DOC-20231208-WA0013..pptx mba hrd for business law

Queuing Theory
Concept, single server [M/M/1, infinite, FIFO]
Introduction of multi server[M/M/C, infinite,
FIFO] ( Numerical on single server model
expected)[8+2]

• Objectives
• After studying this unit, you will be able to:
• Identify and examine situations that generate
queuing problems
• Describe the trade-off between cost of service
and cost of waiting line
• Analyze a variety of performance measures of a
queuing system
• Understand the concept of single server Queuing
model

• Introduction
• Queuing theory deals with problems that involve waiting (or
queuing). It is quite common that instances of queue occurs
everyday in our daily life. Examples of queues or long waiting
lines might be
• Waiting for service in banks and at reservation counters.
• Waiting for a train or a bus.
• Waiting for checking out at the Supermarket.
• Waiting at the telephone booth or a barber’s saloon. Whenever
a customer arrives at a service facility, some of them usually
have to wait before they receive the desired service. This forms
a queue or waiting line and customers feel discomfort either
mentally or physically because of long waiting queue.

• We infer that queues form because the service facilities are
inadequate. If service facilities are increased, then the question arises
how much to increase?
• For example, how many buses would be needed to avoid queues?
• How many reservation counters would be needed to reduce the
queue?
• Increase in number of buses and reservation counters requires
additional resource. At the same time, costs due to customer
dissatisfaction must also be considered.
• In designing a queuing system, the system should balance service to
customers (short queue) and also the economic considerations (not
too many servers).
• Queuing theory explores and measures the performance in a queuing
situation such as average number of customers waiting in the queue,
average waiting time of a customer and average server utilization.

• Queuing System The customers arrive at
service counter (single or in groups) and are
attended by one or more servers. A customer
served leaves the system after getting the
service. In general, a queuing system
comprises with two components, the queue
and the service facility. The queue is where
the customers are waiting to be served. The
service facility is customers being served and
the individual service stations.

History Of Queuing Theory
• Queuing theory was first introduced in the early 20th
century
by Danish mathematician and engineer Agner Krarup Erlang.
• Erlang worked for the Copenhagen Telephone Exchange and
wanted to analyze and optimize its operations. He sought to
determine how many circuits were needed to provide an
acceptable level of telephone service, for people not to be “on
hold” (or in a telephone queue) for too long. He was also
curious to find out how many telephone operators were
needed to process a given volume of calls.
• His mathematical analysis culminated in his 1920 paper
“Telephone Waiting Times”, which served as the foundation of
applied queuing theory. The international unit of telephone
traffic is called the Erlang in his honor.

• Characteristics of Queuing
• System In designing a good queuing system, it is
necessary to have a good information about the
model. The characteristics listed below would
provide sufficient information.
• 1. The arrival pattern.
• 2. The service mechanism.
• 3. The queue discipline.
• 4. The number of customers allowed in the system.
• 5. The number of service channels.

• The Arrival Pattern: The arrival pattern describes how a customer may
become a part of Notes the queuing system. The arrival time for any
customer is unpredictable. Therefore, the arrival time and the number of
customers arriving at any specified time intervals are usually random
variables. A Poisson distribution of arrivals correspond to arrivals at random.
In Poisson distribution, successive customers arrive after intervals which
independently are and exponentially distributed. The Poisson distribution is
important, as it is a suitable mathematical model of many practical queuing
systems as described by the parameter “the average arrival rate”.
• 2. The Service Mechanism: The service mechanism is a description of
resources required for service. If there are infinite number of servers, then
there will be no queue. If the number of servers is finite, then the customers
are served according to a specific order. The time taken to serve a particular
customer is called the service time. The service time is a statistical variable
and can be studied either as the number of services completed in a given
period of time or the completion period of a service.

• The Queue Discipline: The most common queue discipline is the “First Come
First Served” (FCFS) or “First-in, First-out” (FIFO). Situations like waiting for a
haircut, ticket-booking counters follow FCFS discipline. Other disciplines
include “Last In First Out” (LIFO) where last customer is serviced first, “Service
In Random Order” (SIRO) in which the customers are serviced randomly
irrespective of their arrivals. “Priority service” is when the customers are
grouped in priority classes based on urgency. “Preemptive Priority” is the
highest priority given to the customer who enters into the service,
immediately, even if a customer with lower priority is in service. “Non-
preemptive priority” is where the customer goes ahead in the queue, but will
be served only after the completion of the current service.
• 4. The Number of Customers Allowed in the System: Some of the queuing
processes allow the limitation to the capacity or size of the waiting room, so
that the waiting line reaches a certain length, no additional customers is
allowed to enter until space becomes available by a service completion. This
type of situation means that there is a finite limit to the maximum queue size.

• The Number of Service Channels: The more the number of service
channels in the service facility, the greater the overall service rate of the
facility. The combination of arrival rate and service rate is critical for
determining the number of service channels. When there are a number of
service channels available for service, then the arrangement of service
depends upon the design of the system’s service mechanism.
• Parallel channels means, a number of channels providing identical service
facilities so that several customers may be served simultaneously. Series
channel means a customer go through successive ordered channels before
service is completed. The arrangements of service facilities are illustrated
in Figure 11.2. A queuing system is called a one-server model, i.e., when
the system has only one server, and a multi-server model i.e., when the
system has a number of parallel channels, each with one server.

• Waiting Line Process :Waiting in lines is a part of our everyday life.
Waiting in lines may be due to overcrowded, overfilling or due to
congestion. Any time there is more customer demand for a service
than can be provided, a waiting line forms. We wait in lines at the
movie theatre, at the bank for a teller, at a grocery store. Wait time
depends on the number of people waiting before you, the number
of servers serving line, and the amount of service time for each
individual customer. Customers can be either humans or an object
such as customer orders to be process, a machine waiting for
repair. Mathematical analytical method of analyzing the
relationship between congestion and delay caused by it can be
modelled using Queuing analysis. A waiting line process or queuing
process is defined by two important elements:
• the population source of its customers and
• the process or service system.

• The customer population can be considered as finite or infinite.
The customer population is finite when the number of
customers affects potential new customers for the service
system already in the system. When the number of customers
waiting in line does not significantly affect the rate at which the
population generates new customers, the customer population
is considered infinite. Customer behaviour can change and
depends on waiting line characteristics. In addition to waiting, a
customer can choose other alternative. When customer enters
the waiting line but leaves before being serviced, process is
called Reneging. When customer changes one line to another to
reduce wait time, process is called Jockeying. Balking occurs
when customer do not enter waiting line but decides to come
back latter.

• Another element of queuing system is service system. The number of waiting lines, the number
of servers, the arrangements of the servers, the arrival and service patterns, and the service
priority rules characterize the service system. Queue system can have channels or multiple
waiting lines. Example: Single waiting line : bank counter, airline counters, restaurants,
amusement parks. In these examples multiple servers might serve customers. In the single line
multiple servers has better performance in terms of waiting times and eliminates jockeying
behaviour than the system with a single line for each server. System serving capacity is a function
of the number of service facilities and server proficiency. In waiting line system, the terms server
and channel are used interchangeably. Waiting line systems are either single server or multiple
servers. Example: 1. Single server: gas station, food mart with single checkout counter, a theatre
with a single person selling tickets and controlling admission into the show. 2. Multiple server:
gas stations with multiple gas pumps, grocery stores with multiple cashiers, multiple tellers in a
bank. Services require a single activity or services of activities called phases. In a single-phase
system, the service is completed all at once, such as a bank transaction or grocery store checkout
counter. In a multiphase system, the service is completed in a series of phases, such as at fast-
food restaurant with ordering, pay, and pick-up windows. The process of waiting line is
characterized by rate at which customers arrive and are served by service system. Arrival rate
specifies the average number of customers per time period. The service rate specifies the
average number customers that can be serviced during a time period. The service rate governs
capacity of the service system. It is the fluctuation in arrival and service patterns that causes wait
in queuing system. A general waiting line system with parallel server is shown in Figure 8.1.

• Poisson and Exponential Distributions Both
the Poisson and Exponential distributions play
a prominent role in queuing theory.
Considering a problem of determining the
probability of n arrivals being observed during
a time interval of length t, where the following
assumptions are made.

• Probability that an arrival is observed during a small time
interval (say of length v) is proportional to the length of interval.
Let the proportionality constant be ƛ, so that the probability is ƛ
v.
• 2. Probability of two or more arrivals in such a small interval is
zero.
• 3. Number of arrivals in any time interval is independent of the
number in non overlapping time interval. These assumptions
may be combined to yield what probability distributions are
likely to be, under Poisson distribution with exactly n customers
in the system. Suppose function P is defined as follows:
• P (n customers during period t) = the probability that n arrivals
will be observed in a time interval of length t

• P (n, t) = (ƛ t)^n e^-ƛt /n!(n = 0, 1, 2,……………)
This is the Poisson probability distribution for the discrete
random variable n, the number of arrivals, where the length of
time interval, t is assumed to be given. This situation in queuing
theory is called Poisson arrivals. Since the arrivals alone are
considered (not departures), it is called a pure birth process.
• The time between successive arrivals is called inter-arrival time.
In the case where the number of arrivals in a given time interval
has Poisson distribution, inter-arrival times can be shown to
have the exponential distribution. If the inter-arrival times are
independent random variables, they must follow an exponential
distribution with density f(t) where, f(t) = ƛ e^-ƛt , t > 0

• Thus for Poisson arrivals at the constant rate ƛ
per unit, the time between successive arrivals
(inter-arrival time) has the exponential
distribution. The average Inter - arrival time is
denoted by I . By integration, it can be shown
that E(t) = I/ ƛ , If the arrival rate ƛ = 30/hour,
the average time between two successive
arrivals are 1/30 hour or 2 minutes. For example,
in the following arrival situations, the average
arrival rate per hour, ƛ and the average inter
arrival time in hour, are determined.

• One arrival comes every 15 minutes. Average
arrival rate, l =60/15 = 4 arrivals per hour.
Average inter arrival time = 15 minutes = ¼ or
0.25 hour.
• Three arrivals occur every 6 minutes. Average
arrival rate, l = 30 arrivals per hour. Average
Inter-arrival time, =6/3 = 2 minutes =1/30 or
0.33 hr.

Little’s Law
Little’s Law connects the capacity of a queuing system, the average time spent in
the system, and the average arrival rate into the system without knowing any
other features of the queue. The formula is quite simple and is written as
follows:
• L = λ W
or transformed to solve for the other two variables so that:
• λ = W / L
Where:
• L is the average number of customers in the system
• λ (lambda) is the average arrival rate into the system
• W is the average amount of time spent in the system
• Project management processes like Lean and Kanban wouldn’t exist without
Little’s Law. They’re critical for business applications, in which Little’s Law can be
written in plain English as:

• Little’s Law Applied to Real-World Situations
• Little’s Law gives powerful insights because it lets us solve for
important variables like the average wait of in a queue or the
number of customers in queue simply based on two other inputs.
• A line at a cafe
• For example, if you’re waiting in line at a Starbucks, Little’s Law
can estimate how long it would take to get your coffee.
• Assume there are 15 people in line, one server, and 2 people are
served per minute. To estimate this, you’d use Little’s Law in the
form:
• L/ λ = W
• Showing that you could expect to wait 7.5 minutes for your coffee.

M/M/1 Queuing System (∞/FIFO)
• It is a queuing model where the arrivals follow a
Poisson process, service times are exponentially
distributed and there is only one server. In other
words, it is a system with Poisson input,
exponential waiting time and Poisson output with
single channel.
• Queue capacity of the system is infinite with first in
first out mode. The first M in the notation stands
for Poisson input, second M for Poisson output, 1
for the number of servers and ∞ for infinite
capacity of the system.

• Formulas
• Probability of zero unit in the queue (Po) =1− λ
Idle rate or service facility is idle. -----
μ
• Average queue length (Lq ) = λ 2
Expected no of customers --------
in queue μ (μ - λ )
• Average number of units in the system (Ls) = λ
Expected no of customers in the system --------
μ - λ
• Average waiting time of an arrival (Wq) = λ
Expected waiting time in Queue ----------
μ(μ - λ )
• Average waiting time of an arrival in the system (Ws) = 1
Expected time spent by customer in system ---------
μ - λ

• Example 1
• Students arrive at the head office of Universal Teacher Publications
according to a Poisson input process with a mean rate of 40 per hour.
The time required to serve a student has an exponential distribution
with a mean of 50 per hour. Assume that the students are served by a
single individual, find the average waiting time of a student.
• Solution.
• Given
λ = 40/hour, μ = 50/hour
• Average waiting time of a student before receiving service (Wq)
• = 40
---------
50(50 – 40)
• 4.8 minutes

• Example 2
• New Delhi Railway Station has a single ticket counter. During the rush hours,
customers arrive at the rate of 10 per hour. The average number of customers that
can be served is 12 per hour. Find out the following:
• Probability that the ticket counter is free.
• Average number of customers in the queue.
• Solution.Given
λ = 10/hour, μ = 12/hour
• Probability that the counter is free =1 -10
-----
12
• =1/6
• Average number of customers in the queue (Lq ) =(10)2
--------
12 (12 - 10)
• =25/6

• Example 3
• At Bharat petrol pump, customers arrive according to a Poisson process with an average time of 5 minutes between arrivals. The service time is
exponentially distributed with mean time = 2 minutes. On the basis of this information, find out
• What would be the average queue length?
• What would be the average number of customers in the queuing system?
• What is the average time spent by a car in the petrol pump?
• What is the average waiting time of a car before receiving petrol?
• Solution.
• Average inter arrival time =1
---
λ= 5minutes =1
---
12hourλ = 12/hour Average service time =1
---
μ= 2 minutes =1
---
30hourμ = 30/hour Average queue length, Lq =
• (12)2
-----------
30(30 - 12)=4
---
15 Average number of customers, Ls =12
-------
30 - 12
• =2
----
3 Average time spent at the petrol pump =1
----------
30 - 12=3.33 minutes Average waiting time of a car before receiving petrol =12
---------
30(30 - 12)=1.33 minutes

• Example 4
• Universal Bank is considering opening a drive in window for customer service. Management estimates that customers will
arrive at the rate of 15 per hour. The teller whom it is considering to staff the window can service customers at the rate of
one every three minutes.
• Assuming Poisson arrivals and exponential service find
• Average number in the waiting line.
• Average number in the system.
• Average waiting time in line.
• Average waiting time in the system.
• Solution.
• Given
λ = 15/hour,
μ = 3/60 hour
or 20/hour
• Average number in the waiting line =(15)2
----------
20(20 - 15)=2.25 customers Average number in the system =15
----------
20 - 15=3 customers Average waiting time in line =
• 15
------------
20(20 - 15)=0.15 hours Average waiting time in the system =1
---------
20 - 15=0.20 hours

• The number of arrivals in a given time interval has Poisson distribution,
• Inter-arrival times can be shown to have the exponential distribution.
• ƛ = Average arrival rate : Average no. of customers arriving in one unit of
time.
• 1/ ƛ = Average time between arrivals.= Average inter arrival time
• E.g: 0 min (hr) 1min(hr)
• |------2---------| average arrival rate= ƛ= 2
• average time arrivals = 1/ ƛ =1/2 min per customer
E.g: One arrival comes every 15 minutes.
Average arrival rate, ƛ =60/15 = 4 arrivals per hour.
Average inter arrival time = 1/ƛ = ¼ or 0.25 hour.
E.g: Three arrivals occur every 6 minutes. i.e 1 arrival every 2 mins
Average arrival rate, ƛ =60/2 = 30 arrivals per hour. .
Average Inter-arrival time, = 1/ƛ = 1/30 or 0.3333 hour. .

• Ex. Determine average arrival rate ƛ and average inter arrival time in hours.
1. one arrival occurs every 10 mins
Sol: ƛ = average arrival rate =60/10 = 6 arrivals per hour.
1/ ƛ = average inter arrival time = 1/6 hour. = 0.1666 hour
2. 2 arrival occurs every 2 min
Sol: i.e 1 arrival every 1 min
ƛ = average arrival rate = 60/1= 60 arrivals per hour
3. Number of arrivals in a 30 min period is 10
Sol
; 10 in 30 min i.e 1 arrival in 3 mins
: ƛ = average arrival rate =60/3 = 20 arrivals per hour.

• µ= (mue) Average number of customers being served in one unit of time
at a service station.
• 1/µ = Average time taken to service a customer .

DOC-20231208-WA0013..pptx mba hrd for business law

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