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.