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A QUEUING MODEL SIMULATION

WITH DYNAMIC GRAPHICAL DISPLAY

Cyrus Michael Riddel 1

Library,

Naval Postgraduate School

Monterey, California 93940

TGRADUATE SCHOOL

Monterey, California

THESISA QUEUING MODEL SIMULATION

WITH DYNAMIC GRAPHICAL DISPLAY

Cyrus Michael Riddell

Thesis Advisor R. W. Butterworth

March 1973

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T153548

A Queuing Model Simulation

with Dynamic Graphical Display

Cyrus Michael RiddellLieutenant, United States Navy

B.A., University of Washington, 1965

Submitted in partial fulfillment of therequirements for the degree of

MASTER OF SCIENCE IN OPERATIONS RESEARCH

from the

NAVAL POSTGRADUATE SCHOOLMarch 19 7 3

Library

Naval Postgraduate SchoolMonterey, California 93940

ABSTRACT

The purpose of this thesis is to provide the student

of queuing systems with a vehicle through which a better

understanding of the interrelationships between stochastic

processes and queuing systems can be achieved.

TABLE OF CONTENTS

I. INTRODUCTION ------------------ 7

II. QUEUING MODEL SIMULATION ------------ 9

A. THE QUEUING MODELS ------------- 10

1. G/G/C- ----------------- 10

2. G/G/C with Losses- ----------- 11

3. G/G/C with Finite Source --------124. G/G/C with Feedback to the Queue - - - - 13

5. Two Simultaneous G/G/C Queues- ----- 14

B. ARRIVAL/ SERVICE DISTRIBUTIONS ANDQUEUE DISCIPLINES- ------------- 14

1. Distributions- ------------- 15

a. K-ERLANG -------------- 15

b. Hyperexponential ----------15c. Degenerate -------------16

2. Queuing Disciplines- ---------- 16

a. FIFO ---------------- 16

b. LIFO ---------------- 16

c. SSTF ---------------- 16

d. Random Draw- ------------173. Length of the Simulation --------17

III. THE METHOD ------------------- 18

A. THE COMPUTER SIMULATION- ---------- 18

B. THE GRAPHICAL DISPLAY- ------ 18

IV. STATISTICS FROM THE SIMULATION --------- 20

V. PARAMETER INPUT- ---------------- 22

A. SAMPLE INPUT CARD- ------------- 22

B. NAMELIST INPUT VIA THE GRAPHICS TERMINAL - - 23

C. INPUT VARIABLES- -------------- 23

VI. THE PROGRAM- ------------------ 25

APPENDIX A - ANALOG SCHEMATIC- ------------ 28

COMPUTER PROGRAM LISTING - - - _______ 29

INITIAL DISTRIBUTION LIST- ------ - - 87

FORM DD 1473 - - - - 89

LIST OF PICTURES

1. G/G/C Queuing System --------------- 10

2. G/G/C Queuing System with Losses --------- n3. G/G/C Queuing System with Finite Source- ----- 12

4. G/G/C Queuing System with Feedback --------135. Two Simultaneous G/G/C Queuing Systems ------ 14

6. Statistics Page- -----------------207. Parameters Page- ----------------- 22

ACKNOWLEDGEMENTS

The Hybrid Computer at the Naval Postgraduate School

is a self-operated system; because many of the complexities

of the system were unfamiliar to the programmer, the techni

cal assistance of Mr. R. Limes, Mr. W. Thomas, and Mr.

A. Wong was invaluable.

I. INTRODUCTION

A queuing system is comprised of action facilities

(servers) and waiting facilities (queues or lines) . The

relationship between the components can be of many forms,

each with their own characteristics. Consider a tool

crib in a machine shop where the machinists are obliged

to check out their tools; or consider the telephone system,

in which callers request communication capacity. Both

situations involve requests for action from limited resources,

in these cases, servers or telephone circuits. In order to

allocate resources properly, it is necessary to understand

the interaction between requests for action and the actions

themselves. Individually these can be modeled as stochastic

processes, but in order to predict flow through the queuing

system, it is necessary to understand the interrelation

between the patterns of event occurrences. When demands

occur at regular intervals and actions are of fixed dura-

tion, it is relatively easy to predict flow; if there are

more requests per unit of time than actions over the same

time span, the queue will grow without bound unless the

event relationship is changed. When variation is added to

the timing of events, the flow is not as obvious. For

example, if the average number of requests for action

equals the average number of actions, per unit of time, but

both occur with variability, then it would be useful to be

able to predict such quantities as expected delay and action

facility idle time. Queuing models are used for this

purpose

The purpose of this thesis is to provide the student

of queuing systems with a vehicle through which a better

understanding of the interrelationships between stochastic

processes and queuing systems can be achieved.

This is done through a combination of a computer Simula

tion and a computer-driven graphical display.

II. QUEUING MODEL SIMULATION

This computer program simulates queuing systems in two

stages. First, a next-event type computer simulation creates

a time history of the queuing system. Second, a dynamic

graphical display depicts the occurrence of the queuing

process in real time.

Five basic, academically interesting, queuing models,

each with parameter options, can be simulated. Queuing

models are generally described in terms of the distribution

of request occurrences (arrival distribution) , the distribu-

tion of action occurrences (service distribution) , and the

number of action facilities (servers). Specifically, the

times between request occurrences are taken to be indepen-

dent and identically distributed realizations from the

arrival distribution. The duration of each action occur-

rence is also taken to be an independent and identically

distributed realization from the service distribution.

Variations on these assumptions occur for some models and

are described in Section A.

The abbreviation G/G/C describes a queuing system in

which the arrival and service distributions are general

(unspecified) and there are C servers.

The five options in this simulation are:

(1) G/G/C

(2) G/G/C with losses

(3) G/G/C with feedback

(4) G/G/C with finite source

(5) Two simultaneous G/G/C queues for comparison.

The specific choices of arrival distribution and service

distribution which can be simulated are fully described

in Section B.

A. THE QUEUING MODELS

1. G/G/C

Picture 1

This is the simplest of the queuing models considered

here. Each arrival waits in the queue until a server is

available, enters service for a calculated period of time,

then departs the system.

This model may be simulated with from one to six

servers

2. G/G/C with Losses

Picture 2

Consider making a long distance telephone call on

Mother's Day. If all the trunk lines are busy, your request

for service will be rejected. The G/G/C model with losses

reflects this by rejecting arrivals who find the queue

full. The queue limit used is specified as an input

parameter. Note that rejected arrivals are lost and are

not assumed to return later. The fraction of all arrivals

who are served (not rejected) is recorded for subsequent

display.

3. G/G/C with Finite Source

Picture 3

Consider a closed-circuit educational television

system at a school. The finite population consists of all

the televisions in the system. When one fails, it enters

the queue to be serviced. After repair, it re-enters the

population pending the next failure.

This situation is modeled by assuming that each

action source, upon entering the idle population, is

assigned a waiting time from the arrival distribution.

When this time expires, the source requests service,

joining the queue if necessary. As each action source

terminates its service, it returns to the idle population.

The user specifies the number of servers (one to six) and

the size of the population.

4 . G/G/C with Feedback to the Queue

Picture 4

This is an extension of the G/G/C model, with from

one to six servers, allowing customers leaving service to

depart the system or to return to the queue. At each

service completion, the customer leaves the system with

probability P or returns to the queue with probability 1-P,

The user specifies P as an input option.

5. Two Simultaneous G/G/C Queues

Picture 5

This option allows comparison of two standard

G/G/C queues, each with up to three servers. All parameters

of each system are specified separately, and, if desired,

identical arrival streams may be utilized. This allows

comparisons between different arrival and/or service distri-

butions and associated parameters, and between a different

number of servers. Also, the effect of the various queuing

disciplines can be compared.

B. ARRIVAL/SERVICE DISTRIBUTIONS AND QUEUE DISCIPLINES

As well as options already described, the following

arrival and service distributions are available to the

user of the program. Any of the following probability

distributions can be used as an arrival or service

distribution.

1 . Distributions

a. K - ERLANG (exponential when K=l)

F (t) = 1 - V i#*i e~^J=0 J -

RATE A =|

CV = —

Input parameters are:

(2) Rate - in terms of mean number of events

per minute.

b. Hyperexponential

F(t) = pU-e"^) + qCl-e"^ 1 ^

CI) p + q - 1

(2) CV = —/p

(3) A = Rate is specified and then

A,, A?chosen to satisfy--

= (2 + /7)A

12 + /2 - q

Input parameters are:

(1) Rate, specified in terms of mean number

of events per minute.

(2) Coefficient of variation (must be greater

than 1.0)

c. Degenerate (Fixed Rate)

t < 1/XF(t) =

1 t > 1/A

Input parameter:

(1) Rate A specified in terms of mean number

of events per minute.

2 . Queuing Disciplines

For comparison purposes, there are four queuing

disciplines to choose from. These determine the priority

by which members of the queue will enter service.

a. FIFO - the first into queue is the first out.

Over all disciplines which are independent of service times,

this one minimizes the variance of delay.

b. LIFO - the last into the queue is the first

out. Some systems, by their nature, function in this manner

c. SSTF - the queue member with shortest service

time is served first; this minimizes the mean delay over

all other disciplines, but the variance of delay is greater

than that of FIFO.

d. RANDOM DRAW - applies to a system in which

insufficient information is known to implement any other

discipline.

3. Length of the Simulation

The period of simulation is specified in minutes

by the user. All queues are empty and servers idle at the

start of the simulation.

III. THE METHOD

A. THE COMPUTER SIMULATION

The computer simulation is a modified next event type.

There are two types of events, arrivals to the queue and a

completion of service. With the first there will be an

associated entry to service if there is an idle server,

and with the last an associated entry to service if the

queue is not empty.

The simulation is begun with the generation of the first

arrival time and its associated service time. The times are

generated via the transformation of random numbers to stochas-

tic variates of the type specified. At the occurrence of

an arrival, the next arrival time and associated service

time are generated. When a server is available, a waiting

member of the queue enters service with completion time

equal to the sum of his assigned service time and the time

he departs the queue. Thus with the clock starting at zero,

a time history of events is created which tells what happened

and when.

At the same time, the appropriate values are retained

from which statistics of the simulation can later be

calculated.

B. THE GRAPHICAL DISPLAY

At the completion of the simulation phase, the time

history is depicted dynamically on the cathode ray tube

in real time. Pictures 1 to 5 show the form of the display,

with the man-like images in motion along the appropriate

path, and the number in the queue box showing the queue's

current length.

The timing and motion of the display are controlled

by the analog computer via the hybrid connections to the

digital computer. The first connection is for the purpose

of timing until the next event is to be initiated. In this

case, the analog computer integrates from zero to a preset

time value in volts, at the rate of one volt per second.

The interrupt to the appropriate subroutine then occurs,

and the analog computer is reset with the next time value,

after which the integration is reinitiated.

The second connection causes the dynamic motion to

occur by continually checking events in progress and causing

the position of the corresponding images to be incremented

appropriately; this interrogation and incrementation occurs

approximately 10 times per second depending on the settings

of the analog computer. Movement increments are between

one-half inch and two inches (depending on the path)

causing a motion of between five inches and 20 inches per

second.

IV. STATISTICS FROM THE SIMULATION

STATISTICS

101 OS 01 OS

!° T .51 3.99 DC .31 3.67BT a. 3i .95 BC 2.27 2.33LT a. at 4. 94 LC 2.55 6.11

DC 1 . 15 22.24 DV 5,65 1 1 1 . 16sc 5.38 1.41 sv 26.59 1 1 .27WC 5.47 24 . 63 DV 3.67 31.14

DM 5. If 57. ff OH 3.11 9.11Dt .55 .92 Of .35 .51DX .46 .92 OL .14 .76

NC 26 24 PL .ftENTER 1

CONTINUE2. PARAMETER LISTING

At the end of the simulation period, the following

statistics are calculated and displayed, as shown above

AT = time - average length of the queue

BT = time - average number of busy servers

LT = time - average number in the system = QT + BT

QC = customer - average length of the queue

BC = customer - average number of busy servers

LC = customer - average number in the system = QC + BC

DC = customer - average delay in the queue

SC = customer - average service time

WC = customer - average wait in the system = DC + SC

DM = maximum delay experienced by any customer

DO = probability that a customer's delay is greater than

DX = probability that a customer's delay is greater than

XD. (The value of XD is an input option.)

QM = maximum queue length reached during the simulation

QO = probability that the queue length will be greater

than zero.

QL = probability that queue length will be greater than LQ

(the value of LQ is an input option)

DV = standard deviation of the delay in queue

SV = standard deviation of the service times

QV = standard deviation of the queue length

NC = total number of arrivals

PS = percentage not lost (losses model only)

V. PARAMETER INPUT

I TYPE

RUNTIME

1 3 3 4

IDIST 1 ! 1

KK 1 t 1

RATE 31. ft 31 ** 14.11 14 .11

CV 1 . f

1 .ft 1 .ft 1 .11

IDSPLN 1

LD MAXO 1

XD 1 .ft NPOP 1

I DUAL P 1 .11

ENTER 1 . CONTINUE2. STATISTICS

Picture 7

Parameters can be input by the card reader or by the

teletype terminal. In either case, the namelist form is

used, where the entry form is NAME = X, NAME being the

variable title and X being the value being assigned. For

real variables, the decimal point must be entered. Name-

list input is terminated with a *.

A. SAMPLE INPUT CARD

IDEV = 2, ITYPE = 3, RATE (2) = 3.0 *

B. NAMELIST INPUT VIA THE GRAPHICS TERMINAL

If this mode of entry is specified, one variable at

a time is entered, followed by a carriage return. Entries

are displayed, as entered, on the graphics screen. Termina

tion occurs with the entry of a * followed by a carriage

return.

C. INPUT VARIABLES

The following conventions are employed for specifying

the options and parameters to be used.

ITYPE = 1 gives G/G/C

2 gives G/G/C with losses

3 gives G/G/C with finite source

4 gives G/G/C with feedback

5 gives two simultaneous G/G/C queues

RUNTIME = desired length of simulation (minutes)

IC = number of servers for queue 1

IC1 = number of servers for queue 2 (only applicable if

ITYPE = 5 is chosen)

IDIST(I) = 1. K - ERLANG

2. HYPEREXPONENTIAL

3. DEGENERATE

KK(I) = K in K-ERLANG (use K=l for exponential)

RATE (I) = Rate in number per minute

CV(I) = Coefficient of variation (specify for IDIST(I) = 2

only. This variable is computed otherwise.)

where 1=1 for arrivals to queue 1 (except finite model)

2 for arrivals to queue 2 or finite population

idle parameters

3 for service for queue 1 (except finite model)

4 for service for queue 2 or finite population

service parameters

IDSPLN(I) = 1. FIFO

2. LIFO

3. Shortest service time first

4. Random draw

where 1=1 for queue 1

2 for queue 2

LQ = parameter in the statistic P(Q > LQ)

XD = parameter in the statistic P(wait > XD)

IDUAL = 0. Independent arrival streams

1. Identical arrival streams

(applicable to ITYPE = 5 only)

MAXQ = maximum queue size (for G/G/C with losses only)

NPOP = population size (for finite source option only)

P = completion probability (for feedback option only)

VI. THE PROGRAM

The program requires the use of the XDS 9300 digital

computer, the CI 5000 analog computer, and a graphics

terminal using the GATED monitor.

The analog computer wiring diagram necessary to drive

the display sequence appears in APPENDIX A. Before

beginning, set Delay Flop Switches DFOO, DFOl, and DF02

to .1 MS and set counter FOO to 1 and F04 to 0. Then select

RUN and DIGITAL COMPUTER on the analog control panel. This

provides for movement interrupts to occur at a rate of

100/11 hertz. If desired, the timing values will be dis-

played on the RATIOMETER by selecting TRUNK 420 prior to

DIGITAL COMPUTER.

The program consists of a FORTRAN program and a META-

SYMBOL 9300 program. Because the history is buffered out

to the device assigned the logical unit value 7, an addi-

tional control card is needed.

Job cards for deck run should be

-iASSIGN 7 = DF1A

-nFORTRAN LS,GO

FORTRAN" DECK

-nMETA 9300 SI,LO,GO

META- SYMBOL DECK

i LOAD XR,MAP

i DATA

PROGRAM STEPS

SET-UP ANALOGCOMPUTER

RESET GATED(GRAPHICS MONITORp

( STARTJ

SPECIFYGRAPHICS TERMINAL

INPUT PARAMETERSVIA TERMINAL N / INPUT PARAMETERS

VIA CARDSN/V

DISPLAYSTATISTICS

/ \ DISPLAYPARAMETERS

DISPLAYSTATIC PICTURE

SIMULATIONCREATE HISTORY

QUEUING SYSTEMDYNAMIC DISPLAY

WRITE PARAMETERSAND STATISTICS

\DISPLAY \ DISPLAYPARAMETERS S TATISTICS

APPENDIX A

ANALOG SCHEMATIC

TO CPS

100 CPS

SCOPE Y2

-oN201

PSET & 4

CLKENB

RESET,

-oSCOPE Yl

o- 10 io, IV1.0

-1 V/SEC

\23CI 1

o—T420

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INITIAL DISTRIBUTION LIST

No. Copies

1. Defense Documentation CenterCameron StationAlexandria, Virginia 22314

2. Library, Code 0212Naval Postgraduate SchoolMonterey, California 93940

3. Chief of Naval PersonnelPers libDepartment of the NavyWashington, D. C. 20370

4. Naval Postgraduate SchoolDepartment of Operations Research

and Administrative SciencesMonterey, California 93940

5. Professor R. W. Butterworth, Code 55 BdDepartment of Operations Research

and Administrative SciencesNaval Postgraduate SchoolMonterey, California 93940

6. Professor W. M. Raike, Code 55 Rj

Department of Operations Researchand Administrative Sciences

7. Professor P. A. W. Lewis, Code 55 LwDepartment of Operations Research

and Administrative SciencesNaval Postgraduate SchoolMonterey, California -93940

8. Professor D. P. Gaver, Code 55 GvDepartment of Operations Research

9. Professor A. F. Andrus , Code 55 AsDepartment of Operations Research

10. Lieutenant Cyrus Michael Riddell1615 92 North EastBellevue, Washington 98004

UNCLASSIFIEDSecurity Classification

DOCUMENT CONTROL DATA -R&DfSecurily clas sitication of title, body of abstract and indexing annotation ritual be entered when the overall report la classified)

1 originating activity (Corporate author)

Za. REPOR T JECURI T Y CLASSIFICATION

Unclassified2b. GROUP

3 REPOR T TITLE

A Queuing Model Simulation with Dynamic Graphical Display

4 DESCRIPTIVE NOTES (Type of report and. Inclusive dates)

Master's Thesis; March 1973S. au THORISI (Fust name, middle initial, 1ft name)

Cyrus Michael Riddell

6. REPOR T DATE

March 19737«. TOTAL NO. OP PAGES

7b. NO. OF REFS

la. CONTRACT OR GRANT NO.

b. PROJEC T NO.

»e. ORIGINATOR'S REPORT NUMBER(S)

Sb. OTHER REPORT NOISI (Any other nwvbc re that may be aealgnedthlt report)

10. DISTRIBUTION STATEMENT

Approved for public release; distribution unlimited.

II. SUPPLEMENTARY NOTES 12. SPONSORING MILI TARY ACTIVITY

13. ABSTR AC T

The purpose of this thesis is to provide the student ofqueuing systems with a vehicle through which a better under-standing of the interrelationships between stochastic processesand queuing systems can be achieved.

DD ,r:„1473 <

PAGE "S/N 0101 -807-681 1 89

UNCLASSIFIEDSecurity Classification

A- 3 M0

UNCT.ASSTFTFnSecurity Classification

KEY WO HOI

Queues

Simulation

Computer Graphics

DD .T,"..1473 <backi

S/N 0101 -807-682 t

UNCLASSIFIED

90Security Classification A- 31 409

2 J0M76 m 2Uk ?5

.esis ±.<*6Z$<iR425 RJddell

c.l . A queuing model simu-

lation with dynamicgraphical display.

2 JUN76 2<4U25

Thesis -Lfc<3254

R425 Riddel I

c.3 . A queuing model simu-lation with dynamicgraphical display.

thesR425

A queueing model simulation with dynamic

3 2768 000 99057 6

DUDLEY KNOX LIBRARY

Cyrus Michael Riddel - core.ac.uk · CyrusMichaelRiddell Lieutenant,UnitedStatesNavy...

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