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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
External
Schedules employees
Secondary
Services customers
Other
Wait for service
Numbers coming into system
Numbers exiting into system
Manager(primary)
Schedules employees
Employee(secondary)
Servicescustomers
Customer(other)
Waits in line
$$Cash registers
(external hardware)
1, 2, 3, ….Counting device
(external hardware)
Record sales;provide info
Count customers
EMPLOYEES Not a singleton. Entity because they are withinthe modeled system. Behaviors: Service customers; follow a schedule.
CUSTOMERS
QUEUE
MANAGER
Not a singleton. Boundary because they enterinto business from outside and determine queue lengths. Behaviors: Determine length of queue.
Not a singleton. Entity because they are within the modeled system.
Behaviors: Length determined by no. of customers.Singleton. Boundary because they enter intobusiness from outside to only review numbersfor scheduling. They will probably not beprogrammed. Once the program is in place,you may not need a manager!Behaviors: Schedules employees.
Employees
Name or SSN
Service customers Wait for customersEnd their shift
Customers
Assigned number
Queue
No. of queues determined by no. of employeesLength determined bynumber of customers
Manager
Singleton
Schedules employees
Get in line Wait in lineReceive service
EMPLOYEES
QUEUES MANAGER
CUSTOMERS
1
* service >
< receive service from *
* Determ
ine n o. of >
< N
o. depends on * < len
gth d
epen
ds on
no. o
f *
* N
umbe
rs de
term
ine l
engt
h of
>
customerQueue position
n
Queue position0
Queue position3
Queue position2
Queue position1
employeeRegister ()
X Services customers ( )
Gets in line ( )
EMPLOYEE
Servicing CustomersWaiting for Customers
[Customers appear or
leave][Customers are present]
[No customers]
Ending of Shift
[End of Shift]
[End of S
hift]
CUSTOMER
Getting in line Waiting
[Purchasing][Moving Ahead]
Receiving Service[No
pur
chas
ing]
[Front of queue]
[Leaving]
[Fro
nt o
f qu
eue]
Wants to Buy
customer
Purchase
No purchase
[lines too long!]
Leaves store
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
arrival rate
• µ service rate traffic intensity ( / µ ) [< 1]
• n number of customers in system (queue + service)
• p steady state probabilities
• p(0) = 1 - • p(n) = (1 - ) ( ^ n), n = 1, 2, 3….
Variables - continuedW(q) expected time in queue
E [ time in queue]W(q) = / [ ( -1)]
W time in systemE [time in system]
W = 1 / ( - )
L (q) line lengthE[line length]
L (q) = ^ 2 / [ ( - )]
L Number of customersL E[number in system]L = / ( - )
Variables - continued percentile (q) high percentiles for time in queue(w) high percentiles for time in system
ln 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
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 workflow in production facilities. Total Quality Management.
Glass, Victor and Cahn, Ellen S. (1997). A queueing model of organizationstructure. Journal of Business and Economic Studies 3, 13-28.
He, Qu-Ming & Marcel F. Neuts. (1997). On Episodic Queues. Society forIndustrial and Applied Mathenatucs.
Bolch, G., Greiner, S., de Meer, H., and K.S. Trivedi. (1998). Queuing Networksand Markov Chains: Modeling and Performance Evaluation with ComputerScience Applications.
Standard Template Library. (1999). http://www.la.unm.edu:8001/cs259/stl_doc/
Robertazzi, Thomas G. (1994). Computer Networks and Systems: QueueingTheory and Performance Evaluation.
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.