Mb0048 Unit 09-Slm

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Operations Research Unit 9

Sikkim Manipal University Page No. 174

Unit 9 Infinite Queuing Models

Structure:

9.1 Introduction

Objectives

9.2 Queuing Theory

9.3 Operating Characteristics of a Queuing System

9.4 Constituents of a Queuing System

Arrival pattern

Completely random arrivals

9.5 Service Facility

9.6 Queue Discipline

Customer behaviour

Server behaviour

9.7 Summary

9.8 Glossary

9.9 Terminal Questions

9.10 Answers

9.11 Case Study

9.1 Introduction

In the previous unit, you learnt the Integer Programming Problem (IPP) andthe Gomory’s all-IPP method. You also learnt all IPP algorithm and the

branch and bound technique. This chapter introduces the idea of queuing

theory. The formation of waiting lines or queues is common and is usually

formed by elements, people or events that are awaiting some form of

service. Who isn’t familiar with queues in our daily existence? On the way to

your workplace, you have to wait at the bus stop, wait for the traffic light to

turn green, wait for your turn to enter the lift. You come across cars in line at

petrol pumps for service, queues to deposit cash at the bank, mobile

subscriber waiting for a new connection, customers waiting at the checkout

counter, goods in production/warehouse waiting to be shipped and evenaircrafts waiting for a free runway to take off.

The one thing common to all the above mentioned examples is that

customers arrive at a service centre and wait for their turn to receive the

service. Though the arrival of customers is irregular and the time taken for

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the service is not consistent, queues build up during hours of demand and

disappear during the lull period.In commercial or industrial situations, it may not be economical to have

waiting lines. On the other hand, it may not be feasible or economical to

totally avoid queues. An executive dealing with the system then would like to

find the optimal facilities to be provided. In this unit, you will learn about

queuing theory, operating characteristics and constituents of a queuing

system. You will also learn about service facility and queue discipline.

Objectives:

After studying this unit, you should be able to:

describe the queuing process

evaluate the Queuing Theory

differentiate the customer’s behaviour.

9.2 Queuing Theory

Queuing theory had its beginning in the research work of a Danish engineer

named A. K. Erlang. In 1909, Erlang experimented with fluctuating demand

in telephone traffic. Eight years later he published a report addressing the

delays in automatic dialling equipment. At the end of World War II, Erlang’s

early work was extended to more general problems and to business

applications of waiting lines.

Queuing theory is a collection of mathematical models of various queuing

systems. It is based on probability concepts. It gives an indication of the

capability of a given system and the possible changes in its performance

with modification to the system. All the constraints of the process are not

taken into account in the formulation of a queuing model. The application of

queuing theory cannot be viewed as an optimisation process as there is no

maximisation or minimisation of an objective function.

Formation of Queues

Queues or waiting lines arise when the demand for a service facilityexceeds the capacity of that facility, that is, the customers do not get service

immediately upon request but must wait, or the service facilities stand idle

and wait for customers.

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For example

Supermarkets must decide how many cash register checkout positionsshould be opened.

Gasoline stations must decide how many pumps should be opened and

how many attendants should be on duty.

Manufacturing plants must determine the optimal number of mechanics

to have on duty in each shift to repair machines that break down.

Banks must decide how many teller windows to keep open to serve

customers during various hours of the day.

Self Assessment Questions

1. Customers arrive at a bank at regular intervals. (True/False)

2. Queuing identifies the optimal service facilities to be provided.

(True/False)

3. Queuing theory is based on the deterministic model. (True/False)

9.3 Operating Characteristics of a Queuing System

In the previous section, you learnt about queuing theory. You will now learn

the operating characteristics of a queuing system. A queuing model has the

following operating characteristics which enables us to understand and

efficiently manage a queue: Queue length: The number of customers in the waiting line reflects one

of the two conditions. Short queues could mean either good customer

service or too much capacity. Similarly, long queues could indicate

either low server efficiency or the need to increase capacity

Number of customers in system: The number of customers in queue

and also those being served in the queue relates to the service

efficiency and capacity. Large values imply congestion, potential

customer dissatisfaction and a need for more capacity.

Waiting time in queue: Long lines do not reflect long waiting times if

the service rate is fast. However, when waiting time seems long to

customers, they perceive that the quality of service is poor. Long waiting

times may indicate a need to adjust the service rate of the system or

change the arrival rate of customers.

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Waiting time in system: The total elapsed time from entry into the

system until exit from the system may indicate problems with customers,server efficiency or capacity. If some customers are spending too much

time in the service system, there may be a need to change the priority

discipline, increase productivity or adjust capacity in some way.

Service facility utilisation: The collective utilisation of the service

facilities reflects the percentage of time the facilities are busy.

Management is interested in maintaining high utilisation but this

objective may adversely impact the other operating characteristic.

A queuing system is said to be in transient state when its operating

characteristics are dependent upon time. If the operating characteristics

become independent upon time, the queuing system is said to be in a

steady state.


4. _________ represents number of customers waiting in the queue.

5. ________ ________ times may indicate a need to adjust the service

rate of the system or change the arrival rate of customers

6. _____________ represents the percentage of time the facilities are

busy.

9.4 Constituents of a Queuing System

In the previous section, you learnt the operating characteristics of a queuing

system. You will now learn the constituents of a queuing system. The

constituents of a queuing system include arrival pattern, service facility and

queue discipline.

Arrival pattern: It is the average rate at which the customers arrive.

Service facility: Examining the number of customers served at a time

and the statistical pattern of time taken for service at the service facility.

Queue discipline: The common method of choosing a customer for

service amongst those waiting for service is ‘First Come First Serve’.

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Figure 9.1 depicts the components of a queuing system.

Fig. 9.1: Components of a Queuing System

Figure 9.2 depicts an example showing the components of a queuing

system.

Fig. 9.2: Car Wash System Showing Components of a Queuing System

9.4.1 Arrival pattern

The arrival of customers can be regular as in case of an appointment

system of a doctor or flow of components on a conveyor belt. The regular

pattern of arrivals is neither very common nor very easy to deal with

mathematically. The following are the important arrival characteristics:

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1. Size of the population: Unlimited (infinite) or limited (finite)

2. Pattern of arrivals (statistical distribution)3. Behaviour of arrivals

Our primary concern is the pattern of completely random arrivals.

9.4.2 Completely random arrivals

If the number of potential customers is infinitely large, then probability of an

arrival in the next interval of time will not depend upon the number of

customers already in the system. (The assumption is valid by and large,

except for queues involving a small finite number of customers.) When the

arrivals are completely random, they follow the Poisson distribution, which

equals to the average number of arrivals per unit time.

Sometimes it is necessary to distinguish between groups of customers, such

as male and female callers, or large and small aircrafts during the arrivals.

There are several other types of arrival patterns which shall not be dealt with

due to their restricted applications.


7. Every queuing process has an arrival pattern, a service facility and a

queue discipline as its constituents.(True/False)

8. If the arrivals are completely random, then it follows Poisson

distribution. (True/False)

9.5 Service Facility

In the previous section, you learnt the constituents of a queuing system. You

will now learn the service facility. Service Facility is based on three

parameters – Availability of service, number of service centres and duration

of service.

i) Availability of service

It is necessary to examine if there are any constraints that reduce the

number of customers to be served at a time, apart from specifying the timespan of the availability of service. For example, in a waiting line for a

suburban train, apart from the timings of the train services, the probability

distribution of the number of passengers that can be accommodated in a

train that arrives is relevant.

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ii) Number of service centres

If only one service centre is referred to as a service channel, obviously onlyone customer can be served at a time. There will definitely be more than

one service centre and the behaviour of the queues will vary with the

number of channels available for service. Multiple service channels may be

arranged in series or in parallel.

Multiple service channels are arranged in series when a customer has to go

through several counters one after another with each providing a different

part of the service. For instance, bank counters where a customer has to go

to at least two counters to withdraw is an example of arrangement in series.

On the other hand ticket booths in a railway station have multiple channels

with parallel arrangement.

iii) Duration of service

This is the length of time taken to serve a customer. This can be constant or

varying.

(a) Constant service time: Though not in practice, an assumption that

service time is constant holds true, if the pattern of arrivals is very

irregular.

(b) Completely random service time: The service time can be considered

completely random when:

The server does not distinguish between the various arrivals. The server does not deliberately change the duration of service on

the basis of the time taken to serve the previous arrival.

The server forgets the time for which he/she has been serving a

customer.

Under the above mentioned conditions, the service time follows

exponential distribution which means it is equal to reciprocal of the

average rate of service.

(c) Service time following Erlang distribution

Sometimes, the assumption of an exponential distribution for service time isnot valid. Hence Erlang family of service time distributions is used.

Table 9.1 depicts examples of queuing discipline in daily life.

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Table 9.1: Examples of Queuing Discipline in Daily Life

Sr.No.

Situation ArrivingCustomers

Service Facility

1. Passage of customers throughsupermarket checkout

Shoppers Checkout counters

2. Flow of automobile traffic througha road network

Automobiles Road Network

3. Transfer of electronic messages Electronicmessages

Transmission lines

4. Banking transactions Bank patrons Bank tellers

5. Flow of computer programmersthrough a computer system

ComputerProgrammers

Central processingunit

6. Sale of theatre tickets Theatre visitors Ticket bookingwindows

7. Arrival of trucks to carry fruits andvegetables from a central market

Trucks Loading crews andfacilities

8. Registration of unemployed atemployment exchange

Unemployedpersonnel

Registrationassistants

9. Occurrences of fires Fires Firemen andequipment

10. Flow of ships to the seashore Ships Harbour anddocking facilities

11 Calls at police control room Service calls Policemen


9. Multiple service channels may be arranged in ______________ or in

_________.

10. The service time can be __________ or _________.

9.6 Queue Discipline

In the previous section, you learnt the service facility. You will now learn

about queue discipline. The queue discipline is the order or manner in whichcustomers from the queue are selected for service. There are a number of

ways in which customers in the queue are served:

Static queuing disciplines are based on the individual customer’s status in

the queue like:

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I. First In First Out (FIFO) – If the customers are served in the order of

their arrival. E.g., Toll roadII. Last In First Out (LIFO) – Where the last in items are operated first.

E.g., Cargo handling situations, getting down from the bus, etc.

Dynamic queuing disciplines are based on the individual customer attributes

in the queue:

I. Service in Random Order (SIRO): Here customers are selected for

service at random irrespective of their arrivals in the service system.

II. Priority service: Under this rule, customers are grouped in priority

classes on the basis of some attributes such as service time or

urgency, and FIFO rule is used within each class to provide service.

E.g., premium queues in temples like Tirupati.III. Pre-emptive priority (or Emergency): Under this rule, an important

customer is allowed to enter into the service immediately after entering

into the system even if a customer with lower priority is already in

service. E.g., Emergency service in hospitals, ambulance and fire

brigades in traffic signals

9.6.1 Customer behaviour

a) Balking: Arriving customers are said to “balk” if they do not join a queue

because of their reluctance to wait or some customers even before

joining the queue get discouraged by seeing the number of customers

already in service system or estimating the excessive waiting time for a

desired service, decide to return for service at a later time. In queuing

theory this is known as balking.

b) Collusion: Customers may be in collusion, meaning that only one

person would join the queue, but demand service on behalf of several

customers.

c) Reneging: Impatient customers who would not wait beyond a certain

time and leave the queue are said to renege.

For example, a customer who has just arrived at a grocery store and

finds that the salesmen are busy in serving the customers already in the

system, will either wait for service till his patience is exhausted or

estimates that his waiting time may be excessive and so leaves

immediately to seek service elsewhere.

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d) Jockeying: Some customers keep on switching over from one queue to

another in a multiple service centres. This is called jockeying.9.6.2 Server behaviour

Although the service timings have been specified, the server may not be

available through the entire span of time. For instance, in every one hour the

server may disconnect from the service centre for 5 minutes so that it can

go through its daily updates or cleanup routine. The server is characterised

by:

The arrangement of service facilities

The distribution of service times

Server’s behaviour

Management policies


11. When customers keep on switching over from one queue to another

then it is called ________.

12. _____ ________ ______ ______ are the types of customer behaviour.

9.7 Summary

Let us recapitulate the important concepts discussed in this unit:

The waiting line theory or queuing theory analysis suggests the numberof facilities required and the cost of customer’s waiting time and the

optimum service level.

The queuing theory contributes vital information required for balancing

the cost of service and cost associated with waiting time of the

customer.

A queuing system is said to be in transient state when its operating

characteristics are dependent upon time.

9.8 Glossary

Erlang family of service time distr ibutio ns: two constraint generalization

of exponential distribution.

Statist ical pattern: set of probability distributions

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Probabil i ty distr ibutio n: a table containing the outcomes of an experiment

with its likelihood of occurrence.

9.9 Terminal Questions

1. Explain the operating characteristics of a queuing system.

2. Which are the important arrival characteristics?

3. State the ways in which customers in a queue are served.

9.10 Answers


1. False

2. True

3. False

4. Queue length

5. Long waiting

6. Collective utilisation

7. True

8. True

9. Series, parallel

10. Constant, varying

11. Jockeying

12. Balking, Collusion, reneging, jockeying

Terminal Questions

1. The operating characteristics of a queuing system are queuing length,

number of customers in system,waiting time in queue, waiting time in

system and service facility utilisation. For more details, refer section 9.3

2. Size of population, pattern of arrivals and behavior of arrivals. For more

details refer section 9.4.1

3. Static queuing disciplines and dynamic queuing disciplines. For moredetails refer section 9.5

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9.11 Case Study

Drive-in Photo ShopSunder is planning to open a drive-through photo developing booth with a

single service window that will be open approximately 200 hours per month

in a busy commercial area. Space for a drive-through lane is available for a

rental of Rs. 8000 per month per car length. Sunder needs to decide how

many car lengths of space to provide for his customers.

Excluding this rental cost for the drive-through lane, Sunder believes that he

will average a profit of Rs 160 per customer served(nothing for a drop off of

film and Rs 320 when the photographs are picked up). He also estimates

that customers will arrive randomly at a mean rate of 20 per hour, although

those who find the drive-through lane full will be forced to leave. Half of the

customers who find the drive-through lane full wanted to drop off the film

and the other half wanted to pick up the photographs. The half who wanted

to drop off film will take their business elsewhere instead. The other half of

the customers who find the drive-through lane full will not be lost because

they will keep trying later until they can get in and pick up their photographs.

Sunder assumes that the time required to serve a customer will have an

exponential distribution with a mean of 2 minutes.

Discussion Questions:

a) Find L(expected no. of customers in queuing system) and the mean rateat which customers are lost when the number of car lengths of space

provided is 2, 3, 4 and 5.

b) Calculate W (waiting time in the system) and L for the cases considered

in (a).

c) Use the results from part(a) to calculate the decrease in the mean rate

at which customers are lost when the number of car lengths of space

provided is increased from 2 to 3, from 3 to 4, from 4 to 5. Then

calculate the increase in expected profit per hour(excluding space rental

costs) for each of these three cases.

d) Compare the increases in expected profit found in part(c) with the cost

per hour of renting each car length of space. What conclusion do you

draw about the number of car lengths of space that Sunder should

provide?

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References:

Kapoor V. K. (2005). Operations Research. Sultan Chand and Sons.

Sharma J. K. (2006). Operations Research. Macmillan India Limited.

Taha H. Operations Research. Prentice Hall.

Kanti Swarup & Gupta P. K., & Hira D. S., & Manmohan (2004).

Operation Research. Sultan Chand and Sons.

E-References:

http://www.mrt.ac.lk/maths/TMJAC/Operational%20Research

%20Techinques%20II/OR%20Lecture%203.pdf, accessed on Nov 9,

2009

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