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THEORETICAL ARTICLE
An analytical study of queues in medical sector
Rashmi Agarwal1 • B. K. Singh2
Accepted: 22 August 2017
� Operational Research Society of India 2017
Abstract In general, we do not like to wait but each of us has spent a great deal of
time waiting in lines even in medical sector also and queuing has become a symbol
of inefficiency of a hospital. Managing the length of the line is one of the challenges
facing most hospitals. A few of factors that are responsible for long waiting lines or
delays in providing services are lack of passion of hospital staff and overloading of
available doctors as they are attached in more than one clinic etc. This paper is
based on the consideration that most of these problems can be managed by using
queuing model to measure the waiting line performances as average arrival rate of
patients, average service rate of patients, system utilization etc. The purpose of this
study is to provide insight of general background of queuing theory, and how
queuing theory can be used by policy makers to increase efficiency of services and
to improve the quality of care of patients in hospitals, also understanding cost factor
for getting optimum profit.
Keywords Queuing theory � Priority queues � Waiting time � Performance
measures and cost factor of queuing system
& Rashmi Agarwal
agarwal.rashmi.rashmi8@gmail.com
B. K. Singh
dbks68@yahoo.com
1 DAV Inter College, Bulandshahr, U.P, India
2 Head of Department of Mathematics, School of Engineering and Technology, IFTM University,
Moradabad, U.P, India
123
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DOI 10.1007/s12597-017-0324-7
http://orcid.org/0000-0002-5246-6676
http://crossmark.crossref.org/dialog/?doi=10.1007/s12597-017-0324-7&domain=pdf
http://crossmark.crossref.org/dialog/?doi=10.1007/s12597-017-0324-7&domain=pdf
1 Introduction
The existence of any nation is a function of the survival of its citizens and sufficient
healthcare programs of its citizenry. Hence several health policies and laws are
formulated and adopted by the government. Despite of these efforts, it should be
noted that, still there are some problems which challenge the success of the
healthcare programs applied in medical sector. One of the most common of them is
the problem of waiting lines i.e. queues found in hospitals.
In general, queues form when the demand for service exceeds its supply. In
medical sector, the effect of queuing in relation to the time spent by patients for
treatment is increasingly concerned with modern society. Keeping patients waiting
and their time wasted in the queues in hospitals often have severe consequences.
Hence there is need for a significant evaluation of patient waiting time as well as
reducing or eliminating it and for that queuing theory can be applied.
2 Queuing system
To describe a queuing system, an input process and an output process must be
specified. The input process is usually called arrival process. Arrivals are called
customers and in our case ‘patients’. We assume that no more than one patient can
occur at a given instant and if more than one patient can occur at a given instant, we
say that ‘bulk arrivals’ are allowed.
To describe the output process, we usually specify a probability distribution i.e.
service time distribution, which governs a customer’s service time i.e. a patient’s
service time. The uncertainties involved in the service mechanism are the number of
servers i.e. doctors and the number of customers i.e. patients getting served at any
time and leave the service system. Networks of queues consist of more than one
server arranged in series and/or parallel. The facilities given to the customers can be
classified as:
• Single channel facility i.e. one queue-one service station facility: This mean that
there is only one queue in which the customer waits till the service point is ready
to take him for servicing. Here the customers are served by one service station
only i.e. only one customer can be served at a time and the standard assumption
will be made that one server can completely serve a customer. The following
figure shows the single channel facility:
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• One queue—several service stations in parallel: In this case, customers wait in a
single queue until one of the server stations is ready to take them for servicing.
Here the customers are served by several i.e. more than one, service stations.
Hence more than one customer can be served at a time as all servers provide the
same type of service and each of them can completely serve the customer. Such
type of service facilities is illustrated in the following figure:
• Multi-channel facility i.e. many queues—many service station facilities: In such
a situation, there are several queues and the customer can join any one of these,
because the servers are in parallel and all servers provide the same type of
service and a customer need only pass through one server to complete service.
The customer joins the queue of his choice in front of service facilities and may
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be served by any service facility. The parallel arrangement is shown in the
following diagram:
• Multi-stage channel facilities: In this case, customers require several types of
services and different service stations are there, each station providing a
specialised service and the customer passes through each of the service station
before leaving the system. In this arrangement, the servers are in series i.e. the
series arrangement consists of a sequence of a number of service facilities such
that a customer must go through one facility after another in a particular
sequence before the whole service is completed. Each service facility may,
however, work independently of the others, having its own rules of service. The
following figure illustrates the arrangement of service facilities in series:
• The mixed arrangement of service facilities: This arrangement consists of
service facilities arranged in series as well as in parallel. For example: in a
hospital, an incoming patient may go to any of the OPD counter (service
facilities in parallel), where he is checked (served) by various doctors (facilities
in series) one by one.
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Then the queue discipline is specified. The ‘queue discipline’ describes the
method used to determine the order in which patients are served. The most common
queue discipline is the FCFS discipline (i.e. first come, first served) in which
patients are served in the order of their arrival. Under the LCFS discipline (i.e. last
come, first served) the most recent arrivals are the first to enter for service. The
customer i.e. patient to enter for service in randomly chosen from those patients
waiting for service, it is known as the SIRO discipline (i.e. service in random order).
In medical sector we mostly consider ‘Priority Queuing Disciplines’. In a priority
queue discipline, each patient classifies into one of several categories. Each category
is then given a priority level, and within each priority level, patients enter for service
on FCFS basis.
There are two types of ‘Priority Models’-Pre-emptive priority (i.e. emergency)
and Non-Pre-emptive priority. In a pre-emptive queuing system, a lower priority
patient can be interrupted from service whenever a higher priority patient arrives.
Once no higher priority patients are present, the interrupted patient returns to
service. In a pre-emptive resume model, a patient’s service continues from the point
at which it was interrupted.
In a non-pre-emptive model, a patient’s service cannot be interrupted. After each
service completion, the next patient to enter, service is chosen by given priority to
lower numbered patients types. Lower numbered are given on the basis of higher
priority. Another factor that has an important effect on the behaviour of queuing
system is the method that patients use to determine which line to join.
2.1 System capacity
The system capacity is the maximum numberof customers, both those in service
and those in the queue(s), permitted in the service facility at the same time.
Whenever a customer arrives at a facility that is full, the arriving customer is denied
entrance to the facility. Such a customer is not allowed to wait outside the facility
(since that effectively increases the capacity) but is forced to leave without
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receiving service. A system that has no limit on the number of customers permitted
inside the facility has ‘infinite-capacity’; a system with a limit has ‘finite-capacity’.
2.2 Kendall’s notation
We have developed enough terminology to illustrate many queuing systems. Now,
we describe standard notation used in different models in queuing theory which is
initially given by D.G.Kendall. Kendall’s notation for specifying a queue’s
characteristics is a/b/c/d/e, where the first characteristic ‘a’ specifies the nature of
the arrival process and the second characteristic ‘b’ specifies the nature of service
pattern. The following abbreviations are used for the arrival and service pattern to
replace the notation ‘a’ and ‘b’:
M = Markovian (or Exponential) inter-arrival time or service-time distribution.
D = Deterministic (or constant) inter-arrival time or service-time.
G = General distribution of service time (departure), i.e. no assumption is made
about the type of distribution with mean and variance.
Ek = Erlang-k distribution for interarrival or service time with parameter k (i.e. if
k = 1, Erlang is equivalent to exponential and if k = ?, Erlang is equivalent to
deterministic).
GI = General probability distribution—normal, uniform or any empirical
distribution, for inter-arrival time.
The third characteristic ‘c’ specifies the number of service channels (servers), the
fourth characteristic ‘d’ describes the maximum number of customers allowed in the
system (in queue plus in service) and the fifth characteristic specifies the queue-
discipline:
FCFS = First come, first served
LCFS = Last come, first served
SIRO = Service in random order
GD = General queue discipline.
• In general, arrivals in a queuing system do not occur at regular intervals, but
tend to be scattered in some manner. The Poisson assumption specifies the
behaviour of arrivals, by postulating the existence of constant k, which is
independent of time, queue length, or any other random property of the queue,
such that
p the probability of an arrival occurs between time ‘t’ and ‘tþ ot’ð Þ ¼ kot
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If the interval qt is sufficiently small and ‘n’ be a random variable (discrete)
representing the number of arrivals in some time interval of length ‘t’, then ‘n’
obeys a Poisson-distribution. The probability of ‘n’ arrivals in time ‘t’ is defined as:
pn tð Þ ¼ ktð Þ
n�e�kt=n!; where kt is a parameter: ð1Þ
In the similar way, where ‘l’ is the mean servicing rate for a particular service
station then ‘n’ the number of units in the service station could serve in time ‘t’, if
the service is going on throughout ‘t’ will follow the Poisson distribution, i.e.
ft nð Þ probability of ’n’ servicing in time ’t’; if there were no enforced unused time½ �
¼ ltð Þn�e�lt=n!;
Case-1: when n = 0. From (1), the probability for n = 0, i.e. no arrival in time qt
is given by:
p0 otð Þ ¼ kotð Þ
0�e�kot=0!
¼ e�kot
¼ 1� kotþ ðk2=2!Þ otð Þ2�ðk3=3!Þ otð Þ3þ � � �
¼ 1� kotþ 0 otð Þ;
where 0 (qt) denotes a measure which is of smaller order of magnitude than qt as it
contains higher powers of a small quantity qt.
If qt is very small then 0 (qt) = 0
Hence; p0 otð Þ ¼ 1� kot
i.e. the probability of no arrival in time qt = 1 - kqt.
Similarly; fot 0ð Þ ¼ 1� lot
i.e. the probability of no service in time qt = 1 - lqt.
Case-2: when n = 1 From (1), the probability for n = 1, i.e. one arrival in time
qt is given by:
p1 otð Þ ¼ kotð Þ
1�e�kot=1!
¼ kot 1� kotþ ðk2=2!Þ otð Þ2�ðk3=3!Þ otð Þ3þ � � �
n o
¼ kotþ 0 otð Þ;
where 0 (qt) denotes a measure which is of smaller order of magnitude than qt as it
contains higher powers of a small quantity qt.
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Hence; p1 otð Þ ¼ kot: If ot is very small then 0 otð Þ ¼ 0:f g
i.e. the probability of one arrival in time qt = kqt.
Similarly; fot 1ð Þ ¼ lot
i.e. the probability of one service in time qt = lqt.
Case-3: when n = m > 1 From (1), the probability for n = m[ 1, i.e. more than
one arrival in time qt is given by:
Pm otð Þ ¼ kotð Þm�e�kot=m!
¼ kotð Þm=m! 1� kotþ ðk2=2!Þ otð Þ2�ðk3=3!Þ otð Þ3þ � � �
n o
¼ km=m! otð Þm�k otð Þmþ1þ � � �
n o
¼ 0 otð Þ;
where 0(qt) denotes a measure which is of smaller order of magnitude than qt as it
contains higher powers of a small quantity qt.
Hence; pm otð Þ ¼ 0: If ot is very small then 0 otð Þ ¼ 0:f g
i.e. the probability of more than one arrival in time qt = 0.
Similarly; fot mð Þ ¼ 0
i.e. the probability of more than one service in time qt = 0.
Hospitals are like service industry with different branches of services as doctors,
nurses, and other paramedical services. The treatment process consists of a set of
activities and procedures that the patient must go through in order to receive the
required treatment.
In order to represent the workflow in a hospital, it is important to understand how
the departments of the hospital operate and how data flows. The following flow-
diagram shows the hospital system:
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2.3 Hospital-system
Now for calculations, we define k to be the ‘arrival rate’, which will have units of
arrivals per minute and l to be ‘service rate’. Ws is the expected time that a patient
spends in the queuing system, including time in line (i.e. queue) plus time in service,
and Wq is the expected time that a patient spends waiting in line.
We have assumed that the arrival rate, service rate and number of servers has
stayed constant over time, this is known as steady state of the system. In many
situations, the arrival rate, the service rate, and number of servers may vary over
time, and then we say that the queuing system is not stationary.
We are using the queuing model (M/M/1): (?/FCFS) which is also known as
Birth and Death Model as here is Poisson arrival and Poisson service. We are
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assuming that servicing is through single channel with infinite capacity (because
mostly hospitals give 24-h service). The service discipline is first come first serve.
Case-1: there is no patient in the queue in a hospital at time (t 1 qt)
This follows in two ways:
1. The probability of no patient in the queue in a hospital at time ’t’ = p0 (t)
No patient arrives in time qt, with probability p0 (qt) = 1 - kqt.
Hence, the probability in this case is p0 (t)�(1 - kqt).
2. There is one patient in the queue in a hospital at time ‘t’, with probability p1 (t)
No patient arrives in time qt, with probability p0 (qt) = 1 - kqt.
One service in time qt, with probability fqt (1) = lqt.
Hence, the probability in this case is p1 (t)�(1 - kqt)�lqt.
Since above two ways are mutually exclusive, therefore, p0 (t ? qt), the
probability of no patient in the queue in a hospital at time (t ? qt) is obtained by
adding the above two probabilities.
i:e:; p0 tþ otð Þ ¼ p0 tð Þ � 1� kotð Þ þ p1 tð Þ � 1� kotð Þ � lot
or; p0 tþ otð Þ ¼ p0 tð Þ þ �kp0 tð Þ þ lp1 tð Þ½ � � otþ �klð Þ � p1 tð Þ � otð Þ
2
or; p0 tþ otð Þ ¼ p0 tð Þ þ �kp0 tð Þ þ lp1 tð Þ½ � � otþ 0 otð Þ
or; p0 tþ otð Þ � p0 tð Þ½ �=ot ¼ �kp0 tð Þ þ lp1 tð Þ½ � þ 0 otð Þ=ot
Taking limit as qt ? 0, we get
d=dt p0 tð Þ ¼ �kp0 tð Þ þ lp1 tð Þ½ � ð2Þ
Case-11: there are n > 0 patients in the queue in a hospital at time (t 1 qt)
This occurs in the following ways—
(a) There are (n - 1) patients in the queue in a hospital at time ‘t’, with
probability pn-1 (t)
One patient arrives in time qt, with probability p1 (qt) = kqt.
No service in time qt, with probability fqt (0) = 1 - lqt.
Hence, the probability in this case is pn-1(t)�kqt�(1 - lqt).
(b) There are ‘n’ patients in the queue in a hospital at time ‘t’, with probability pn
(t)
No patient arrives intime qt, with probability p0 (qt) = 1 - kqt.
No service in time qt, with probability fqt (0) = 1 - lqt.
Hence, the probability in this case is pn(t)�(1 - kqt).(1 - lqt).
(c) There are (n ? 1) patients in the queue in a hospital at time ‘t’, with
probability pn?1(t)
No patient arrives in time qt, with probability p0 (qt) = 1 - kqt.
One service in time qt, with probability fqt (1) = lqt.
Hence, the probability in this case is pn(t)�(1 - kqt)�(lqt).
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Since above ways are mutually exclusive, therefore, p0 (t ? qt), the probability
of ‘n’ patients in the queue in a hospital at time (t ? qt) is obtained by adding the
above probabilities.
i:e:; pn tþ otð Þ ¼ pn�1 tð Þ � kot � 1� lotð Þ þ pn tð Þ � 1� kotð Þ � 1� lotð Þ þ pnþ1 tð Þ � 1� kotð Þ � lot
or; pn tþ otð Þ ¼ kpn�1 tð Þ� kþ lð Þpn tð Þ þ lpnþ1 tð Þ
� �
� otþ pn tð Þ þ 0 otð Þ
or; pn tþ otð Þ � pn tð Þ½ �=ot ¼ kpn�1 tð Þ� kþ lð Þpn tð Þ þ lpnþ1 tð Þ
� �
þ 0 otð Þ=ot
Taking limit as qt ? 0, we get
d=dt pn tð Þ ¼ kpn�1 tð Þ� kþ lð Þpn tð Þ þ lpnþ1 tð Þ
� �
ð3Þ
Now when the system reaches steady state,
lim
t!1
pn tð Þ ¼ pn and lim
t!1
d=dt pn tð Þ ¼ 0
Therefore, under steady state of the system, the Eq. (2) reduces to
�kp0 þ lp1 ¼ 0
Or; p1 ¼ k=lð Þp0; where k=l\1ð Þ ð4Þ
And the Eq. (3) reduces to—
kpn�1� kþ lð Þpn þ lpnþ1 ¼ 0
Putting n = 1, 2… etc. in the above equation, we get –
p2 ¼ �k=lð Þp0 þ k=lþ 1ð Þp1
¼ �p1 þ k=lð Þp1 þ p1; from 4ð Þf g
¼ k=lð Þp1
¼ k=lð Þ2p0
and p3 ¼ k=lð Þ
3
p0; . . .and so on:
Proceeding in this way, we have—
pn ¼ k=lð Þ
n
p0; for n� 0:
But
X1
n¼0
pn ¼ 1
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or; p0 þ p1 þ p2 þ p3 þ � � � ¼ 1
or; 1þ k=lð Þ þ k=lð Þ2þ k=lð Þ3þ � � �
h i
p0 ¼ 1
or; 1= 1�k=lð Þ½ �p0 ¼ 1 wherek=l\1ð Þ
or; p0 ¼ 1� k=lð Þ
Therefore, the probability of ‘n’ patients in the queue in a hospital at any time
pn = (k/l)
n[1 - (k/l)].
2.3.1 Cost factor of the queuing system
In today’s throat cut competition for all healthcare organizations, they always look
for more and more customers for generating maximum profit and providing the best
available service to the patients. Also the large segments of our organizations
operate in such a way that consumers have no real capability to learn about price
and quality. We shall discuss mainly two types of costs associated with the service
providing activities, they are:
• The cost of waiting in line The problem in almost every queuing situation is
traffic and the supervisor must think about the additional cost of providing more
rapid services such as more checkouts, more attending staff etc. against the
intern cost of waiting. For example, if patients are walking away disgusted
because of insufficient customer support persons, the business could compare
the cost of hiring more staff to the value of increase of revenues and maintaining
customer loyalty. Waiting time cost consists of all the costs which incurred to
the hospital caused by the dissatisfaction of the patients due to wastage of their
valuable time as standing in queue for getting the service.
TotalWaitingCost ¼ Number of arrivalsð Þ � Averagewait per arrivalð Þ
� Cost of waitingð Þ
¼ k�Wq � Cw:
• Service cost Service cost is defined as all the costs associated with service
providing activities as we know that any providing service is not free of cost.
These costs are directly related with the facility provided to the patients arrived
for treatment. It is a substantial cost incurred for the service provider; this is the
combination of both fixed as well as variable cost for any service. For example,
fees of the doctor, salary of the staff, rent of the room, electricity and telephone
bills, required stationary items etc.
Total Service Cost ¼ Number of channelsð Þ � Cost per channelð Þ ¼ m� Cs:
The person who is responsible for the management of all the activities in a
hospital has to analyze these two types of costs as both of them are inverse in nature,
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on reducing one type of the above mentioned two costs the other cost increases and
vice-versa. Initially, the cost of waiting in line is at the maximum, when the hospital
is at the minimum service capacity. As service capacity increases, there is a
reduction in the number of patients in the line and in their wait times, which
decreases queuing cost. The relationship between these can be expressed graphically
as follows:
The optimal total cost is found at the intersection between service capacity and
waiting line curves. The upper curved line in the graph represents the total cost
associated with the service providing operations.
Total Cost ¼ TotalWaiting TimeCost þ Total Service Cost
3 Review of literature
The application of queuing theory within the medical sector has been in existence
for over 50 years. Waiting time is considered to be an important part in a hospital
system. Now an overview of previous work falling within the scope of the paper is
presented.
Gupta et al. [1], Adeleke et al. [2], and Brohma [3] planned man power as how
large the staff is required to give adequate service. According to them more doctors
should be deployed to convert the single-channel queuing units to multi-channel
units.
Marvasti [4] and Mishra [5] were used survey methodology to avoid delays at
medical processes and compute the total optimum cost of independent queuing
systems with controllable arrival rates.
Fomundan and Herrmann [6], Wang et al. [7], Fazlul et al. [8], Obamiro [9],
Foaster et al. [10], Soni and Sexena [11] and Mehendiratta [12] proposed to use
queuing theory models in healthcare process and developed some new methods.
Barlow [13] makes some recommendations for removing the waiting problems.
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Verma [14] evaluates all the costs, profits and utilization factor for an optimum
profit.
4 Objectives of the study
The objectives of this work are as follows:
• To obtain probability of patients in a queue in a small time interval according to
gender of respondents.
• To compare waiting time of patients in the system according to gender of
respondents.
• To compare waiting time of patients in the queue according to gender of
respondents.
• To determine the length of queue according to gender of respondents.
• To obtain probability of patients in a queue in a small time interval on the basis
of age-groups of respondents.
• To compare waiting time of patients in the system on the basis of age-groups of
respondents.
• To compare waiting time of patients in the queue on the basis of age-groups of
respondents.
• To determine the length of queue on the basis of age-groups of respondents.
• To obtain the cost of the queuing system based on time spends in the queue
according to gender of respondents.
• To obtain the cost of the queuing system based on time spends in the queue on
the basis of age-groups of respondents.
5 Methodology
The admitting department is one of the most congested hospital services, and faces a
great deal of pressure compared with other components of the hospital system. In
this study, field observations are conducted to determine the current operation of the
admitting department. The author collects actual data for arrivals, waiting times and
service times. The data is collected from 60 peoples of Meerut through the
questionnaire.
The actual workload within the hospital is difficult to measure due to its
multifunctional systems; therefore, in order to obtain a reliable figure, the data is
collected for different factors related to a hospital system. This data will enable us to
obtain the arrival rate, the service rate and performance measures regarding queue
lengths.
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6 Work
The data collected by the questionnaire with the sample size 60 is analysed as under:
0
10
20
30
40
50
60
70
80
90
100
MALE FEMALE
N
O
. O
F 
RE
SP
O
N
DE
N
TS
 
→
OPTIONS → 
Sex of Respondents
56.7% 
43.3 % 
Table 1 and the above graph show that out of 60 (100%) respondents, 34 (56.7%)
respondents were male and 26 (43.3%)were female and the findings are shown in
the following table according to gender of respondents:
S.no. Performance measures Male
respondents
Female
respondents
1. Mean arrival rate of patients [k (approx. value)] 14 15
2. Mean service rate of patients [l (approx. value)] 15 16
3. Average time of patients spend in the system [Ws = 1/
(l - k)]
60 s 60 s
4. Average time of patients spend in the queue [Wq = (k/l)(1/
l - k)]
56 s 57 s
5. Average length of queue = [l/(l - k)] 15 16
By calculation, the author finds that by the opinion of male respondents, the
average time of patients spend in the system is 60 s and that according to female
respondents also, it is 60 s. The average time of the patients spend in the queue is
approximately 56 s by the opinion of male respondents and that according to female
respondents is 57 s while the average length of queue according to male respondents
is 15 patients and by the opinion of female respondents, it is 16 patients. Hence the
author finds that there is a little difference between the opinions of male and female
respondents.
Table 1 Distribution of sample
according to sex
Sex No. of respondents Percentage
Male 34 56.7
Female 26 43.3
Total 60 100
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6.1 To obtain the probabilities of patients in the queue in a hospital
according to gender
As we know that, p0 = 1 - (k/l), p1 = (k/l) p0, p2 = ((k/l)
2 p0 (where k/l\ 1)
The following table shows their values:
Probabilities Male Female
p0 0.067 0.062
p1 0.062 0.058
p2 0.058 0.055
The above table shows that the probability of no male in the queue is 0.067 and
that of female is 0.062. The probability of one male in the queue is 0.062 and that of
female is 0.058, also the probability of two males in the queue is 0.058 and that of
female is 0.055.
Hence, we find that the probability of females is less than that of males for
waiting in the queue, also probability of waiting in the queue decreases as the
number of patient increases in the queue, this may be depend upon speed of the
service.
Table 2 and the above graph show that out of 60 (100%) respondents, the age of 8
(13.3%) respondents were below 18 years, 34 (56.7%) were within the range of
18–35 years, 13 (21.7%) were within the range of 35–50 years and 5 (8.3%) were
above 50 years. After the calculations for performance measures, the findings on the
basis of age-groups of respondents are shown in the following table:
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S.no. Performance measures Results according to age-group of respondents
Below
18 years
18–35 years 35–50 years Above
50 years
1. Mean arrival rate of patients [k (approx.
value)]
13 15 16 11
2. Mean service rate of patients [l (approx
value)]
14 16 17 13
3. Average time of patients spend in the
system [Ws = 1/(l - k)]
60 s 60 s 60 s 30 s
4. Average time of patients spend in the
queue [Wq = (k/l)(1/l - k)]
56 s 56.25 s 56.5 s 25.4 s
5. Average length of queue = [l/(l - k)] 14 16 17 7 (approx.)
Hence the author finds that the average time of the patient spend in the system is
60 s by the opinion of respondents of the age-groups below 18 years, 18–35 years
and 35–50 years while according to the respondents of age above 50 years, it is
30 s. Average time of the patient spend in the queue is approximately 56 s while the
average length of queue is 14 patients according to the respondents of the age below
18 years. According to the respondents of age-group 18–35 years and 35–50 years,
average time of the patient spend in the queue is 56.25 and 56.5 s respectively and
average length of queue is 16 and 17 patients respectively, but the opinion of the
respondents of the age above 50 years is that the average time of the patient spend
in the queue is 25.4 s and average length of queue is 7 patients.
6.2 To obtain the probabilities of patients in the queue in a hospital
according to the age-groups
Probabilities Below 18 years 18–35 years 35–50 years Above 50 years
p0 0.071 0.062 0.059 0.154
p1 0.066 0.058 0.056 0.130
p2 0.061 0.055 0.052 0.110
Table 2 Distribution of sample
according to age
Age No. of respondents Percentage
Below 18 years 08 13.3
18–35 years 34 56.7
35–50 years 13 21.7
Above 50 years 05 8.3
Total 60 100
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The above table shows that the probability of no patient in the queue for the
patient below 18 years, 18–35 years, 35–50 years and above 50 years are 0.071,
0.062, 0.059 and 0.154 respectively. The probability of one patient in the queue in a
hospital for the patient below 18 years, 18–35 years, 35–50 years and above 50
years are 0.066, 0.058, 0.056 and 0.130 respectively, also the probability of two
patients in the queue for the patient below 18 years, 18–35 years, 35–50 years and
above 50 years are 0.061, 0.055, 0.052 and 0.110 respectively.
Hence, we find that the probability of patients waiting in the queue decreases as
age of the patient increases but for old patients this increases, also probability of
waiting in the queue decreases as the number of patient increases in the queue as we
see the same in the case on the basis of gender. The probability for old patients is
high; hence this should be reduced by given special facilities to them in a hospital so
that they do not suffer and get the treatment in time.
From the above comparisons based on gender and different age-groups of the
respondents, the author finds that the average waiting time is not much but if this
became negligible by the application of queuing theory then the average length of
queue can be reduced and the system will be an ideal system, also there requires to
pay attention on old patients in a hospital so that they could get treatment in time.
6.3 To obtain the cost of the queuing system based on time spends
in the queue
For this, we assume that the hospital’s facility can handle only one emergency at a
time as we are using (M/M/1):(?/FCFS) queuing model. Suppose that it costs the
hospital Rs. 100 per minute and each minute of waiting costs by Rs. 10.
As we know that,
Total waiting cost ¼ k�Wq � Cw and Total service cost ¼ m � Cs
The following tables show their values:
According to gender of respondents:
Costs (in Rs.) Male Female
Waiting cost 130.70 142.50
Service cost 100.00 100.00
Total cost 230.70 242.50
On the basis of age-groups of respondents:
Costs (in Rs.) Below 18 years 18–35 years 35–50 years Above 50 years
Waiting cost 121.30 140.60 150.70 46–60
Service cost 100.00 100.00 100.00 100.00
Total cost 221.30 240.60 250.70 146.60
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A final approach is evaluation of ‘‘cost of illness’’ which includes ‘direct costs’
(where money actually charges), ‘indirect costs’(the value of lost output from time
off work due to illness) and ‘intangible costs’(the disvalue to an individual of pain
and suffering). The optimal level of investment in health occurs where the ‘marginal
cost’ of health capital is equal to the ‘marginal benefit’.
7 Conclusion and recommendations
Unnecessary waste of time in the hospitals or healthcare centres may lead to
patients’ health complications and in some cases eventual death which may be
avoided. There are several probable ways of improving patient flow, and thus
reducing waiting time for the patients, such as:
• By increasing the number of servers,
• By controlling the arrival rate; and
• By optimizing the service rate, etc.
The number of servers can be increased by hiring more admitting staff which will
present an immediate improvement in services of registration system. Pre-registering
a large number of patients and visiting the staff to serve directly the patient will
eliminate the crowd in the visiting-room.More doctors and more paramedical officers
should be deployed to healthcare centres for taking patients’ preliminary tests or
service before they spot the doctor. This will reduce the service time of the doctor in
attending to patients and thus increase the service efficiency. This study attempted to
analyze actual operations of hospitals andproposed modifications in the system to
reduce waiting times for patients, which should show the way to an improved vision of
the quality of services provided in the medical sector.
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http://dx.doi.org/10.1186/1471-2288-10-60
	An analytical study of queues in medical sector
	Abstract
	Introduction
	Queuing system
	System capacity
	Kendall’s notation
	Hospital-system
	Cost factor of the queuing system
	Review of literature
	Objectives of the study
	Methodology
	Work
	To obtain the probabilities of patients in the queue in a hospital according to gender
	To obtain the probabilities of patients in the queue in a hospital according to the age-groups
	To obtain the cost of the queuing system based on time spends in the queue
	Conclusion and recommendations
	References

Outros materiais