one-day census prediction errors, Gainesville
Veterans Administration Hospital ......................... 141

4.21 Frequency of seven-day census prediction errors,
Gainesville Veterans Administration Hospital .............. 142

4.22 Prediction errors for unscheduled admissions, Gainesville
Veterans Administration Hospital ......................... 146

4.23 Prediction errors for discharges, Gainesville Veterans
Administration Hospital ................................... 147

4.24 Fraction errors of one-day census prediction, Gainesville
Veterans Administration Hospital ........................ 148









FIGURE PAGE

4.25 Fraction errors for seven-day census prediction,
Gainesville Veterans Administration Hosptial .............. 149

4.26 One-day census prediction, Alachua General Hospital ....... 155

4.27 One-day census prediction errors, Alachua General
Hospital ............................................... 156

4.28 Seven-day census prediction, Alachua General Hospital ..... 157

4.29 Fraction errors of one-day census prediction, Alachua
General Hospital .......................................... 158

4.30 Fraction errors of seven-day census prediction, Alachua
General Hospital .......................................... 159









Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy


MODELS FOR HOSPITAL CENSUS PREDICTION AND
ALLOCATION

By

Khanh-Luu Thi Nguyen

June 1977

Chairman: Dr. Kerry E. Kilpatrick
Major Department: Industrial and Systems Engineering

This dissertation presents models for hospital census prediction

and allocation. Models to minimize the penalty costs for not having

sufficient beds for individual services are used to describe the hos-

pital bed allocation problem. An algorithm using a queueing theory

approach for a system with a Poisson arrival process, an Erlang k

service time distribution and a random number of servers is used to

derive the bounds on the probability of the number of beds occupied

in each service. The minimum expected cost allocation is evaluated

by a heuristic algorithm.

The census prediction model is based on probability theory with

the assumption that patients' lengths of stay are statistically in-

dependent. The mean residual length of stay function is used to

evaluate the conditional probability of staying in the hospital given

the number of days the patient has already spent in the hospital.

A method using the correlation between unscheduled and scheduled ad-

missions is applied to predict the number of daily unscheduled ad-

missions for hospitals with a large percentage of scheduled admissions.

A time series model is used to estimate the unscheduled admissions by









day of the week for hospitals with a significant number of unscheduled

admissions. Models are tested at various hospitals with different

operating settings. Effects of hospital characteristics on the census

prediction are identified.















CHAPTER ONE

INTRODUCTION


The objective of this research is to develop and test models

related to the allocation of hospital beds between services and models

for the prediction and control of inpatient admissions. The purpose of

these models is to develop bed allocations and census predictions

which improve the operation of the hospital with respect to specified

criteria. Among the general objectives of this research are the

reduction of health care costs due to inefficient resource allocation

and the improvement of patient care by fostering timely assignments of

a patient to the hospital and service best suited for his needs.

Efficiency of hospital operation is directly dependent on the skill

with which management utilizes available resources. One of the major

operational controls on resource utilization is the control of inpatient

admissions (49) because these admissions trigger the usage of virtually

every resource in the hospital from cotton swabs to nuclear scanners.

The occupancy level of a hospital not only affects its financial

viability but also influences the effectiveness with which it can deliver

its services. Low occupancy results in high costs per patient day and

eventually leads to a reduction in staff and other resources required for

high quality patient care. Excess occupancy stresses available resources,

overloads staff and facilities, and causes congestion and delays through-

out the hospital. Also, as the occupancy reaches the maximum, patients









seeking admissions must be turned away. This results in further

disruption of hospital operations as scheduled procedures are cancelled,

causing inconvenience for the patient and the physician, and, in some

cases, aggravating the patient's medical condition.

In addition to the problems caused by the level of occupancy (either

too high or too low), high variance in occupancy creates its own problems.

With high census variance, it is necessary to maintain staff and other

resources at a level sufficient to handle peak demand. This means that

90-95% of the time, depending on the risk level the administration adopts,

these resources are under-utilized. This leads to further inefficiencies

in operation.

Thus, admissions must be controlled to achieve an appropriate census

level and to minimize census variance. However, achieving this control

is difficult due to the random nature of requests for admissions and the

random length of stay of the patients. Further, while the hospital

administration can influence the rate and number of admissions, it cannot

influence the discharge process which lies wholly within the physician's

domain. Thus, control can be exercised on input but not on process or

output. Further, control can be affected only over the elective portion

of incoming patients. Emergency patients must be admitted immediately

and urgent patients must be admitted (usually) within 24 hours. Only

elective patients, who do not require immediate hospitalization, can be

scheduled sufficiently far into the future to be useful as control

variables. The problem of determining occupancy level and reducing

occupancy variance therefore becomes one of finding optimal policies for

scheduling elective admissions.









It is further recognized that many hospitals do not operate as

monolithic units but as confederations of individual 'services', each

controlled by an area chief. Services such as pediatrics, OB/GYN,

medicine, psychiatry, and others, often function as hospitals within

hospitals. Each service has an allocation of beds and admits patients

to its own beds. If these beds are filled the service must borrow beds

from services with empty beds or turn patients away. The probability

that a service has sufficient beds is a function of the allocation of beds

to each service and the patient demand for the service. Because it

affects the response of the total hospital, bed allocation is considered

as a sub-problem in this research. Models are developed to assist in

decisions concerning the allocation of beds to services. Once the

allocation decision has been made, the hospital can be compartmentalized

and models of the admissions process can be tailored to each service.

There are five chapters in this dissertation. Chapters One and Five

contain the introduction and conclusions of the research. Chapter Two

reviews the relevant literature concerning the allocation of hospital

beds, control of admissions, and related questions such as length of

stay estimation and the patient arrival process. Chapter Three focuses

on the allocation of hospital beds among services. Models are constructed

to find an allocation that can minimize hospital operating costs such as

the costs of turning patients away, of having patients in a common pool,

and of having patients in borrowed beds. In these models, the state

probability distribution function of the number of patients in each

service is derived. These state probabilities, together with the relative

costs of turning a patient away, of having a patient in the common pool or

in a borrowed bed, make up the objectives of the allocation model. A sample









hospital is used for evaluating the relative costs and determining the

allocation.

Chapter Four concentrates on the control of elective admissions.

A census prediction model is developed based on the principal components

of the admissions process. The model uses probability theory to estimate

the expected census on any day in the future based on the current census,

the scheduled reservations, the emergency arrivals and the patient length

of stay. Different methods for predicting emergency arrivals and daily

discharges are described in Chapter Four. The census prediction models

are tested at various hospitals. From the results of these tests, the

characteristics of each hospital and their effects on census prediction

are identified. An ideal operation setting for the census prediction model

is defined based on these characteristics. A brief look at policies for

scheduling elective patients is also presented with suggested questions

for future research.














CHAPTER TWO

LITERATURE REVIEW

The problem of admissions control has received considerable attention

in the Operations Research literature. This chapter briefly surveys the

relevant literature dealing with bed allocation,admissions control, length

of stay estimation and related questions. A summary of the models and

their assumptions is presented in Table 2.1. Section 2.1 discusses the

literature related to the problem of bed allocation. Section 2.2 reviews

research concerning the admissions process, control and modeling.

Sections 2.3 and 2.4 study the literature on the length of stay and the

patient arrival process, respectively.

2.1 Bed Allocation

Blumberg (11) and Weckwerth (70) predicted bed needs for a distinctive

patient facility using the assumption that the daily census was Poisson

distributed. Blumberg developed a table, based on the Poisson distributed

daily census, that could determine the number of beds assigned to services

to result in a fully occupied facility on an average of one day in 10,

one day in 100, or one day in 1,000. Weckwerth formulated the proportion

of time that an exact number of beds is filled as follows:

-ADC
P[b] = (ADC)b e--

where ADC is the average daily census. Thus, both Blumberg and Weckwerth

have used the Poisson distribution for planning the size of a hospital




































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facility to accommodate a given average daily census with a predetermined

probability of overflow. The assumption of a Poisson distributed daily

census requires that every admission to the hospital is a random occurrence

that is independent of every other admission, the length of stay has a

negative exponential distribution, and the bed capacity is infinite.

These requirements have restricted the applicability of this methodology.

Thompson and Fetter (65), recognizing the difficulty of assuming a

Poisson distributed daily census, introduced a computer simulation to

predict bed requirements for a maternity ward, for any given patient load

and any desired service level. Service level was defined as the proportion

of patients for which no extraordinary action would have to be taken as

they progressed through the various facilities. Thompson and Fetter used

accumulated data as input to the model and compared the output statistics

to the actual ones for a period of 30 days to validate the simulation model.

The simulation output on the number of beds occupied for different patient

input rates was used to determine the number of beds required for a service.

Thompson and Fetter also studied the sensitivity of the model with respect

to an increase in admission rate, to elective inductions and to a change

in length of stay. They found that the service level decreased with an

increasing admission rate. By introducing an elective induction policy

that allows a certain number of patients to be admitted to the hospital at

two specific times of the day, they found an increase in bed utilization

when compared to utilization without an elective induction policy. They

concluded the method of scheduling was an important determinant of average

occupancy. The simulation also showed that the number of beds occupied

decreased as length of stay decreased, but the variance in the number of

beds occupied remained the same.









Another simulation for bed allocation was done by Goldman,

Knappenberger, and Eller (27). Goldman et al. investigated the effects

of various beds-to-service and beds-to-room policies using simulation.

Goldman considered two kinds of services: unrestricted services which

were permitted to use beds in other services, and restricted services

which were only allowed to use their own beds. Goldman introduced policies

to allocate sufficient beds to restricted services to meet demand 95%,

85% and 75% of the time; the remaining beds were allocated to unrestricted

services by an average demand for beds. The average bed utilization was

tested for significant difference between these policies. It was found

that the bed utilization of restricted services decreased significantly

as the number of beds increased. The unrestricted service showed no

significant difference in bed utilization for the above policies. Goldman

suggested the following criteria for evaluating these policies: total bed

utilization, patients waiting for admission, transfer problems resulting

from any type of patient segregation, emergency patients placed in

temporary beds, and patient care quality for patients placed in another

service.

Zaldivar (76) used a Markovian model to derive the probability

distribution of the number of beds occupied for each system state, with a

state representing a different bed allocation. In his model, Zaldivar

assumed patient arrivals were Poisson distributed, and patient lengths

of stay were negative exponentially distributed. Zaldivar attempted to

define bed allocation policies to minimize the expected number of borrowed

beds in the system. A heuristic algorithm was used to solve the minimizing

problem. The assumptions on patient arrival and patient length of stay

distributions allowed Zaldivar to achieve analytic results. However, the









assumption of negative exponentially distributed length of stay does not

generally hold for patients of all hospital services.

Jackson (36) used the M/M/x/x queueing system (Poisson arrivals to

x identical servers with a negative exponential service distribution

and no waiting line) to develop bed allocation policies such that the

penalty of turning patients away was minimized. Jackson constructed an

optimizing problem which can be solved by dynamic programming methods as

follows:

N
Min I wiiPi(xi)
1=1

N
s.t. xi = B ,
i=l

where w. is a weighting factor for turn-away cost of service i, A. is the

patient arrival rate to service i, and Pi(x.) is the probability of all x.

beds of service i full. The probability function Pi(xi) is given by the

well-known Erlang loss formula. In his research, Jackson assumed that

each service operated independently from one another. This assumption

has restricted the applicability of Jackson's model. Jackson's approach

to the bed allocation problem will be used in this research for developing

different models where the independence assumption can be relaxed.

Singh (61) developed a procedure to allocate beds to services such that

the operating costs were minimized. Three types of costs were considered:

a fixed cost associated with the initial set-up cost of building the hospital,

a holding cost associated with maintaining empty beds, and a shortage cost

associated with refusing admission to a patient. Sinqh presented the

following nodel that minimized the expected cost of providing) 11k I 'd' to

the kth disease.








N k mk1
Min Ak k + C v (0 (-v )PV + C k ( v-mk (v
k=mk+l
N
s.t. m mk= B,
k=l

where Ak = the fixed cost for disease k,

6k = 0 mk = 0 ,

1 mk > 0

k
Ci = the holding cost for disease k,

C2 = the shortage cost for disease k,

vk = the random variable for the demand of beds of disease k,

and

p(vk) = the probability function of the demand of beds.

Singh used the assumption that the probability function of the demand of

beds followed a Poisson distribution. Singh also gave a detailed analysis

to evaluate the shortage costs in his research. Singh studied all possible

courses of action available to the patient who faced a shortage situation

such as being assigned a bed in a different ward, being sent to another

institution, and going home to wait for admission. For each case, Singh

considered the costs involved including both the cost to the hospital and

the cost of the patient. The assumption of Poisson distributed demand for

beds has restricted the applicability of this model as in the models of

Blumberg and Weckwerth presented earlier. Other papers related to bed

allocation are (4,10,18,31,32,45,53,68).


2.2 Census Process

Three fundamental factors of hospital census processes which have

appeared in most of the literature are hospital bed occupancy levels,

patient lengths of stay, and patient arrivals.









Early research on the number of patients in a hospital was done by

Bailey (3), Thompson and Fetter (66), and Blumberg (11). In these

studies, daily census levels were assumed to follow the Poisson dis-

tribution. Bailey assumed that the number of admissions was Poisson

distributed and the patient length of stay was negative exponentially

distributed. Under these assumptions, Bailey used the statistical theory

of queues to show that the daily census was Poisson distributed. Thompson

and Fetter compared the theoretical occupancy as obtained from a Poisson

distribution, and the actual number of patients in a delivery suite over a

period of 30 days. Thompson found that there was sufficient agreement

between the two series of values and concluded that the number of patients

in the delivery suite facilities was Poisson distributed. Blumberg based

the assumption on his previously unpublished studies of data from several

hospitals which indicated that daily census levels on a distinctive patient

facility were generally Poisson distributed. Blumberg also showed that

the daily census of over-crowded facilities with long waiting lists are

not Poisson distributed. Therefore, the assumption of Poisson distributed

daily census only applies to a certain distinctive patient facility.

Balintfy (5) predicted census analytically. Using a probabilistic

approach, he predicted a hospital census one day in advance. Balintfy

assumed that patient admissions followed a negative binomial distribution,

and patient length of stay a lognormal distribution. The next day's census

was derived from the current census and the convolution between the

admission and discharge distributions. An approximation of the census

distribution was made by assuming a normal distribution with the estimated

census mean and variance. Balintfy's approach for predicting census is

most useful when theoretical distributions can be shown to adequately

approximate the admission and length of stay distributions. Frequently,








this is not the case. Also, a longer horizon for predicting census levels

would be more useful for hospital planning.

Young (74,75) proposed a way to control census levels in a typical

hospital ward. He developed queueing models, where hospital beds were

considered as identical servers, patients as customers, and patient length

of stay as service time. The hospital care unit then resembled a queueing

system with finite number of servers, and two independent arrivals streams--

one elective and one emergency. The emergency arrival process was assumed

to be Poisson and the patient length of stay negative exponentially

distributed. Young introduced different assumptions for the elective

arrival process in two models: the rate control model and the adaptive

control model. In the rate control model, the elective arrival process

was assumed to follow an L-stage Erlang distribution with no waiting line.

Steady state probability distributions of the number of patients in the

hospital were found to be independent of L, the Erlang parameter. In the

adaptive control model, elective arrivals were controlled by means of a

control bed occupancy level B. When the number of occupied beds dropped

below level B, elective arrivals were scheduled to bring the census level

up to B. When the census levels were above B, only emergency patients were

admitted. In the adaptive control model, Young assumed that patients formed

a sufficiently long waiting queue and would be available immediately

to enter the hospital upon scheduling. Steady-state probability

distributions of the number of patients in the hospital were developed for

the adaptive control model. The assumptions for the patient arrival process,

the patient length of stay distributions, and the instantaneous availability

of scheduled patients have limited applicability of the models. Moreover,

the census control of the rate control model uses only the mean interarrival









time between two admissions and does not take advantage of any information

about the state of the hospital system.

Resh (56) developed a scheduling approach for surgical patients using

a probabilistic model to estimate the census mean and variance over a

planning horizon period. Resh treated the scheduling problem in two stages.

The first stage was concerned with the optimal type and number of patients

to admit during the scheduling horizon, given a forecast of bed and surgical

suite capacities, and considering prescheduled and nonscheduled admissions.

This stage was described by a linear programming problem which minimized

the expected waiting time per patient subject to the requirements that all

admission requests had to be scheduled, total admissions of prescheduled

and scheduled patients should not exceed the bed capacity, and the total

surgical operation time should not exceed the number of available hours of

the surgery suite with a probability of at least P. The second stage was

concerned with daily optimal allocation of inpatients to operating rooms,

given the admission data from the first stage decisions. This allocating

program attempted to minimize overtime and idle time in the operating room.

Both stages of the scheduling problem were planned to be executed daily.

Resh's model has the advantage that no assumption is made on the types of

distribution for patient arrivals and patient length of stay. However,

Resh's batch method of scheduling requires an existing queue of elective

patients and a deterministic number of patients scheduled for each day.

Resh's model does not take into account the effect of additional scheduled

patients on bed occupancy levels of the succeeding days.

Eberle (21) extended the work of Young in the adaptive control model to

a more general formulation. Eberle was able to consider day-to-day effects

on admissions and discharges. Eberle assumed that the number of elective









patients was constant for a given day of the week unless the number of

unoccupied beds was less than the number of patients scheduled for

admission. She also assumed that the distribution of the number of

discharges per day, conditional on the census of the previous night, was

Poisson distributed with mean values taken from historical averages for

discharge data. Eberle then estimated the census-after-discharge

distribution and derived the probability that some emergency patients

would be turned away. These two quantities were suggested for measuring

the effects of a policy change on the number of elective admissions

admitted daily. Eberle's model as Young's model, requires an existing

patient queue. Furthermore, the assumption that the discharge distribution,

conditional on the census, is Poisson has restricted the generality of the

model.

Thomas (64) developed a Markovian model for predicting the recovery

progress of a particular class of patients--the coronary patients. States

of the system corresponded to the states of the patients' health. Since

the probability of transition to one recovery state depends on how long

patients have been in the former state and is not memoriless as in

Markovian systems, Thomas established three recovery phases within each of

the recovery states. The probability of transition to one recovery phase

has Markovian characteristics. With these phases, the number of patients

in each state were approximated from the Markovian model. Thomas's model

uses a Markovian assumption which is not realistic since a patient's future

movements depend on his past history and not just on his current state of

health. Moreover, the states of the system which are the states of the

patients' health are not well-defined and will cause difficulty in

implementing the model.








Robinson et al. (58) developed a simulation dealing sequentially

with requests for admission, scheduling,and cost evaluation of three basic

scheduling systems. Each elective patient entered the scheduling system

with information on a desired admission date, flexibility of the possible

admission day, potential length of stay in the hospital, and hospital

service demand. The first scheduling system was termed the "filled page"

method in which patients were scheduled to enter the hospital until a page

had been filled or a set number of patients had been scheduled. The

second scheduling system was based on the estimated patient length of stay

to carry the expected census in the hospital out to some fixed horizon.

The third method was called the "PT" (probability table) version where

information about the conditional probability of the actual length of

stay was used. The simulation evaluated the performance of these three

different scheduling rules using the relative costs of empty beds, of

hospital overflow and of turning prospective patients away. Robinson

concluded that the estimated length of stay was the best technique. In

the simulation, Robinson explicitly excluded emergency admissions which

eliminated the random nature of the system. Thus, if emergency admissions

were included, Robinson would probably find the PT method more useful.

Bithell (8) presented a class of discrete time models: nonhomogeneous

Markov chains. The model is a discrete analog of Young's adaptive control

model. The census was represented by a discrete time Markov chain, where

the census levels were the states of the chain. The length of stay was

distributed according to the geometric distribution. Bithell also

generalized the length-of-stay distribution from a geometric to a Pascal

distribution and expanded the scheduling horizon from a one-day to a

several-day period. Bithell's model has two weaknesses: the restricted









assumption for the length of stay distribution and the Markovian assumption

for the census movements.

Kolesar (42) formulated a Markovian decision model for treating the

problem of scheduling elective admissions. Kolesar's model is a discrete

time generalization of Young's adaptive control model and is almost

identical to Bithell's model. Kolesar also used a Markov chain to represent

census movements with random inputs and a geometric length of stay

distribution. The only difference in Kolesar's model is that he gave

the optimal control decisions for admissions scheduling, which were derived

from the Markovian model, by controlling xt, the number of beds occupied at

time t. The problem was formulated as a linear programming problem with the

objective function in terms of xt. The Markovian decision model has

predictive characteristics for a finite planning horizon by introducing

more variables for states and decision rules for the problem. Kolesar's

model has the same problems as Bithell's model: the restricted assumption

for length of stay distributions and the Markovian census movements and

unreasonable size.

Connors (16) used an admission scheduling algorithm in a real-time

simulation model to schedule patients into a hospital. He used a probabi-

listic approach almost identical to the one developed by Resh to calculate

the census mean and variance, and approximated the census distribution by a

normal distribution. In the model, Connors excluded emergency admissions

by assuming that a certain number of beds had been reserved for emergency

use. The objective function of the model attempted to balance the patient's

preference for day of admission and the type of facility with the hospital's

desire to achieve an allocation providing the most revenues generated by

the patient's admission.









Finarelli (23) developed a probabilistic model, based on that of

Resh, for surgical patients. Finarelli included additional constraints

on room and service capacity. Finarelli used hospital-oriented objectives

instead of Resh's patient-oriented objectives since he argued that

elective patients were those whose admission can be delayed at no

medical cost. Finarelli introduced a heuristic algorithm for scheduling

elective surgical patients to optimize the utilization of surgical beds

and operating suite facilities. In summary, Finarelli's model is a

modification of Resh's model, and still has Resh's weakness in the non-

stochastics aspect of the model.

Kao (37,38,39,40,41) introduced a semi-Markovian model to describe

patient movements through various hospital care zones. Kao's model

modifies Thomas' model to handle system states to obtain Markovian

characteristics. In Kao's semi-Markovian model, whenever a patient enters

a state i, he will stay Tij days in state i before making a transition

to state j with the probability pij. Using this model, Kao predicted the

number of patients in each hospital care zone which could be used for

planning nursing staff and hospital facilities. Kao's model has the same

weakness as Thomas' model in the definition of the states of health which

limits the usefulness of this model. Moreover, the semi-Markovian model

requires even more information than the Markovian model in order to

construct holding time distributions.

Briggs (12) concentrated on inpatient scheduling to stabilize occupancy,

thus reducing the probability of under and over-staffing in the service.

A heuristic scheduling algorithm to minimize the discrepancy between actual

and desired census for a given level of delays in admission was presented.

The forecast of census depended on the expected number of non-emergent and








of emergent admissions for each day of the week, the number of scheduled

reservations, the midnight census, and the estimate length of-stay of

patients. Using a recursive relation to estimate census after discharge

for successive days, the model sequentially assigned a number of admissions

each day to bring the expected census to some desired level after allowing

for emergencies. Briggs' use of physicians' estimates of length of-stay for

predicting the number of patients discharged each day requires a high degree

of cooperation from the medical staff. Moreover, Briggs' method of

estimating the census does not take into account the fact that the probability

of a patient remaining in a hospital for a certain number of days depends

on the number of days that he has already spent in the hospital.

Shonick and Jackson (60) developed a queueing model combining Young's

two models. The emergency arrival process and elective arrival process

were both assumed to be Poisson distributed. The length of stay was

assumed to be distributed negative exponentially for all patients. When

the number of occupied beds reached a given level B, all elective arrivals

were queued and only emergency arrivals were admitted. Shonick and Jackson

derived the expected occupancy and the expected number of lost patients for

steady-state solutions. Shonick and Jackson's assumption on the distribution

of patient length of stay limits the usefulness of the model.

Kushner and Chen (44) built simulation models for studying different

scheduling policies. The length of stay data were collected for emergencies

and electives by the day of the week of admission. The policy used in

admission was to admit scheduled patients until a bed capacity N was

attained and urgent patients until B bed capacity was obtained (B > N).

Emergency patients were always admitted. The simulation was run with

different policies for the number of reservations accepted on each day of









the week. Kushner and Chen evaluated these policies by the number of

admissions and average bed utilization.

Hancock et al. (30) tested different scheduling policies for an

overbedded hospital. The system consisted of a set of policy rules,

called "allowances." Each service had an "allowance" (or limit) on the

number of scheduled patients. Weekend allowances were also considered in

the system. Simulation was used to evaluate how a particular set of values

for allowances met the objectives of the hospital.

Swain (62) has proposed a more elaborate predictive model for census.

Statistical analyses were done on the empirical length of stay distributions

for each service. The predicted census was derived from the current number

of patients together with their length of time already in the hospital,

the scheduled reservations for elective patients, and the expected number

of emergency admissions for each day of the week. Since Swain dealt with

a large population, he approximated the census distribution by a normal

distribution with the estimated mean and variance derived from three above

elements. Swain used the objective of maximizing average occupancy subject

to bed overflow constraints based on the normal census approximation to

derive policies on the number of scheduled elective patients for each day.

Rubenstein (59) introduced a model almost identical to that Swain's.

The analyses on length of stay data were done for admitting diagnoses, age,

and sex of patients. Rubenstein also found the exact census distribution

by using the convolution of probability distributions of the three elements

cited in Swain's model. Rubenstein concluded that the exact and normally

approximated results did not differ significantly, even though the

computational efforts for an exact result were 10 times more than that for

an approximate one.









Barber (7) developed a model for census estimation and elective

patient scheduling. He considered the case of dynamic decisions and that

the requests for admission are processed on a continuous basis. The model

consists of a two-step algorithm: 1) estimation of future census, and

2) optimal scheduling of elective patients. The census estimation is based

on a mathematic development of the equations similar to the one developed

by Swain (62). The census estimator consists of all census components:

current patients, previous elective patient reservations, emergency

admissions, transfers, and additional reservations. Barber introduced

a method where stochastic admission requests may be processed. He considered

the actual additional admissions, s., instead of the allowed additional

admissions, e i.e.,


s. = e. for requests greater than availability, and
1 1

s. < e. for requests less than availability.
1 1

Barber assumed that the probability distribution f[st let] was known and

proceeded from this assumption to derive the optimal scheduling for

maximizing census levels. Barber separated the optimal scheduling into

two steps: a short-term sub-optimization for the decision period (1, T),

and a modification on these decisions using a discount for future to

account for the sequential nature of the decision process. Barber has

considered the problem of scheduling patients a step closer to the real

world situation. However, the probability function f[st et] on the

actual admissions conditioned on the allowed admissions is difficult, if

not impossible, to obtain.

Additional papers dealing with census prediction and admissions

control include (20,22,24,33,50,52,69,71,73).








The models for census prediction used by Swain, Rubenstein, and

Barber are based on probabilistic assumptions and provide a general

approach to the census prediction problem. The use of empirical data on

length of stay has made the models applicable to any hospital. However,

the empirical data can also introduce too much noise which may effect the

accuracy of the prediction. The approach of Swain, Rubenstein, and Barber

will be used in this research to predict census along with a method to

smooth out noise in empirical data on length of stay.


2.3 The Length of Stay Estimation

Early studies (2) on the length of stay indicated that the average

length of stay was longer for males than females, for low-income groups

than high-income groups. The average length of stay was also longest

among persons aged 75 years and older. Another study (64) showed that the

average length of stay for persons discharged by death was nearly double

that for those discharged alive. In both discharge status groups, the

average length of stay was greatest for persons 65 years of age and older.

Robinson et al. (57) studied various methods for predicting patient

lengths of stay with less uncertainty, including use of diagnostic infor-

mation, physicians' estimates, and nurses' discharge predictions. Physicians'

estimates prior to admission, revised by additional estimates after

admission, were found to be a useful method for reducing uncertainty in

the length of stay of patients. The difficulty with this method of

prediction is that physicians' estimates of length of stay are not usually

available.

McCorkle (47) studied potential factors that would affect the length

of stay. McCorkle compared the length of stay of patients who had a

physician directly responsible for their hospital care (private patients)









with those who did not (staff patients). McCorkle found that the

differences in the duration of hospitalization between private and staff

patients varied considerably among hospital department and did not appear

to be explained by variations in age, race, sex, or method of paying for

care (Blue Cross vs. non-Blue Cross patients). McCorkle (48) examined

the reason for the prolonged preoperative stay of a group of patients who

underwent initial surgery two or more days following admission. He found

that neither age nor classification as a private or staff patient was as

important as admission status in predicting whether or not surgery would

occur without delay. In one-third of the cases studied, delays were due

to the fact that patients had been admitted to a nonsurgical department.

Gustafson (29) demonstrated five methodologies for predicting hospital

length of stay. Three of these methodologies gave a point estimate of

length of stay based on physicians' subjective opinions, while two gave a

probability distribution for length-of-stay based on empirical data. In

a statistical comparison of these five prediction methods, Gustafson

concluded that the Baysian model, which combined the impacts of data

complexes such as day of admission, diagnostic information, and age and

sex of patients on the hypothesized length-of-stay, was the most accurate

technique, especially when attending physician's estimates on length of

stay were not available.

Bithell and Delvin (9) studied the lengths of stay of a group of

surgical patients to ascertain the extent to which discharges could be

predicted. The estimates on the residual lengths of stay of individual

patients were made by responsible clinicians two or three times each week

and were recorded together with a degree of certainty attached to each

estimate. Bithell found that the initial length of stay estimates together









with continuous predictions of discharges reduced variability in the

discharge predictions. The problem with this method is that the

estimates of length of stay are not always available especially with

"hard cases" where uncertainty in length of stay is acute.

A report (15) on the length of stay of patients in short-term

hospitals indicated that the average stay of patients varied considerably

with age. The average length of stay of patients under one year of age

was significantly greater than that of patients greater than one-year-old.

Thirteen and fourteen-year-olds had a longer average stay than patients

in the surrounding age group. From age eighteen to age twenty-six the

average stay increased steadily with increasing age. This study was done

with a sample taken from patients with all diseases in PAS (Professional

Activity Study) hospitals during the first six months of 1970.

Another study (34) was done on short-term hospitals registered by

the American Hospital Association of the seasonal effects on the

length of stay of patients. Fluctuations caused by seasonality were found,

tending to obscure long-term trends and cyclical movements. The study

suggested that seasonally adjusted data, obtained through seasonal indexes,

would be helpful for short-range and intermediate-range planning. Each

monthly index characterizes the level of activity during that month as a

percentage of the annual average.

Altman et al. (2) investigated the effects of different variables on

length of stay such as mental status factors, diagnoses, sex, race, and

marital status. They found that diagnostic categories were the best

predictor of length of stay. The mental status of patients (depression

or anxiety), marital status (married or widowed), and sex were also good

predictors for length of stay.









Posner and Lin (54) studied the effect of patient age on length of

stay. Their findings suggested that the variation in length of stay was

not well accounted for by differences in the ages of individual patients.

Patients of the same age had lengths of stay which varied much more than

patients of different ages, even when diagnostic variables were taken

into account.


2.4 Arrival Process

Other studies have concentrated on patient arrivals. Balintfly (5),

as discussed previously, developed a negative binomial distribution to

model inpatient admissions. Young (75), Thompson and Fetter (65), and

Weckwerth (70) assumed a constant Poisson arrival process for emergency

admissions. Swartzman (63), in a statistical analysis of patient arrivals

in a Michigan hospital, concluded that the arrival processes were Poisson

with arrival rates that differed significantly from segment to segment of

the day but not from day to day over weekdays. The weekend arrival

process was dropped from consideration since the greater part of patient

arrivals occurred on weekdays.

Chen (13) and Fries (26) provide additional information concerning

aspects of the admissions system.














CHAPTER THREE

THE BED ALLOCATION PROBLEM


3.1 Introduction

A hospital consists of many professional services; each deals with a

medical specialty such as internal medicine, psychiatry, ophthalmology, and

orthopedics. A practice that many hospitals use is to group hospital beds

into sections or wards, each assigned to a service. The grouping of patients

into a service section has the advantage of placing common types of

patients together. Moreover, the physicians can concentrate on working

within a section which can save their time and travel. Each section can

also be equipped for a specialty which affords better quality of care for

the patients. Due to the stochastic nature of the demand for beds, however,

the grouping of patients in services often causes a situation where some

services under-utilize their allotment of beds, while others have more

patients than the assigned number of beds. The underbedded services with

the demand for beds generally higher than their bed availability can resolve

this problem in three different ways: 1) refuse admission to overflow

patients; 2) borrow beds from under-utilized services for their overflow

patients; or 3) use beds in a central bed pool, if such a pool exists, for

their overflow patients. A central pool is a common ward with a fixed number

of beds which can be used to accommodate the patients of any underbedded

service. Solutions 2 and 3 cause a dilution in the grouping of common type

patients. The control of elective admissions can be used to keep the

occurrence of this situation to a minimum. However, with an unbalanced









allocation, the control of admissions might delay patient admissions. If

a deferred patient chose to go to another hospital, hospital revenue

would be reduced. Therefore, the allocation of hospital beds to services

should be examined before any control is imposed on elective admissions.

There three solutions to the problem of underbedded services are discussed

in detail in following paragraphs.

"Bed borrowing" creates administrative problems such as additional

record keeping, additional patient transfers, a lack of grouping common type

patients, and inconvenience for physicians. "Bed borrowing" also introduces

an extra operating cost. This cost is due to the fact that many services

equip their rooms for special care. Having a patient in a borrowed bed

necessitates moving specific equipment while equipment already in the room

becomes idle. Moreover, patient grouping into professional services allows

hospital staff to develop specialized skills in the performance of patient

care functions. Having a patient in a borrowed bed demands extra skills

from the nursing staff, while their specialized skills are not used.

An alternative to "bed borrowing" is to have a central pool in the

hospital. The existence of a central pool is justified when the cost of

having patients in central pool beds is lower than the cost of having patients

in borrowed beds for some services. The central pool beds are usually

equipped at a minimum level so that the operating cost due to idle equip-

ment is negligible.

When an overflow service cannot find a bed to accommodate its patients,

the patients will be turned away. Turned away patients can be either sent

home to wait for admission on another date or admitted to another hospital.

Thus, turning patients away creates an immediate loss to the hospital and

a long range loss when physicians start admitting their patients to other

hospitals.








It may be desirable for the hospital to allocate beds to services

to minimize the costs of turning patients away and of having patients in

borrowed or central pool beds. Another objective of bed allocation can be

to minimize the probabilities of turning patients away and of having

patients in borrowed or central pool beds. In this research, descriptive

models of bed allocation systems will be used to determine the optimal bed

allocation policies using the minimum cost objective. The optimal bed

allocation policies with minimum probabilities of turning patients away and

of having patients in borrowed or central pool beds can be constructed

similarly. Four different hospital systems will be studied: 1) a system

with no interaction among its services; 2) a system with interaction among

its services; 3) a system with a central pool and no interaction among its

services; 4) a system with a central pool and interaction among its services.

For the four hospital systems presented here, methodology will be

developed for the allocation of B beds among N professional services.

Each professional service has its own arrival and length-of-stay distri-

butions, and a fixed number of beds assigned to it. The costs associated

with having patients in central pool beds, in borrowed beds, and with

turning patients away will be used as weighting factors in the models. The

turnaway cost for service j, W., represents the service's subjective

judgment about the acceptable number of patients turned away. The cost,

Vij, of having a patient of service j in a service i bed and the cost, Uj,

of having a patient of service j in a central pool bed represent a

relative cost due to the inconvenience of displacing the patient from

service j.

It is very difficult to assign any monetary value to these costs, It

would be best to have these costs in terms of general parameters which can









be changed by the administrators. One service can be chosen as the basis

for assigning values to these parameters. Values of the parameters for

other services can be considered as multiples of the ones of the chosen

service. The opinions of the administrators and medical staff will he

used to determine these parameters, especially when the three costs are

weighted against each other. A method for evaluating these relative costs

is presented and applied to a sample hospital.

Section 3.2 describes the four different models for bed allocation.

The state probability distribution function for these systems is derived

in Section 3.3. Solution to the bed allocation problem is presented in

Section 3.4 and the evaluation of relative costs in Section 3.5.


3.2 Development of Descriptive Models

Consider an N-service hospital with a number of beds, B. Each service

N
j is assigned s. beds, Y s. = B. The patient arrival process to each
S j=1 J

service j is Poisson distributed with rate A.. In the bed allocation study,
J
the arrival process is assumed not to be under any kind of admissions control,

that is,all requests for admission are satisfied, or even if it is

controlled, the number of admissions to each service follows a Poisson

distribution as shown in the study of Swartzman (63). In addition to the

beds assigned to individual services, a hospital system with a central bed

pool also has a number of beds sN+1 assigned to the central pool, i.e.,

N+1
Ssj = B, sj 0j .
j=1

Let C. be the random variable for the number of beds occupied by

patients of service j. P.(m), for m=0,1,2,...,B is the probability

distribution function of the random variable C..
J








3.2.1 Model 1. A model of a hospital system with no interaction among
services.


Jackson (36) developed this model based on the assumption that all

services operate independently. In other words, all overflow patients of

a service are turned away. This simplified model does not accurately

represent the real world where services can borrow beds. However, this

model can be used to gain insight into more realistic models which will be

discussed later.

The objective of this model is to minimize the expected cost of turning

patients away where the turnaway cost, as discussed previously, is

measured by a weighting factor W..

Let P (s.) be the probability that a patient arriving for service

j will find all s. beds of service j filled, i.e., the probability that

an arriving patient for service j will be turned away:


Pj(s.) = Pr[Cj s ]


The flow of patients in this system is summarized in the flow chart in

Figure 3.1.

The expected rate at which patients are turned away from service j,

given the mean arrival rate of patients to service j is Aj patients/day,

is

A P (sj)


The expected penalty cost of turning patients away from service j is

W.A.P. (s.)


The total expected penalty cost of turning patients away from the hospital

is
















Arriving Patients to Service j


Figure 3.1 The flow of patients in Model 1.









I w .P.(s) .
j=1


The bed allocation problem for this model can be written as

N
Min I W..jP(s )
j=l (s

N
s.t. I s. = B
j=l


s. integer, j=1,2,.. .,N.


If the P.(sj) functions are known, the problem is easily recognized as a

simple dynamic programming problem of the knapsack variety.


3.2.2 Model 2. A model of a hospital system with interactions among
services.


The assumption for this model is that the overflow patients of a

service can be absorbed by any other service, if there is an available

bed. If there is no bed available in the hospital, the overflow patient

is turned away. In this model, it is assumed that a service can borrow

beds from any other service, and the penalty cost of having its patient

in any borrowed bed is independent of the service borrowed from, i.e.,

V.. = V. Vi.
Uj j

In this system, the overflow patient is turned away only when the

hospital is completely full. Therefore, the turnaway rate is independent

of the allocation of beds among service. The objective of this system is

assumed to be to minimize the expected cost of having patients in

borrowed beds.








The flow of patients in service j is described in the flow chart

in Figure 3.2.

If the probability distribution function, P.(m), of the number of

beds occupied by service j is known, the expected rate of borrowed

beds can easily be found:


Pj j(sj) Pr[CI+ ... + C + ... + CN < B].

The bed allocation problem for this model can be written as

N

j=

N
s.t. I s. B
j=1


s. integer, j = 1,2,...,N

The probability distribution function P.(m) will be derived in Section 3.3,

together with solution techniques for the above optimization problem.


3.2.3 Model 3. A model of a hospital system with a central pool and
without interaction among services.

In this system, each of the N professional services of the hospital

is assigned a fixed number of beds. There is also a central pool (considered

to be the (N+1)th service) which can provide a fixed number of beds to

overflow patients from any service. Overflow patients of a service can

be absorbed by the central pool if there are beds available, otherwise,

they will be turned away. The objective is assumed to be to allocate beds

to services so that the costs of having patients in central pool beds and

of turning patients away from the hospital are minimized. Overflow patients















Arriving Patients to
Service j


Figure 3.2 The flow of patients in Model 2.








of service j become the arriving patients of service j to the central
pool. These patients will be turned away from the central pool if the
central pool is full.
The general flow of patients of service j is summarized in the
flow chart in Figure 3.3. The probability that an overflow patient of
service j will find the central pool full can be written as

sN+l
Pr[CN+1 = sN+1Cj sj] = mL Pr[CN1 = sNICj = s +m]Pr[C =sjm] ,
N31 m Ni Nm=O* lIC1 = 3

where CN+1 is the number of occupied beds in the central pool. The
probability function of CN+1 is the convolution of overflow probabilities
of all services. The right hand side of the above equation can be easily
found if the probability distribution functions, P.(m), for all services
are known.

The expected rate of overflow patients from service j that are
turned away is

ijPj(s ) Pr[CN+1 = SN+lCj 2 sj] .

The expected rate of patients from service j to central pool beds is

SN+1
Aj m0 P (s +m) [I-Pr[CN+1 = sN+1lC = s +m]]


The expected costs of having patients in central pool beds and of
turning patients away are
N SN+1
i{U m f Pj(s +m) [l-Pr[CN+l=SN+lICj=s +m]l
j=~ m 0 i


+ Wj.jPj(s ) Pr[CN+1 = SN+ICj s]} .
















Arriving Patients to
Service j


Figure 3.3 The flow of patients in Model 3.








The bed allocation problem for this model can be written as

N sN+l
Min 1 {/ j j o=0 Pj^sm
Minjl UjA m 0 P (s +m) [-Pr[CN+=sN+ICj =s +m]]


+ WjijPj(s)Pr[CN+1 = sN+lJC s j]}

N+1
s.t. Z s. = B s. integer, j=1,2,...,N+l
j=1 J j

The probability distribution function P.(m) will be derived in Section 3.3.

Solution techniques for the above optimization problem will also be
presented.

3.2.4 Model 4. A model of a hospital system with a central pool and
interaction among its services.
Model 4 is an extension of Model 3. When all central pool beds are
filled, an overflow patient can be assigned to a bed borrowed from some
other service. Patients are turned away from the hospital only when all
beds in the hospital are filled. Therefore, the rate of turned-away
patients does not depend on the allocation of beds to services. For this
hospital system, the objective is assumed to be to allocate beds to
services so as to minimize the total costs of having patients in central
pool beds and in borrowed beds.

The general patient flow of service j is summarized in the flow chart
in Figure 3.4.

The expected rate of overflow patients from service j that can be
accommodated in central pool beds is

sN+1
j m= Pj(s+m) [I-Pr[CNl=sN+ C=s +m]]











Arriving patients of
Service j


Figure 3.4 The flow of patients in Model 4.








The expected rate of overflow patients from service j that find all
central pool beds filled but are able to obtain a borrowed bed is

AjPj(Sj)Pr[CN+1 SN+1|Cj s ]Pr[C+......+C+...+CN
The bed allocation problem for this model can be written as

N
Min J { U.jPj(s.) [1-Pr[CN+1 N+Cj s]]
j=l jj N+l N+13 3
N
+ VXjPj(sj)Pr[CN+1 sN+lIC s] iPr[

N+1
s.t. I s. = B
j=1 J

s. integer, j = 1,2,...,N+ .

If the probability function P.(m) is known, the objective of the above
optimization problem can be evaluated and the solution found. The deri-

vation of this probability function will be presented in Section 3.3.

3.3 Development of Probability Distribution Functions
Before proceeding further, notational definitions for the queueing
systems are given for later use in the analysis.

M/Ek/s /si : The queueing system with Poisson arrivals to s.+s.
identical servers with Erlang k service time
distribution and no waiting line. P.(m),m=O,1,2,..,s.+s.
are the state probabilities for the system of service j.

M/Ek/Sj/si : The queueing system with Poisson arrivals to sj
continuously available servers plus si servers
available on a random basis, each of the s +si servers
possessing the same Erlang k service time distribution,









and no waiting line. P (m),m=O,l,... ,s +si are the

state probabilities for the system of service j.

Model 1 is easily seen to be equivalent to well-known queueing

systems with Poisson arrivals to identical Erlang k servers and no waiting

line. The state probabilities of service i in this system are the same

as the ones of the M/Ek/si/O queueing system.

In the following section, the state probability distribution for

the systems in Models 2, 3, and 4 will be examined. The determination

of state probabilities for these systems appears to be extremely difficult

at best. In this research, no attempt is made to solve for state

probabilities explicitly. Instead, portions of the system are related to

queueing systems for which solutions are well known. In so doing, it is

possible to determine upper and lower bounds for certain elements of the

state probabilities. The bounds may then be successively tightened by a

procedure which takes advantage of the relationship existing between

services.


3.3.1 The M/Ek/sj/si Queueing System

The M/Ek/s /si queueing system (Poisson arrivals to (s.+si) identical
k J 3 J 1
servers with an Erlang k service distribution and no waiting line) is the

basic vehicle of comparative analysis. The solution for the M/Ek/s /si

system is well-known (43,17,28,55):

Pj(m) = P (O)(-)m I- where

Pj(m) = the probability of m customers in the system, m=0,1,...,s.+s.,

Pj(0) =i s ( I the probability of no customer in the system,
n=i X)n
n=0 j









A = the arrival rate of customers to the system, and

1/i = the mean service time of customers in the system.


3.3.2 The Single Service M/Ek/si/s Queueing System

The hospital system, if considered as a whole, is extremely complex.

However, individual services of the hospital have identical structures.

Overflow patients of a service can be absorbed by a central pool, if

there are beds available, otherwise, the patients are turned away. There-

fore, the single service system can be considered as an M/Ek/sj/sN+l

queueing system, where sj beds are assigned to service j and sN+l beds

are assigned to the central pool. In the following analysis, advantage is

taken of this characteristic.


Property 1. The state probabilities P (m) for the M/Ek/sj/s,+l queueing

system retain the Poisson characteristic for states m = 0,1,2,...,sj, i.e.,

X.
P (0)( )m m=0,1,2,...,s

Pj(m) = unknown m=s +1,...,sj +sN+l

0 m > sj+sN+I


Pj(0) cannot be defined precisely, since the availability of central pool

beds depends on the overflow probabilities of other services in the

hospital system. However, due to the fact that the relationships between

the first s. states are unchanged, it follows that the relative rela-

tionships between state probabilities are of the form

Pj(m+l) = P (m) m = 0,1,2,...,s- .
j p.ml j









Property 2. The sum of state probabilities over states m = s +1,...,s +sN+1

for the M/Ek/sj/s*+1 queueing system is bounded above by that of the

M/Ek/Sj/sN+l queueing system, i.e.,

s +sN+1 sj+sN+1
S P(m) Z P (m)
m=s.+1 m=s.+1
J J

Proof. Consider the M/Ek /sj/N+1 queueing system, whose state probabilities

are described by the truncated Poisson distribution. It is assumed that

servers s.+1 through sj+sN+1 are always available. In the M/Ek/sj/sN+l

system, servers s +l through s.+sN+1 are not necessarily always available

when requested by an arriving customer. When a server is not available,

the arriving customer is rejected from the system. It follows that the

probability of having s.+1 through si+sN+1 customers in the system is

reduced from that of the M/Ek/s /sN+1 system.
Q.E.D.

Property 3. The state probabilities P*(m) for the M/Ek/sj/sN+1 system

are bounded below by the state probabilities Pj(m) for the M/Ek/sj/sN+l

system for m = 0,1,2,...,sj, i.e.,

P.(m) > P.(m) m = 0,1,2,...,s


Proof. Since state probabilities sum to 1.0, it follows from Properties

1 and 2 that the state probabilities 0 through sj are no less for the

M/Ek/Sj/sN+1 system than the M/Ek/sj/sN+1 system.
Q.E.D.

Property 4. The state probabilities P.(m) for the M/Ek/s /sN+1 queueing

system are bounded above by the state probabilities P.(m) for M/Ek/sj/O








queueing system for states m = 0,1,2,...,s., i.e.,

P (m) < P.(m) 0,1,2,...,s

Proof.


3 P.(m) 1
m=O j
or

P*(0) J ()m 1m I 1
S m=0 M!

p( < 1
Pj(O) 1s.
J( J)m 1
m=O j

therefore

P (0)( 1 )m 1 j)m 1
m j m! s ()n j m!
3 n 1 j
n=O ~j nJ
Q.E.D.

Property 5. The state probability Pj(s +sN+1) of the M/Ek/sj/sN+
system is bounded from above by the state probability Pj(s +sN+1) of the
M/Ek/sj/sN+l queueing system, i.e.,

Pj(s +sN+1) Pj(s +sN+l)


Proof. For the M/Ek/s /sN+l system, sj+sN+l servers are continuously
available for use. For the M/Ek/s /sN+1, sj servers are available on a
random basis. Since for the M/Ek/sj/sN+l system, there would be, on the
average, less than sj+sN+l total servers available, it follows that the
probability of having the system full, (i.e., s +sN+l customers in the
system) would be less than for the M/Ek/sj/sN+l system.
Q.E.D.









By Property 3, the probability mass function for the M/Ek/sj/sN+1

system is point by point greater than that of the M/Ek/s./sN+l system

for states m=0,1,...,s.. By Property 2, the probability mass Functions

for the M/Ek/ /s N+l and the M/Ek/s /sN+1 systems must cross at some

point in region sj, s.+sN+1. A sketch of the probability mass function

for the M/Ek/s./s N+ queueing system in comparison to the probability mass

functions for the M/Ek/s /O and the M/Ek/sj/sN+1 systems is in Figure 3.5.

Continuous curves are used to approximate the discrete probability mass

functions.


3.3.3 Improving the Bounds on State Probabilities of the M/Ek/sj/sN+1
Queueing Systems

Let the states of the queueing systems be divided into two regions:

Region 1 for states m = 0,1,2,...,s which includes all states

where service j does not use any beds other than its own, and

Region 2 for states = s .+1,...,s +sN+l, which includes all states

where service j has to use beds in the central pool.

Consider an N-service hospital system. The lower and upper bounds on

the state probabilities in Region 1 of a service can be obtained from

Properties 2 and 3. The relationships between a single service and all

the others can be used to tighten the bounds on the state probabilities for

the single service M/Ek//sN/sN+ queueing systems. For instance, the upper

bound on the probability that any of the first N-l services use the central

pool can be used to derive a lower bound on the probability that the Nth

service uses the central pool. Similarly, the lower bound on the

probability that the first N-1 services use central pool beds can be used

to derive a new upper bound on the probability that the Nth service uses














































0





0
4-'

4-


-E
4-'


.0 m,



-o
0~




4-'n
:3







viCZ
0 -)
+


o uw







'U u
-L..









central pool beds. The increased value of the lower bounds for central

pool use decreases, in turn, the values of upper bounds of the state

probabilities of Region 1 of each service. Continuing the procedure, the

lower and upper bounds on the probability that a service uses central pool

beds can be successively tightened, and the lower and upper bounds on the

state probabilities in Region 1 can be improved accordingly. The properties

necessary to support the above general procedure are now presented.

Let 4j(m) be the probability of having m or more central pool beds
L u
available to service j, and Lij be the corresponding lower and upper

bounds, respectively, i.e.,


> 0*(m) for m=1,2,...,sN+1
L< *

j (m) for m=l,2,...sN1

L u
Property 6. The lower and upper bounds, j and p., on the probability

cj(m) for m=1,2,...,sN+1 are


L = Lower bound {probability that all the other services (fj)

use their own beds only} .

S Lower bound {probability that service i uses all
ifj
sN+l central pool beds}

Proof. For service j, by definition

C*(m) = the probability that there are m or more central pool beds

available for service j

= the probability that other services (Cj) use no more than

(sN+l-m) central pool beds








= 1 the probability that other services (fj) use more
than (SN+l-m) central pool beds.

Since the probability that other services (fj) use more than (sN+l-m)
central pool beds is no greater than the probability that other services
(/j) use any central pool beds, for m=1,2,...,sN+1, it follows that

>
(j(m) 1 the probability that other services (fj) use central
pool beds.
Thus
(4(m) 1 upper bound )the probability that other services (fj)
use central pool beds(
or,
4j(m) lower bound probability that other services (fj) use
their own beds$
i.e.,
*(m) lower bound m=O Pi(m)
itj m=O0

j*(m) 2T lower bound{ fY P (m)}
ifj m=0

Also, the probability that other services (fj) use more than (sN+1-m)
central pool beds is no less than the probability that other services (Cj)
use all central pool beds, for m=l,2,...,sN+, it follows that

m.(m) < 1 probability that other services (fj) use all central
pool beds
thus
.C(m) < 1 1 lower bound the probability that service i uses
itj
all central pool beds








i.e.,

Sj(m) 1 i lower bound {P*(si+s )} Q.E.D.
{Pi (iS Q.E.D.


Let the M/Ek/s./sN+1 queueing system be the system with two arrival rates

Arrival Rate X. for states m=0,1,2,...,s. .
J J
Arrival Rate X j for states m=s .+,...,s.+s .
J J J J N+1

Similarly, define the M/Ek//s /sN+ queueing system to be the system with
two arrival rates X. and A .
J J J
Property 7. The upper bounds PH(m) on state probabilities in Region 1
of service j are


P m)= j(o)( L)m for m=O,1,2,... ,

where

Pj(O) =

.Z J () 1 s +sN+1 ( m -s.
z= V (.J)m Mi+ Y IJi 1!
m=O m= j+1 S m

L = the lower bound on the probability that m or more central
pool beds are available for service j (Property 6), for
m=1,2,...,sN+1 '

Proof. Using the relationships between state probabilities P*(m) presented
in Appendix A the total probability for the queueing system M/E /s /sN+1 is

sj SN+1 *(m) = P (0) i )m 1 +sjN+l l)m m-s.
m=O m=2O s.+l +1 m 1 ( n=l


Using the lower bound p in place of pt(n), for all n, we have
2sn h oe on 2j








+ > 1 j + N+L! m m-s
(m m=m0 j m=s .+1 j m .

Since the total probability is 1, we have

1 > P*(O)
s. m-ss +
Ij (X)m + SnSN+l Ij. m (j M-j
M=0 3 m=s.+1


Let Z denote the value within the brackets. Then

S(Aj~m A.
z (j)m m Pj(0) ( )m for m=0,1,2,...,sN+1
Q.E.D.
Property 8. The lower bound on the sum of the state probabilities of those
states where service j uses central pool beds can be found accordingly:

sjSN+ X
s j+l1 Pj(n) 1 pj ) () n
n=s +1 n=0 j "

Proof. Property 8 follows directly from Property 7.

Property 9. The lower bound on the state probability Pj(s +sN+1) for
service j is

(s +sN+ ) = P() (X)sj+sN+1 1 L SN+l


L
N+I l~j i (sj1+sm+I

Proof. By definition, the probability 0 that central pool beds will be
available for patients of a serivce in the M/Ek/s /SN+l queueing system
is the lower bound of the probability j of that for the M/Ek/sj/sN+l system.
Thus, the probability of finding all central pool beds occupied by patients
of a service for the M/Ek/sj/sN+l queueing system is smaller than that








of the M/Ek/sj/sN+1 system, i.e.,


P (s +sN) > P.() (')s +sN+1 1 SN+1
Jj (s +sN+)!
S N+ Q.E.D.

Property 10. The lower bound P.(m) on state probabilities in Region 1 of

service j are

L P ( J) 1
Pj(m)= Pj(0) () m for m=0,l,2,...,sj

where

Pj(0) =


X I 3 m + S N + l (J ) m l ( U) 3- j
Z = ()m 5S+sN+1 m-s
m0 m=s .+1 j
J

pu = the upper bound on the probability that m or more central

pool beds are available for service j, from Property 6, for

m=1,2,...,s N+

Proof. The proof for Property 10 is similar to that for Property 7 with

the M/Ek/sj/sN+1 system in place of M/Ek/sj/sN+1 and rates A;,jj .

Property 11. The upper bound on the sum of the state probabilities of

those states where service j uses central pool beds can be found accordingly:

si + < AX.
s N+ P(m) 1 J P(0) (' )m 1
m=s.+1 m=0 j

Proof. Property 11 follows directly from Property 10.

Property 12. The upper bound on the state probability Pi(s +sN+,) for

service j is








SX. ( s +s 1
P (s + N(O) (_I ) i N-l
3 ij 3"N )= p P


1 u SN+1
(sj +sN+ (J


Proof. The proof for Property 12 is similar to that of Property 9.

Property 13. The lower bounds on the state probabilities in Region 2
of service j are


P (m) = Max PS (s)( -s.
P ( 3 J s j 3


, P (s + ,s+sN+1-m
' j i(sj N+1 Ul
jj


Proof. Using the relationships developed in Appendix A and the upper bound
probability 4i we have
3


P.(m+l) A )
-3--
Pj (m) j (m

ore

P*(s +s )
P*(m)


u
i- for m=s.+..,s+s -1
+1) 3 3 N+1




(j) s SN+1- m!
Pj (s +sN+1)


Using the lower bound probability, P(s +s we obtain
j ji N+1)' we obtain


P (s +N+ u)s +SN+l-m (sj+SN+1)!
N j4j m!


Similarly

Sm L
P.(m)
P (m- ) 1


P.(m)
P.(s.)


- (m)


for m=s +1l...,s +sN+l


A m-sj


mi N+)!


Theref(








Using the lower bound probability P (sj), we obtain


> n m-s s.
P (m) PL(s.) (-J j) J-
mj i

Therefore

P (m) Max P(sJ)( J )m-s P (s +sN ( ) j N+1-m (sjN+
m! NJJ Pi J J N+l 1
J J

for m=sj+l....sj+sN+1

Intuitively, the lower bound probabilities, P.(m), are found by
J
decreasing the probabilities with a higher rate from a lower bound state
probability at sj, or by increasing at a lower rate toward sj from a lower
bound state probability at s j+N+1 This method is illustrated in
Figure 3.6.

Property 14. The upper bounds on state probabilities in Region 2 of
service j are

PY(m) = Min (s)( )m-s Pj(s +SN+)( sj+N+-m 5(s N+l)
J (*sj ) m! P + At m!


Proof. The proof for this property is similar to that of Property 13.
Intuitively, the upper bound probabilities Pu(m) are found be decreasing
the probabilities at a lower rate from an upper bound state probability
at s., or by increasing at a higher rate toward sj from an upper bound
state probability at s +sN+1 This method is illustrated in Figure 3.7.




























s N + NAoher of fb-d.
or. oP.d


Figure 3.6. The lower bounds on the state probability

for the M/Ek/sj/s+1 queueing system.












Probability








/k// ~N+s






+ + Nunh, r of beds
occupied



Figure 3.7. The upper bounds on the state probability

for the M/E /s /s*N+ queueing system.








3.3.4 An Algorithm for Evaluating the State Probability Bounds
Using the above properties, the upper and lower bounds on the state
probabilities over Regions 1 and 2 can be successively tightened. Thus,
good approximation of the state probabilities of the M/M/s /sN+l system
can be obtained. The following algorithm summarizes the steps of the
procedure.
Step 1. Evaluate the state probabilities (the truncated Poisson probabilities)
for the M/Ek/Sj/SN+1 queueing system.

P.(m) = 1m
P s sjN+l(S )i l j
JC--r
i=0 "

for m=0,1,2,...,sj+sN+ and j=1,2,...,N

Step 2. Evaluate the lower bounds on the probability that a service uses
m beds of its own, (Property 3)

PL(m) = Pj(m)
3 3

for m = 0,1,2,...,s. and j = 1,2,...,N

Step 3. Evaluate the lower bound on the probability that m or more
central pool beds are available for a service, (Property 6), for
m=1,2,...,sN+1

L f= sT i P (n)
J ifJ n=O

Step 4. Improve the upper bounds on state probabilities in Region 1,
(Property 7)

PU(m) = Pj(O) ( J)m for m = 0,1,2,...,s., and

for j = 1,2,...,N,








where


Pj(0) =

(m=o

Step 5. Improve the

(Property 9):


A -s.+s m-s.
(-)m L+ sJ N+1 ()m m1 L
S m=s.+l m 'j

lower bound P (sj+sN +) on the state probability
p obb it


P(s +s -) P(0) ( -)sj+sN+1
J rj+1 PJ Wj


1 LsN+1
(s +sN+I )


Step 6. Evaluate the upper bound on the probability that m or more

central pool beds are available for a service, (Property 6), for

m=1,2,...,sN+1:

U = 1 (si+N+


Step 7. Improve the

(Property 10).


lower bounds on state probabilities in Region 1,


P (m) = P ) ( )m i
.3 i !


for m=0,1,2,...,s., and

for j=1,2,...,N


where


Pj(0) =


Si j)m sL SN+l (j m ( m-s
m=0 3 m=sj+.1 ~j


Step 8. Test for improvement on the lower and upper bounds on the

probabilities over Region 1. If improvements are less than some Epsilon,

for all states m=0,1,...,s. then go to Step 9; otherwise, go to Step 3.
J








Step 9. Evaluate the lower bounds on the state probabilities in Region 2

(Property 13)

L t i \.- m-s. sJ! L p, s +s -m (sJ+s N
P:(m) Max P P ((s- + P N+
j m j +SN+l) ml


for m=s.+l1,...,sj +SN+1, and

for j=l,2,... ,N.

Step 10. Evaluate the upper bounds on the state probabilities in Region 2

(Property 14)

X, m-s. ", 'y
P (m) = Min P (s )( J Lms P(sj +sN+1)( us +sN+lm (s +sN+
in P(sJJ j m! J N A m+



for m=s.+l,... ,s.+sN+1, and

for j=1,2,...,N.

Stop.

3.3.5 Verification of the Bounds on the State Probabilities

A simulation model was used to evaluate the state probabilities for

the M/Ek/sj/si queueing system. A comparison of the analytic results

and the simulation results was made to evaluate the usefulness of the

analytic bounds on the state probabilities.

3.3.5.1 A Simulation Model of a Three-Service Hospital

A simulation model of a three-service hospital was constructed using

the GPSS language under the following assumptions:

The arrival process is Poisson distributed.
The patient length of stay is distributed as an Erlang-2
distribution.








A central pool with beds which can be used by overflow patients

of any of these three services.

Patients in central pool beds have to return to their original

service beds as soon as there are available beds.

The flow chart of the simulation model is presented in Appendix B.


3.3.5.2 Comparison of the Analytic and Simulation Results

An example of a three-service hospital system with a central pool is

used. The parameters for the system are in Table 3.1. The output of the

bounds on state probabilities of the analytic results is plotted against

the state residence time distribution function obtained from the

simulation results in Figures 3.8, 3.9, and 3.10.

The simulation was run for a simulated period of seven years, before

a state residence time distribution function was obtained that was within

the analytic bounds. The cost in CPU time of the simulation is 569 seconds

on an IBM-370-165. The analytic bounds on the state probabilities were

obtained using a FORTRAN program which ran in 1.06 seconds. The agreement

between the simulation and analytic provides additional assurance that the

procedure for determining the analytic bounds on the state probabilities are

of value in evaluating the system operating characteristics. Moreover,

for a given level of accuracy, finding the analytic bounds cost considerably

less than the simulation results. The costs for obtaining the analytic

bounds increase slowly as shown in Table 3.2.


3.3.6 Analogous Analysis for Models 2 and 4

For Model 2, the analysis for the bounds on state probabilities of

the N-service hospital is similar to that of Model 3. The probability

that a service uses its own beds can be used to improve the bounds on




























__ analytic l bound

- simulation reoslt


occupied


Figure 3.8. The state probabilities
service 1.


and bounds for


10 15 20


Figure 3.9. The state
service 2.


probabilities and bounds for


Pr-o ability


'I i

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62





Table 3.1

Parameters for a Three-Service Hospital System
Used in the Example for State Probability Bounds.



Service Number of Mean arrival rate Length of stay
beds (patients/day) (days)

1 10 2 4.5

2 20 3 6.0

3 25 3 8.0

Central pool 5











Table 3.2

Computational Costs for Analytic Results.
(Stopping Criteria e = 10-7)


*Each service has 25 beds.


Number of services* Number of beds CPU time in
in central pool seconds


5 5 1.19

6 5 1.23

7 10 1.34

9 10 2.37








state probabilities of other services. The upper bound on the probability

that a service uses m or less beds of its own can be used to derive the

lower bound on the probability that other services use (B-m) beds.

Similarly, the lower bound on the probability that a service uses m or

less beds of its own can be used to derive the upper bound on the

probability that other services use (B-m) beds. These relationships

between the N services can then be used successively to tighten the bounds

on the state probabilities.

For Model 4, the analysis is again analogous to the one presented

for Model 3. The two known probability distributions used as bounds on the

state probabilities are those of the M/Ek/sj/sN+l and M/Ek/sj/(B-sj)

queueing systems, where j=1,2,...,N. The bounds on the probability that a

service has m or less beds occupied (0 m s +sN+) can be used to

improve the bounds on the probability that other services have (B-m) beds

occupied.


3.4 Solution Techniques for the Bed Allocation Problem

The allocation problem for Models 2, 3 and 4 can be written generally

as

Min (costs of allocating B beds among N services)

N+1
s.t. I s. = B (for a system with a central bed
j=1 J pool)

N
or I s. = B (for a system without central bed
j=l J pool)

The expected costs of the allocation problem are not any kind of

special functions such as linear or convex. The constraints are linear,









therefore, the problem can be solved by using a heuristic algorithm

that searches for the minimum costs.

Let a bed allocation to be denoted as n (s,,s2,...,sN) where si

beds are assigned to service i.

Definition: A neighboring point of an allocation (s,s2,.... sN) is any

allocation which can be presented by a perturbation of the above

allocation (s1 + al, s2 + a2,..., sN + aN) such that


N
a. = 0
i=1


and ak = 1, al = -1 for k f 1

am = 0 for all m / k,l


Therefore, there are at most N(N-1) neighboring points for each allocation.

3.4.1 Heuristic Algorithm for Solving the Bed Allocation Problem

Step 1. Choose a set of (sj) that satisfies the constraint.

Let this allocation be n.

Step 2. Find the objective function value, f(no), for the

allocation no

Step 3. Generate all neighboring points of allocation n

Step 4. Find objective function values f(n9) for all neighboring
J
points of allocation n

Step 5. Take the minimum of objective values.

If f(no) = Min f(n9) < f(no), then let no = no go to
x J X
Step 2. Otherwise, n* = no, where n* is a local optimum

solution of the allocation problem; stop.








The allocation problem is solved twice, once for the upper bounds on state

probabilities and once for the lower bounds on state probabilities. The

allocations of the two problems are compared, if they are equal the

heuristic allocation of the problem is found. The approximate allocation

is considered acceptable when the two allocations differ by one or two beds.


3.4.2 Experimental Results for the Bed Allocation Problem for Model 3

An example of the bed allocation problem for Model 3 is shown for a

hospital with two services and a central bed pool using the procedures

presented in previous sections. The parameters for the system are in

Table 3.3.

The computational results are presented in Table 3.4. The allocations

were found after 5 iterations to be the same for the lower and upper bounds

on the state probabilities. Therefore, the allocation (30, 19, 1) is the

best allocation found for the system.


3.5 The Relative Costs in Bed Allocation Models

The costs of having a patient in a borrowed bed, in a central pool

bed or turned away are difficult to evaluate in monetary values. Singh (61)

attempted to evaluate the shortage costs by considering both the costs to

the hospital and the costs to the patient. The absolute costs to the

patient vary with individual cases and depend on many factors such as

medical condition, social and economical inconvenience. Singh surveyed

some sample patients at various economic levels and constructed utility

functions to determine the costs of admission delays to the patient. In

this study, the objective functions of the models only require relative

costs between services, elaboration on the determination of the cost as in

Singh's study is not necessary. The emphasis is on the relative costs to





67



Table 3.3

Parameters for a Three-Service Hospital System
Used in an Example of Model 3


Service Number of Arrival rate Length of Central Turnaway
beds (patients/day) Stay (days) pool cost cost


1 2.15 5.81 6.11 40.16

2 2.24 8.93 1.03 30.36
60
3 1.92 6.35 2.10 60.73

Central
pool










Table 3.4

Computational Results for an Example of Model 3


Iteration Number of beds Expected Costs

Service Service Service Central Lower Upper
1 2 3 pool Bound Bound


1 15 20 20 5 21.28 21.56

2 16 19 20 5 20.21 20.78

3 17 18 20 5 19.63 20.35

4 18 17 20 5 19.53 20.21

5 19 17 19 5 19.32 19.80

6 18 18 19 5 19.03 19.60

7 18 19 18 5 18.93 19.30

8 18 20 18 4 18.69 19.71

9 17 21 18 4 18.60 19.67

10 18 21 18 3 18.43 19.60

11 17 22 18 3 18.24 19.49




- Other initial allocations attempted: (20, 15, 20, 5), (25, 15, 15, 5),
and (20, 25, 13, 2).









the hospital and to the physicians for causing inconvenience to their

patients and their medical practice. The method for evaluating the

relative values of the costs is presented in the following section. The

opinions of the health center officials at each institution can be used

to validate these values. The Gainesville Veterans Administration Hospital

was used as a sample for evaluating the relative costs.


3.5.1 The Method for Evaluating the Relative Costs

The characteristics of the existing allocation of the institution

has to be identified: the number of services, the bed assignment of each

service, and the interaction among services. One of the models presented

in Section 3.2 can be chosen to describe the allocation of the system

and the probability distribution functions of the number of beds occupied

for each service can be determined by the method presented in Section 3.3.

The expected numbers of patients in borrowed beds, in central pool beds,

and turned away can be evaluated from the probability distribution functions.

The relative values of the costs are based on the corresponding

expected numbers and the weight of importance for each service. For

example the relative cost for bed borrowing for service j is found as the

ratio of the weight of importance of service j and the expected number of

borrowed beds service j. In this study, the number of beds assigned to

each service is taken as the weight of importance for the service.

The relative values of the costs can be used to evaluate the cost for

the existing allocation. The relative values of the costs are verified

when the costs for the existing allocation are found to be the minimum

costs with respect to any neighboring allocation.









3.5.2 Implementation Test

The Gainesville Veterans Administration Hospital was chosen for the

test of evaluating the relative costs. A questionnaire was given to

members of the hospital administrative and medical staffs to help in

identifying the characteristics of the existing allocation system. These

opinions of the officials were drawn from the following questions:

1. How many patients from other services can your unit absorb without

impairing care to patients from your service or seriously inconveniencing

your staff, given that beds are available?

2. How many patients assigned to your service can be placed in

beds of other services before patient care is impaired or your staff is

seriously inconvenienced?

3. What is your preference of the allocation of the off-service beds?

4. How many times per month can your serivce tolerate the situation

where an emergency admission request causes a special action such as

discharging a current patient early, holding of the new patient in a

nonstandard area, or referring the patient to another service?

From the replies of the V.A. Hospital officials to the questionnaires, the

existing allocation and its interaction among services are as follows. The

hospital has eleven services grouped into four main services: Psychiatry,

Medicine, Neurology, and Surgery. Psychiatry is completely segregated from

other services; there is no interaction between Psychiatry and any other

service in the hospital. Surgery is further divided into sub-services such as

General Surgery, Thoracic Surgery, Plastic Surgery, Otolaryngoloqy (E.N.T.),

Urology, Neurosurgery, Orthopedics, and Ophthalmology. Medicine includes

three sub-services: General Medicine, Pulmonary, and Cardiology. The

Pulmonary and Cardiology services have been assigned a fixed number of beds









for their own patients recently. However, Pulmonary and Cardiology

services still do not operate independently from the General Medicine

services. In other words, medical services can freely borrow beds from

one another without impairing care to patients. General Medicine,

Pulmonary, and Cardiology therefore can be considered together as one

service. It is totally unacceptable to the hospital officials to have

medical patients placed in beds belonging to the surgical services. The

same situation applies to the surgical services for having patients on

medical floors. Thus, there is no interaction between medical and surgical

services within the V.A. hospital. For the Gainesville Veterans

Administration Hospital, it also appears that each of the four main

services of the hospital have not faced a situation where an emergency

patient is turned away due to no beds available.

In the following analysis, Psychiatry service is considered

separately. The remaining system consists of three main services:

Medicine, Neurology, and Surgery. In order to have some preliminary

knowledge of the behavior of these services, it is assumed at first that

there exists no interaction between these services. The M/G/s /O queueing

system provides the probability distribution of the number of beds

occupied for each service. The distribution functions for Medicine,

Neurology, and Surgery are plotted in Figures 3.11, 3.12, and 3.13

respectively. It can be easily seen from Figures 3.11 and 3.13 that

Medicine and Surgery services have no chance of having more patients than

their assigned numbers of beds. This result agrees with the replies

from Medicine and Surgery services that there have been no problems in

turning away patients for the last five years. Neurology has a highest

chance of overflowing, however Neurology can send its patients to

































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off-service beds with the most preferable ones on Neurosurgery Ward.

From these observations, the V.A. hospital system can be decomposed

further: Medicine as a separate service, Neurology and Surgery services

can be included in one system. A sketch of the decomposition of the

system is illustrated in Figure 3.14.

Neurology and Surgery services are assigned a fixed number of beds

to admit their own patients. If these services are considered

independently, they can experience some overflow problems. Since these

services are allowed to interact, that is borrow beds from one another,

the bed capacity of each service can expand beyond its allocation. There

is no chance of having a patient turned away; therefore,the system for an

individual service is equivalent to the M/G/oo queueing system. The

probability that a service j has m beds occupied by its patients is the

well-known Poisson distribution:
A.

P (m) = e Pj (i)"m for m=0,1,....



where A. = the mean arrival rate for service j, and
J

-= the mean length of stay.
1j


The probability distribution functions for services Neurology, ENT,

General Surgery, Neurosurgery, Ophthalmology, Orthopedics, Plastic

Surgery, Thoracic Surgery, and Urology are plotted in Figures 3.15, 3.16,

3.17, 3.18, 3.19, 3.20, 3.21, 3.22, and 3.23, respectively.

The number of beds occupied for each service is divided into two

regions. Region 1 consists of all beds which are assigned to the





















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service, i.e., for m=0,1,2,...,sj. Region 2 includes all beds that

service j borrows from other services, i.e., for m>s.. The expected rate

of bed borrowing is equivalent to the rate of overflow in this case and

can be found as


s.-l
[j [1 J Pj(m)] .
m=0

The expected number of borrowed beds for service j is the product of the

expected rate of bed borrowing and the mean length of stay for patients

of service j:

A. s.-l
[I i P.(m)] .
"j m=0


The relative costs of bed borrowing can be determined as the ratio of the

number of beds assigned to service j and the expected number of

borrowed beds for service j:


V =- J
j A. s.-l
-L[1 I P.(m)]
Vj m=O J

The relative costs of bed borrowing for Neurology and Surgery services are

listed in Table 3.5. The relative cost for Ophthalmology is smallest,

1.368, while the relative cost for Neurology is highest, 46,433. This can

be interpreted to mean that the Neurology service is much more sensitive

than Ophthalmology to having its patients in off-service beds. A

Neurology patient in a borrowed bed is perceived to "cost" 33 times

more than one for Ophthalmology. This can be easily seen from the

current allocation where Neurology is separated from all surgical










Table 3.5

Relative Costs of Bed Borrowing


Services Number of Expected number Relative
assigned beds of beds borrowed costs

Neurology 38 .818 46.433

E.N.T. 20 2.611 7.659

General Surgery 50 4.259 11.038

Neurosurgery 30 1.867 16.070

Ophthalmology 10 7.308 1.368

Orthopedics 40 2.740 14.598

Plastic Surgery 20 1.246 16.050

Thoracic Surgery 24 14.419 1.664

Urology 20 9.175 2.180

Total 252




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