Discrete Event Simulation

1
Discrete Event Simulation

• Dr Shamala Subramaniam
• Lecture Notes
• Week 3 - 6

2
Simulation

• Definition A computer program which executes
  the model developed for a particular system

3
Simulation Base System

• Focus of this course is on modeling and
  simulating queuing systems
• What do you understand from the above sentence?
• This means that we WONT be simulating continuous
  process such as a flight trajectory
• We wont be simulating systems such as cars and
  engines, bridges
• We wont be simulating a food cycle system
• We WILL be simulating systems which have DISCRETE
  processes- clear differences between one entity /
  event to another

4
Developing the skills to write a discrete event
simulation

• Steps and Descriptions
• Identify the system under study
• Develop the model of the project under study
• Define an event
• Derive the logical sequence of the events, which
  comes first and second and etc
• Integrate / Correlate time with the events
• Design the scheduler
• Assign the continuous start times for the events
• Define each event Arrival
• Define each event Departure
• Develop the main program
• Colleting the simulation results

5
Link between Discrete and Queuing Systems

• Why?
• The simulation technique you will learn is called
  Discrete Event Simulation
• The link it has with queuing system is every
  queuing system has discrete events which
  influence waiting time (delay ) in a high
  percentage.
• Event ? An occurrence which changes the
  statistical composition of a system

6
Event queuing system

• Step 1 Identify your system (Project)
• Step 2 define Statistical composition of a
  system
• Before you define an event, kindly define what is
  your statistical composition

• Example of Statistical Composition of Systems
• Carre Four Customer
• Bank Customer
• Computer System data/instruction
• Network - packets

7
Event queuing system (cont.)

• Step 3 Define an event (Please do remember this
  is actually a skill and not a fixed method)
• For example
• In a simple queuing system in a payment counter
  we will be able to define two events, the arrival
  and departure of a customer
• In a Medium Access Control (MAC) IEEE 802.11
  protocol, we could define either 8 or 12 events

8
Example Carre Four Payment Counter System

• 1 (one) payment counter
• Model the payment counter system
• Source of customers
• Service Counter / Cashier
• Queue
• Statistical composition
• Customers

Source of Load
System
Service Counter
Queue
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Example Carre Four Payment Counter System
(cont.) defining an event

• Event an occurrence that changes the
  statistical composition of a system
• Strategy to derive an event
• Define the statistical composition in this case
  its the number of customers denoted as noc
• Number of customer (noc) 0 , t 0
• When will this increase (noc) ?
• When a new customer comes in (arrival event) ?
  Yes
• When a customer leaves the system noc will
  decrease (departure event) Yes
• Can we consider queuing to be an event? No,
  because queuing does not change the noc
  (statistical composition)

10
Why are we focused on only Queuing Systems?

• Queuing is a fundamental issue in issues related
  to delay (waiting time), loss (customers leaving
  without being served), throughput (number of
  customers served) and etc.
• Queuing Theory defined by Leonard Kleinrock (he
  published two (2) books on Queuing Theory vol 1
  (1975) and vol. 2 (1976)) analytical
  explanation for the queuing phenomenon.
• In this class, we complement this analytical
  models with discrete event simulation.

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Types of Queuing Systems

• Simple Queuing System (Base Model)
• M/M/1 queuing model
• Notation of a queuing system is done by
• Kendalls Law

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Kendalls Law Notation for Queuing System

• Single (1) Data Sink (i.e. server, CPU, Service
  Counter)
• Single Queue (1 Queue)
• Single Source (i.e. Traffic, Customers Arriving
  from a door , data / instructions from memory)

13
Inter-relate the components in single
source/queue/data sink system
Data Sink represents the processing capacity
Source represents the load placed onto the
system
Load lt processing capacity (lightly loaded) Q
0 Load processing capacity (heavy load) Q 0
Load gt processing capacity (over loaded) Q gt 0
. Delay x
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• At present most queue sizes are fixed .
• However, the Queue Occupancy changes.

15
General observations from reference books on
simulation

• Journal / Conference References contain heavy
  load analysis
• Whereas generally most text books contain
  lightly loaded analysis
• Congestion (over loaded)
• Delay
• Queue size
• Multiple queues
• Packet / customer loss

16
Algorithm to develop a Discrete Event Simulation

• Identify your system under study (Step 1)
• Develop the model (Step 2)
• Develop the events (arrival departure) (Step 3)
  
• Derive the sequence of events (Step 4)
• T0 arrival of customer 1
• T1 if (load lt processing capacity) delay 0
  , q 0 departure ? arrival of c 2
• T1 (case 2 ) if (load gt processing capacity)
  delay x , qy arrival of customer 2 ?
  departure of customer 1

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Simulating the sequence of events

• Integrate TIME (Step 5)
• Difference between time and rate
• Load (customers / second)
• Processing Capacity (customers / second)
• From the events of arrival and departure assign
  time to each event. The assignment of time
  indicates Arrival time , departure time
  (departure time arrival time delay)
• Simulate time
• Can we use the in-built clock? NO, why because
  its time consuming.
• Have you ever used any other types of clock?

18
Try This Exercise !!!

• Assume that today is Monday and you forgot to use
  your watch. Without having to ask anyone What
  time is it ?, manage the day and suggest how you
  would attend all activities in the day.
• Would any other day be easy to do this for you ?

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An example of My Daily Time Advancing Mechanism

• EVENTS TIME
• Mum opens door 5.30am
• Mum leaves for work 7.10am
• UPM traffic Controls 7.45 8.00 am
• Cars more than 10 at the Faculty - gt 8.30 am
• Students entering the class 9.00
• Change of students in the class at Bilik Seminar
  1300
• Neighbor room leaves in the afternoon for class
  1600
• UPM traffic Controls 1700 - 1730

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Analyze the events and time

• The observation is that the virtual clock is
  ticking but not in an interval of 1s. It moves
  based on the difference of time between two
  events

530am
0710 am
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Structure of a Simulation
Main
Scheduler
Main
Functions is an event
Functions
Functions
Programmer
Simulation
Functions
Functions
Functions
Application
Scheduler A scheduler is a list of events and
its respective START time , where it finds the
smallest start time
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Scheduler for M/M/1 Queue

• Event Start Time
• Arrival x
• Departure y

How would we code this ? Declaration Part
Double arrival Double departure
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S6.1 Simulating Arrival and Departure for n
number of customers

• Arrival gt 1
• Departure gt 1
• Double arrival number of sources of customers
• Double departure

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S6.2 initialize starting time for all events

• Initialize
• sim_clock 0.0
• for(i0 ilt number of sources of customers
  i)
• Arrivali random
• Departure 0.0 or random or ????
• Whatever value we select must be larger than
  arrival thus its would be safe to place a large
  value such as (1.0e30) // The key concept is
  to ensure that departure never gets selected
  until an arrival has happened

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Initialization

• sim_clock 0.0
• Servicetime 0.5
• Service_counter 0 // 0 idle 1 busy
• for(i0 ilt numberofdoors i)
• Arrivali random
• Departure 0.0 or random or ???? Whatever value

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S6.3 Scheduler find the smallest start time

• Write a function to find the smallest event among
  all the n-arrivals and 1- departure events.
• Is this a simulation question or is it a data
  structure question? Finding the smallest start
  time is a data structure question ? SEARCHING for
  the smallest item among n arrivals and 1
  departure

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S6.4 Code of Scheduler find the smallest start
time in C language

• Void scheduler (void)
• Smallest 1.0e30
• For (p0 p lt numberofcustomers p)
• if (arrivalp lt smallest)
• smallest arrivalp
• doornumber p

Main
Scheduler
Arrival
Departure

• if (departure lt smallest)
• Smallest departure
• Event_type 1 // 1- departure 0 - arrival
• Else if (departure gt smallest)
• Event_type 0
• Doornumber p

Simulation
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S6.5 Advance The Clock Call the event

• void adv_clock()
• sim_clock smallest
• Void activate_event (void)
• if (event_type 0)
• arrival()
• Else if(event_type 1)
• departure()

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S7 Arrival Event

• Void arrival (void)
• nca // nca number of customers arrived
• arrivaldoornumber sim_clock iat()
• if (service_counter 0) // idle / available
• service_counter 1
• departure sim_clock servicetime

Random pattern distribution (for customers gt
Poisson distribution)
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S7.1 Function arrival
Load Source e.g noc/m one at a time
Load 10/s
Service 10c/s

• Arrival number of customers x
• Can we know for definite the number of customers
  ? No
• The operating time 930 1600

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7.2 Inter arrival times (IAT)
T2 t1 iat
T19
Iat depends on the system which is under
study Text/ Customers- Poisson
distribution Video On/Off Model Iat random
pattern (acquired from the system under study)
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arrivaldoornumber sim_clock iat()
(1) Arrivalnoc sim_clock iat() (2)
10 c/s
Total Load load 1 load 2 load 3 30 c/ s
(Poisson Theory)
10 c/s
10 c/ s
10 c/ s
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• Void arrival (void)
• nca // nca number of customers arrived
• arrivaldoornumber sim_clock iat()
• if (service_counter 0) // idle / available
• service_counter 1
• departure sim_clock servicetime
• Else if (service_counter 1)
• if (qsize MAXQSIZE) // Q is full
• totalcloss

34

• Else if (service_counter 1)
• if (qsize MAXQSIZE) // Q is full
• totalcloss
• Else if (qsize lt MAXQSIZE) // q is not full
• qsize // number of customers in the q
• Etimeqqsize sim_clock

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• Void fdeparture (void)
• // customer leaves the bank
• nocd
• If (qsize gt 0) // there are customers in the q
• // integration of scheduling algorithms
• Delay sim_clock etimeq1 // fifo/lifo-
  delay per customer
• // Which customer is being selected from the
  queue
• Tdelay delay // average delay tdelay/nocd
• Departure simclock servicetime
• Mforward()
• --qsize

36
S8 Define the Departure Event

• Void departure (void) // cont. case when q is
  empty
• // customer leaves the bank
• If (qsize 0) // there are no customers in the
  q
• // please refer handout of complete source code

37

• Void mforward (void)
• // function to move the customers to the front of
  the queue
• // once the customer in the first position of the
  queue has moved to the service counter
• Int t
• for (t0 t lt qsizet)
• etimeqt etimeqt1

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S9 Main () - Simulation Architecture

• Main()
• load 0
• For (i0 ilt 10 i)
• Initialization()
• While (nocd lt TOS)
• Scheduler()
• Advance_clock()
• activate_event()
• Buff_mgt()
• // end for

• Initialization()
• Sim_clock 0.0
• Arrival0 0.3
• Departure 1.0e30
• Load 10
• Scheduler ()
• Smallest 0.3
• Event_type 0
• Advance Clock
• Sim_clock smallest 0.3

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Simulation Architecture

• Main()
• Initialization()
• While (nocd lt 30000)
• Scheduler()
• Advance_clock()
• activate_event()

• Initialization()
• Sim_clock 0.0
• Arrival0 0.3
• Departure 1.0e30
• Arrival
• Arrival0 (0.3 iat)iat
• Departure 0.3 st

40
S10 Simulation Results

• Average delay (y axis)
• (load (customers/s) and service
  time(customers/s))
• Customer Loss Ratio (y axis)
• Average Queue Size (y-axis)
• X-Axis ????
• Time
• Number of service Counters

41
Average Delay

• Load (customers/m)
• random pattern (Poisson Distribution)
• Poisson Distribution
• Largely used for simulation of human queuing
  patterns
• Input interval (load increase) the delay will
  increase exponentially. The input interval is of
  fixed size (i.e. all intervals are of 1ms
• Relationship between interarrival time (iat)
  and load

42
Average Delay

• Load (customers/m)
• Inter arrival time (iat) is inversely related
  to load

Random
Load 6 Customers/s Iat 1/6s Delay a
0
1s
Time Interval 1230pm
Load 3 Customers/s Iat 1/3 , delay b lt a
0
1s
Time Interval 0900am
Pattern load increases, the delay increases as
the inter arrival time decreases.
43
Poisson Distribution

• Pattern ? Load increases iat decreases
• Iat 1/ load (/) iat load (X)
• Iat 1/ load (customers/time frame) ? Pattern
• RANDOM
• Iat 1/load (DETERMINISTIC)

44
Poisson Distribution

• Iat 1/load
• Option 1 iat ? random 1/ load
• Option 2 iat 1? random / load
• Option 3 iat 1/ load ? random
• Iat 1/ 10
• Iat ½
• iat rand()/load O1

delay
Load
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Poisson Distribution

• iat rand()/load Op.1
• Iat (rand() but it must close to value 1)/load
• Eg.
• Iat 10/100 0.1
• Iat 200/2001

delay
Load
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How can u make it close to 1?

• X rand()
• How can we make x as close to 1
• To maintain the 1/load pattern
• 1/x solution
• 1-1/x
• x/very big number (gt x)
• Iat -log(x/very large number)/load

Average delay (s)
Load (pkts/s)
47
Termination of Simulation

• TOS
• Collect Results - beautiful average delay graph
• Question ? What is the average delay of a
  customer if the load is 10pkt/s and the service
  capacity is 5 pkt/s ? delay x
• If we were to run this for different TOS
• 15 nocd
• 5,000 nocd
• 10,000 nocd
• 100,000 nocd
• 500,000 nocd

48
Delay ? load service capacity
Average delay (s)
Transient Period
Steady State
x
Nocd (person)
5k
10K
100K
500K
49
Average Delay ? load service capacity
Average Queue Size
Average delay (s)
100K

Load (customer/s)
10
20
30
100
50
Average Delay ? load service capacity
Service capacity 90 customers/s
Capacity lt 10c/s
Average delay (s)
Capacity gt 100c/s
100K

Load (customer/s)
10
20
30
100
900am
1200noon
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Average Queue Size

• Queue Size Utilization
• Average Queue Size ( x ) / (y )
• Common understanding
• Average qsize totalqsize/nocd
• x total queue size
• Y number of queues (ans.1)
• Y number of customers arrived (ans.2)
• Average qsize totalqsize/time
• Time sim_clock

52
Calculating Total Queue Size
Total delay
A3
Queue Size
Arrival / Departure
q1
Arrival / Departure
A1
A2
A4
0
time
0
5
t2
time
t1
5
Total queue size a1a2a3a4 A1 q1t1 A2
q2t2
T1 t2 ? No Why? Because the time difference
between two events are not the same
53
Calculating Total Queue Size (cont.)
A3
Queue Size
Void buff_mgt() Totalqsize totalqsize (area
of each time interval)
Arrival / Departure
q1
Arrival / Departure
A1
A2
A4
time
0
5
t2
time
t1
Total queue size a1a2a3a4 A1 q1t1 A2
q2t2
Aoeti qsizeinterval (e.g t1)
54
Calculating Total Queue Size (cont.)
A3
Queue Size
Arrival / Departure
q1
Arrival / Departure
A1
A2
A4
0
5
t2
t1
2.3
5.3
Sim_clock 0.3 (arrival)
Sim_clock 1.3(departure)
Sim_clock
time
55

• Interval calculation
• Initialization ? last_check 0.0
• Void buff_mgt(void)
• Totalqsize qsize (sim_clock last_check)
• Last_check sim_clock

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• Main()
• load 0
• For (i0 ilt 10 i)
• Initialization()
• While (nocd lt TOS)
• Scheduler()
• Advance_clock()
• activate_event()
• Buff_mgt() (19/8/2005)
• Results() (19/8/2005)
• // end for

57
Results

• Total delay ? departure
• Total qsize ? buff_mgt
• Total customer loss ? arrival ? q is full
• Void results(void)
• Averagedelay totaldelay/nocd
• AverageQsize totalqsize/sim_clock
• Clossratio totalcustomerloss/noca

58
Assignment 1

• Write a complete program for the following
  scenario
• Giant Hypermarket has the problem of deciding how
  many counters to place on
• A) Monday Friday
• B) Saturday and Sunday
• If the ideal average delay for per customer
  should be between 5 10 minutes decide the
  number of service counters which are needed for
  both the set of cases

59
Case Studies

• Multiple sources Single Queue Multiple
  Servers (Service Counters)
• Example
• Bank Systems
• TGV
• Multiprocessor Systems

60
Case Studies Networking
Store and Forward Share Resources Bandwidth
Switch/Routers
Intermediate Nodes
E
C
D
Source
Z
Destination
A
B

• Routing algorithm
• Shortest Path
• Scheduling Algorithm
• 1) FIFO, LIFO, WFQ,RR

End-to-end Source to destination
61

• Routing Algorithms
• Source to destination
• Scheduling Algorithms
• One node

62
Routing Algorithm
Q1
Q1
Q1
Q1
Z
E
B
C
A
A
63
Incoming Buffer
Outgoing Buffer
Switching Fabric Multiplexing
Queue
p
Queue
p
Queue
p
Queue
Queue
p
Queue