QUEUING ANALYSIS
MNG221 - Management Science –
Queuing Analysis
Introduction
Queuing analysis is the probabilistic
analysis of waiting lines.
• Because time is a valuable resource,
the reduction of waiting time is an
important topic.
• Providing quick service is an important
aspect of quality customer service.
Queuing Analysis
Introduction
• Aware of this, more and more
companies are focusing on reducing
waiting time as an important
component of quality improvement.
• Increased service capacity will reduce
waiting time and provide faster service,
which usually means adding more
servers.
Queuing Analysis
Introduction
• However, increasing service capacity
has a monetary cost, and therein lies
the basis of waiting line analysis: the
trade-off between the cost of
improved service and the cost of
making customers wait.
Queuing Analysis
Elements of Waiting Line Analysis
Elements of Waiting Line Analysis
• Waiting lines form because people or
things arrive at the servicing function,
or server, faster than they can be
served.
• Waiting lines result because customers
do not arrive at a constant, evenly
paced rate, nor are they all served in an
equal amount of time.
Elements of Waiting Line Analysis
• Thus, a waiting line is continually
increasing and decreasing in length
(and is sometimes empty), and it
approaches an average rate of
customer arrivals and an average time
to serve the customer in the long run.
Elements of Waiting Line Analysis
• Decisions about waiting lines and the
management of waiting lines are based
on the average customer arrival and
service time.
• Operating Characteristics are average
values for characteristics that describe
the performance of a waiting line
system:
Elements of Waiting Line Analysis
• Example: The average time a customer
must wait in line, the average number
of customer waiting in line etc.
NOTE: The operating characteristics of a
queuing system are steady states.
Queuing Analysis
The Single-Server
Waiting Line System
The Single-Server Waiting Line System
The Fast Shop Market waiting line system
Components of a waiting line system include arrivals, servers, and the waiting line structure.
The Single-Server Waiting Line System
The most important factors to consider in
analyzing a queuing system:
1. The Queue Discipline - is the order in
which waiting customers are served.
(In what order customers are served?)
The Single-Server Waiting Line System
1. The Queue Discipline
Example:
Customers - can be processed on a First
Come, First Serve, A Predetermine
Appointment, or Alphabetically according
to their last name.
A machine queue discipline, however can
have “last-in, first-out” or “random
selection”.
The Single-Server Waiting Line System
The most important factors to consider in analyzing a
queuing system:
2. The nature of the calling population
(where customers come from).
The calling population is the source of
the customers, which may be finite or
infinite.
 Queuing systems that have an assumed
infinite calling population are more common.
The Single-Server Waiting Line System
The most important factors to consider in analyzing a
queuing system:
3. The arrival rate (how often customers
arrive at the queue). The arrival rate (λ)
is the frequency at which customers
arrive at a waiting line during a specified
period of time according, to a probability
distribution and is most frequently
described by a Poisson distribution.
The Single-Server Waiting Line System
The most important factors to consider in analyzing a
queuing system:
This rate can be estimated from
empirical data derived from studying
the system or a similar system, or it
can be an average of these empirical
data.
The Single-Server Waiting Line System
The most important factors to consider in analyzing a
queuing system:
4. The service rate (how fast customers
are served). The service rate (μ) is the
average number of customers who
can be served during a specified time
period and is often described by the
negative exponential distribution.
The Single-Server Waiting Line System
The most important factors to consider in analyzing a
queuing system:
A service rate is similar to an arrival
rate in that it is a random variable.
However, to analyze a queuing system,
both arrivals and service must be in
compatible units of measure. Thus, service
time must be expressed as a service rate
to correspond with an arrival rate.
The Single-Server Waiting Line System
• Poisson Distribution
a probability distribution that describes the
occurrence of a relatively rare event in a
fixed period of time; often used to define
arrivals at a service facility in a queuing
system
• Exponential Distribution
a probability distribution often used to
define the service times in a queuing system
Single-Server (SS) Model
Assumptions of the basic SS Model
• An infinite calling population
• A first-come, first-served queue discipline
• Poisson arrival rate
• Exponential service times
NOTE: customers must be served faster
than they arrive (λ>μ) , or an infinitely
large queue will build up.
Single-Server (SS) Model
Assumptions of the basic SS Model
Given that
• λ = the arrival rate (average number of arrivals per
time period)
• µ = the service rate (average number served per time
period)
• λ = mean arrival rate;
• µ = mean service rate;
We can state the following formulas for the operating
characteristics of a single-server model.
Single Server Queuing Formulas
The probability that no customer is in the queuing
system (either in the queue or being served)


P0  1  


The probability that n customer are in the
queuing system
n
n

 

Pn    * P0    1  


 
Single Server Queuing Formulas
The average number of customers in the
queuing system (i.e., customers being served
and in the waiting line)
  

L  
 
The average number of customers in the waiting
line is
 2


Lq  
  (   ) 


Single Server Queuing Formulas
The average time a customer spends in the total
queuing system (i.e., waiting and being served) is:
1
L
W

  
The average time a customer spends waiting in
the queue to be served is:




Wq  
  (   ) 
Single Server Queuing Formulas
The probability that the server is busy (i.e., the
probability that a customer has to wait), known
as the utilization factor, is:

U 

The probability that the server is idle (i.e., the
probability that a customer can be served) is:

I 1U 1

Single Server Queuing Formulas
• This last term, 1 - (λ /µ), is also equal to P0.
That is, the probability of no customers in the
queuing system is the same as the probability
that the server is idle.
• We can compute these various operating
characteristics for Fast Shop Market by simply
substituting the average arrival and service
rates into the foregoing formulas.
Queuing Analysis
The Single-Server
Waiting Line System
A Worked Example
Single Server Queuing
A Worked Example
If λ = 24 customers per hour arrive at checkout counter
µ = 30 customers per hour can be checked out
Single Server Queuing
A Worked Example
If λ = 24 customers per hour arrive at checkout counter
µ = 30 customers per hour can be checked out
Single Server Queuing
A Worked Example
If λ = 24 customers per hour arrive at checkout counter
µ = 30 customers per hour can be checked out
Single Server Queuing
A Worked Example
If λ = 24 customers per hour arrive at checkout counter
µ = 30 customers per hour can be checked out
Single Server Queuing
A Worked Example
Important Aspects of General Model and Example
• The operating characteristics are averages.
• They are assumed to be steady-state averages.
• Steady state is a constant average level that a
system realizes after a period of time.
• For a queuing system, the steady state is
represented by the average operating
statistics, also determined over a period of
time.
Queuing Analysis
The Effect of Operating Characteristics
on Managerial Decisions
Effects of Operating Characteristics
on Managerial Decisions
• Alternative 1: The addition of an
employee
• Alternative 2: Addition of a new check
out counter
Effects of Operating Characteristics on
Managerial Decisions
Alternative 1: The addition of an
employee
• The addition of an extra employee will
cost the store manager $150 per week.
• For each minute that average customer
waiting time is reduced, the store
avoids a loss in sales of $75 per week.
Effects of Operating Characteristics on
Managerial Decisions
Alternative 1: The addition of an employee
• If a new employee is hired, customers can be
served in less time, that is the service rate
will increase to
µ = 40 customers served per hour
• The arrival rate will remain the same (λ = 24
per hour)
λ = 24 customers per hour arrive at
checkout counter
Effects of Operating Characteristics on
Managerial Decisions
Effects of Operating Characteristics on
Managerial Decisions
Alternative 1: The addition of an employee
• The average waiting time per customer
has been reduced from 8 minutes to 2.25
minutes.
• The savings (that is, the decrease in lost
sales) is computed as follows:
8.00 min. - 2.25 min. = 5.75 min.
5.75 min. x $75/min. = $431.25
Effects of Operating Characteristics on
Managerial Decisions
Alternative 1: The addition of an
employee
• Because the extra employee costs
management $150 per week, the total
savings will be:
$431.25 - $150 = $281.25 per week
Effects of Operating Characteristics on
Managerial Decisions
Alternative 2: The addition of a new
checkout counter
• Constructing a new checkout counter
will cost $6,000, plus an extra $200 per
week for an additional cashier.
• The new checkout counter would be
opposite the present counter.
Effects of Operating Characteristics on
Managerial Decisions
Alternative 2: The addition of a new checkout counter
• There would be several display cases
and racks between the two lines so that
customers waiting in line would not
move back and forth between the lines.
• Such movement, called jockeying, would
invalidate the queuing formulas we
already developed.
Effects of Operating Characteristics on
Managerial Decisions
Alternative 2: The addition of a new
checkout counter
• We will assume that the customers
would divide themselves equally
between the two lines, so the arrival
rate for each line would be half of the
prior arrival rate for a single checkout
counter.
Effects of Operating Characteristics on
Managerial Decisions
Alternative 2: The addition of a new checkout counter
• Thus, the new arrival rate for each checkout
counter is:
λ = 12 customers per hour arrive at checkout
counter
And the service rate remains the same for
each of the counters:
µ =30 customers per hour can be checked out
Effects of Operating Characteristics on
Managerial Decisions
P0 = .60 probability of no customers in the system
L = 0.67 customer in the queuing system
Lq = 0.27 customer in the waiting line
W = 0.055 hr. (3.33 min.) per customer in the system
Wq = 0.022 hr. (1.33 min.) per customer in the waiting line
U = .40 probability that a customer must wait
I = .60 probability that a server will be idle and a customer can be served
Effects of Operating Characteristics on
Managerial Decisions
• Using the same sales savings of $75 per week
for each minute's reduction in waiting time,
we find that the store would save:
8.00 min. - 1.33 min. = 6.67 min.
6.67 min. x $75/min. = $500.00 per week
• Next we subtract the $200 per week cost for
the new cashier from this amount saved:
$500 - 200 = $300
Effects of Operating Characteristics on
Managerial Decisions
• Because the capital outlay of this project is
$6,000, it would take 20 weeks ($6,000/$300 =
20 weeks) to recoup the initial cost (ignoring
the possibility of interest on the $6,000).
• Once the cost has been recovered, the store
would save $18.75 ($300.00 - 281.25) more
per week by adding a new checkout counter
rather than simply hiring an extra employee.
Effects of Operating Characteristics on
Managerial Decisions
• However, we must not disregard the fact
that during the 20-week cost recovery
period, the $281.25 savings incurred by
simply hiring a new employee would be
lost.
Operating Characteristics for each
Alternative System
Operating
Characteristics
Present
System
Alternative I
Alternative II
L
4.00 customers 1.50 customers 0.67 customer
Lq
3.20 customers 0.90 customer 0.27 customer
W
10.00 min.
3.75 min.
3.33 min.
Wq
8.00 min.
2.25 min.
1.33 min.
U
.80
.60
.40
Queuing Analysis
Undefined and Constant Service Times
Single-Server (SS) Model
Assumptions of the basic SS Model
• An infinite calling population
• A first-come, first-served queue discipline
• Poisson arrival rate
• Exponential service times
NOTE: customers must be served faster
than they arrive (λ>μ) , or an infinitely
large queue will build up.
Single-Server (SS) Model
Assumptions of the basic SS Model
Given that
• λ = the arrival rate (average number of arrivals per
time period)
• µ = the service rate (average number served per time
period)
• λ = mean arrival rate;
• µ = mean service rate;
We can state the following formulas for the operating
characteristics of a single-server model.
Undefined and Constant Service Times
• Constant service times occur with
machinery and automated equipment.
• Constant service times are a special case
of the single-server model with general,
or undefined service times.
Undefined and Constant Service Times
• The basic queuing formulas:
Undefined and Constant Service Times
• Employees arrive randomly to use the
fax machine, at an average rate of 20
per hour, according to a Poisson
distribution.
• The time an employee spends using the
machine is not defined by any
probability distribution but has a mean
of 2 minutes and a standard deviation of
4 minutes.
Undefined and Constant Service Times
λ = 20
µ = 30
σ = 1/15
Undefined and Constant Service Times
• In the case of constant service times,
there is no variability in service times
(i.e., service time is the same constant
value for each customer); thus, s = 0.
• Substituting s = 0 into the undefined
service time formula for Lq results in the
following formula for Lq with constant
service times:
Undefined and Constant Service Times
Undefined and Constant Service Times
• Example - The Petroco Service Station has an automatic car
wash, and can accommodate one car at a time.
• It requires a constant time of 4.5 minutes for a wash.
• Cars arrive at the car wash at an average rate of 10 per hour
(Poisson distributed).
• The service station manager wants to determine the average
length of the waiting line and the average waiting time at the
car wash.
• First, determine λ and µ such that they are expressed as rates:
• λ = 10 cars per hr.
• µ = 60/4.5 = 13.3 cars per hr.
Undefined and Constant Service Times
• Example - Substituting λ and µ into the queuing formulas
for constant service time,
Queuing Analysis
Finite Queue Length
Finite Queue Length
• In a finite queue, the length of the queue is
limited, because space may permit only a
limited number of customers to enter the
queue.
• Such a waiting line is referred to as a finite
queue and results in another variation of the
single-phase, single-channel queuing model.
• The service rate does not have to exceed the
arrival rate (µ > λ) in order to obtain steadystate conditions.
Finite Queue Length
• The resulting operating characteristics, where M is the
maximum number in the system, are as follows:
Finite Queue Length
• Example - Metro Quick Lube, a one-bay service facility
located next to a busy highway has space for only one
vehicle in service and three vehicles lined up to wait for
service.
• There is no space for cars to line up on the busy adjacent
highway, so if the waiting line is full (three cars),
prospective customers must drive on.
• The mean time between arrivals is 3 minutes, that is λ =
20,
• The mean service 2 minutes, that is µ = 30
• The maximum number of vehicles in the system is four,
that is M = 4
Finite Queue Length
• Example
• First, we will compute the probability that the system is
full and the customer must drive on, PM.
• However, this first requires the determination of P0, as
follows:
Finite Queue Length
• Example
• Next, to compute the average queue length, Lq, the
average number of cars in the system, L, must be
computed, as follows:
Finite Queue Length
• Example
• To compute the average waiting time, Wq, the average
time in the system, W, must be computed first:
Queuing Analysis
Finite Calling Population
Finite Calling Population
• For some waiting line systems there is a
specific, limited number of potential
customers that can arrive at the service
facility. This is referred to as a finite calling
population.
• A finite calling population has the following
set of formulas for determining operating
characteristics. λ in this model is the arrival
rate for each member of the population:
Finite Calling Population
Finite Calling Population
• Example - Wheelco Manufacturing operates a shop
that includes 20 machines. When a machine breaks
down, it is tagged for repair, with the date of the
breakdown noted and a repair person is called.
• The company has one senior repair person and an
assistant, who repair the machines in the same
order in which they break down (a first-in, first-out
queue discipline).
• Machines break down according to a Poisson
distribution, and the service times are exponentially
distributed.
Finite Calling Population
• Example The finite calling population for this
example is the 20 machines in the shop, which we
will designate as N.
• Each machine operates an average of 200 hours
before breaking down and a repair person is
called.
• The average time to repair a machine is 3.6 hours.
• The company would like an analysis of machine
idle time due to breakdowns to determine
whether the present repair staff is sufficient
Finite Calling Population
• Example
These results show that
the repair person and
assistant are busy 35%
of the time repairing
machines. Of the 20
machines, an average
of .52, or 2.6%, are
broken down, waiting
for repair, or under
repair. Each brokendown machine is idle
(broken down, waiting
for repair, or under
repair) an average of
5.33 hours. Thus the
system
seems
adequate.
Queuing Analysis
Multiple-Server Waiting Line
Multiple-Server Waiting Line
Multiple-server Models – two or more independent server
in parallel serve a single waiting line
• Examples of this type of waiting line include an airline ticket and
check-in counter where passengers line up in a single line, waiting
for one of several agents for service, and a post office line, where
customers in a single line wait for service from several postal clerks.
Multiple-Server Waiting Line
• The formulas for a Multiple Server model,
like single-server model formulas, have
been developed on the assumption of
–a first-come, first-served queue
discipline,
–Poisson arrivals,
–exponential service times,
–and an infinite calling population.
Multiple-Server Waiting Line
The parameters of the multiple-server model are as
follows:
• λ = the arrival rate (average number of arrivals per
time period)
• μ = the service rate (average number served per
time period) per server (channel)
• c = the number of servers
• cμ = the mean effective service rate for the system,
which must exceed the arrival rate
• cµ > λ : the total number of servers must be able
to serve customers faster than they arrive.
Multiple-Server Waiting Line
• Formulas - The probability that there are no customers in
the system (all servers are idle) is:
• The probability of n customers in the queuing system is
Multiple-Server Waiting Line
• Formulas - The average number of customers in the
queuing system is
• The average time a customer spends in the queuing
system (waiting and being served) is
Multiple-Server Waiting Line
• Formulas - The average number of customers in the
queue is
• The average time a customer spends in the queue,
waiting to be served, is
Multiple-Server Waiting Line
• Formulas - The probability that a customer arriving in the
system must wait for service (i.e., the probability that all
the servers are busy) is:
• In the foregoing formulas, if c = 1 (i.e., if there is one
server), then these formulas become the single-server
formulas
Multiple-Server Waiting Line
• Example - Biggs Department Store customer service
department of the store has a waiting room in which
chairs are placed along the wall, in effect forming a single
waiting line.
• Customers come to this area with questions or
complaints or to clarify matters regarding credit card bills.
• The customers are served by three store representatives,
each located in a partitioned stall.
• Customers are served on a first-come, first-served basis.
• The store management wants to analyze this queuing
system because excessive waiting times can make
customers angry enough to shop at other stores.
Multiple-Server Waiting Line
• Example - Let us assume that a survey of the
customer service department for a 12-month period
shows that the arrival rate and service rate are as
follows:
• λ = 10 customers per hr. arrive at the service
department
• µ = 4 customers per hr. can be served by each store
representative
• In addition, this is a three-server queuing system;
therefore,
• c = 3 store representatives
Multiple-Server Waiting Line
Multiple-Server Waiting Line
• Substituting this value along with l and µ into our queuing
formulas results in the following operating
characteristics:
• P0 = .073 probability that no customers are in the service
department
• L = 3.0 customers, on average, in the service department
• W = 0.30 hr. (18 min.) average time in the service
department per customer
• Lq = 0.5 customer, on average, waiting to be served
• Wq = 0.05 hr. (3 min.) average time waiting in line per
customer
• Pw = .31 probability that a customer must wait for service