MODELS FOR HOSPITAL CENSUS PREDICTION AND
ALLOCATION
By
KHANHLUU 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 cochairman, 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 ThreeService Hospital System Used in
the Example for State Probability Bounds .............. 62
3.2 Computational Costs for Analytic Results .............. 63
3.3 Parameters for a ThreeService 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 ChiSquare 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 Oneday census prediction errors, Shands Teaching Hospital
Pediatrics unit ........................................... 127
4.12 Sevenday census prediction errors, Shands Teaching
Hospital Pediatrics unit .................................. 128
4.13 Frequency of oneday prediction errors, Shands Teaching
Hospital Pediatrics unit .................................. 129
4.14 Frequency of sevenday 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 oneday census prediction, Shands
Teaching Hospital Pediatrics unit ......................... 136
4.18 Oneday census prediction errors, Gainesville Veterans
Administration Hospital ................................... 139
4.19 Sevenday census prediction errors, Gainesville Veterans
Administration Hospital ................................... 140
4.20 Frequency of oneday census prediction errors, Gainesville
Veterans Administration Hospital ......................... 141
4.21 Frequency of sevenday 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 oneday census prediction, Gainesville
Veterans Administration Hospital ........................ 148
FIGURE PAGE
4.25 Fraction errors for sevenday census prediction,
Gainesville Veterans Administration Hosptial .............. 149
4.26 Oneday census prediction, Alachua General Hospital ....... 155
4.27 Oneday census prediction errors, Alachua General
Hospital ............................................... 156
4.28 Sevenday census prediction, Alachua General Hospital ..... 157
4.29 Fraction errors of oneday census prediction, Alachua
General Hospital .......................................... 158
4.30 Fraction errors of sevenday 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
KhanhLuu 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
9095% of the time, depending on the risk level the administration adopts,
these resources are underutilized. 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 subproblem 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 bedstoservice and bedstoroom 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 turnaway 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
wellknown 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 setup 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 ( vmk (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 overcrowded 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 Lstage 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. Steadystate 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 daytoday 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 censusafterdischarge
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 patientsthe 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 welldefined 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 lengthofstay distribution from a geometric to a Pascal
distribution and expanded the scheduling horizon from a oneday to a
severalday 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 realtime
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 hospitaloriented objectives
instead of Resh's patientoriented 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 semiMarkovian 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 semiMarkovian 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 semiMarkovian 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 overstaffing 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 nonemergent and
of emergent admissions for each day of the week, the number of scheduled
reservations, the midnight census, and the estimate length ofstay 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 ofstay 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
steadystate 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 twostep 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 shortterm suboptimization 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 lowincome groups
than highincome 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. nonBlue 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 onethird 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 lengthofstay 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 lengthofstay, 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 shortterm
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 oneyearold.
Thirteen and fourteenyearolds had a longer average stay than patients
in the surrounding age group. From age eighteen to age twentysix 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 shortterm 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 longterm trends and cyclical movements. The study
suggested that seasonally adjusted data, obtained through seasonal indexes,
would be helpful for shortrange and intermediaterange 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 underutilize 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 underutilized 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 lengthofstay 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 Nservice 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) [IPr[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) [lPr[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 turnedaway
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) [IPr[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+1Cj s ]Pr[C+......+C+...+CN
The bed allocation problem for this model can be written as
N
Min J { U.jPj(s.) [1Pr[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 wellknown 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 wellknown (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 Nservice 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 Nl 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 N1 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+lm) central pool beds
= 1 the probability that other services (fj) use more
than (SN+lm) central pool beds.
Since the probability that other services (fj) use more than (sN+lm)
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+1m)
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 ms.
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 ms
(m m=m0 j m=s .+1 j m .
Since the total probability is 1, we have
1 > P*(O)
s. mss +
Ij (X)m + SnSN+l Ij. m (j Mj
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 ms
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 Nl
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+1m
' 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+lm (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 msj
mi N+)!
Theref(
Using the lower bound probability P (sj), we obtain
> n ms s.
P (m) PL(s.) (J j) J
mj i
Therefore
P (m) Max P(sJ)( J )ms P (s +sN ( ) j N+1m (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)( )ms 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 fbd.
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
JCr
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 ms.
()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 ( ms
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 \. ms. 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, ms. ", '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 ThreeService Hospital
A simulation model of a threeservice 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 Erlang2
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 threeservice 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 IBM370165. 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 Nservice 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
Pro ability
'I i
* S
* S
I
I
I
I
I
\\
61
0 C3
01
cli
.0 .
m
(U
(_0
C
0
a0
0 0 C
I 1
o
s0 c=
cl)
I
'\UQ
tt
K 0
'" n
+
in
^1 r
2 CO
LJ
62
Table 3.1
Parameters for a ThreeService 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 = 107)
*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 (Bm) 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 (Bm) 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/(Bsj)
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 (Bm) 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(N1) 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 ThreeService 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 offservice 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 subservices such as
General Surgery, Thoracic Surgery, Plastic Surgery, Otolaryngoloqy (E.N.T.),
Urology, Neurosurgery, Orthopedics, and Ophthalmology. Medicine includes
three subservices: 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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From these observations, the V.A. hospital system can be decomposed
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system is illustrated in Figure 3.14.
Neurology and Surgery services are assigned a fixed number of beds
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individual service is equivalent to the M/G/oo queueing system. The
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A.
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J
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1j
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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 offservice 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
