introduction(packages)1 management of the simulated clock lfixed-time increment lvariable-time...

Introduction(Packages) 1

Management of the Simulated Clock

Fixed-time Increment

Variable-time Increment

Flowchart for arrival routine,queueing model

Arrivalevent

Set delay = 0for this customer

and gather statistics

Isthe server

busy?

Schedule the nextarrival event

Add 1 to thenumber of

customers delayed

Make theserver busy

Schedule adeparture event

for this customer

Add 1to thenumber in queue

Write errormessage and stop

simulation

Store time ofarrival of this

customer

Isthe queue

Full?

Return

NoYes

Yes

No

Departureevent

Subtract 1 fromthe number in

queue

Isthe queueempty?

Compute delay ofcustomer entering service

and gather statistics

Add 1 to the number of customers delayed

Schedule adeparture event

for this customer

Make theserver idle

Move each customerin queue (if any)

up one place

Eliminate departure event from

consideration

Return

NoYes

Flowchart for departure routine,queueing model


Distribution of time between arrivals

Time between Cumulative Random digitArrivals(min.) Probability Probability Assignment

12345678

0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125

0.125 0.2500.3750.5000.6250.7500.8751.000

000-125126-250251-375376-500501-625626-750751-875876-000


Service Time Distribution

Service Time Cumulative Random digit (Minutes) Probability Probability Assignment

1 0.10 0.10 01 - 10 2 0.20 0.30 11 - 30 3 0.30 0.60 31 - 60 4 0.25 0.85 61- 85 5 0.10 0.95 86 - 95 6 0.05 1.00 96 - 00


Time-between-arrival DeterminationRandom Time btwn Random Time btwn Digit Arrivals Digit Arrivals Customer (Minutes) Customer (Minutes)

1 _ _ 11 109 1 2 913 8 12 093 1 3 727 6 13 607 5 4 015 1 14 738 6 5 948 8 15 359 3 6 309 3 16 888 8 7 922 8 17 106 1 8 753 7 18 212 2 9 235 2 19 493 410 302 3 20 535 5


Service Times Generated

1 84 4 11 32 3 2 10 1 12 94 5 3 74 4 13 79 4 4 53 3 14 05 1 5 17 2 15 79 5 6 79 4 16 84 4 7 91 5 17 52 3 8 67 4 18 55 3 9 89 5 19 30 210 38 3 20 50 3

CustomerRandom

Digit

ServiceTime

(Minutes)Customer

RandomDigit

ServiceTime

(Minutes)

Event Type

Work Sheet

ClockTime

CustomerNumber

Arrival 1 0

.

.

.

.

.

.

.

.

.


Findings from the Simulation

(min)8.220

56

1. Average waiting time for a customer

Total time customers wait in queue(minute)Total number of customers


Findings from the Simulation(cont)

2. Prob. that a customer has to wait in a queue

Number of customers who waitTotal number of customers

65.020

13


Findings from the Simulation(cont)3. Proportion of idle time of the server

Total idle time of server(minute)Total run time of simulation(minute)

21.086

18

Thus, the probability of the server being busy is the complement of 0.21, or 0.79


The Able-Baker carhop problem

The purpose of this example is to indicate the simulation procedure when there is more than one channel.

Consider a drive-in restaurant where carhops take orders and bring food to the car.

Cars arrive in the manner shown in the following table.


The Able-Baker carhop problem(Interarrival distribution of cars)

Time between Random

Arrivals Cumulative Digit

(Minutes) Probability Probability Assignment

1 0.25 0.25 01-25

2 0.40 0.65 26-65

3 0.20 0.85 66-85

4 0.15 1.00 86-00


The Able-Baker carhop problem(continued)

There are two car hops -- Able and Baker.

Able is better able to do the job, and works somewhat faster than Baker.

The distribution of service times of Able and Baker is following.


The Able-Baker carhop problem(Service Distribution of Able)

Service Random

Time Cumulative Digit


2 0.30 0.30 01-30

3 0.28 0.58 31-58

4 0.25 0.83 59-83

5 0.17 1.00 84-00


The Able-Baker carhop problem(Service Distribution of Baker)

Service Random

Time Cumulative Digit


3 0.35 0.35 01-35

4 0.25 0.60 36-60

5 0.20 0.80 61-80

6 0.20 1.00 81-00


The Able-Baker carhop problem(Continued)

l Over the 62-minute period Able is busy 90% of the time.

2 Baker was busy only 69% of the time. The seniority rule keeps Baker less busy.

3 Nine of 26 or about 35% of the arrivals had to wait. The average waiting time for all customers was only about 0.42 minute, or 25 seconds, which is very small.


The Able-Baker carhop problem(Continued)

4 Those 9 who did have to wait only waited an average of 1.22 minutes, which is quite low.

5 In summary, this system seems well balanced. One server cannot handle all the dinners, and three servers would probably be too many. Adding an additional server would surely reduce the waiting time to nearly zero. However, the cost of waiting would have to be quite high to justify an additional worker.


GPSS (Process-Oriented) Diagram

PERMANENT-CreatedImplicitly

GPSS Equipment Entities-Facilities, Storages,Queues, switches,...

ELEMENTS-Entities and

Attributes

TEMPORARY- created/Destroyed

Dynamically

TRANSACTIONS-Attributes

ModelStructure

OPERATIONS

BLOCKS- Activatedby Entering

Transactions

NETWORK

Logical LevelGPSS Implementation

Level

FlowingThrough

Connectedto form


GPSS Events

Current-Event Chain: Transaction with BDT <= C1

Future-Event Chain:Transaction with BDT > C1


GPSS Events(continued)Note

Events must be executed chronologically, so chains are maintained in ascending order

Current-Event Chain performs an additional tasks(i.e. maintains by priority)


GPSS Execution Trace

The system. Four transactions enter the system at intervals of 3 time units, starting at time unit 1.

They try to seize facility MACH for 4 time units and then try to enter storage BUFFE (with capacity 2) before releasing MACH.

After 9 time units in BUFFE, they leave storage and terminate. In this model the number of transactions must be limited because it has been devised to cause catastrophic congestion.


GPSS Sample instructionsBUFFE STORAGE 2

GENERATE 3,,1,4QUEUE WAITSEIZE MACHDEPART WAITADVANCE 4ENTER BUFFERELEASE MACHADVANCE 9LEAVE BUFFETERMINATE 1START 4


GPSS (Process-Oriented) Diagram

no1 no2 no3 no4

Queue WAIT

no1 no2 no3 no4

1 5 9 14

no1

no2

no4

no4

FacilityMACH

StorageBUFFE

18

5 9 14 18 23

27

no1 no2 no3 no4

0

no1 no2 no3

no3

1 4 7 10 15 2520 30


GPSS/H Block Diagram, Queueing Model

GENERATERVEXPO(1,1.0)

QUEUE

SEIZE

DEPART

ADVANCERVEXPO(2,0.5)

RELEASE

TERMINATE

SERVERQ

SERVERQ

SERVER

SERVER

1

LVEQ

CreateArriving

Customer

Enter theQueue

Seizethe

Server

Leavethe

Queue

Test for theterminationof the run

Delay forService

CustomersDepart

Releasethe

Server(STOP)

N$LVEQL

TEST

(STOP)

1000


GPSS/H - Queueing Model 1 * SIMULATION OF THE M/M/1 QUEUE 2 * 3 SIMULATE 4 GENERATE RVEXPO(1,1,0) Create Arriving

Customer 5 QUEUE SERVERQ Enter the Queue 6 SEIZE SERVER Seize the Server 7 LVEQ DEPART SERVERQ Leave the Queue 8 TEST L N$LVEQ,1000,STOP Test for Termination 9 ADVANCE RVEXPO(2,0.5) Delay for service10 STOP RELEASE SERVER Customers Depart11 TERMINATE 112 *13 * CONTROL STATEMENT14 *15 START 1000 Make 1 simulation run16 END


GPSS/H standard output report, Queueing Model

RELATIVE CLOCK: 1014.1565 ABSOLUTE CLOCK:1014.1565

BLOCK CURRENT TOTAL1 10002 10003 1000LVEQ 10005 10006 999STOP 10008 1000

-- AVG-UTIL-DURING--FACILITY TOTAL AVAIL UNAVL ENTRIES AVG CURRENT TIME TIME TIME TIME/XACT STATUSSERVER 0.516 1000 0.523 AVAIL


GPSS/H standard output report, Queueing Model(Continued)

QUEUE MAXIMUM AVERAGE TOTAL ZERO PERCENT CONTENTS CONTENTS ENTRIES ENTRIES ZEROSSERVERQ 8 0.605 1000 454 45.4

QUEUE AVERAGE $AVERAGE QTABLE CURRENT TIME/UNIT TIME/UNIT NUMBER CONTENTS

SERVERQ 0.614 1.124 0

RANDOM ANTITHETIC INITIAL CURRENT SAMPLE CHI-SQUARE STREAM VARIATES POS. POS. COUNT UNIFORMITY 1 OFF 100000 101001 1001 0.71 2 OFF 200000 200999 999 0.69


SIMSCRIPT II.5 Preamble,Queueing Model

1 PREAMBLE 2 PROCESSES INCLUDE ARRIVAL.GENERATOR, 3 CUSTOMER, AND REPORT 4 RESOURCES INCLUDE SERVER 5 DEFINE DELAY.IN.QUEUE,

MEAN.INTERARRIVAL.TIME, 6 AND MEAN.SERVICE.TIME AS REAL VARIABLES 7 DEFINE TOT.DELAYS AS AN INTEGER VARIABLE 8 DEFINE MINUTES TO MEAN UNITS 9 TALLY AVG.DELAY.IN.QUEUE AS THE AVERAGE AND10 NUM.DELAYS AS THE NUMBER OF DELAY.IN.QUEUE11 ACCUMULATE AVG.NUMBER.IN.QUEUE AS THE12 AVERAGE OF N.Q.SERVER13 ACCUMULATE UTIL.SERVER AS THE AVERAGE OF14 N.X.SERVER15 END


SIMSCRIPT II.5 Main programQueueing Model

1 MAIN 2 3 READ MEAN.INTERARRIVAL.TIME, 4 MEAN.SERVICE.TIME, AND TOT.DELAYS 5 6 CREATE EVERY SERVER(1) 7 LET U.SERVER(1) = 1 8 9 ACTIVATE AN ARRIVAL.GENERATOR NOW1011 START SIMULATION1213 END


SIMSCRIPT II.5 Process routineARRIVAL.GENERATOR

1 PROCESS ARRIVAL.GENERATOR 2 3 WHILE TIME.V >= 0.0 4 DO 5 WAIT EXPONENTIAL.F(MEAN.INTERARRIVAL.TIME, 6 1) MINUTES 7 ACTIVATE A CUSTOMER NOW 8 LOOP 910 END


SIMSCRIPT II.5 Process routineCUSTOMER

1 PROCESS CUSTOMER 2 3 DEFINE TIME.OF.ARRIVAL AS A REAL VARIABLE 4 LET TIME.OF.ARRIVAL = TIME.V 5 REQUEST 1 SERVER(1) 6 LET DELAY.IN.QUEUE = TIME.V - TIME.OF.ARRIVAL 7 IF NUM.DELAYS = TOT.DELAYS 8 ACTIVATE A REPORT NOW 9 ALWAYS10 WORK EXPONENTIAL.F (MEAN.SERVICE.TIME, 2)11 MINUTES12 RELINQUISH 1 SERVER(1)1314 END


SIMSCRIPT II.5 Process routineREPORT

1 PROCESS REPORT 2 3 PRINT 5 LINES THUS

SIMULATION OF THE M/M/1 QUEUE 9 PRINT 8 LINES WITH MEAN.INTERARRIVAL.TIME, 10 SERVICE.TIME, AND TOT.DELAYS THUS

MEAN INTERARRIVAL TIME **.**MEAN SERVICE TIME **.**NUMBER OF CUSTOMERS *****


SIMSCRIPT II.5 Process routineREPORT(Continued)

19 PRINT 8 LINES WITH AVG.DELAY.IN.QUEUE,20 AVG.NUMBER.IN.QUEUE(1),

ANDUTIL.SERVER(1) 21 THUS

AVERAGE DELAY IN QUEUE ***.**AVERAGE NUMBER IN QUEUE ***.**SERVER UTILIZATION *.**

29 STOP3031 END


SIMSCRIPT II.5 Output ReportQueueing Model

SIMULATION OF THE M/M/1 QUEUE

MEAN INTERARRIVAL TIME 1.00

MEAN SERVICE TIME .50

NUMBER OF CUSTOMERS 1000

AVERAGE DELAY IN QUEUE .43

AVERAGE NUMBER IN QUEUE .43

SERVER UTILIZATION .50


SLAM II network forsingle-server queue simulation

1000

1 1

0.

EXPON(4,5)

CREATE node QUEUE node ACTIVITY TERMINATE node

1

0RNORM (3.2, 0.6)


SLAM II Model of Single-Server Queue

GEN, BANKS CARSON, NELSON SINGLE SERVER QUEUE EXAMPLE, 1/20/95

LIMITS,1,0,30; MODEL CAN USE 1 FILE, MAX NO. OF SIMULTANEOUS ENTRIES 30

NETWORK; BEGINNING OF MODEL

CREATE, EXPON(4.5) CUSTOMERS ARRIVE AT CHECKOUT

QUEUE(1); CUSTOMERS WAIT FOR SERVICE IN

QUEUE FILE ONE (1)

ACTIVITY(1)/1,RNORM(3.2,.6); CHECKOUT SERVICE TIME IS N(3.2,0.6)

TERMINATE, 1000; SIMULATE UNTIL 1000 CUSTOMERS

ARE CHECKED OUT

ENDNETWORK; END OF MODEL

END OF SIMULATION


CSIM Sample Code(1)/* simulate an M/M/1 queue

(an open queue with exponential service times and interarrival intervals)

*/

#include "lib/csim.h"

#define SVTM 1.0 /*mean of service time distribution */

#define IATM 2.0 /*mean of inter-arrival time distribution */

#define NARS 5000 /*number of arrivals to be simulated*/

FACILITY f; /*pointer for facility */

EVENT done; /*pointer for counter */

TABLE tbl; /*pointer for table */

QTABLE qtbl; /*pointer for qhistogram */

int cnt; /*number of active tasks*/


CSIM Sample Code(2)sim() /*1st process - named sim */{

int i;set_model_name("M/M/1 Queue");create("sim"); /*required create statement*/

f = facility("facility"); /*declare facility*/done = event("done"); /*declare event*/tbl = table("resp tms"); /*declare table */qtbl = qhistogram("num in sys", 10); /*declare qhistogram*/

cnt = NARS; /*initialize cnt*/for(i = 1; i <= NARS; i++) {

hold(expntl(IATM)); /* hold interarrival*/cust(); /*initiate process cust*/}

wait(done); /*wait until all done*/report(); /*print report*/theory(); /*print theoretical res*/

}


CSIM Sample Code(3)cust() /*process customer*/{

float t1;create("cust"); /*required create statement*/

t1 = clock; /*time of request */note_entry(qtbl); /*note arrival */reserve(f); /*reserve facility f*/

hold(expntl(SVTM)); /*hold service time*/release(f); /*release facility f*/record(clock-t1, tbl); /*record response time*/note_exit(qtbl); /*note departure */cnt--; /*decrement cnt*/if(cnt == 0)

set(done); /*if last arrival, signal*/

}


CSIM Sample Code(4)theory() /*print theoretical results*/{

float rho, nbar, rtime, tput;printf("\n\n\n\t\t\tM/M/1 Theoretical Results\n");

tput = 1.0/IATM;rho = tput*SVTM;nbar = rho/(1.0 - rho);rtime = SVTM/(1.0 - rho);

printf("\n\n");printf("\t\tInter-arrival time = %10.3f\n",IATM);printf("\t\tService time = %10.3f\n",SVTM);printf("\t\tUtilization = %10.3f\n",rho);printf("\t\tThroughput rate = %10.3f\n",tput);printf("\t\tMn nbr at queue = %10.3f\n",nbar);printf("\t\tMn queue length = %10.3f\n",nbar-rho);printf("\t\tResponse time = %10.3f\n",rtime);printf("\t\tTime in queue = %10.3f\n",rtime - SVTM);

}


CSIM Results(1)Tue Dec 1 09:25:18 1987 CSIM Simulation Report Version 12

Model: M/M/1 Queue

Time: 10041.661

Interval: 10041.661

CPU Time: 32.183 (seconds)

Facility Usage Statistics

+----------------------+---------------means----------------+---counts----+

facility srv disp serv_tm util tput qlen resp cmp pre

facility 0.992 0.494 0.5 0.991 1.989 5000 0


CSIM Results(2)Table 1

Table Name: resp tms

mean 1.989 min 0.000

variance 3.813 max 14.273

Number of entries 5000

QTable 2

QTable Name: num in sys

Mean queue length 0.991 Max queue length 13

Mean time in queue 1.989 Number of entries 5000


CSIM Results(3)Queue Table Histogram

Length % of Elapsed Time Cumulative Count Mean Time

0 0.506 0.506 2516 2.020

1 0.242 0.748 3694 0.657

2 0.123 0.871 1845 0.671

3 0.067 0.938 1014 0.659

4 0.035 0.972 510 0.686

5 0.015 0.988 234 0.652

6 0.007 0.995 99 0.700

7 0.002 0.997 40 0.627

8 0.001 0.998 21 0.469

9 0.001 0.999 13 0.823

10 0.000 0.999 6 0.519

over 0.001 1.000 8 0.807


CSIM Results(4) M/M/1 Theoretical Results

Inter-arrival time = 2.000

Service time = 1.000

Utilization = 0.500

Throughput rate = 0.500

Mn nbr at queue = 1.000

Mn queue length = 0.500

Response time = 2.000

Time in queue = 1.000

introduction(packages)1 management of the simulated clock lfixed-time increment lvariable-time...

Documents

arrivals time

introductionpackages6

customer total time

model slide

bdtc1 slide

average waiting time

proportion of idle time

queue number of customers