Team Members
Clement Hudson --Tularosa High School
Albert Simon -- Alamogordo High School
Margaret Suzukida--Alamogordo High School
Clare Riker Tinguely--Alamogordo High School
THE QUEUEING THEORY
Tulie Basin Dream Team
Alamogordo and Tularosa High
Schools
Executive Summary
• Queueing Theory - mathematical technique
cin >>
• Low traffic intensity - # of customers is low
• Bursty process - long queues build up
cout <<
• Output - analysis of scheduling matrices
Problem Statement
• Standing in line (queue) = fact of life!
• Managers need a way to optimize
scheduling of employees
• Program results in a number assigned to
Time in system AND Time in Queue
Number of customer in line AND in system
based on arriving and exiting
Secondary
Other
Services
customers
Wait for service
External
Numbers exiting into
system
Schedules
employees
Numbers coming into
system
Schedules
employees
Manager
(primary)
Waits in line
Services
customers
Employee
(secondary)
Customer
(other)
Count customers
Record sales;
provide info
$$
Cash registers
(external hardware)
1, 2, 3, ….
Counting device
(external hardware)
EMPLOYEES
Not a singleton. Entity because they are within
the modeled system.
Behaviors: Service customers; follow a schedule.
CUSTOMERS
Not a singleton. Boundary because they enter
into business from outside and determine
queue lengths.
Behaviors: Determine length of queue.
QUEUE
Not a singleton. Entity because they are
within the modeled system.
Behaviors: Length determined by no. of customers.
MANAGER
Singleton. Boundary because they enter into
business from outside to only review numbers
for scheduling. They will probably not be
programmed. Once the program is in place,
you may not need a manager!
Behaviors: Schedules employees.
Employees
Customers
Name or SSN
Assigned number
Service customers
Wait for customers
End their shift
Get in line
Wait in line
Receive service
Queue
Manager
No. of queues determined
by no. of employees
Length determined by
number of customers
Singleton
Schedules employees
*
EMPLOYEES
service >
< receive service from *
CUSTOMERS
< No. depends on *
QUEUES
MANAGER 1
customer
Gets in line ( )
Queue position
n
Queue position
3
Queue position
2
Queue position
1
Queue position Register () employee
0
X
Services customers ( )
EMPLOYEE
[No customers]
[Customers
appear or
leave]
[Customers
are present]
Servicing Customers
[End of Shift]
Ending of Shift
Waiting for
Customers
CUSTOMER
[Moving Ahead]
[Purchasing]
Receiving Service
[Front of queue]
Waiting
[Front of queue]
[No purchasing]
Getting in line
[Leaving]
Leaves store
customer
Wants
to Buy
No purchase
Purchase
[lines too long!]
Gets in a queue
Position n
Position 3
Position 2
Position 1
Gets rung up
Method
• Code performs a summation using a loop
until the numbers converge or diverge
• C++, Markovian equations, Poisson arrival
equations
• Computes such statistics as: time customer
spends in the queue and in the system,
expected line length and no. of customers
Variables - 1
•
•
•
•

µ

n
• p
• p(0) =
• p(n) =
arrival rate
service rate
traffic intensity ( / µ ) [< 1]
number of customers in system
(queue + service)
steady state probabilities
1- 
(1 - ) ( ^ n), n = 1, 2, 3….
Variables - continued
W(q)
W(q)
=
expected time in queue
E [ time in queue]
 / [ ( -1)]
=
time in system
E [time in system]
1 / ( - )
=
line length
E[line length]
 ^ 2 / [ ( - )]
=
Number of customers
E[number in system]
 / ( - )
W
W
L (q)
L (q)
L
L
L

 (q)
(w)
ln
Variables - continued
percentile
high percentiles for time in queue
high percentiles for time in system
the natural logarithm
 (q) [90] = W * ln (10 * )
 (q) [95] = W * ln (20 *  )
 (w) [90) = W * ln (10)
 (w) [95] = W * ln (20)
Original Achievement
• Meeting the challenge of doing the project
in two weeks
• Very useful in understanding what our
students experience
• Outcomes from the queueing calculations
may help in lessening waiting times
Strengths and Weaknesses
• Weaknesses:
– Time constraints; project can be expanded to
look at other scenarios.
– Poisson arrival rate - exponential
– Deterministic
• Strengths: Team-building skills which
resulted from cooperative learning
environment of the class.
Results of Our Program
• Using a series of math formulae, enhance
decision-making processes for employee
scheduling
• M/M/1 steady state model
• Wq, W, Lq, L were calculated to give time in
a queue, time in system, line length, and
number in system
Conclusions
• Organizations are ubiquitous.
• Anytime you are serviced, you must queue.
• We wanted to know what happened during
peak periods (high traffic intensity) in order to
properly schedule employees.
• The program outputs expected length of queue,
time in system, length of line (hours), and the
amount of time waiting to be serviced.
Resources/Bibliography
Bolch, G., Greiner, S., de Meer, H., and K.S. Trivedi. (1998). Queuing Networks
and Markov Chains: Modeling and Performance Evaluation with Computer
Science Applications.
Davis, William S. and Yen, David C. (1999). Systems Analysis and Design:
Information System Consultant’s Handbook.
Enns, S.T. (1999). A simple spreadsheet approach to understanding work
flow in production facilities. Total Quality Management.
Glass, Victor and Cahn, Ellen S. (1997). A queueing model of organization
structure. Journal of Business and Economic Studies 3, 13-28.
He, Qu-Ming & Marcel F. Neuts. (1997). On Episodic Queues. Society for
Industrial and Applied Mathenatucs.
Robertazzi, Thomas G. (1994). Computer Networks and Systems: Queueing
Theory and Performance Evaluation.
Standard Template Library. (1999). http://www.la.unm.edu:8001/cs259/stl_doc/
Many Thankx!!!
• Nolan Gray, Mike
Fisk, Shaun Cooper
• for support and
because they’re the judges :).
• NMSU & LANL for
venue, labs & $$$$.
and especially:
• Karin, Sharon, Chris,
Gina, and David without
whom this extraordinary fun
would not have been possible.
• All the little people
(students), and Miles,
too, with whom we
now empathize and
who will provide the
coding to the next
generation’s problems.
We can now help them
more intelligently.