Queuing
Theory
Queuing Theory
Waiting Line Theory
Developed by A K Erlang
Analysis for Telephone Congestion(1909)
Applicable when:
(i) Costumer
(ii) Service Station
(iii) Wait occasionally
(iv) Leave the system
“Arrival-Departure” Problem
Queuing Theory Examples
• Deposit Electricity Bill
• Electronic Message
• Computer Programme
• Machine Repair
• Aeroplanes Landing
• Hospital patients
• Dentist Clinic
• Flow of Ships to offshore
• Fire fighting
• Calls to police helpline
Causes
• Waiting line Develop as service may not be
rendered immediately
• Increase in service capacity required but costly
affair
• Achievement of an economic balance
between cost of service providing and waiting
cost
General Structure
Arrival Process
a) Acc. to Source: finite or infinite
b) Acc. to Numbers: individual or group
c) Acc. to Time: Deterministic or
Probabilistic
Service Systems
1.Structure of Service System
a) A single service system
b) Multiple, parallel facilities with single Queue
c) Multiple, parallel facilities with multiple Queue
d) Service facilities in a series
2.Speed of Service
a) Service time
b) Service rate
Queue Structure
First Come- First Served
Last Come-First Served
Service in Random Order
Priority Service
Kinds of Costumers
• Jockeying- Switch to other queue
• Reneging- standing in queue and leaves
• Balking-come but not joins & leaves
Operating Characteristics
i) Queue Length
ii) System Length
iii) Waiting time in the Queue
iv) Total Time in the System
v) Server Idle Time
Models of Queuing Theory
Deterministic Queuing Model
• Customers arrival regularly and the
service time is known and constant
• µ: service rate
• λ: arrival rate
• If λ> µ waiting line infinitely increase
• If λ≤µ no waiting, idle time will be (1- λ/µ)
• λ/µ=ρ
is called average utilization or traffic intensity
• If ρ>1 system would ultimately fail
• If ρ ≤1 it works & ρ is the time it is busy
Probabilistic Model
• Prob. of n customers in the system
Pn = ρn(1-ρ)
P4 = ρ4(1-ρ)
P6 = ρ6(1-ρ)
λ/µ=ρ
• Expected no. of Customers in the System
Ls= ρ/1-ρ = λ/(µ-λ)
• Expected no. of Customers in the queue
Lq= ρ2/1-ρ =λ2/µ(µ-λ)
Exp. Length of non empty queue
Lq' = 1/1-ρ = µ/(µ-λ)
• Exp. Waiting time in queue
Wq = λ/µ(µ-λ)
• Exp. time a customer spends in the
system
Ws = 1/(µ-λ)
Utilization Factor
λ µ λ/µ Lq=ρ2/1-ρ
=λ2/µ(µ-λ)
5 10 0.5 0.50
6 10 0.6 0.90
7 10 0.7 1.63
8 10 0.8 3.20
9 10 0.9 8.10
9.5 10 0.95 18.05
9.8 10 0.98 48.02
Computer maintenance
Contract
Cost of unavailable Rs. 8000
Arrival 3 comp per month
Anne Comp. Safi Comp.
Rs. 3000 Rs. 5000
Repair per month 5 comp 3 comp
Ls 1.5 1
Cost of off road1.5*8000 1*8000
Cost of Contract Rs. 3000 Rs.5000
Total Cost Rs.15000 Rs.13000
• If λ=6 µ=10
• P(0-5 customers during period 15 min)=
e-m( mn/n!)
• Prob. of 0-5 customers in the queuing
system Pn = ρn(1-ρ)
• e= 2.7183
• m= λT
• n = number of customers
• e-m( mn/n!)
On an average 5 customers reach a dentist
clinic every hour. Determine the probability
that exactly 2 customers will reach in 30
minutes.
• T=30 mins. = 0.5 hours
• m= λT
• λ =5
• e= 2.7183
• e-m( mn/n!)
• P(2 customers)=0.257
• The Burger Dome fast-food restaurant operates a single-
channel service facil-ity for its customers. A single Burger
Dome employee takes the customer’s order, determines the
total cost of the order, takes the money from the cus-tomer, and
then fills the order.
• Once a customer’s order is filled, the employee takes the order
from the next customer waiting for service. The Poisson
probability distribution describes the customer arrivals.
• The mean arrival rate is 0.75 customers per minute. The
exponential probability distribution describes the service times.
• The mean service rate is one customer per minute.
• For economic considerations, Burger Dome evaluates customer
waiting time at $10 per hour while the service channel can be
operated for $7 per hour.
• Management is interested in developing a better
understanding of the operating characteristics of the
food-service waiting line.
Specifically, management would like to know :-
The probability that no customers are in the system,
the average number of customers waiting for service,
The average time a customer waits for service,
The probability that an arriving customer has to wait, and
The hourly cost of the service operation.
