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
by
Cyrus Michael Riddell
Thesis Advisor R. W. Butterworth
March 1973
Appiovtd fioi puhLLc si&l&aAz; dUtAAJbiLtioYi wnJUjnitz.d.
T153548
A Queuing Model Simulation
with Dynamic Graphical Display
by
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
7^^
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.
-V-.
.
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
.
10
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.
11
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.
12
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.
13
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
14
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:
(1) K
(2) Rate - in terms of mean number of events
per minute.
b. Hyperexponential
F(t) = pU-e"^) + qCl-e"^ 1 ^
where
:
CI) p + q - 1
(2) CV = —/p
(3) A = Rate is specified and then
A,, A?chosen to satisfy--
A2
= (2 + /7)A
15
12 + /2 - q
A
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
16
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.
17
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
18
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.
19
IV. STATISTICS FROM THE SIMULATION
t
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
20
DM = maximum delay experienced by any customer
DO = probability that a customer's delay is greater than
zero.
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)
21
V. PARAMETER INPUT
I TYPE
IC
5
3
RUNTIME
IC1
1 II
1
1 3 3 4
IDIST 1 ! 1
KK 1 t 1
RATE 31. ft 31 ** 14.11 14 .11
CV 1 . f
1
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 *
22
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.)
23
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)
24
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
:
-iJOB
-iAGT
-iASSIGN 7 = DF1A
-nFORTRAN LS,GO
FORTRAN" DECK
-nMETA 9300 SI,LO,GO
META- SYMBOL DECK
25
iEOF
i LOAD XR,MAP
i DATA
26
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
27
APPENDIX A
ANALOG SCHEMATIC
TO CPS
o—
RESET
o-
Cll
O-
100 CPS
O
N402
SCOPE Y2
-oN201
PSET & 4
CLKENB
IMS
DF
2
N104
DF
1
INT
-o53
RESET,
-oSCOPE Yl
-O
.IMS
DF
INT
o52
-100
o- 10 io, IV1.0
-1 V/SEC
\23CI 1
DAC 1
o—T420
28
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86
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
Naval Postgraduate SchoolMonterey, California 93940
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
and Administrative SciencesNaval Postgraduate SchoolMonterey, California 93940
9. Professor A. F. Andrus , Code 55 AsDepartment of Operations Research
and Administrative SciencesNaval Postgraduate SchoolMonterey, California 93940
87
10. Lieutenant Cyrus Michael Riddell1615 92 North EastBellevue, Washington 98004
88
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)
Naval Postgraduate SchoolMonterey, California 93940
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
90
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
Naval Postgraduate SchoolMonterey, California 93940
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
8
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