1

IMPROVING SYSTEM PERFORMANCE THROUGH

QUEUE MODIFICATIONS.

BY

NITAH ELEANOR AMBANG

PG/M.Sc/08/50332

A PROJECT SUBMITTED TO THE

DEPARTMENT OF STATISTICS,

UNIVERSITY OF NIGERIA,

NSUKKA

IN PARTIAL FULFILLMENT OF THE

REQUIREMENT FOR THE AWARD OF MASTER

OF SCIENCE (M.Sc) IN STATISTICS

SUPERVISOR: PROFESSOR P. I. UCHE

AUGUST 2012

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Improving system performance through queue modifications.

By

Nitah Eleanor A.

PG/M.Sc./08/50332

A Project submitted in partial fulfillment of the requirement for the

award of Master of Science (M.Sc) in Statistics.

Department of Statistics,

University of Nigeria,

Nsukka.

Supervisor: Prof P. I. Uche

August 2012

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Certification

Nitah, Eleanor Ambang, a postgraduate student of the Department of Statistics, University of

Nigeria, Nsukka with registration number PG/M.Sc./08/50332, has satisfactorily completed the

research requirements for the award of a Master of Science (M.Sc.) in Statistics. The work

embodied in this project is original and to the best of our knowledge has not been submitted in

part or in full for another degree in this or any other university.

………………………………………….. ………………………………………….

Prof. P.I Uche Mr W.I. E. Chukwu

(supervisor) (Head of Department)

………………………………………………….

(External Examiner)

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DEDICATION

This work is dedicated to God almighty and to my loving parents and siblings for

their complete care and support towards me.

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ACKNWOLEDGMENT

I want to state here that I would not have completed this work on my own. Many individuals

contributed in one way or another in seeing me through this work.

My greatest appreciation and thanks goes to my Lord and Savior, Jesus Christ for his Love, care,

guidance and protection. With a deep sense of appreciation, I also wish to acknowledge my

supervisor, Professor P.I. Uche for his wonderful encouragement, guidance and supervision of

this work. Without his fatherly supervision, this work would not have been completed. I also

wish to show my appreciation to all the staff and postgraduate students of the department of

Statistics.

My profound gratitude goes to my dear parents Mr and Mrs Nitah Samuel who stood by me and

helped me financially and otherwise throughout this period. My wonderful siblings Leonard,

Mildred, Vennessa and Stacey are not left out. To all my friends who have been there for me, Dr

Luc Ngongeh, Dr Fobella Dominic, Ogbonna Decency, Emma Ngongeh, Pharm Nancy Njilele,

etc I say a big thank you .

August 2012.

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ABSTRACT

Many establishments which make use of queues are under increasing pressure to improve on the

quality of services and decrease cost by becoming more efficient. Efficiently organizing the

delivery of services is one way to improve performance while decreasing cost. One way of

efficiently improving on these performances is by modifications of the mechanics of the system.

It is possible that a queue operating under the M/M/C (C>1) say, may be inefficient in the sense

that queues maybe unnecessarily long. So a queue may transit from M/M/C (C>1) to M/M/S

(S>C). Also a system of parallel queues may be collapsed into one with many servers. This work

examines the merit or otherwise of going from multiple queues with many, say C servers to a

single queue with C servers. The University of Nigeria Medical Centre was used as a case study.

It was found that a pooled queue with C servers (C>1) performs better than C single parallel

queues with a server each. Our results show that the optimal level of staffing required to reduce

cost for this particular system studied was found to be three servers. Also pooling is not always

beneficial for urgent customers in situations where we have urgent or critical and non urgent or

non critical customers.

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TABLE OF CONTENTS

Chapter one: Introduction

1.0 Background of the study 1

1.1 Measures of performance of a queuing systems 6

1.2 Statement of the problem 7

1.3 Objectives of the study 7

1.4 Significance of the study 8

1.5 Scope of the study 9

Chapter two: Literature Review

2.0 Overview of queuing theory 10

2.1 Waiting time 11

2.2 Pooling in queuing systems 15

Chapter three: Methodology

3.0 Theoretical framework 19

3.1 Operations of the General Outpatient Department of the University of

Nigeria Medical Center. 20

3.2 Queuing representation of the operations of the Outpatient Department

of the University of Nigeria Medical Centre. 21

3.3 Performance measures for the three parallel queues 24

3.4 Performance measures for the pooled queue with many servers 28

3.5 Determination of the optimum number of servers to minimize cost 29

Chapter four: Analysis

4.0 Application 32

4.1 Determination of service time distribution 33

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4.2 Evaluation of the queues 35

4.3 Results of the parallel and pooled queue 36

4.4 Further use of M/M/S 37

4.5 Input for the cost model 38

4.6 Results of the cost model 40

4.7 Determination of the conditions under which there would be no gain in pooling 41

Chapter five: Conclusions and Recommendations

5.0 Discussion 43

5.1 Conclusion 44

5.3 Recommendations 44

References 45

Appendix 48

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CHAPTER ONE

Introduction

1.0 Background of the study

A wide variety of establishments make use of queues to offer certain services to their

customers. These establishments range from large organizations such as banks, hospitals, post

offices, super markets etc, to smaller scenarios such as students queuing up to see a lecturer. The

different establishments vary in scope and complexity but they all consist of a set of activities

and procedures that require queuing, in which a customer must undergo in order to receive the

needed services. The resources (or servers) in these systems, (queuing system) are the trained

personnel and specialized equipment that these activities and procedures require. Often,

customers get to these servers to receive the needed services only to find that they are not been

attended to as soon as they get there due to one reason or the other. This causes the customers to

wait for the services for usually an unknown period of time. According to David (1985), waiting

is frustrating, demoralizing, agonizing, aggravating, annoying, time consuming and incredibly

expensive. The truth of this assertion cannot be denied: there should be just a few consumers of

services in modern society who have not felt, at one time or another, each of the emotions

identified above. What more, each of us who can recall such experiences can also attest to the

fact that the waiting line experience in a service facility significantly affects our overall

perceptions of the quality of service provided. Once we are being served, our transaction with the

service organization may be efficient, courteous and complete; but the bitter taste of how long it

took to get attention pollutes the overall judgment that we make about the quality of service.

According to John, (2010), if customers are attended to almost as soon as they join the queue,

queuing is minimized. If not, then customers could suffer considerable queuing delays. Olaniyi

(2004) warned that the danger of keeping customers in a queue is that their waiting time could

become a cost to them because the time wasted on the queue would have been judiciously

utilized elsewhere. Failing to take account of the customer‟s waiting time leads to an

exaggeration of productivity of the organization. On the other hand, Marek et al, (2009) asserted

that if so many service channels are employed the movement of customers would be very fast

and customer delay would be minimized but warned that with such a system, some servers might

be underutilized. If this happens, it means the organization would have incurred expenses in

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getting the servers which were not fully utilized hence wasted money which would have been

used to solve other pertinent issues. Hence there is need to think of a way of solving this problem

in such a way that would satisfy both parties (the customer and the organization providing the

service. This is where queuing comes into play.

Increasing criticisms, cost pressure and increasing demand of quality and efficiency from

highly aware and educated customers have started putting more pressure on the many

organizations urging them to improve on the quality of service they offer. The urge to study

queues is prompted by two obvious features. Owing to the ebb and flow of customers, there

would be some occasions when the service facility is not fully employed, i.e where there are

more servers and fewer customers such that the servers wait idly for a period of time and others

where it is under continuous pressure, with a long queue of customers waiting to be attended to.

Costs are involved when the service is under-employed (low productivity), and in the congested

period, loss of productive time for queuing members. According to Singh (2006), if the

organization decides to increase the level of service provided, cost of providing service would

increase, if it decides to limit the same, costs associated with waiting for service would increase.

So the manager has to balance the two costs and make a decision about the provision of optimum

level of service. Hence it is one of the tasks of queuing theory to try to see how these costs can

be reduced by modifications to the mechanics of the system.

Some organizations operate with multiple parallel queues each having a server while a

few others have a single pooled queue with many servers. Multiple queue and multiple server

queuing situations are frequently encountered in the telecommunications sector, customer service

environment and in the Outpatient Department of most hospitals. The single queue where

customers arrive at the service station in a single line could offer better performance in certain

conditions and the multiple queue could be better in other situations. However, no system fits

both the multiple and single queues. For many systems having multiple queues, it is possible to

reduce the number of queues (pooling them into a single queue) in order to achieve better

efficiency. A single queue with many servers could be more efficient compared to parallel

queues with the same number of servers. Pooling of several queues seems to be beneficial in the

sense that if these multiple queues are pooled together, none of the servers would likely be idle

waiting for customers to arrive at their queue when customers to be served are still waiting on

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other queues, hence resources are less likely to be wasted. It also eliminates the annoying habit

of jockeying from one queue to another in a bid to get a faster queue. Scientists like Rothkopt et

al (1987) argued that customers might not react kindly to this because the combined queues

actually look longer than the single parallel queues. Our minds have issues thinking of the fact

that say, four lines with four people waiting each is the same as one line with sixteen people

waiting. Some people also feel that with multiple queues, in a place like a grocery shop for

example, we can look at the quantity of products in the baskets of people ahead of you on the

queue, the perceived speed of the cashier, if there is a second employee at the register helping to

bag the groceries etc, in order to make smart choices on which queue to join thereby reducing

his/her waiting time.

According to Mandelbaum (1998), pooling is the replacement of several ingredients

(elements) by a single functional equivalent ingredient. He said the decision where to or not to

pool is a fundamental problem in the design and management of stochastic service systems. He is

of the opinion that the value of pooling can only be assessed by comparing the steady state

mean-sojourn time of the two systems being compared. He advised that care must be taken in

pooling because pooling can be helpful or disastrous and its effects (both good and bad) can be

devastating. According to him, pooling could take one or more of these forms, pooling of queues

(demand), pooling of tasks (processes), pooling of servers (resources). He concluded by saying

that pooling is most efficient in a system that has light traffic since a customer at the service

system typically enjoys a service rate that is the total capacity of the specialized system.

Rothkopf et al (1987) was of the opinion that the multiple queues is of importance in that it

makes the customers feel happier in the sense that the queues look shorter since there are many

queues with many servers but he had doubts if pooling of these queues would not yield

operational benefits. According to him, contrary to common calculation, there are reasons to

believe that combining queues, especially queues of people may at times be counterproductive.

He said the reasons include customer‟s reaction to long queues and increased costs etc. He insists

that if queues are to be combined, then the customers not just the managers should be educated

on the benefits of this scheme. Peter et al (2010), however advised that pooling is not always

beneficial for the waiting times of urgent customers and for the total number of first

consultations required to meet the waiting time performance target for urgent as well as regular

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customers. Van Dijk (2002), wondered if combining separate queues into a single queue would

always improve operational benefits since in a multiple queuing system, the servers may tend to

be idle for some queues while waiting for customers to arrive meanwhile a similar server could

be packed with lots of jobs to handle. He urged that research should be carried out in this area.

Both queuing systems would have their merits and demerits, but one of them could certainly

ensure a better performance than the other and that is what this study tries to find out. Waiting

lists and waiting line greatly determine the efficiency of a queuing system. Long queues or

waiting lists is something that is usually frowned at. Any arrangement that helps to reduce

waiting time would be of immense help.

There are basically four types of queuing systems and different combinations of the same

can be used for very complex networks.

Single channel- single phase system: Here there is a single queue of customers waiting

for service and only one phase of service is involved. An example of this type of queuing

is a flu vaccination camp where a nurse practitioner is the server who does all the work.

Single channel-multiple phase system: In this case there is still a single queue but the

service involves multiple phases. For example in some hospitals, patients arrive at the

registration counter, get the registration done and then wait in a queue to be seen by the

physician.

Multiple channel- single phase system: In this type of queuing system, customers form

multiple queues, waiting for the service which involves only one phase. Customers also

have the liberty to switch from one line to the other. An example is customers waiting in

a pharmacy store.

Multiple channel- multiple phase system: This type of system has numerous queues and a

complex network of multiple phases of services involved. This type of service is typically

seen in a hospital setting, Emergency Departments (ER), multi-specialty outpatient

clinics, etc. For example, in a hospital outpatient clinic, patients first form the queue for

registration, then they are triaged for assessment, then for diagnostics, review, treatment,

intervention or prescription and finally exit from the system or triage to different

provider.

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The second part of this study tries to strike a balance between the two costs involved. This is

because having many servers is likely to decrease the waiting time of customers but this is not

without acquiring a high cost on the system (the more servers there are, the more expensive the

resources), also, waiting time could become a cost to customers because the time wasted on the

queue would have been judiciously utilized elsewhere. Therefore, there is the need to strike a

balance between getting an appropriate number of servers while minimizing the cost of operation

on the system while also minimizing the cost of waiting. We proceed to obtain the optimum level

of service that would minimize cost as we vary the number of servers for the pooled queue, also

the condition under which pooling would no longer be effective is also given.

The data used for this study were obtained from the Outpatient Department of the

University of Nigeria Medical Centre. The Outpatient Department of most hospitals including

the University of Nigeria Medical centre carries a significant workload. This Medical centre runs

a multiple queuing system. It is possible to assess the performance of the Medical centre by

modifying the queuing system. According to Singh, (2006), the management of healthcare

facilities such as Outpatient clinics are very complex and demanding to manage. According to

Yeon et al, (2010) the Outpatient Departments in large hospitals carry a significant amount of

load in the healthcare delivery system for non-urgent customers. In the Outpatient Department of

most hospitals, including the University of Nigeria Medical Centre, several doctors work

independently with customers simultaneously. The customers form multiple-parallel queues,

each having a single server (the doctor). In this study, we consider three separate parallel queues

with each queue having its own server, we analyze the performance of each of these individual

parallel queues. After which these single queues are theoretically pooled into a single queue, i.e

the three streams of queues are merged into a single queue with three servers, the performance

of the pooled queue is also measured thereafter the two systems are compared to see if there is

any gain or loss in pooling. In the General Outpatient Department of the University of Nigeria

Medical centre, customers come, their files are sorted out from a file cabinet, they queue up to

see a doctor and a patient consults with one as soon as he is available. The doctor may refer the

patient to the laboratory or prescribe drugs or book an appointment for another time, or refer the

patient to another hospital, all this depends on the patient‟s condition. There is usually more than

one doctor available and customers go to consult on a first come first served (FCFS) basis.

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Recently, according to SERVICOM, (2011) the Federal government of Nigeria embarked

on a mission to reduce waiting times for customers attending the General Outpatient Department

of Federal owned hospitals. The University of Nigeria Teaching Hospital, Ituko-Ozalla in Enugu

State of Nigeria is one of such hospitals where this plan is already being embarked on.

SERVICOM in 2007 visited this hospital and reported that standards were not set for waiting

times at the GOPD and that delays have led to decreased customers satisfaction and adverse

clinical outcomes. The hospital in its service improvement plan is targeting reduction of waiting

time to only 45minutes. They intend to achieve this by ensuring that the clinic commences on

time, by carrying out a reorientation program on the staff of the department and the hospital is

also considering the employing more medical staff and reducing tallies to streamline

appointments.

Policies to reduce waiting times will yield clinical benefits beyond decreasing the length of time

customers spend in poor health, it would go further to the sorting of staffing problems, moreover,

short waiting lists would improve efficiency by eliminating periods of reduced activity. This

applies to all other organizations which make use of queuing system to offer services to their

clients.

1.1 Measures of performance of a queuing system

The success or failure of any model is determined by its performance. It also enables the

managers or policy makers to analyze the results, identify the needs or gaps and carryout

necessary interventions or modifications. The salient measures of performance which are usually

used in assessing the merits of pooling are:

Expected number of people waiting in the queue or system

Expected waiting time

Capacity utilization

Cost of given level of capacity

Probability that an arriving patient would have to wait.

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1.2 Statement of the problem

Relatively little technical talent and material resources have been devoted to improving the

overall performance of most queuing systems. We intend to examine if there is any gain or loss

in the pooling of several parallel queues together to form one single functional queue with C (c >

1) servers for an organization. It is worth mentioning here that most establishments have the

capacity to serve customers in a shorter time and than they actually do. At times the employees

who serve (servers) customers may be frequently inactive while they wait for customers to

arrive. Most big establishments have queuing systems that can be visualized as a complex

queuing network in which delays can be reduced through:

Synchronization of work among service stages e.g (coordination of tests, discharging of

customers

Scheduling of resources (doctors, nurses, clerks) to match arrival patterns.

Constant system monitoring like tracking number of customers waiting to reach a server.

Use of alternative systems like going from one server to another.

In Summary, this work is aimed at analyzing and comparing two queuing systems; a

series of parallel single queues and a system of these same parallel queues pooled together to

form one single queue. To determine if there is any gain or loss in moving from one of these

queuing system to the other. Also, to determine the optimal level of staffing (servers) needed

which would enable the system to obtain an optimum trade-off between the two types of costs

involved and making appropriate suggestion to the modifications of the mechanics of the system

based on the findings in order to improve on the overall operational benefits. The Outpatient

Department of the University of Nigeria Medical Centre is used as our case study.

1.3 Objectives of the study

1. Evaluate the performance of a system with n parallel queues.

2. Evaluate the performance when n queues are pooled into one queue with n servers.

3. Compare the performance of the pooled and the unpooled systems using relevant

performance measures.

4. Determine the number of servers that would balance the two conflicting costs involved

(cost of waiting and cost of providing service).

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5. Determine the condition(s) under which there will be no gain in pooling.

1.4 Significance of the study:

Overcrowding and long waiting lines in many organizations are a common occurrence. Any

arrangement that alters the system and reduces waiting time would go a long way to alleviate

customers‟ anxiety as they wait to get services. This work would provide an assessment of the

performance of pooled queues versus unspooled queues. The findings would be useful to systems

which make use of multiple queues in a bid to reduce their waiting times. e.g Hospitals, banks

etc. In addition to this, if this arrangement helps in minimizing cost to the establishment, it would

not have only improved on the customer‟s general satisfaction with the organization, but it would

have improved on the smooth running of the organization and increase system‟s revenue. Thus

results of this research would provide innovations that would assist organizations which make

use of multiple queues to improve on their services while minimizing costs.

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1.5 Scope of the study:

This work covers the operations of the Outpatient Department of the University of Nigeria

Medical Centre. Patients come in, get their files prepared for them and queue up to see a doctor.

There is usually more than one doctor available and patients queue up in multiple queues waiting

to go in and consult on a first come first served (FCFS) basis. The data was collected over a

period of three weeks and covers all the working days from Mondays through Fridays during a

period when the students were in session.

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CHAPTER TWO

Literature Review

2.0 Overview of queuing theory

According to Gupta and Hira (2007), queues or waiting lines or queuing theory was first

analyzed by a Danish Engineer, A.K Erlang in 1903, in the context of telephone facilities. He

started with the problem of the congestion of telephone traffic and later on extended to business

applications and waiting lines. This development was however not possible until World War II.

Queues are found everywhere in our daily lives, businesses of all types, schools, industries

hospitals, post offices, banks etc. Queuing theory is used extensively to analyze production and

service processes exhibiting random variability in market demand, arrival times and service

times. It has been used in a wide variety of application. Queuing methods have been used for the

problem of machine breakdowns and repairs There are a number of machines that breakdown

individually and at random times. The machines that break down form a waiting line for repairs

by maintenance personnel and it is required to find the optimum number of repair personnel

which makes the sum of the cost of repairmen and the cost of production loss from downtime, a

minimum. Also, queuing theory has been applied in analyzing the performance of computational

systems.

According to Johnson, (2007), much has been written about queuing theory and its powerful

applications. But only recently have professionals discovered the benefits of applying queuing

theory techniques in the reduction of waiting time to their organizations. He advised that queuing

theory has a lot of benefits if properly applied in the management of organization in areas of

waiting line and service capacities.

Emergency room arrivals: Singh (2006) put it that emergency room arrivals is one of the areas

where most of the research and applications of queuing theory has been done. Closure of several

Emergency departments in last few years and significant variation in patient arrival rates has led

to increased crowding and prolonged waiting time. In a recent study conducted by Alan (2004),

he found out that in 2001, 7.7% of the 36.6 million adults in the United States who sought care in

a hospital ED reported trouble in receiving emergency care, and that more than half of these

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people cited long waiting times as the cause. So waiting time in real systems has always been a

problem.

Outpatient clinics and outpatient surgeries: There have been studies on Outpatient clinics with

the most common objectives of these studies being the reduction of customers waiting time in

the system, improvement on customer service, better resource utilization and reduction of

operating costs. Gordon (1970) agreed that analysis of such cases should involve an in depth

analysis of the patient arrival and flow, structure of the system, manpower characteristics and the

scheduling of the system. Appropriate queuing models should then be applied for process

modification, appropriate staffing, scheduling or facility changes.

2.1 Waiting time: This is basically the period of time which one must wait in order for a specific

action to occur, after which that action is requested or mandated.

Potential reasons for a long waiting time at any service points are

High Workload: if the servers are overworked, then customers have to wait

longer as the servers have too many customers to attend to. This problem can be

rectified by reducing service time or by shifting staff from areas where there is

low workload for staff.

Patients arriving in a big batch: if many customer arrive in a big batch: if many of

these customers arrive at the same time then most of these customers as the

servers would be busy attending to the people who were the first in the batch

while the rest would be waiting. This problem can be solved by encouraging

customers to come at less busy times. If its in a hospital, the patients should be

encouraged to make appointments for quieter times and quieter days of the week.

Lack of efficiency: Customers are not effectively attended to while the servers or

those providing servers are busy with something else such administrative work or

preparing to start work. This means that servers are not prioritizing attending to

customers. This problem can be reduced by making attending to customers the

number one priority.

A mismatch: a mismatch occurs when patients arrive to be seen but staff are not at

the service points. This typically happens before the opening time of the service

point when customers arrive before those providing the services. However it

could occur at anytime if staff are away from their service point due to outreach

activities, meetings, breaks etc. This problem can be solved by encouraging

patients to arrive later in the day. Meetings could be held at quieter times and

breaks could be taken at quieter times whenever possible.

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A logistical problem: customers could be waiting and the staff/ servers are

available but due to lack of equipments, waiting rooms or logical needs, staff are

unable to attend to customers. This can be solved by always making sure that all

the equipments or resources needed are available.

Queuing problem: This occurs when customers are attended to by servers in an

illogical manner i.e the customers are not attended to in the order in which they

arrive. This means that those who arrive first are not necessarily seen first but are

made to wait. Illogical queuing does not usually affect the median waiting time

although it had a large effect on individual customer waiting time. This problem

can be solved by maintaining a proper queue order using queue numbers or queue

seating arrangements.

Scholars like Fomundam and Jeffery (2007), studied waiting time in organizations agree that

long waiting lists are not desirable and for healthcare it generates stress and dissatisfaction for

customers, increases the cost of seeking medical care and can even constitute a barrier to

healthcare services. Many scholars are of the opinion that prioritizing healthcare services would

go a long way to reduce waiting time, some of their works are here by explained.

McQuirre (1983) shows that it is possible, when utilization is high, to reduce waiting times by

giving priority to clients who require shorter service time. This rule is in the form of a shortest

processing time rule that is known to minimize waiting times. He was of the opinion that life-

saving treatments should not be allocated using first-come first-served queues. Instead,

customers should be placed in prioritization categories and

Francisco and Daniel (2004) carried out a study on locating emergency services with priority

rule. They insisted on the fact that calls that involve danger to human life deserve higher priority

over calls for more routing accidents. They were so much against the idea of placing emergency

calls in the same category of importance. He put forward a model that allows prioritizing the

calls for service. The problem which he called priority queuing covering location problem

defines separate allocations for the different priorities with different time constraints imposed

for different priorities. According to him, the average waiting time for priority class K can be

divided into three components; Wo is the expected remaining time in service for the user who

occupies the server at the time when the new user (of priority class k) arrives at the queuing

system, iL is the expected number of users in priority class i who are already waiting in queue at

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the instant when the new user arrives and iM is the expected number of users in priority class i

who will arrive while the newly arrived user waits and iS is the service time of customers of

priority i. With the expected waiting time of customers of priority k given as:

1

0 1 1

k kk i i i iW W S L S Mi q

They were also of the opinion that optimized location of all facilities and allocation of customers

to those facilities is a vital factor in improving on-time performance in addition to that,

appointment systems should be put in place to reduce waiting time without greatly increasing

server idleness.

One resonating theme proposed by David and Benjamin (2005) is that doctors should be

effectively involved in prioritizing for the prioritizing scheme to be effective. However, he said

the major problem was the objection to doctors being involved in priority settings or in

enforcing prioritization.

Qi-Ming, et al (2009) proposed that in an emergency department, customers should be classified

into critical and non-critical groups. Those in the critical group should be given a higher priority

as they arrive if there is a doctor while those in the non-critical group should wait on a queue to

get medical attention. The condition of the patient in the non-critical group may deteriorate while

waiting and become critical, then the patient has to be attended to as soon as a doctor is

available. In a fire department, cases should also categorized, the dispatch of fire trucks and

ambulances is arranged accordingly. Allocation of resources in such systems is a key issue,

especially the allocation of limited resources.

David (2006) was not of this school of thought is, he argued against the need for prioritization in

healthcare. According to him other methods that would create a more efficient healthcare system

could be employed. He advised that policies that would remove beaurocratic inefficiencies,

streamline staffing and cut the high wages of most medical professionals should be employed

and when this is done, a greater level of duty consciousness would be seen in the hospital staff

and patients waiting time would be significantly reduced.

Frank et al (1997) carried out a study on prioritizing cardiac surgery waiting list, he wanted to

find out how clinical and demographic factors affect prioritization of waiting list of cardiac

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customers and came out with the suggestion that it was generally inappropriate to take account

of demographic or lifestyle factors when making prioritization decisions for cardiac surgery and

concluded that one of the hallmarks of a valid clinical guidelines is the involvement of all key

stakeholders including customers. The interest in waiting time, queue length and inter-arrival

time of priority queue stems from the fact that priority queues occur frequently in hospital

settings, telecommunications, computer networks and elsewhere. According to him, one of the

reasons for allocation of resources is to ensure that the system is stable in the sense that the

queue length does not grow too long. In cases where customers simply queue up for treatment,

measurement of the waiting time-outcome relationship is straightforward. Life-saving

treatments, however, are rarely allocated using first-come first-served queues. Instead, customers

are placed into prioritization categories , and customers in urgent need of treatment are advanced

to the top of the list. Situations occur where a given service facility is made use of by k(>1)

classes of units or customers which are distinguished according to some “measure of

importance”. We associate with each priority index i (1<i<k) where 1 denotes the highest

“measure of importance” and k the lowest. The discipline according to which the sever gets the

next unit to serve is termed “priority discipline”. Any priority discipline must, therefore specify

the rules for making the following decisions

Which unit to select for service once the server is free

Whether to allow for preemptive or non preemptive service.

When the decision to select the next unit for service depends only upon the priority class, a unit

of the ith class if present is always taken. From the above literature, we can see what researchers

have done to minimize the waiting time in queues. this study will consider how pooling of

queues can be of use in assessing performance through the waiting time.

23

2.2 Pooling in queuing systems

Peter et al (2009), in his paper „efficiency evaluation for pooling resources in healthcare‟ shows

that a pooled clinic that serves all customer types may achieve shorter waiting times than a

number of unpooled clinic focuses on a more limited range of customer types. According to him,

hospitals are struggling with the question of whether to become a more centralized to achieve

economies of scale or more decentralized to achieve economies of focus. He used quantitative

queuing theory and simulation models to examine where service and patient group characteristics

to examine where decentralization and centralization would be properly applied, He concluded

that generally, a pooled system reduces the length of time a patient waits for a doctor and

improves on the efficiency of the system generally.

Mandelbaum (1998) carried out a research on pooling in queuing networks. According to him,

pooling can be done in either of two ways: there could be a complete pooling of queues into a

single queue and servers into a single server, giving rise to an M/PH/1 queue where the server is

flexible in the sense that it processes all tasks. According to him, the benefits of the complete

pooling can be assessed by comparing the mean sojourn time of the system before pooling and

after pooling. An alternative pooling scenario is that which involves the complete pooling of

only queues which results in an M/PH/S system. He quantified the effects of pooling in terms of

an efficiency index and showed that pooling always helps in light traffic but in a heavy system,

the effects of pooling can go either ways.

Joustra et al (2007) carried out a research on pooling of queues. According to him, pooling two

or more separate queues is generally perceived to be more efficient. He asserts that when two or

more queues of the same type of service are pooled together, none of the servers can be idle

while a customer is still waiting.

This reasoning however relies upon the assumption of two identical servers or two identical

service characteristics. According to him, pooling two servers is not beneficial to all customers

(e.g urgent and regular customers). For 2 sufficiently large and 1 sufficiently small, it is

advised not to pool. Where 2 is the traffic intensity or utilization rate for urgent customers and

1 is the utilization rate for non urgent patients. Where

.

24

Wikipedia (2010) proved with an example that a pooled single queues offer a shorter waiting

time compared to multiple queues. He showed this using the following example, he considered a

system having 8 input lines, single queues and 8 servers. The output line has a capacity of

64kbits/sec. Considering the arrival rate at each input as 2 packets/sec, the total arrival rate is 16

packets/sec. With an average of 2000bits per packet, the service rate becomes 64kbits/s/2000b

which gives 32 packets/s. Hence the average response time of the system is

1 10.0625sec

32 16onds

.Now considering a second system with 8 parallel queues,

one for each server. Each of the 8 output lines has a capacity of 8kbits/s. the calculation yields

the response time as 1

=

10.5sec

4 2onds

. The average waiting time in the queue for

the first case (pooled case) is a lot shorter than that for the parallel case.

Ward (1992) advised that it is important to determine the appropriate number of server to use

including the server utilization (the proportion of time each server should be working).

According to him, in a multi-server queue system with unlimited waiting space, the level of

server utilization typically increases as the number of servers (and the arrival rate) increases. He

also showed that utilization increased variability in the arrival and service processes tends to

reduce utilization with a given grade of service.

i.e 1s where is a constant giving a rough indication of the grade of service. More

specifically, he suggested that if the number of servers is increased from s1 to s2, then the

utilization would increase from 1 to 2

where 1 11s for i = 1,2

Fomundam and Jeffrey (2007) surveyed the contributions and applications of queuing theory in

the field of healthcare. They summarized a range of queuing theory results in areas of waiting

time, system design, utilization analysis and appointment systems. They provided sufficient

information to analysts who wish to model healthcare processes using queuing theory.

Worthington (1991) suggests that increasing service capacity (the traditional method of

attempting to reduce long queues) has little effect on queue length because as soon as customers

realize that waiting time would reduce, the arrival rate increases which increases the queue again.

25

Green, et al (2006) carried out a study on the patient arrival rates in an emergency department

with a view to adjusting the staffing patterns to optimize the timely care of customers using

queuing models. She used the lag SIPP queuing analysis and discovered that the arrival pattern

of customers during weekdays varied significantly with that of the weekends. She decided to use

queuing theory to develop two different schedules, one for weekdays and the other for weekends.

Richard (1999) studied the consequences of long waiting lists. He said that long waiting lists are

not part of the solution to the crises in health care but they are part of the problem and exist

because too few resources have been directed towards quality control and resource management.

Marek (2009) did a comparative study on agent based models (ABM) and queuing models to

investigate patient access and patient flow through emergency departments. The models were

developed independently, with a view to compare their suitability to emergency department

simulation. The models were developed independently with a view to compare their suitability to

emergency Department simulation. He discovered that both models are oriented to augmenting

simulation with empirical data when available.

Gordon (1970) studied the arrival process in hospitals. He studied the hourly variation in

weekday arrival rates and used Time series analysis to come out with a models describing the

process with a considerable high degree of predictivenes. He tested this model by comparing

predictive values with data from the follow-up survey.

Robert and Rassul (2007) carried out a comparative study of parallel and sequential priority

queues algorithms. They used the classic hold, the Markov model and an Up/Down access

pattern to measure performance and look at both the average access time and the worst-case time

that are of vital interest to real time applications. Their results suggest that the best choice for

priority queue algorithm depends heavily on the application. For queue sizes smaller than 1000

elements, the splay tree, the skew heap and the Henriksen‟s algorithm show good average access

times. For large queue sizes of 5000 elements or more, the calendar queue offers a good average

access times but have very long worst-case access times.

Singh (2006) studied the use of queuing models in healthcare. He came up with the fact that

queuing models are basically used to achieve a balance between capacity and service delays

which all aim at minimizing costs (both tangible and intangible) to the organization and

individual.

Johanna (2007) carried out a research on the effect of waiting time on health outcomes and

service utilization. He did a randomized study on customers admitted to hospital for hip or knee

Replacement. He came up with the conclusion that people who wait longer than necessary end

up spending more simply because they use the services for longer period.

Ward (1992) In his paper “ Understanding the efficiency of multi-server service systems said

server utilization increases as the number of servers (and the arrival rate) increases. He also

26

showed how increasing variability in the arrival and service processes tends to reduce server

utilization with a given grade of service.

Johnson (2007) used the classic hold, the Markov model and an Up/Down access pattern to

measure performance, he also looked at both the average access time and the worst-case time

that are of vital interest to real time applications. Their results suggest that the best choice for

priority queue algorithm depends heavily on the application. For queue sizes smaller than 1000

elements, the splay tree, the skew heap and the Henriksen‟s algorithm show good average access

times. For large queue sizes of 5000 elements or more, the calendar queue offers good average

access times but have very long worst-case access times.

Green et al (2006) carried out a research using queuing theory to increase the effectiveness of

emergency department provider staffing. He used the lag SIPP queuing analysis to gain insights

on how to change mechanics of the system so as to provide appropriate staffing patterns to

reduce the fraction of customers who leave without being seen by the doctor. Their conclusion

was that queuing models can be extremely useful in most effective allocation of staff.

John (2010) studied queuing theory and patient satisfaction in an Anti-natal Care unit. He

evaluated the effectiveness of a queuing model in identifying the ante-natal queuing system

efficiency parameters. He used TORA optimization system to analyze the data and he discovered

that there are less average pregnant women in the queue and system in the first week than in the

other three weeks.

Banks et al (2004) advised that the concept of simulation modeling should not been overlooked.

He went further to add that discrete even simulation models should be carried out to investigate

various scenarios and their impacts on customers outcome and quality of care. In recent years,

due to the dramatic increase in healthcare cost, and development of use-friendly and more

functional simulation software, more and more simulation studies have been applied to

healthcare management as an effective decision-making tool. Typically, such studies include

patient flow analysis, capacity sizing and staff planning.

Jared et al (2009) reported that using these models, one can have a better understanding of the

relationship between various variables in healthcare systems so that costs would be minimized

and the quality of service would be improved upon.

27

CHAPTER THREE

Methodology

3.0 Theoretical framework

As mentioned earlier, different organizations make use of different queuing methods, it could be

the single queue with many servers or multiple parallel queues with a corresponding number of

servers. These multiple queues could have the same arrival and service rates or different service

or arrival rates. Each of these queuing methods have their queuing distributions. As the

mechanics of a particular queuing system differs from another so are the methods of calculating

their performance measures.

According to singh (2006), the goal of queuing analysis and its application in healthcare

organization is to “minimize costs” both tangible and intangible. This can be done by using

queuing models to determine the waiting line performance such as; average arrival rate of

customers, average service rate of customers, system utilization factor, cost of service, the

probability of finding a specific number of people in the system and also, the probability of not

finding any patient in the queue . Hence queuing theory basically tries to see how these costs can

be reduced by modifications to the mechanics of the system. He added that the management of

healthcare services such as outpatient clinic is very complex and demanding to manage.

Different policies to address long waiting times have been tried, some of them being more

efficient than others. However, according to Green et al, (2001) it is of great importance to the

management of waiting lists and actions against long waiting times to have reliable and valid

information on waiting time.

28

3.1 Operations at the General Outpatient Department of the University of Nigeria Medical

Centre.

The university of Nigeria, Nsukka Medical centre is the only medical facility located on the

University of Nigeria, Nsukka Campus. They have an approximately 14000 patient visits

annually (Records Department, University of Nigeria Medical Centre, 2011). The general

Outpatient Department of this facility is where new customers are diagnosed, referred or treated

of their diseases, the Outpatient Department carries a significant load in the healthcare delivery

system. It is generally an entry point into a healthcare system for non-urgent customers and a

major channel for delivering cure. In this study, we would get the service time and arrival rates

of customers in an n parallel queuing system with single servers, pool this together into a single

queue with multiple parallel servers and compare performance measures of n parallel queues and

the pooled queue to see if there is any loss or gain in the performance measures. Also, an optimal

level of staffing which would minimize the costs involved would also be obtained. For the

purpose of this study, it is assumed that all the customers have completed the necessary

registration with the hospital and are ready for the services they came for. Majority of the

customers arrived between the hours of 9am-2pm hence, we focus our analysis within this

interval. Luigi (2004) defined the waiting time as the period of time from when a patient is first

referred to a general practitioner or doctor for assessment till when the patient starts consultation

with the doctor and the service time as the length of time a patient actually spends in consultation

with the doctor. Measurements of time spent from registration until consultation by a doctor

were made using a stopwatch and this when done when the university was fully in session. The

service time, inter-arrival time and number of servers were the data used for the study which was

collected by direct observation of the subjects involved, with this method, this analysis system

was carried out without interfering with the activities of the system. This research instrument

was adopted so that the queuing system can be examined naturally.

In the general Outpatient Department of the University Medical Centre, the customers

come in, get their files from the file cabinet and register with the nurses after which they are

scheduled to see a doctor. As the patient reaches the service point, if service is rendered

immediately, he/she leaves the queue, if not, the patient(s) waits in the queue until service is

rendered. During the consultation with the doctor, he decides whether the patient needs to be

sent to the laboratory for some tests or drugs should be prescribed for the patient. If the patient is

29

sent to the laboratory, he/she has to consult with the same doctor for a second time but on

different dates so on the second day, he comes in as a new case and queues up just as others do.

The doctor may also decide to refer the patient to another hospital or hospitalize the patient

depending on the severity of the illness.

3.2 Queuing representation of the operations of the Outpatient Department of the

University Medical Centre.

Components of a basic queuing system.

Fig 3.1. A multiple server queuing system, M/M/C

Fig 3.20. n parallel queues, with a single server each

INPUTS

Service

facility

OUTPUTS

server

Pooled

queues

Parallel

queues

Single pooled

queue

server

Parallel queues

30

The service point has multiple servers but access to the servers is such that each server has

dedicated homogenous queue.

Assumptions of M/M/S:

Inter arrival and service times are independent and identically distributed

The system is in a steady state

Infinite queue capacity or buffer

Customers are served on a first come first served (FCFS) basis.

Model description: The Poisson process and exponential processes are used to model the arrival

rates. This is due to the following reasons:

There is a certain amount of regularity in the arrivals of customers in the Outpatient

Department. Although individual arrivals are impossible to predict, there is perhaps some

statistical regularity in the sense that when we observe customers arrivals during a period

of one month say, without knowing the absolute time frame, then we have no way to

decide whether we observe the time period of January, February or June. In probability

terms, the process is stationary in time. In other words, the course of time should not

change the probability properties of the process.

The fact that there is an occurrence at the particular time, says nothing about the

probability of an occurrence at or around a later or earlier time. In other words, there

seems to be some kind of independence with respect to various occurrences.

The next occurrence cannot be predicted from past or current information. In other

words, the process of occurrences seems to have no memory. The fact that something

happened in the past has no effect on the probabilities for future occurrences.

There is no accumulation of occurrences at anytime. That is, in each finite time interval,

there are only finitely many occurrences.

The above characteristics pave the way to the approximate model and we‟ve seen how. Doing

computations in a given model is one thing, but to make an appropriate model is of the highest

importance. We consider the distribution of waiting times (exponential distribution) the first

31

point above suggests that the waiting time distribution should be the same at all times i.e the

waiting time between the 2nd

and 3rd

events should have the same distribution as the waiting time

between the 7th

and 8th

event. The 3rd

point above suggest that the process should have no

memory, and the exponential distribution has no memory i.e /p X s t X t p X s

.All these make the exponential distribution a serious candidate for the waiting time between

successive occurrences.

The service time of customers would be tested to see if it follows the exponential distributions.

The poisson distribution provides a realistic model for many random phenomena. A poisson

distribution is a discrete probability distribution which predicts the number of arrivals in a given

time. It involves the probability of occurrence of an arrival. In healthcare this could be arrival

rate to the emergency department or arrivals at the Outpatient Department. According to Gupta

and Hira (1979), the mean arrival rate (i.e number of arrivals per unit of time) λ is assumed to be

constant over time and is independent of the number of units already serviced, queue length or

any other random property of the queue. This distribution is characterized by the mean and

variance being equal.

Probability of n arrivals in time t = 𝑒−λt (λt)n

𝑛 !, n= 0,1,2,…..

Ivo and Jacques (2002) agreed that the exponential distribution allows for a very simple

description of the state of the system at any point in time t. It is an appropriate model for time

between events when a process exhibits a Poisson distribution for the occurrence of that event.

The exponential distribution is characterized by mean and standard deviation being equal and the

model for this distribution is given as:

f(t) = λ te , t> 0

P{T≤ t}= 0

t

te dt

= 1- 𝑒−λT.

32

Queuing models are usually constructed to represent the steady state of a queuing system. That is

the typical long run or average state of the system. As a consequence, these are stochastic models

that represent the probability that a queuing system will be found in a particular state.

This mean and standard deviation of the exponential distribution is 1/arrival rate i.e 1/ λ. The

model assumes a single queue with an unlimited waiting room that feeds into S identical servers.

Arrivals occur according to a time homogenous Poisson process with a constant service which

has an exponential distribution. These two properties are referred to Markovian hence the use of

the two M‟s in the notation used for the model.

The arrival and service rates for 3 parallel queues with three servers were obtained, these were

later pooled to form a single queue having a different arrival rate but the service rate of the

servers remain the same. The utilization rate and other performance measures of the system

were calculated using a standard queuing formulae and software. In the pooled system,

customers form a single queue and are served by any of the available servers in a first come first

served manner.

It is worth mentioning here that the key word in queuing models is “average”, it takes the

average of random number of customers arriving randomly, of the service time, arriving intervals

etc. It is important to note that the model‟s delay predictions pertain only to waiting time due to

provider unavailability and do not include any other possible delays before seeing a doctor such

as registration, etc.

Some of the notations that would be used in this work are enumerated below. Queuing models

are generally constructed to represent the steady state of a queuing system, that is the typical

long run or average state of the system. Basically, the arrival rate, service rate for the different

number of servers was the data used for the study and this data is collected by observation with

the aid of a stopwatch.

3.3 Performance Measures: The success or failure of any model is determined by its

performance measure. It enables one to analyze the results, identify the gaps and carryout

necessary modifications. Analysis of the relevant queuing models allows the cause of queuing

problems to be identified and the proposed changes to be effected.

33

The probabilities of queue length, Pn are determined by using the transition-state diagram below.

The queuing system is in state n when the number of customers in the system is n. The

probability of more than one event occurring in a small time interval h, tends to zero as h tends

to zero.

This means that for n> o, state n can change only to two possible states, to state n-1 when a

departure occurs at the rate un and n+1 when an arrival occurs at the rate λn. State 0 can only

change to state 1 when an arrival occurs at the rate λ0. No departures can occur when the system

is empty hence state µo is not defined. state n can be changed to states n-1 and n+1 only, so we

have :

Movements into state n = λn-1pn-1 + un+1 pn+1 similarly,

Movements out of state n= (λn + un) pn

Under steady –state conditions, the expected rates of flow into and out of the state n must be

equal. i.e (λn + un) pn = λn-1pn-1 + un+1 pn+1

Under the assumptions of the M/M/1 the queue, the queue length is given by

1 2....... 0

1........ 1opn nPn

n n

(3.10)

where Po is obtained from the knowledge that 0

1n

n

p

hence

Having known the expression for the value of Pn, we proceed to get the various characteristics of

the system.

µn+1

µn

µ2

µ1

λn λn-1 λ1 Λ0

... n-1

n+1

1 0

n

n+1

34

3.3 Performance measures for the 3 parallel queues.

Gupta and Hira (1979), give the following measures of performance.

Each of the three queues has an arrival rate, i and service rate, i where {i =1, 2, 3)

0iL nPs n

n

10

i i

ii i

n

L nsn

10

i

i ii

ii i

n

L nsn

2 3

0 2 3 ...1 i i i i

ii i i i

Ls

2

1

1

i

i i

ii

i

i

Ls

i

ii i

Ls u

(3.11)

The expected number of units in the queue, iqL

iqL = expected number of units in the system – expected number in service.

i

i ii

L Lq s

.i i

ii i i

Lq

(3.12)

The expected time a unit spends in the system, iSW

35

iSW = 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑢𝑛𝑖𝑡𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑦𝑠𝑡𝑒𝑚 ,

𝑎𝑟𝑟𝑖𝑣𝑎𝑙 𝑟𝑎𝑡𝑒

i

ii

LsWs

i

ii i i

Ws

The Traffic intensity or utilization factor, i determines the degree to which the capacity of the service

facility is busy.

ii

i

(3.13)

The expected waiting time per unit time in the queue, iqW

iqW = expected time in the queue – time in the service

1

i ii

W Wq s

1.i

ii i i

Wq

(3.14)

36

3.4 Performance measures for the pooled queue with many servers.

Here, all the queues have been pooled into a single queue with arrival and service rates given

as and respectively. The single queue has multiple parallel servers, M/M/Si. i = (2,3,4,5)

According to Gupta and Hira (1979, p 952), the performance measure for this pooled model

is given by:

1.Expected (average) number of customers in the system, p

Ls .

p

L = p +s 021 !

i

i i

s

s s

(3.15)

2. Expected (average) number of customers waiting in the queue, p

Lq

L Lq s average number being served

pLq L ss i s

i

02

1 !

i

p

i i

s

L Pqs s

(3.16)

3. Average time a customer spends in the system, psW

p

LsWs

02

1

1 !

i

pi i

s

W Pss s

(3.17)

4. Average waiting time of a customer in the queue, p

Wq

p

LqWq

02

1 !

i

p

i i

s

W Pqs s

(3.18)

37

5. Pr ( that a customer has to wait)

2

01 !i i

P n c ps s

(3.19)

6. Pr ( that a customer gets service without waiting)

1- P n c

2

101 !i i

ps s

(3.20)

7. Utilization rate, p

p

is

(3.21)

Where

1

0 1 00

1 2 1

1 ...P

3.5 Determination of the optimum number of servers needed to minimize cost.

This section presents a decision model for determining “suitable” service levels for a pooled

queuing system with many servers. The cost model tries to balance two conflicting costs, cost of

the service provided by the queuing system and the cost of making the customers wait in the

system. According to Taha (2008), providing too much service capacity causes excessive cost on

the organization. Providing too little causes excessive waiting and the two types of costs are

conflicting because an increase in one automatically causes reduction in the other. The cost

models below aim at striking a balance between the conflicting factors of service level and

waiting.

There are two ways of seeking this trade-off, one is to establish one or more criteria for a

satisfying level of service in terms of how much waiting would be acceptable. The criterion

might be that the expected waiting time in the system would not exceed a certain number of

minutes. It may also be stated in terms of the number of customers in the queue. Once these

criteria are established, a trial and error method is used to find the least costly design that

satisfies all the criteria. Another approach for seeking the trade-off involves assessing the cost

38

associated with the consequences of making customers to wait. Making customers wait at the

queue causes loss of productive time which we would call the waiting cost associated with the

queuing system. By expressing this waiting cost as a function of the amount of waiting, the

probability of determining the best design of the queuing system can now be posed as the

minimization of the expected total cost (service cost + waiting cost) per unit time.

The objective is to minimize the total cost which is the sum of the operational cost and the

waiting cost.

The total cost of this waiting line system is the sum of the cost of waiting and cost of service.

Total cost = cost of waiting + cost of service

T C = WC2 + OC1 where

WC2 = C2i Lsi {i= 2,3,4,5}

Where C2i = average cost of waiting per hour for all the waiting customers for channel i

{i= 2,3,4,5}

Lsi = average number of people in the system for channel i {i= 2,3,4,5}.

i

L = p +s 021 !

i

ii i

i i

ii i i i

s

s s

OCi = cost of operating a server = {i= 2,3,4,5}.

For the purpose of this study, it was assumed that each of the servers costs the same.

The operational cost per hour (cost of providing service per hour) was obtained from the yearly

salaries of medical officers at the University of Nigeria, Nsukka medical Centre. From this

yearly income, an hourly estimate of the income was obtained. The salary structure of these

officers was obtained from the consolidated medical salary structure (CONMESS) per annum,

which was obtained from the national salaries, income and wages commission, Abuja, Nigeria.

In estimating the waiting cost, the average total expenditure in terms of school fees was gotten

from students, of all categories, an estimate of the income of workers was also made ( since the

pooled queue was made up of both workers and students). From these, an hourly estimate of the

39

cost of waiting was obtained. These values were further used as the operational cost per server

and the hourly cost of waiting by patients, C1 and C2 respectively.

40

CHAPTER FOUR

Analysis

4.0 Application

The data used for this study were collected from the General Outpatient Department of the

University of Nigeria, Nsukka Medical Centre. Three separate queues were used for the study;

that is three doctors were used, one queue for each doctor. In the general OPD of the

establishment, patient‟s arrival time is at random and it is assumed that the nurses assign the

patients randomly to any of the three doctors. Patients wait for a doctor from a queue. Three

different observers were used for the study with each observer recording for a single queue. The

observers used a stopwatch to record the time each patient arrived the queue and the time it took

for each patient to consult with the doctor. The average of the arrival rate and service rate for

each of these three queues was obtained. The arrival rate (number of patients/hr) was obtained by

considering the number of patients who arrived within the duration of the day‟s consultation

with the doctor. From there, the rate of arrival of patients per hour was obtained by dividing the

total number of patients by the total number of hours used for consultations for that day. Also for

the service rate, the amount of time used by each patient in consulting was added up and the

average number of people who consulted within one hour was obtained. This was done for each

of the three queues and three doctors and the information was used as the input to get the

performance for the three separate queues.

Using the arrival time of the patients, these three queues were later merged according to the

arrival time to form one pooled queue with three servers i.e using the arrival times earlier noted

for these patients, they were later pooled theoretically according to their arrival times, thus

creating a single queue with three servers. With this single pooled queue, customers queue up

and go to any available server in a first come first serve (FCFS) manner. The performance for

this pooled queue was also obtained and a comparative study was made on these two queuing

systems to determine the queuing system which offered a better performance.

41

A descriptive statistic of the service times for the three queues is presented in the table below.

Descriptive Statistics of the service times

N Minimum Maximum Mean

Std.

Deviation

queue 1 71 .15 20.31 7.29 7.09

queue 2 67 1.00 21.09 6.05 6.35

queue 3 59 1.06 22.42 7.77 7.92

Table 4.1 A descriptive statistic of the three queues.

Let X1 X2….. Xn be the observations of service time (in minutes). Queue one had an average of

71 customers with the highest duration of service lasting 20.31 minutes while the shortest lasted

for 0.15 minutes and had a mean service time of 7.29 and standard deviation of 7.09, queue two

an average queue size of 67, the highest duration of service of a patient in that queue was 21.03

minutes while the lowest was 1.00 minute and the mean service time and standard deviation

being 6.05 and 6.354 respectively while for queue three, the average queue size was 59, the

highest duration of service in that queue was 22.42 minutes while the shortest time was 1.06

minutes and the mean service time and standard deviation was 7.77 and 7.29 minutes

respectively.

4.1 Determination Of Service Time Distribution

One of the assumptions of the models chosen was that the service time for the three service

points should be exponentially distributed. Hence we proceed to examine if our service time

distribution is exponential. The Kolmogorov-Smirnov test was used to determine if the

exponential distribution fits the data. This test was carried out using SPSS software version 16.0

for windows. The choice of the Kolmogorov-Smirnov test is due to the fact that the

distributional form of the random variable under consideration is unknown. The following null

and alternative hypotheses was used for the three single queues.

42

Ho: The distribution of service time is exponential

H1: The distribution of the service time is not exponential

Decision rule: Reject Ho if the P-value is less than 0.05, accept otherwise.

The P values for the three queues are larger than 0.05 (.059, 0.06 and .051) hence we conclude

that the random observations of the service time come from the exponential distribution.

.

Table 4.2: Test analysis of the service time for the three service points.

Since the P value for each of the three service points is greater than 0.05, (0.059, 0.060 and

0.051) for queues 1,2 and 3 respectively we accept the null hypothesis and conclude that the

distributions of service time are exponential.

The following null and alternative hypotheses for the pooled queues was given as follows:

Ho: The distribution of service time for the pooled queue is exponential

H1: The distribution of the service time for the pooled queue is not exponential

Decision rule: Reject Ho if the P-value is less than 0.05, accept otherwise

One-Sample Kolmogorov-Smirnov Test

queue 1 queue 2 queue 3

N 71 67 59

Exponential

parameter.a

Mean service time 7.29 6.05 7.77

Kolmogorov-Smirnov Z 2.247 2.210 2.190

P-value .059 .060 .051

43

One-Sample Kolmogorov-Smirnov Test

Pooled

queue

N 197

Exponential

Parametera

Mean 7.03

1.39539

Kolmogorov-Smirnov Z

P- value .058

Table 4.3: Test analysis of the service time for the pooled queue.

Since the P-value for pooled queue is greater than 0.05, we accept the null hypothesis and

conclude that the distribution of service time for the pooled queue is exponential.

4.2 Evaluation of the queues

In the first queuing system, (single queue) each customer stream has its own single server. In the

second situation,(the pooled case), the streams are merged into a single stream with three servers.

Since the service times follow the exponential distribution, we would proceed to use the

M/M/1model and consider a single queue with a single server to analyze the performance

measures for each of the three single queues. Thereafter, the three queues are pooled together to

form a single queue with three servers, hence the M/M/3 model is used to analyze the pooled

system.

The table below shows the arrival and service rates for the pooled and unpooled queues as well

as the number servers as they would be used for the analysis.

parameters Queue one Queue two Queue three Pooled queue

Arrival rate 4customers/hr 4customers/hr 4customers/hr 11customers/hr

Service rate 7.03customers/hr 6.29customers/hr 7.77customers/hr 7.03customers/hr

Number of

servers

1 1 1 3

Table 4.3: Arrival and service rate for the pooled and unpooled queues.

44

The above inputs were used to assess the performance of the three queues and the pooled queue.

4.3 Results for the parallel and pooled queues

Queue one Queue two Queue three Pooled queue

Arrival

rate

4customers/hr 4customers/hr 4customers/hr 11customers/hr

Service

rate

7.03customers/hr 6.29customers/hr 7.77customers/hr 7.03customers/hr

Rho 0.56 0.635 0.514 0.523

Ls 1.32 1.746 1.061 1.84

Ws 0.33 0.43 0.265 0.168

Lq 0.75 1.11 0.546 0.75

Wq 0.18 0.277 0.136 0.026

Table IV: Results of some measures for both the single and pooled queue systems.

From the table above, we see that waiting time in the queues 1, 2 and 3 are 10.8minutes

(0.18x60), 16.6minutes (0.277x60) and 8.16minutes (0.136x60) respectively while in the pooled

queue is 1.56minutes (0.026x60). The waiting time in the system for queue 1,2,3 and the pooled

queue is 19.8, 25.8, 15.9 and 10.08minutes respectively (all have been converted to minutes by

multiplying the time specified on the table by 60). The average number of patients in queues

1,2,3 and the pooled queue is 0.75, 1.11, 0.546 and 0.75 respectively meanwhile the average

number of patients in the system for queue 1,2,3 and the pooled queue is 1.32, 1.74, 1.06 and

1.84 respectively. The utilization factor for queue 1,2,3 and the pooled queue are 0.56, 0.63,

0.51 and 0.523 respectively.

45

4.4 Further Use Of M/M/S

In this section, we proceed to prove the results of equation 3.17. We show that as the number of

servers increase, there is a reduction in the average waiting time in the system, as well as a

general improvement in the performance of the queue. We use the M/M/S where S=2,3,4,5 to

show this. Using the characteristics of the pooled queue, that is the arrival rate of 11 patients/hr

and the service rate of 7.03, assuming that all the servers considered here have this service rate.

The results are presented on table 4.5 in the next page.

Input: Value

Arrival rate: 11 customers/hr

Service rate: 7.03

Number of servers: c= 2,3,4,5. { we increase the number of servers from 2,3,4 and 5 and

compare the performance in each case. This is to show that the waiting time decreases with an

increase in the number of servers}.

46

After using the above as input for the cost model, the following results were obtained

Servers Arrival

rate

Service

rate

Po Ls Lq Ws Wq

2 11.000 7.03 0.12211 4.03372 2.46900 0.36670 0.22445

3 11.000 7.03 0.19518 1.84870 0.28398 0.16806 0.02582

4 11.000 7.03 0.20671 1.61921 0.05449 0.14720 0.00495

5 11.000 7.03 0.20873 1.57554 0.01082 0.14323 0.00098

Table 4.5: Showing the improvement of performance.

From the analysis above, we see that the performance measures of the system gets better as the

number of servers increase from two to five. For example considering the column for the waiting

time on the queue, qW , we see a considerable reduction from 13.22 minutes (0.22x60), to 0.05

minutes (0.00098x60). There is also a considerable reduction in the length of the queue in the

system from 4.03372 to 1.57554. Hence we conclude that the performance improves remarkably

with an increase in the number of servers. This confirms the theory of equation 3.17 seen earlier.

If the organization continues to get more servers so as to improve on its performance measures,

there would be an increase in its cost. Hence we have to find the service rate that would

minimize the organization‟s cost.

4.5 Input for the cost model:

As seen above, having many servers decreases the waiting time of patients remarkably but this is

not without acquiring a high cost on the system (the more servers there are, the more costs the

organization incurs by way of paying the servers). Also if very few servers are used, the lower

the cost on the organization but the greater the waiting time costs to the patients because the time

wasted could have been used judiciously in doing something else. Therefore we try to strike a

balance between an appropriate number of servers while minimizing the cost of operation in the

system while also minimizing the cost of waiting.

The cost model is given by:

ET C =EOC+ EWC where

ETC = total cost per unit time

47

EOC = cost of operating the facility per unit time

EWC = cost of waiting per unit time

The waiting and operating costs C1 and C2 are constants, which represent the hourly estimate of

the amount of money which is associated with the cost of providing service/ server and waiting

respectively. They remain constant throughout the analysis. We now consider our pooled queue

with the arrival rate of 11patients/hr and the service rate for each of the servers is 7.03patients/hr.

We vary the number of servers from 2 to 5 while calculating the expected total cost of operations

which is a function of the service level and the expected total cost of waiting which is a function

of the queue length.

Cost of operating the servers: According to the consolidated medical salary structure

(CONMESS), a medical officer receives an average amount 3,024,806 naira annually. This

implies the cost of service is about 350 naira hourly. The cost of waiting was obtained in a

similar manner but in this case, the hourly estimated of the waiting cost was calculated by taking

into consideration all the parties involved since the pooled queue was made up of both students

and workers (greater weight was assigned to the workers because they occurred more frequently

compared to students since some of them came on account of themselves and on other occasions

on account of their sick children.

After careful calculations, the annual average cost of waiting was estimated at 1,296,000 naira.

This implies the hourly cost of waiting is about 150 naira.

Cost of operation, C1 = 350 naira

Cost of waiting, C2 = 150 naira

48

4.6 Results Of The Cost Model:

Servers 2 3 4 5

Arrival rate 11.0 11.0 11.0 11.0

Service rate 7.03 7.03 7.03 7.03

Lsi 4.0337 1.847 1.619 1.575

Operating cost, OCi 700 1050 1400 1750

Cost of waiting, C2i Lsi 655.5 277.05 242.85 236.25

Total cost 1355.5 1327.05 1642.85 1986.85

Table 4.6: Results Of The Cost Model

From the table we see that the cost of providing service increases with an increase in the number

of servers i.e 700, 1050, 1400 and 1750 for 2,3,4 and 5 servers respectively. Also, we see from

above that the waiting cost for the patient decreases 655.5, 277.05, 242.85 and 236.25 with an

increase in the number of servers from 2,3,4 and 5 servers, while the cost of providing service

increases with an increase in the number of servers i.e 700, 1050, 1400 and 1750 with 2, 3, 4 and

5 servers respectively. We then sum up the two categories of costs involved (cost of operations

and costs of waiting) to get the total cost involved. The column with three servers has the least

cost (1327.05) for the system under study. Hence we observe from these cost models that it

would be more efficient to have 3 servers with one pooled queue, M/M/3, which does not only

minimize the total waiting cost incurred but also minimizes the total cost. So for the queuing

system considered in this study, the option of three servers with a single pooled queue is an

optimal solution with optimum trade-off between the two types of costs involved in queuing

models.

49

4.7 Determination of the conditions under which there will be no gain in pooling.

There may be situations where the pooling of customers does not improve efficiency. In this

section we tried to look at some of the conditions which may not allow for pooling. First it is

assumed that all the servers can offer the needed services. This flexibility may be expensive and

may actually cause inefficiencies if servers are no longer able to focus on a single customer type.

The ideas presented above rely on the assumption of identical servers. Statistically, the

advantage of pooling is credited to the reduction in variability due to the „portfolio effect‟ (the

theory that a diversified portfolio will minimize risk and maximize returns). There are also

situations where the pooling of customers actually adds undesired variability to the system thus

offsetting any gains in efficiency, Peter T. et al (2010). More precisely, with WP and WA, being

the mean waiting time for the pooled and separate queues,1

and

the traffic load per

server, by straight forward calculations from standard M/M/1 and M/M/3 expressions, pooling

two parallel exponential servers would lead to a reduction factor of at least 50% of the mean

waiting time, (Joustra, 2007) as shown by

2 2/ 1

/ 1 1

P

A

W

W

(4.1)

This reasoning however relies upon the assumption of two identical servers or two identical

service characteristics. However, customers of type 1 may not benefit from pooling when the

expected waiting time for the pooled system, WP exceeds the expected waiting time W1 for the

customers of type 1. By standard M/M/1 and M/M/3 expressions with = 1 2

2

1

2

1

211 1

PW W

(4.2)

It is easy to see that the above inequality holds for

11 2 12

(4.3)

50

From the above inequality, trade-off values for 1 and 2 can be computed. Therefore pooling

is not always beneficial to all customers. When you have urgent (emergency cases) as well as

non urgent cases, it is advised not to pool.

51

CHAPTER FIVE

DISCUSSION AND CONCLUSION

5.0 Discussion

This work compares the performance of two queuing systems (the parallel queues and the pooled

queue) as applied to the General Outpatient Department of the University of Nigeria Medical

centre. The aim is to carry out a comparative study of two queuing systems with a view to

finding out which arrangement has a better performance. From the results for the unpooled

queues we see that the utilization rate, which is the fraction of time the server is busy working is

low, (56%). The average waiting time in the system for the unpooled queues is

0.34(20.4minutes) meanwhile that of the pooled is 0.168 (10.08minutes). We can see that there is

a significant difference in the waiting time between the pooled and unspooled queues. The

customers under the pooled queue experienced a significant reduction of waiting time (49.4 %).

The queue length also decreased from 0.802 for the single queue to 0.75 for the pooled queue.

We therefore conclude that there was an overall improvement in the performance measures of

the pooled queues of the Outpatient Department of the University of Nigeria, Nsukka Medical

Centre. The pooled queue was then modified for varying number of servers so as to examine

how cost varies with the number of servers and to determine the optimal level of staffing that

would be needed so that a balanced or optimal solution between the two types of costs would be

obtained.

From the M/M/S model, we see that there is a drop in waiting time as more servers are added.

There is also a drop in queue length, this would probably come with some degree of happiness

and satisfaction to the customers, but heavy cost would be incurred by the hospital if they have to

employ many doctors for the Outpatient Department. For this study, given the queue

characteristics, the optimal number of servers which would minimize operational cost is three for

this particular establishment with time saving and the reduction of operational costs as the only

factors considered.

From our analysis, we see that it would be more efficient to have three servers with a single

pooled queue for the Outpatient Department of the University of Nigeria, Nsukka because this

52

arrangement minimizes not only the cost of providing service, but also minimizes the waiting

time for customers.

5.1 Conclusion

Providing customers with timely access to the needed services while minimizing cost to the

establishment is one of the major goals of every organisation. Generally, an excess of demand

over supply (more customers than doctors) causes waiting. Since we obviously have more

patients than doctors, the complete elimination of waiting lists is impossible, we only try to

minimize this waiting. With the results obtained from this study, we see that the pooling of

queues helps to improve the operations of the University of Nigeria Medical Centre and having

three servers in a single pooled queue is best for the University of Nigeria Medical Centre. This

work is in line with the results of Joustra et al (2007), who said that for systems with one or

similar type of service, a single queue with multiple servers performs better than each server

having their own queue. Finally, we advice that a similar study should be carried out for systems

which may have different service times or service rates in order to ascertain if pooling could also

be beneficial to such systems as this work is limited only to the Outpatient Department of the

University of Nigeria, Nsukka Medical Centre.

5.2 Recommendation

We recommend that the University of Nigeria, Nsukka Medical Centre should operate a system

whereby three servers or doctors consult with patients at the General Outpatient Department. As

this would not only minimize the cost of operations in the system, but would also minimize the

patient waiting time hence giving satisfaction to both the relevant authorities and the patients.

We also strongly advice that a pooled system of queuing be introduced whereby patients come

and form one pooled queue and consult with any doctor as soon as he/ she is available because

this method reduces the waiting time of patients remarkably.

53

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Appendix

Queue one (service

time/mins)

Queue two (service

time/mins)

Queue three (service

time/mins)

1. 4.2 3.13 6.22

2. 3.07 6.15 7.29

3. 2.09 2.17 4.02

4. 1.2 6.32 4.37

5. 4.09 6.01 11.05

6. 3.58 10.09 5.49

7. 3.21 7.29 1.59

8. 3.14 4.27 4.16

9. 5.4 5.05 7.0

10. 11.34 1.0 3.48

11. 5.46 2.55 1.28

12. 4.29 4.17 4.09

13. 4.0 2.12 8.43

14. 4.35 4.01 6.09

15. 7.06 18.14 3.13

16. 3.01 9.36 2.0

17. 0.15 7.1 13.01

18. 3.03 11.4 13.51

19. 10.42 10.22 11.02

20. 3.09 18.07 7.44

21. 10.02 10.55 4.09

22. 4.12 21.0 17.15

23. 7.15 8.1 7.54

57

24. 5.04 2.0 13.21

25. 3.29 21.09 8.02

26. 5.28 8.52 6.19

27. 7.32 8.25 5.59

28. 8.16 8.57 8.44

29. 11.13 3.5 5.46

30. 4.21 8.09 22.42

31. 16.01 18.35 11.42

32. 8.58 9.19 1.06

33. 5.05 11.56 10.2

34. 8.15 7.34 4.51

35. 7.57 11.44 12.45

36. 10.49 6.16 13.45

37. 6.15 9.07 3.04

38. 8.53 8.55 10.54

39. 6.39 5.32 7.58

40. 9.01 10.29 10.08

41. 10.01 12.51 8.23

42. 7.04 4.44 5.46

43. 6.05 7.56 7.56

44. 8.16 11.09 7.04

45. 12.15 4.58 6.58

46. 20.27 6.53 12.0

47. 7.15 12.06 7.29

48. 9.5 5.08 6.42

49. 11.17 14.51 9.57

50. 5.31 10.31 11.01

51. 7.15 6.22 7.03

52. 11.26 6.11 5.06

53. 17.09 5.49 7.07

54. 7.41 9.37 8.53

55. 6.55 6.22 5.23

56. 9.27 3.07 7.34

57. 1.56 6.03 10.21

58. 0.53 9.33 9.29

59. 8.58 5.08 10.56

60. 8.1 4.42

61. 12.18 9.52

62. 9.22 4.53

63. 11.13 7.01

64. 9.01 8.15

65. 6.21 12.1

66. 11.0 10.1

67. 6.2 7.58

68. 20.31

58

69. 6.29

70. 3.0

71. 11.44

72.