Queuing Theory
Introduction
Queuing theory is one of the most widely used
quantitative analysis techniques. The three basic
components are:
 Arrivals
 Service facilities
 Actual waiting line
Waiting Line Costs
Queuing analysis includes:
 Determining the best level of service for an
organization.
 Analyzing the trade-off between cost of providing
service and cost of waiting time.
 Finding the service level that minimizes the total
expected cost.
Queuing Costs and
Service Levels
Optimal Service
Cost of Operating
Service Facility
Level
Total
Expected
Cost
Cost of
Providing
Service
Cost of
Waiting
Time
Service Level
Characteristics of a Queuing System
 Arrival Characteristics
Size of the calling population
Pattern of arrivals
Behavior of arrivals
 Waiting Line Characteristics
Queue length
Queue discipline
 Service Facility Characteristics
Configuration of the queuing system
Service time distribution
Arrival Characteristics of a Queuing
System
 Calling Population:
 Unlimited (infinite)
 Limited (finite)
 Arrival Pattern
 Randomly
 Poisson Distribution
Arrival Characteristics: Poisson
Distribution
0 1 2 3 4 5 6 7 8 9 10 1
1
X
.30
.25
.20
.15
.10
.05
.00
P(X),  = 4
P(X)
.35
.30
.25
.20
.15
.10
.05
.00
P(X)
P(X)
P(X),  = 2
e  x

X!
0 1 2 3 4 5 6 7 8 9 10 1
1
X
Arrival Characteristics of a Queuing System
(continued)
Behavior of arrivals:
 Join the queue, and wait till served.
 Balk; refuse to join the line.
 Renege; leave the line.
 Jockeying; move from one queue to another
queue.
Waiting Line Characteristics of a Queuing
System
Waiting Line Characteristics:
 Length of the queue
 Limited
 Unlimited (assumed)
 Service priority/Queue discipline
 FIFO (assumed)
 Other
Waiting Line Characteristics of a Queuing
System (continued)
Service Facility Characteristics
 Number of channels (servers)
 Single
 Multiple
 Number of phases in service system (customer
stations)
 Single (1 stop)
 Multiple (2+ stops)
 Service time distribution
 Negative exponential
 Other
Service Characteristics: Queuing System
Configurations
Queue
Service
facility
arrivals
Single Channel, Single Phase
Service Facility
Queue
Facility
1
arrivals
Single Channel, Multi-Phase
Facility
2
Service Characteristics: Queuing System
Configurations
Service
facility 1
Queue
arrivals
Service
facility 2
Multi-Channel,
Single Phase
Queue
arrivals
Multi-Channel,
Multiphase Phase
Service
facility 3
Type 1
Service
Facility
Type 2
Service
Facility
Type 1
Service
Facility
Type 2
Service
Facility
Service Characteristics of a Queuing
System
Service Time Patterns:
 Negative exponential probability distribution
 Other distribution
Service Time Characteristics: Exponential
Distribution
f(x)  μe μx for x  0, μ  0
Probability
(for Intervals of 1 Minute)
μ  Average Number Served Per Minute
Average Service Time of 20 Minutes
Average Service Time of 1 Hour
30
60
90 120 150 180
X
Kendall Notation for Queuing Models
Kendall notation consists of a
basic three-symbol form.
Arrival
Service Time Number of Service
Distribution Distribution
Channels Open
Where,
M = Poisson distribution for the number of occurrences
(or exponential times)
D = Constant (deterministic rate)
G = General distribution with mean and variance known
Single channel with Poisson arrivals and
exponential service times
M/M/1
Assumptions: M/M/1 Model
1. Queue discipline: FIFO
2. No balking or reneging
3. Independent arrivals; constant rate over time
4. Arrivals: Poisson distributed
5. Service times: average known
6. Service times: negative exponential
7. Average service rate > average arrival rate
Operating Characteristics of Queuing
Systems





Average time each customer spends in the queueW 
Average length of the queue L
Average time each customer spends in the system Ws
Average number of customers in the system
Probability that the service facility
will be idle L s

q
s
P0  1 

 Utilization factor for the system
 Probability of a specific number of customers inthe
system
Pn

   




1










n
Operating Characteristic Equations:
M/M/1
Average number in system, L s 
Average time in system, Ws 

 -
1
 -
2
Average number in queue, L q 
  -  

Average time waiting, Wq 
  -  
Utilizatio n Factor,  
Percent Idle, P0  1 
Probability the number of customers is > k,


Pn  k


 

 



k 1
Car Wash Example: M/M/1
 Assume you are planning a car wash to raise money
for a local charity.
 You anticipate the cars arriving in a single line and
being serviced by one team of washers.
 Based on historical data, you believe cars will arrive
every 30 minutes, and the team can wash a car in
about 20 minutes.
 The arrival rates follow a Poisson distribution and the
service rates are exponentially distributed.
 What are the operating characteristics for this
system?
Car Wash Example: Operating Characteristics

= 2 cars arriving per hour
μ = 3 cars serviced per hour
Ls
Ws
Lq
Wq

Po
=>
=>
=>
=>
=>
=>
? cars in the system on average
? hour that an average car
spends in the system
? cars waiting on average
? hours is average wait
? percent of time car washers are busy
? probability that there are 0
cars in the system
Car Wash Example: Operating Characteristics
Solution
 = 2 cars arriving per hour
μ= 3 cars serviced per hour
Ls =2/(3-2) =>
Ws
= 1/(3-2) =>
Lq = 2^2/3(3-2) =>
Wq = 2/3(3-2) =>
 = 2/3 =>
P(0) = 1 – (2/3) =>
2 cars in the system on average
1 hour that an average car spends in the
system
1.33 cars waiting on average
.67 hours is average wait
.67 percent of time washers are busy
.33 probability that there
are 0 cars in the system