HAWASSA UNIVERSITY - INSTITUTE OF TECHNOLOGY

Faculty of Manufacturing

Department of Industrial Engineering

IEng4162 Research methodology

Project 1

Group members Id number

1. ALAZAR KEBEDE ………………………………………………..1688/09

2. ABENET FEKADU ………………………………………...…….0906/09

3. NIKODIMOS TAMEN …………………………………………0709/09

4. GETU ZELEKE …………………………………………., .…./09

5. AMANUEL GIRMA.………………………………………’’……/09

6. EYERUSALEM BERHANU………………………………………../09

7. REDIT MOLA ……………………………………………………0052/09

8. BERKET DAMEN……………………………………………………/09

Preference
It gives us great pleasure and satisfaction to studies this clinic queuing model because to know
how data collected, analysis and organized in waiting line.

An attempt has been to ignore the emergence patients, While to collect data, due consideration
has been given to the valuable suggestion made by the nurse of the clinic.

Chapter 2(literature view) of this studies has be taken from different text books and reference
guide lines this chapter need more briefly explanation about queuing model.

We hope the instructor gives us more valuable and any constructive suggestions for further how
queuing model form in different system will be highly appreciated and thank full acknowledged.
 ACKNOWLEDGEMENT

First of all we want thank GOD, we would like to take this opportunity to express our profound
gratitude and deep regard to our guide DR FANTAHUN for his perfect guidance, valuable
feedback and constant encouragement throughout the duration of the project. His valuable
suggestions were huge help throughout this research work. Knowledgeable experience for us.
And we would like to thanks the zowi clinic manager and doctor YOHANES for their support.
 ABSTRACT

Healthcare systems differ intrinsically from manufacturing systems. As such, they require a
distinct modeling approach. In this article, we show how to construct a queuing model of a
general class of healthcare systems. We develop new expressions to assess the impact of service
outages and use the resulting model to approximate patient flow times and to evaluate a number
of practical applications.

In a clinic system, increasing resource utilization to reduce costs and decreasing patients’ waiting
time to provide timely care and improve patient satisfaction are important but conflicting goals.
Queuing models can provide reasonably accurate evaluations of system performance and are
popular among researchers and system designers because of their analytical nature and their
ability to provide quick solutions for “what-if” analyses. There is a considerable amount of
published research on using queuing to analyze and design hospital facilities. We review and
categorize this literature in an attempt to motivate Further research in applying queuing models
in the healthcare domain.

The purpose of this project was to analyze time that patients can spend waiting for service in
ZOWI clinic. The main objective for this research was to provide necessary Information to
policy makers aimed to contribute in wellbeing of population by reducing waiting time for
service because in excessive cases, long queues can delay appropriate decision for a specific
disease that can cause occurrence of death while patient still waiting for service. This project
examined, first the waiting time of patients in outpatient department by using queuing model
after calculating the mean number of arrivals per hour and the mean number of patients served
per hour. Further results from questionnaire from staff were analyzed in order to know their
opinions about the waiting time in outpatient department. The results showed that the system
utilization factor is greater than one. This means that the queue grew without bound. There were
a big number of patients waiting in the queue and they waited for a long time before being seen
by a physician. The correlation analysis revealed a significant negative correlation between days
and patient arrivals which means that there were many patients on Monday more than Friday.
The reasons given by staff interviewed were the big number of patients visiting this department
and the shortage of staff. To reduce the waiting time, we suggested increasing the number of
physicians and nurses in outpatient department and strengthening the capacity building of health
care providers. The clinic should develop a staffing plan and put more effort in the beginning of
the week for efficient use of available resources.
Table of Contents

1.1 Background...........................................................................................................................8

1.2 Problem Statement.....................................................................................................................9

1.3 Objectives................................................................................................................................10

1.3.1 General Objective.............................................................................................................10

1.3.2 Specific Objectives.......................................................................................................10
1.4 Research Questions..................................................................................................................11

1.5. Research Hypothesis...............................................................................................................11

1.6. Scope.......................................................................................................................................11

1.7 Limitations of the study...........................................................................................................12

2.1 Literature..................................................................................................................................13

2.2 Theoretical Review..................................................................................................................14

2.2.1 Queuing Theory................................................................................................................14

i) Essential components to describe a phenomenon of waiting line.......................................14
ii) Birth and Death Process Queuing Models.........................................................................15
iii) M/M/1 Queuing System...................................................................................................16
2.3 The M/M/1/K Queuing System.............................................................................................18

2.3.1 Probability of having no customers in M|M|1|N Queuing Model...............................18

2.4 Application of Queuing Theory in patient Departments.........................................................22

3.1 Research methodology.............................................................................................................24

3.2 Instruments..............................................................................................................................24

3.3 Data Collection Procedure.......................................................................................................25

4.1 Introduction..............................................................................................................................26

4.2 System performance measure of the zowi clinic................................................................31

 (I) The Mean Number of Arrivals per Hour...........................................................................32
(II) System Utilization Factor (ρ)...........................................................................................32
(III) Probability that there is no patient existing in the System..............................................32
(IV) Average number of patients in the waiting line (Lq).....................................................33
(VI) Average number of patients in system (L):....................................................................33
(VII) Average time patients spends waiting and being served (W).......................................33
(VIII) Average time patients spends waiting in the queue (Wq)...........................................33
4.3 Data Processing and Analysis..................................................................................................35

5.1 Summary..................................................................................................................................36

5.2 Conclusion...............................................................................................................................37

5.3 Recommendations....................................................................................................................37
 CHAPTER ONE

Introduction

1.1 Background
In order to respond to the demand of service on time and efficiently, many institutions use
queuing models. However the use of queuing models is not widespread in hospitals. Regarding
the efforts made by health facilities to prevent the harms that can be caused by delay of service,
queuing models can contribute to allocate efficiently the available resources in their institutions.

There are queues as long as the subjects that request service, known as customers, are not
immediately served when arrive at a service facility. In hospitals, patients are considered as
customers and different departments such as laboratory, diagnostic imaging, pharmacy or
outpatient department can be referred as service facilities. A service facility can have one or
more service stations where customers request service and each station can also have either one
or more servers. For example, in laboratory department, a patient who needs a certain test can be
requested to pass through two types of servers; cashier who receive money for the prescribed test
and a nurse who takes the requested sample test.

Queuing models have been applied in numerous industrial settings and service industries. The
number of applications in healthcare, however, is relatively small. This is probably due to a
number of unique healthcare related features that make queuing problems particularly difficult to
solve. In this section, we will review these features and where appropriate we will shortly discuss
the methodological impact. Before we dig into this issue, let’s first discuss two important
modeling issues in healthcare: the performance measures and the issue of pooled capacity.

The performance measures in healthcare systems focus on internal and external delays. The
internal delay refers to the sojourn time of patients inside the hospital before treatment. The
external delay refers to the phenomenon of waiting lists. Manufacturing systems may buffer with
finished goods inventory, service systems rely more on time buffers and capacity buffers.
Another important performance measure is related to the target occupancy (utilization) levels of
resources. Average occupancy targets are often preferred by government and other institutional
agents. Hereby, higher occupancy levels are preferred, but this results in longer delays. We are
often confronted with conflicting objectives. Instead of determining capacity needs based on
(target) occupancy levels, it is preferable to focus on delays. The key issue in delay has to do
with the tail probability of the waiting time. The tail probability refers to the probability that a
patient has to wait more than a specified time interval. Capacity needs (e.g. staffing) of an
emergency department should be based on an upper bound on the fraction of patients who
experience a delay of more than a specific time interval before receiving care from a physician .
The second modeling issue has to do with pooling. In general, pooling refers to the phenomenon
that available inventory or capacity is shared among various sources of demand (well known
examples are location pooling, commonality or Flexible capacity). Pooling is based on the
principle of aggregation and mostly comes down to the fact that we can handle uncertainty with
less inventory or capacity.

A common characteristic of the majority of queuing models is that customers are discrete, and
the number of customers waiting in the service facility is an integer value. Regarding the risks
behind the queues in clinic, the following questions can be asked:

(i) Why do queues form in the clinic?

(ii) Why must patients wait to be served?

(iii)Which characteristics of the clinical system affect queuing and by how much?

1.2 Problem Statement

The service facilities whose customers are patients vary generally in capacity and size, from
small patient clinics to large, urban hospitals to referral hospitals. The servers in hospital queuing
systems are the trained staff and equipment required for specific activities and procedures.

Almost all of us have waited for many hours, many days or many weeks to get an appointment
with a medical doctor and at arrival we are obliged to wait for a long time until being seen. In
Clinic, it is not strange to get patients waiting for radiologist for imaging diagnosis and delays
for surgery appointment. So in zowi clinic the main problem is this the waiting of the patients to
be served or to get the doctor. So lengthy queues are unfavorable for patients Because delay in
accessing needed services often cause prolonged pain and economic failure when patients are
not able to work and potential deterioration of their medical conditions that can augment
consequent treatment expenses and poor health outcomes. In excessive cases, long queues can
delay appropriate decision for a specific disease that can cause occurrence of death while patient
still wait for service. So this need to be addressed in order to reduce it we have to measure the
performance of the zowi clinic.
This project is based on the perceptive that most of these challenges can be managed by using
queuing model to determine the waiting line performance such as: average arrival rate of
patients, average service rate of patients, system utilization factor and the probability of a
specific number of patients in the system. The resulting performance variables can be used by the
policy makers to increase competence, improve the quality of patient care and reduce cost in
hospital institutions as well.

1.3 Objectives:-

1.3.1 General Objective

The main objective of this project was to apply a queuing model for healthcare services in zowi
clinic. The outcome of this study is better understanding of the queuing theory and monoqiment
service .In addition the health care providers can make decisions that increase the satisfaction of
all relevant group besides optimizing the resources. This study further established the queuing
theory.

Modeling is an effective tool that can be used to make decisions on staffing needs for optimal
performance with records queuing. Challenges in clinics, this study therefore can be replicated in
other clinics or other counties.

In order to inform clinic administrators more on the usefulness of the applications with queuing
theory and modeling as a tool for improved decision making.
1.3.2 Specific Objectives

This project had the following specific objectives:

 To determine the mean number of arrivals per hour (λ) in zowi clinic.
 To determine the mean number of patients served per hour (µ) in zowi clinic.
 To compare the mean number of arrivals and the mean number of patients served per
hour (λ and µ) in zowi clinic.
 To determine the average time a patient spends waiting in the queue before being seen b
a physician in zowi clinic.
 To analyze the waiting line of patients in zowi clinic.

1.4 Research Questions

 What is the mean number of arrivals per hour (λ)?

 What is the mean number of patients served per hour (µ) ?
 What is the relationship between the mean number of arrivals and the mean number of
patients served per hour (λ and µ)?
 What is the average time a patient spends waiting in the queue before being seen by a
physician?
 What resources needed to reduce the length of queues in clinic and increase patients’
satisfaction?

1.5. Research Hypothesis

 There is a long waiting time of patients before being seen by physicians in

outpatient department of zowi clinic.
 The system utilization factor is less than one.
 There is a significant negative correlation between days and patient arrivals in
patient department of zowi clinic.
1.6. Scope

Due to the financial and time constraints, this research has been conducted only to the patients
visiting zowi clinic in Outpatient Department (OPD) for consultation by a physician.

A period of 3 days has been covered in week from Monday to Friday were considered because
they are the working days of the week, from 08:00 A.M to 12:00 AM and from 01:00 P.M to
05:00 P.M. A questionnaire has been conducted to the nurses and physicians of the Outpatient
Department to collect their opinions about causes and proposed solutions of queues.

1.7 Limitations of the study

 It is difficult to distinguish the actual service time of the patient, as the patient may be
with the doctor but the doctor is attending phone calls (personal/official/not relating to
patient).
 The study is limited to M/M/1 single server queuing model.
 There may be difference in the process flow of the different activities from one hospital
to other, but, on the whole, network analysis can be applied to study the hospital process
flows.
 Scrap value or resale value cannot be exactly established for medical equipment. So
only estimated value of medical equipment is considered for calculating the optimum
replacement period.
 Approximate cost of medical equipment is considered for finding out the optimum
replacement period. The cost of the medical equipment depends on the features available
and the technology existing at that time.
Though LPP is applied in formulating optimal balanced diet problem, availability of
food items and the cost is subjected to seasonal variations
 Chapter -2

Literature Review

2.1 Literature

Waiting in lines seems to be a component of a human daily life. Queues form when the demand
for a service exceeds its supply. In clinic, patients can wait a certain period of time (minutes,
hours, days or months) to receive healthcare service. For many patients or customers, waiting in

lines or queuing is annoying or negative experience. The disagreeable experience of waiting in

line can often have a negative consequence on the rest of a customer’s experience with a firm.
The way in which managers address the waiting line issue is critical to the long term success of
their firms.

Literature on queuing models indicates that waiting in line or queue causes problem to economic
expenses to persons and institutions. Hospitals, banks, airline companies, industrialized firms
etc., attempt to decrease the total waiting price, and the cost of service provided to their
customers. Therefore, speed of service is increasingly becoming a very important competitive
parameter. Davis (2003) assert that providing ever-faster service, with the ultimate goal of
having zero customer waiting time, has recently received managerial attention for several
reasons. First, in the more highly developed countries, where standards of living are high, time
becomes more valuable as a commodity and consequently, customers are less willing to wait for
service. Second, this is a growing realization by organizations that the way they treat their
customers today significantly impact on whether or not they will remain loyal customers
tomorrow. Finally, advances in technology such as computers, internet etc., have provided firms
with the ability to provide faster services.

For these reasons hospital managers and health providers are always finding way to deliver more
rapidly services, believing that the waiting will negatively affect the organization performance
evaluation. Cochran and Bharti (2006) also argue that higher operational efficiency of the
hospital is likely to help to control the cost of medical services and consequently to provide more
affordable care and improve access to the public. Researchers have argued that service waits can
be controlled by two techniques: operations management or perceptions management (Hall,
2006).The operation management feature deals with the organization of how customers
(patients), queues and servers can be coordinated towards the goal of rendering efficient service
at the minimum cost. The act of waiting has significant impact on patients’ satisfaction. The
amount of time customers must spend waiting can significantly influence their satisfaction.
Additionally, research has demonstrated that customer satisfaction is affected not just by waiting
time but also by customer expectations or attribution of the causes for the waiting. Consequently,
one of the issues in queue management is not only the actual amount of time the customer has to
wait, but also the customer’s perceptions of that wait. Clearly, there are two approaches to
increasing customer satisfaction with regard to waiting time: through decreasing actual
waiting time, as well as through enhancing customer’s waiting experience.

2.2 Theoretical Review

2.2.1 Queuing Theory

Queuing theory is the mathematical study of waiting lines, or queues (Sundarapandian, 2009). In
Queuing theory a model is constructed so that queue lengths and waiting times can be predicted
(Sundarapandian, 2009). Queuing theory is generally considered as branch of operations research
because the results are often used when making business decisions about the resources needed to
provide a service. Queuing theory has its origins in research by Agner Krarup Erlang when he
created models to describe the Copenhagen telephone exchange (Sundarapandian,2009).The
ideas have since seen applications including telecommunication, traffic engineering, computing
and the design of factories, shops, offices and hospitals (Schlechter, 2009).

i) Essential components to describe a phenomenon of waiting line:-

The following components are essential to describe a phenomenon of waiting line: the
population source, the arrival, queues, queue discipline, service mechanism, departure or exit.
a. Population source:-

The population source serves as where arrivals are generated. Arrivals of patients at the hospital
may be drawn from either a finite or an infinite population. A finite population source refers to
the limited size of the customer pool. Alternatively, an infinite source is forever.

b. Queue discipline:-

The queue discipline is the sequence in which customers or patients are processed or served.

The most common discipline is first come, first served (FCFS). Other disciplines include last
come, first served (LCFS) and service in random order (SIRO). Customers may also be selected
from the queue based on some order of priority (Taha, 2005).

c. Service mechanism:-

The service mechanism describes how the customer is served. It includes the number of servers
and the duration of the service time-both of which may vary greatly and in a random fashion.
The number of lines and servers determines the choice of service facility structures. The
common service facility structures are: single-channel, single-phase; single-channel, multiphase;
multichannel, single phase and multi-channel, multiphase.

d. Departure or exit:-

The departure or exit occurs when a customer is served. The two possible exit scenarios as
mentioned by Davis (2003) are: (a) the customer may return to the source population
and immediately become a competing candidate for service again; (b) there may be a low
probability of re-service.

ii) Birth and Death Process Queuing Models:-

A number of important queuing theory models fit the birth-and-death process. A queuing system
based on the birth-and–death process is in state En at time t if the number of customers is then n,

that is, N(t)=n. A birth is a customer arrival, and a death occurs when a customer leaves the
system after completing service. Thus, given the birth rates {ʎn} and death rate {µn}, and
assuming that.
iii) M/M/1 Queuing System

M|M|1 Queuing model following Poisson distribution of arrival (λ) and service rate (μ) with
single server is presented in Figure 9.4. There is no balking and reneging in this model. The
numbers of customers in a queue as well as in the system are represented by Lq and Ls
respectively.

To analyze M|M|1 Queuing model we require information regarding the number of customers
currently present in the system, which is represented by the state of the system.

State of the system:-

 The state of queuing system is represented by a single number n, the number of

customers currently in the system.
 It utilizes memory-less property of exponential distribution. As per this property the time
since the last arrival and the time the current customer has been in the service process are
irrelevant to the future behavior of the system.
 Consider the system to be in steady state, which means that the system has been running
for so long that the current state doesn’t depend on the starting condition.
 By computing the long run probabilities of being in each state, we will determine the
performance measures of queuing models as long term steady state performance
measures hence, the customers arrive only one customer at a time. The system state can
change only by one unit at a time.

Transition from one state to another state in a queuing system:-

  If, currently there are n customers in the system, then the following changes can happen
in the system
 The state of the system increases from n to n +1, if arrival occurs in to the service system.
The rate of increase is represented by λ, the arrival rate.
 The state of the system decreases from n to n-1, if departure to the system
occurs. The rate of decrease is represented by µ, the service rate.

Figure Transition of state in a queuing system

Performance measures of M|M|1 queues

The memory less property is utilized to define the state of the queuing system. To determine the
performance measures, first we will find the probability of having n number of customers in the
queuing system. Probability of having 1 customer (i.e. n=1) in the service system is:
2.3 The M/M/1/K Queuing System:-

In real cases, queues never become infinite, but are limited due to space, time or service
operating policy. Such queuing model falls under the category of finite queues.

Examples:-

 Parking of vehicles in a supermarket is restricted to the space of the parking are

 Limited seating arrangement in a restaurant

Finite queue models restrict the number of customers allowed in service system. That means if
N represents the maximum number of customers allowed in the service system, then the (N+1)th
arrival will depart without being part of the service system or seeking service.

2.3.1 Probability of having no customers in M|M|1|N Queuing Model:-

The service system can accommodate N customers only. The (N+1)th customer will not join the
queue We know that, probability of having one customer in the queuing system is as mentioned
below.

Similarly, probability of having N customers in the service system will be

Probabilty of having no customer in finite single server

Using effective arrival rate and Little’s law we can determine other performance measures of M
Average number of customers in the system,

Ls can be determined using probability of having finite, N, customers in the service system.|M|1|
N queuing system, which are given below:-

We know that for a finite queue system, probability of having n customers with (n>0) can be
written as:-

P (n>0) = 1-P0

Hence, the number of customers in queue can determined as given below.

Lq=Ls – (1-P0)

Average waiting time in the queuing system, Ws, comes out to be as mentioned below.

Average waiting time in the queue, Wq, can be written as given below.
The M/M/c Queuing System:-

For this model we assume random (exponential) inter arrival and service times with C identical
Servers (Asmussen, 2003). This system can be modeled as a birth-and-death process with the
Coefficients

Multi- channel queuing theory treat the condition in which there are several service station in
parallel and each customer in the waiting line can be served by more than one station. Each
service facility is prepared to deliver the same type of service. The new arrival select one station
without any external pressure. When a waiting line is formed. A single line usually breaks down
into shorter lines in front of each service station. The arrival rate  and service rate  are mean
values from Poisson distribution and exponential distribution respectively. Service discipline is
first come, first served and customer are taken from single queue i.e., any empty channel is filled
by the next customer in line.

n= number of customer in the system

pn=probability of n customer in the system

C= number of parallel service channels(c>1)

= arrival rate of customer

= service rate of individual channel.

When n < c , there is no queue because all arrivals are being serviced, and the rate of servicing
will be n as only n channels are busy, each at the rate of . When n=c, all channels will be
working and when n>c, there will be (n-c) customer in the queue and rate of service will be c as
all the c channels are busy. There will be three cases in this system. To determine the properties
of multi-channels system, it is necessary to find an expression for the probability of n customer
in the system at time t i.e. pn(t).
2.4 Application of Queuing Theory in patient Departments

The health system’s ability to deliver safe, efficient and smooth services to the patients did not
receive much attention until mid-1990. Several key reimbursement changes, increasing critiques
and cost pressure on the system and increasing demand of quality and efficacy from highly
aware and educated patients due to advances in technology and telecommunications, have started
putting more pressure on the healthcare managers to respond to these concerns (Singh, 2011).
Queuing theory manages patient flow through the system. If patient flow is good, patients flow
like a river, meaning that each stage is completed with minimal delay, when the system is
broken, patients accumulate like a reservoir (Hall, 2006). Healthcare systems resemble any
complex queuing network in that delay can be reduced through:

(1) Synchronization of work among service stages

(2) Scheduling of resources (e.g. doctors and nurses) to match patterns of arrival and
Constant system monitoring (e.g. Treating number of patients waiting by location,
diagnostic grouping) linked to immediate actions (Hall, 1991). Recently, application of
stochastic methods has increased in analyzing clinical problems (Kandemir, 2007).
 CHAPTER 3

METHODOLOGY

3.1 Research methodology

In this study, the patients are coming from infinite population and the system was enough to
receive. All the patients coming in Outpatient consultation. There were also one consultation
rooms (1 server) to receive patients (customers). From above conditions the model to be used is
M /M/1: FCFS/∞/∞ where;

M=Markovian (or Poisson) arrivals and exponential service time.

FCFS = First come, first served:-

∞ = Infinite system limit;

∞ = Infinite source limit.

For the purpose of modeling, the arrivals (n) are the outpatients. As each reaches the hospital,
He/she books for service. If service is rendered immediately he/she leaves the hospital or
otherwise joins the queue. The doctors are the servers (c).

The arrival rate, service time and number of servers were the data used for the study that have
been collected using observation method. The data collection covered a period of 3 days in week
from Monday to Friday were considered because they are the working days of the week. By
using this data we can check the performance of the zowi clinic by calculating the give data.

3.2 Instruments

In our work, different documents have been used such as books, reports and electronic sources.
All these documents helped us to make the conceptual and theoretical framework of our work as
well as to analyze the data and interpret the results. Also, we have used a register to record
discrete time for patient arrival and service. For collection of staff opinions, a questionnaire has
been used.
3.3 Data Collection Procedure

In this project the observation technique has been used where we registered the time when every
patient enters in outpatient department and a time when he/she comes out from outpatient
department. This helped to draw a table used in estimating the average number of patients
entered in the system and average number of patients served in one hour. From this we have
estimated the remaining performance parameters of the system. These data have been collected
for a period of 3 days from Monday to Wednesday, from 08:00 A.M to 12:00 and from 01:00
P.M to 05:00 P.M. A questionnaire has been used to collect staff opinions about causes and
proposed solutions of queues.
 CHAPTER 4

DATA ANALYSIS AND DISCUSSION OF THE FINDING

4.1 Introduction

In this chapter, we have described how data analysis was done and the findings have
been presented. The main results are presented and finally the model is built that is used to
respond to the research questions of this study and address the main objectives. The general
objective of this project was to apply a queuing model for healthcare services in Muhima District
Hospital. In this study we have used a multiple queuing model M/M/1: FCFS/∞/∞ where we
have multiple servers represented by one physicians, infinite system limit and infinite source
limit of patients.

Friday May 25 /2019

Morning afternoon
Arrival time Arrival time
2:00 - 2:10 10 7:00 – 7:18 18
2:10 - 2:17 7 7:18 – 7:30 2
2:17 - 2:30 13 7:30 – 7:41 11
2:30 - 2:33 3 7:41 – 8:00 19
2:33 - 2:40 7 8:00 – 8:17 17
2:40 - 2:41 1 8:17 – 8:27 10
2:41 - 2:44 3 8:27 – 8:39 12
2:44 - 2:48 4 8:39 – 8:51 3
2:48 – 2:51 3 8:51 – 8:54 3
2:51 - 2:58 7 8:54 – 8:55 1
2:58 – 3:00 2 8:55 – 9:13 18
3:00 – 3:03 3 9:13 – 9;22 9
3:03 – 3:06 3 9:22 – 9:23 1
3:06 – 3:10 4 9:23 – 9:25 2
3:10 – 3:17 7 9:25 – 9:37 12
3:17 – 3:21 4 9:37 – 9:48 11
3;21 – 3:27 6 9:48 – 9:56 8
3:27 – 3:30 3 9:56 – 10:03 7
3:30 – 3:31 1 10:03 – 10:13 10
3:31 – 3:39 8 10:13 – 4:25 12
3:39 – 3:45 6 10:25 – 10:30 5
3:45 – 3:55 10 10:30 – 10:41 11
3:55 – 4:07 12 10:41 – 10:51 10
4:07 – 4:17 10 10:51 – 10:53 2
4:17 – 4:22 5 10:53 – 10:55 2
4;22 – 4:28 6
4:28 – 4:37 9
4:37 – 4:49 12
4:49 – 4:57 8
4:57 – 5:09 12
5:09 – 5:18 9
5:18 – 5:27 9
5:27 – 5:34 7
5:34 – 5:44 10
5:44 – 5:49 5
5:49 – 5:51 2
5:51 – 5:54 3
5:54 – 5:56 2
5:56 – 5:57 1

Monday may 28/2019

Morning Afternoon
Arrival time Arrival time
2:00 - 2:18 18 7:00 – 7:03 3
2:18 - 2:28 10 7:03 – 7:10 7
2:28 - 2:40 12 7:10 – 7:14 4
2:40 - 2:48 8 7:14 – 7:17 13
2:48 - 2:59 11 7:17 – 7:25 8
2:59 - 3:17 18 7:25 – 7:31 6
3:17 - 3:30 13 7:31 – 7:38 7
3:30 - 3:44 14 7:38 – 7:44 6
3:44 – 4:00 16 7:44 – 7:45 1
4:00 - 4:15 15 7:45 – 7:48 3
4:15 – 4:25 10 7:48– 7:52 4
4:25 – 4:40 15 7:52 – 8;00 8
4:40 – 4:55 15 8:00 – 8:13 13
4:55 – 5:10 5 8:13 – 8:23 10
5:10 – 5:20 10 8:23 – 8:28 5
5:20 – 5:32 12 8:28 – 8:39 11
5;32 – 5:42 10 8:39 – 8:47 8
5:42 – 5:50 8 8:47 – 9:00 13
5:50 – 6:00 10 9:00 – 9:07 7
9:07 – 9:20 13
9:20 – 9:31 11
9:31– 9:37 6
9:37 – 9:38 11
9:38 – 9:50 12
9:56 – 10:01 5
10:01 – 10:19 18
10:19 – 10:33 14
10:33 – 10:44 11

Thursday June 1/2019

Morning Afternoon
Arrival time Arrival time
2:00 - 2:19 19 7:00 – 7:09 9
2:19 - 2:27 8 7:09 – 7:17 8
2:27 - 2:29 2 7:17 – 7:22 5
2:29 - 2:39 10 7:22 – 7:33 11
2:39 - 2:41 2 7:33 – 7:41 8
2:41 - 2:44 3 7:41 – 7:47 6
2:44 - 2:57 13 7:47 – 8:05 18
2:57 - 3:00 3 8:05 – 8:19 14
3:00 – 3:13 13 8:19 – 8:26 7
3:13 - 3:19 6 8:26 – 8:37 11
3:19 – 3:24 5 8:37– 8:41 4
3:24 – 3:36 12 8:41 – 8;52 11
3:36 – 3:44 8 8:52 – 9:01 9
3:44 – 3:49 5 9:01 – 9:11 10
3:49– 3:56 7 9:11 – 9:19 9
3:56 – 4:02 6 9:19 – 9:27 8
4;02 – 4:16 14 9:27 – 8:31 4
4:16 – 4:17 1 9:31 – 9:44 13
4:17 – 4:22 5 9:44 – 9:51 7
4:22 – 4:31 9 9:51 – 10:00 9
4:31 – 4:40 9 9:20 – 9:31 11
4:40 – 4:48 8 10:00– 10:12 12
4:48 – 5:00 12 10:12 – 10:25 13
5:00 – 5:07 7 10:25 – 10:31 6
5:07 – 5:17 10 10:31 – 10:33 2
5;17 – 5:23 6
5:23 – 5:31 8
5:31 – 5:34 3
5:34 – 5:41 7
5:41 – 5:47 6

Friday May 25 /2019

Service time morning Service time after noon

2:20 - 2:24 7:10 - 7:13

2:28 – 2:31 7:18 - 7:22
2:31 - 2:35 7:23 - 7:28
2:40 - 2:45 7:34 - 7:40
2:45 - 2:47 7:42 - 7:46
2:47 - 2:50 7:48 - 7:51
2:58 – 3:02 8:06 – 8:10
3:02 – 3:07 8:20 – 8:23
3:14 – 3:20 8:27 – 8:34
3:20 – 3:24 8:38 – 8:42
3:25 – 3:28 8:42 – 8:46
3:37 - 3:41 8:53 – 8:56
3:45 - 3:49 9:02 – 9:07
3:50 - 3:55 9:12 – 9:15
3:57 – 4:00 9:20 – 9:27
4:03 – 4:07 9:28 – 9:33
4:17 – 4:21 9:33 – 9:37
4:26 – 4:30 9:45 – 9:48
4:32 – 4:39 9:51 – 9:55
4:41 – 4:46 10:01 – 10:06
4:49 – 4:52 10:13 – 10:17
5:01 – 5:06 10:26 – 10:30
5:08 – 5:10 10:32 – 10:37
5:18 – 5:22
5:24 – 5:30
5:32 – 5:36
5:36 – 5:40
5:42 – 5:44
5:48 – 5:51
5:53 – 5:59
Monday may 28/2019
 Service time Service time

2:13 – 2:18 7:18 – 7:23

2:18 – 2:21 7:31 – 7:34
2:30 – 2:34 7:42 – 7:45
2:34 – 2:39 8:01 – 8:07
2:41 – 2:45 8:18 – 8:23
2:45 – 2:48 8:28 – 8:35
2:48 – 2:53 8:40 – 8:43
2:53 – 2:56 8:52 – 8:57
2:56 – 3:00 8:57 – 8:59
3:00 – 3:04 8:59 – 9:04
3:04 – 3:08 9:14 – 9:18
3:08 – 3:12 9:23 – 9:26
3:12 – 3:16 9:26 – 9:29
3:16 – 3:19 9:29 – 9:33
3:19 – 3:24 9:38 – 9:40
3:24 – 3:28 9:49 – 9:54
3:28 – 3:32 9:57 – 10:00
3:32 – 3:38 10:04 – 10:08
3:38 – 3:41 10:14 – 10:17
3:41 – 3:45 10:26 – 10:30
3:46 – 3:49 10:31 – 10:34
3:56 – 4:00 10:42 – 10:47
4:08 – 4:12 10:52 – 10:56
4:18 – 4:23 10:56 - --------
4:23 – 4:26
4:29 – 4:31
4:38 – 4:44
4:50 – 4:55
4:55 – 4:58
4:58 – 5:01
5:10 – 5:14
5:19 – 5:23
5:28 – 5:31
5:35 – 5:39
5:45 – 5:51
5:51 – 5:58
5:58 – 6:02
6:02 – 6:06
6:06 – 6:11
6:11 – 6:15
6:15 - ------

service time morning

 Thursday June 1/2019 Service time after noon
2:20 – 2:28 7:05 – 7:07
2:29 – 2:39 7:11 – 7:15
2:43 – 2:49 7:16 – 7:18
2:52 – 2:55 7:19 – 7:24
3:02 – 3:07 7:27 – 7:30
3:20 – 3:29 7:32 – 7:34
3:31 – 3:35 7:39 – 7:44
3:48 – 3:58 7:45 – 7:46
4:04 – 4:08 7:49 – 7:52
4:19 – 4:24 7:54 – 7:58
4:28 – 4:34 8:02 – 8:08
4:43 – 4:48 8:15 – 8:20
4:57 – 5:01 8:25 – 8:29
5:13 – 5:19 8:29 – 8:30
5:22 – 5:25 8:41 – 8:47
5:34 – 5:36 8:50 – 8:55
5:44 – 5:47 9:00 – 9:02
5:53 – 5: 54 9:08 – 9:13
5:54 - ------ 9:24 – 9:28
9:32 – 9:37
9:40 – 9:42
9:42 – 9:48
9:53 – 9:58
10:02 – 10:05
10:22 – 10:29
10:30 – 10:36
10:37 – 10:39
10:48 – 10:54
10:54 - -------

4.2 System performance measure of the zowi clinic

(I) The Mean Number of Arrivals per Hour

Table 1. M/M/1 Queue
Parameters
Unit of time Hour

Arrival rate () 11.89  12 persons

Service rate () 14.27  14 persons

Minute
Results

Mean time between arrivals 8.64 min/person

Mean time per service 4.44 min/person

(II) System Utilization Factor (ρ)

This factor has been calculated to compare the mean number of arrivals and the mean number of
patients served per time period (λ and µ) which can give an idea on the system performance and
show that there is a probability that a queue can be formed or not. The system utilization factor
(ρ) has been calculated using the following formula.

❑ 11.89
= ❑ = 14.27 =0.84 84 % utilizes or busy time

(III) Probability that there is no patient existing in the System.

po=1- ❑ =1-0.84= 0.16 16 % idleness
❑

(IV) Average number of patients in the waiting line (Lq)

2 11.89 2 141.7
Lq= ¿¿ = 14.27(14.27−11.89) = 33.96 =
❑ 4.16 patient
(VI) Average number of patients in system (L):
11.89 11.89
L= ¿❑¿ = ( 14.27−11.89 ) = 6.93 = 4.99 patient

(VII) Average time patients spends waiting and being served (W)
1 1
W= ¿1¿ = (14.27−11.89) = 2.38 =0.42 hr = 25.2 min

(VIII) Average time patients spends waiting in the queue (Wq)

11.89 11.89
Wq= ¿❑¿ = 14.27(14.27−11.89) = 33.96 = 0.35 hr= 21 min

Performance measure

Utilization rate of server 84%

Average no of patients waiting in line 4 patient

(Lq)
Average no of patients in system (L) 5 patient

Average time waiting in line(Wq) 21 min

Average time in the system (W) 25 min

Probability of no customer in the system 16

(Po)

The mean number of arrivals has been calculated from the data collected during 3 days of field
Visit in zowi clinic in patient department.
 Chart Title of the three days of arrivals in range
9

0
2:00 - 2:30 2:30 - 3:00 3:00- 3:30 3:30 - 4:00 4:00 - 4:30 4:30 - 5:00 5:00 - 5:30 5:30 - 6:00

DAY 1 DAY 2 DAY 3

Chart Title of the three days on service

9

0
2:00 - 2:30 2:30 - 3:00 3:00 - 3:30 3:30 - 4:00 4:00 - 4:30 4:30 - 5:00 5:00 - 5:30 5:30 - 6:00

DAY 1 DAY 2 DAY 3

4.3 Data Processing and Analysis

For analysis of our data and interpretation of the results, different computer tools have been used
especially Microsoft word and SPSS. The data collected using observation technique has been
the recorded time in interval time and then imported in SPSS for analysis where descriptive
statistics and significance test have been carried out as well as estimation of different
performance parameters describing the behavior of the system.

The data from questionnaire has been directly entered in a designed SPSS sheet for cleaning and
analysis. The figures and tables were interpreted in scope predefined objectives in order to make
Data meaningful and come out with conclusions and recommendations. The system performance
parameters used in this study were defined as follows:

λ : - Arrival rate of patients per hour;

μ : - Service rate ( Length of stay) of patients per hour;

C:- Number of doctors (servers) working.

In this model, there are only one physicians

:- system utilization factor = λ/μ,

Lq: - Average number of patients in the queue

LQ= L - /
L: Average number of patients in the system = Lq + λ/μ,

Wq: Waiting time of patients in the queue = Lq/ λ

W: Waiting time of patients in the system = L/ λ

Pn = probability of n patients existing in the system.

 CHAPTER 5

SUMMARY, CONCLUSION AND RECOMMENDATIONS

5.1 Summary

The purpose of our research was to analyze the waiting line of patients and propose solutions
for needed resources to reduce the length of queues in zowi clinic and increase patients’
satisfaction. The reason for our research is to provide necessary information to policy makers
aimed to contribute in wellbeing of population by reducing waiting time for service
because in excessive cases, long queues can delay appropriate decision for a specific disease that
can cause occurrence of death while patient still wait for service.

The research is composed of five chapters, each of them dealing with different aspects of
queuing model for healthcare in public health facilities. Chapter One is introductory and deals
with the background of the research, statement of the problem, objectives, research questions,
justification and scope of the research. Chapter Two consists of the literature review of the
queuing model for healthcare. The chapter defines basic terminology used in the project and
explores the theoretical review of different types of queuing model of interest.

Chapter Three provides the methodological framework of our project. This chapter describes all
the methods and instruments used to collect the information we need as well as the procedure of
the analysis and interpretation of the information gathered. Chapter Four concentrates on
research findings and discussion on results found after analysis. This chapter shown that system
utilization factor is 0.84. The average number of patients waiting in the queue is 4 patients. Here
we have also seen that a patient can wait in the queue around 21 minutes before being seen by a
physician and 25 minutes in the system. Conclusions are drawn in Chapter Five. The main aim of
the research is to evaluate and improve the system performance for zowi clinic and propose
solutions on needed resources to improve the quality of service offered to the patients visiting
this hospital and it has been reached. We suggest that the number of physicians and nurses be
increased and a staffing plan to be developed in the clinic in order to manage an efficient shifting
which can provide more efforts at the beginning of the week.
5.2 Conclusion

In this research a single server channel model was developed for patient department at the public
health clinic with a focus on the patient waiting time having a treatment. We have determined
waiting time of patient flow which can be used to improve the operating performance and also
improving the services provided to the patients.

We have also determined the waiting, arrival time and service time of patients at the clinic. The

λ
result stated that the average waiting time of patient in queues is wq = ( =21min) and the
(μ−λ) λ
average service time is 4 minutes to be seen by the physician. The average service time is less
than the average waiting time in the queue so the patient spent most of their time in the queue.

5.3 Recommendations

Develop a staffing plan that put more effort in the beginning of the week the staffing plan
should be developed depending on the trend of patients during the working days. This means that
the Outpatient department should provide a big number of nurses and physicians in the beginning
of the week in order to respond to the big number of patients coming in these days. The
consultation in Outpatient department should also begin in the morning as early as possible so
that the patients who are coming in the morning can be served without spending a long
time on the queue.
To increase number of staff in patient department there is need to increase number of
physicians and nurses in Outpatient department in order to respond to the needs of a big number
of patients visiting this department. The increase of staff will also need to increase the number of
consultation rooms to provide a place to be used by additional staff. Finally we conclude to add
one physician in the outpatient department and we improve the performance of the clinic.
 REFERENCES

 Allen, A.O. (1978) Probability, Statistics and Queuing Theory: With Computer
Applications,( for chapter 3 overview of queuing theory)
 Academic Press, Florida.Asmussen, S. (2003). Applied probability and queues (2nd ed).
New York: Springer,( for chapter 3 overview of queuing theory).
 A View of Performance Analysis of UML Specifications
based on the 2002 Performance Profile Queuing Models Murray Woodside Carleton
University, Ottawa... Nov 2002
 Introduction to Probability Models Ninth Edition Sheldon M. Ross University of
California Berkeley, California
 Guy L. Curry · Richard M. Feldman Manufacturing Systems Modeling and Analysis
 Gupta, Diwakar, and Brian Denton. "Appointment scheduling in health care:
Challenges and opportunities." IIE transactions 40.9 (2008): 800-819.
 Cayirli, Tugba, and Emre Veral. " Outpatient Scheduling in Health Care: A Review of
Literature ." Production and Operations Management 12.4 (2003): 519.
 De Vuyst, Stijn, Herwig Bruneel, and Dieter Fiems. "Computationally efficient
evaluation of appointment schedules in health care." European Journal of Operational
Research 237.3 (2014): 1142-1154.
 Harper, Paul Robert, and H. M. Gamlin. "Reduced outpatient waiting times with
improved appointment scheduling: a simulation modelling approach." Or Spectrum 25.2
(2003): 207-222.
 Syi Su, Chung-Liang Shih. “Managing a mixed-registration-type appointment system in
outpatient clinics” International Journal of Medical Informatics (2003) 70, 31-40..
APPENDICES

Appendix I: Research Questionnaire

This questionnaire is used to collect information from physicians and nurses working in
outpatient department at Muhima district hospital.

N.B: The information that will be given is only for academic purpose and will be kept

confidentially.

1. Sex

a. Male

b. Female

2. What is your position?

a. Physician

b. Nurse

c. Other

(Specify)……………………………………………………………………………

3. How long do you think a patient can spend waiting before being seen by a physician?

(Select only one answer)

a. Less than 5 minutes

b. Between 5 minutes and 10 minutes

c. Greater than 10 minutes and 30 minutes

d. Greater than 30 minutes and 1 hour

e. More than 1 hour (Specify time in hours)………………….

4. Do you think that this time is convenient and comfortable for the patients?

a. Yes

b. No

5. If No, what are causes of the long waiting time in this department? (Many answer can be

selected)

a. Shortage of staff (physicians)

b. Shortage of staff (nurses)

c. Big number of patients

d. Delay of personnel in service delivery

e. Shortage of consultation rooms (Infrastructure issue).

f. Other(Specify)…………………………………………………………………

6. What solutions do you think can be proposed to reduce the waiting time of patients in this

department? (Many answers can be selected)

a. Increase number of physicians in outpatient department

b. Increase number of nurses in outpatient department

c. Increase number of outpatient consultation rooms

d. Capacity building of health care providers.

e. Reduce the number of steps where a patient has to pass before being seen by a

physician.

f. Other (Specify)
…………………………………………………………………………………………

…………………………………………………………………………………………

Thanks for your kind participation.