Presented by :Shinki jalhotra

Queuing Theory • Queuing theory is the mathematics of waiting

lines.

• It is extremely useful in predicting and evaluating system performance.

• Queuing theory has been used for operations research, manufacturing and systems analysis. Traditional queuing theory problems refer to customers visiting a store, analogous to requests arriving at a device.

Queuing problems arises because either …There is too much demand on the facilities

There is too less demand

(Much waiting time or inadequate Number of service facilities)

(Much idle facility time or too Many facilities)

The problem is to either schedule arrivals or provide extra facilities or both so as to obtain an optimum balance between costs associated with waiting time and idle time .

Basic elements of Queuing System • Entries or customers• Queue (waiting lines) • Service channels or service facility

Basic Structure

Arrival pattern •Customers arrive in scheduled or random fashion.

• The time duration between each customers’ arrival is known as inter arrival time. We assume it to follow Poisson Distribution

Poisson Distribution p

roba

bili

ty

Arrival per unit time(λ)

Service Pattern

Exponential Distribution

•Number of servers and speed of service to be considered.

•The time taken by a server to service a customer is known as Service Time. It is represented by Exponential Distribution

pro

babi

lity

Customer served per unit time (μ)

Assumption:λ<μ

Service Channels

• Single channel queuing system

• Multi channel queuing system

• Single channel multi phase system • Multi channel multi phase system

Service Discipline

• FCFS (First-Come-First-Served)• LCFS (Last-Come-First-Served)• Service in random order • Priority service

System Capacity

• Maximum number of customers that can be accommodated in the queue.

• Assumed to be of infinite capacity.

Customer’s Behavior

• Balking- When a customer leaves the queue because it is too long, has no time to wait, no space to stand etc.

• Reneging- When a customer leaves the queue because of his impatience.

• Jockeying- When a customer shifts from one queue to another.

Service Facility Behavior•Failure: A server may fail while serving a customer, thereby interrupting service untill a repair can be made.•Changing service rate: A server may speed up or slow down, depending on the number of customers in the queue.•Batch processing: A server may service several customers simultaneously.

Waiting and Idle time costs

Cost of waiting customers Cost of idle service facility

• Indirect cost of business loss • Direct cost of idle equipment

or person.

• Payment to be made to the servers(engaged at the facilities),for the period for which they remain idle.

The optimum balance of costs can be made by scheduling the flow of units or providing proper number of service facilities .

Tot

al e

xpec

ted

cost

of

oper

atin

g fa

cilit

y

Increased service

Total expected cost

Cost of providing service

Waiting time cost s

Total expected cost = waiting time cost + cost of providing service

Let Cw = expected waiting cost/unit time Ls= expected no. of units Cf = cost of servicing one unit

Expected waiting cost= Cw. Ls

=Cw.[λ /(μ-λ )]Expected service cost/unit time= Cf.μ Total cost , C = Cw. [λ/(μ-λ)] +μCf

This will be minimum if d/dμ(C)=0Or if -Cw.[λ/(μ-λ)²]+Cf=0 ,Which gives ..μ=λ±sqrt(Cw .λ/Cf)

Transient & Steady State of the system• When the operating characteristics are

dependent on time, it is said to be a transient state.

• When the operating characteristics are independent of time, it is said to be a steady state.

Traffic intensity

The ratio λ/μ is called the traffic intensity or the utilization factor and it determines the degree to which the capacity of service station is utilized.

Applications of Queuing Model

• Telecommunications• Traffic control • Determining the sequence of computer

operations.• Predicting computer performance• Health services (e.g.. control of hospital bed

assignments)• Airport traffic, airline ticket sales• Layout of manufacturing systems.

Thank you