Computer Simulation and Modeling
Solution of Unit Test I
Q.1
A) A Simulation is the imitation of the operation of a
real-world process or system over Time.
When Simulation is the Appropriate Tool:
Simulation
enables the study of and experimentation with the internal interactions of
a complex system, or of a subsystem within a complex system.
Informational,
organizational and environmental changes can be simulated and the effect of
those alternations on the model‘s behavior can be observer.
The knowledge
gained in designing a simulation model can be of great value toward suggesting
improvement in the system under investigation.
By changing
simulation inputs and observing the resulting outputs, valuable insight
may be obtained into which variables are most important and how variables
interact.
Simulation can
be used as a pedagogical device to reinforce analytic solution methodologies.
Simulation can
be used to experiment with new designs or policies prior to implementation,
so as to prepare for what may happen.
Simulation can
be used to verify analytic solutions.
By simulating
different capabilities for a machine, requirements can be determined.
Simulation
models designed for training, allow learning without the cost and
disruption of on-the-job learning.
Animation shows
a system in simulated operation so that the plan can be visualized.
The modern
system(factory, water fabrication plant, service organization, etc) is so complex
that the interactions can be treated only through simulation.
1B)
Customer
|
IAT
|
Arrival
Time
|
Service
Time
|
Time
service begins
|
Time
customer waits in queue
|
Time
service ends
|
Time
customer waits in system
|
Idle
time of server
|
1
|
-
|
0
|
4
|
0
|
0
|
4
|
4
|
0
|
2
|
8
|
8
|
1
|
8
|
0
|
9
|
1
|
4
|
3
|
6
|
14
|
4
|
14
|
0
|
18
|
4
|
5
|
4
|
1
|
15
|
3
|
18
|
3
|
21
|
6
|
0
|
5
|
8
|
23
|
2
|
23
|
0
|
25
|
2
|
2
|
|
|
|
14
|
|
3
|
|
17
|
11
|
- Avg. waiting time for a customer= 3/5
= 0.6 min
- Prob. That customer has to wait in
queue= 1/5 = 0.2
- Prob. Of idle server= 11/25 = 0.44
- Avg. service time= 14/5 = 2.8 min
- Avg. time between arrival= 23/4 =
5.75 min
- Avg. waiting time for customers who
wait= 3/1 = 3 min
- Avg. time customer spend in the
system= 17/5 = 3.4 min
1
C)
Definition:
N(t) is a counting function that represents the number of events
occurred in [0,t]. A counting process {N(t), t ≥ 0} is a
Poisson process with mean rate l if:
·
Arrivals occur one at a time
·
{N(t), t ≥ 0} has stationary
increments
·
{N(t), t ≥ 0} has independent
increments
Properties
·
Equal mean and variance: E[N(t)] =
V[N(t)] = lt
·
Stationary increment: The number of
arrivals between time s and time t > s is also
Poisson-distributed with mean l(t – s)
Splitting:
Suppose
each event of a Poisson process can be classified as Type I, with probability p
and Type II, with probability 1-p.
N(t)
= N1(t) + N2(t), where N1(t) and N2(t) are
both Poisson processes with rates l p and l (1-p)
Pooling:
Suppose
two Poisson processes are pooled together
N1(t)
+ N2(t) = N(t), where N(t) is a Poisson processes
with rates l1 + l2
1 D)
Event: An
instantaneous occurrence that changes the state of a system (such as an arrival
of a new customer).
Event notice: A
record of an event to occur at the current or some future time, along with any
associated data necessary to execute the event; at a minimum, the record
includes the event type and the event time.
Event list: A
list of event notices for future events, ordered by time of occurrence; also
known as the future event list (FEL).
Activity: A
duration of time of speci_ed length (e.g., a service time or interarrival
time), which is known when it begins (although it may be defined in terms of a
statistical distribution).
1 E)
Daily Demand
|
Probability
|
Cumulative Probability
|
Random-Digit Assignment
|
0
|
0.2
|
0.2
|
1-2
|
1
|
0.5
|
0.7
|
3-7
|
2
|
0.3
|
1.0
|
8-10
|
Day
|
Beginning Inventory
|
RD for Demand
|
Deamand
|
Ending Inventory
|
Shortage Quantity
|
1
|
4
|
4
|
1
|
3
|
0
|
2
|
3
|
1
|
0
|
3
|
0
|
3
|
3
|
8
|
2
|
1
|
0
|
4
|
1
|
5
|
1
|
0
|
0
|
5
|
0
|
2
|
0
|
0
|
0
|
Therefore,
no shortage condition occurs in the inventory.
1 F)
Step 1. Remove the event notice for the imminent event (event
3, time t\) from PEL
Step 2. Advance CLOCK to imminent event time (i.e.,
advance CLOCK from r to t1).
Step 3. Execute imminent event: update system state, change
entity attributes, and set membership as needed.
Step 4. Generate future events (if necessary) and place
their event notices on PEL ranked by event time. (Example: Event 4 to occur at
time t*, where t2 < t* < t3.)
Step 5. Update cumulative statistics and counters.
2A)
The Calling Population: -
The population of potential customers, referred to as the
calling population, may be assumed to be finite or infinite. In systems with a
large population of potential customers, the calling population is usually
assumed to be finite or infinite. Examples of infinite populations include the
potential customers of a restaurant, bank, etc. The main difference between
finite and infinite population models is how the arrival rate is defined. In an
infinite-population model, the arrival rate is not affected by the number of
customers who have left the calling population and joined the queueing system.
On the other hand, for finite calling population models, the arrival rate to
the queueing system does depend on the number of customers being served and
waiting.
System Capacity: -
In many queueing systems there is a limit to the number
of customers that may be in the waiting line or system. For example, an
automatic car wash may have room for only 10 cars to wait in line to enter the
mechanism. An arriving customer who finds the system full does not enter but
returns immediately to the calling population. Some systems, such as concert
ticket sales for students, may be considered as having unlimited capacity.
There are no limits on the number of students allowed to wait to purchase
tickets. When a system has limited capacity, a distinction is made between the
arrival rate (i.e. the number of arrivals per time unit) and the effective
arrival rate (i.e., the number who arrive and enter the system per time unit).
The Arrival Process: -
Arrival process for infinite-population models is usually
characterized in terms of interarrival times of successive customers. Arrivals
may occur at scheduled times or at random times. When at random times, the interarrival
times are usually characterized by a probability distribution. The most
important model for random arrivals is the Poisson arrival process. If An
represents the interarrival time between customer n-1 and customer n (A1 is the
actual arrival time of the first customer), then for a Poisson arrival process.
An is exponentially distributed with mean I/λ time
Units. The arrival rate is λ customers per time
unit. The number of arrivals in a time interval of length t, say N( t ) , has
the Poisson distribution with mean λt
customers. The Poisson arrival process has been successfully
employed as a model of the arrival of people to restaurants, drive-in banks,
and other service facilities.
A second important class of arrivals is the scheduled
arrivals, such as patients to a physician's office or scheduled airline flight
arrivals to an airport. In this case, the interarrival times [An , n = 1,2,. .
. } may be constant, or constant plus or minus a small random amount to
represent early or late arrivals.
A third situation occurs when at least one customer is
assumed to always be present in the queue, so that the server is never idle
because of a lack of customers. For example, the "customers" may
represent raw material for a product, and sufficient raw material is assumed to
be always available.
Queue Behavior and Queue Discipline: -
Queue behavior refers to customer actions while in a
queue waiting for service to begin. In some situations, there is a possibility
that incoming customers may balk (leave when they see that the line is too
long), renege (leave after being in the line when they see that the line is moving
too slowly), or jockey (move from one line to another if they think they have
chosen a slow line).
Queue discipline refers to the logical ordering of
customers in a queue and determines which customer will be chosen for service
when a server becomes free. Common queue disciplines include first-in,
first-out (FIFO); last-in firstout (LIFO); service in random order (SIRO);
shortest processing time first |(SPT) and service according to priority (PR).
In a job shop, queue disciplines are sometimes based on
due dates and on expected processing time for a given i type of job. Notice
that a FIFO queue discipline implies that services begin in the same order as
arrivals, but that customers may leave the system in a different order because
of different length service times.
Service Times and the Service Mechanism: -
The service times of successive arrivals are denoted by
S1, S2, S3…They may be constant or of random duration. The exponential, Weibull,
gamma, lognormal, and truncated normal distributions have all been used
successfully as models of service times in different situations. Sometimes
services may be identically distributed for all customers of a given type or class
or priority, while customers of different types may have completely different service-time
distributions. In addition, in some systems, service times depend upon the time
of day or the length of the waiting line. For example, servers may work faster
than usual when the waiting line is long, thus effectively reducing the service
times.
2B)
3A)
Daily Demand
|
Probability
|
Cumulative Probability
|
Random-Digit Assignment
|
3
|
0.20
|
0.20
|
01-20
|
4
|
0.35
|
0.55
|
21-55
|
5
|
0.30
|
0.85
|
56-85
|
6
|
0.15
|
1.00
|
86-100
|
Lead Time
|
Probability
|
Cumulative Probability
|
Random-Digit Assignment
|
|
1
|
0.36
|
0.36
|
01-36
|
|
2
|
0.42
|
0.78
|
37-78
|
|
3
|
0.22
|
1.00
|
79-100
|
|
Cycle
|
RD for Lead Time
|
Lead Time
|
RD for Demand
|
Demand
|
Lead Time Demand
|
1
|
21
|
1
|
99
|
6
|
6
|
2
|
19
|
1
|
85
|
5
|
5
|
3
|
30
|
1
|
52
|
4
|
4
|
4
|
78
|
2
|
42
|
4
|
8
|
29
|
4
|
5
|
80
|
3
|
91
|
6
|
16
|
88
|
6
|
35
|
4
|
3 B)
1. Problem formulation
Every study begins with a
statement of the problem, provided by policy makers. Analyst
ensures its clearly understood.
If it is developed by analyst policy makers should understand and
agree with it.
2. Setting of objectives and
overall project plan
The objectives indicate the
questions to be answered by simulation. At this point a
determination should be made
concerning whether simulation is the appropriate methodology.
Assuming it is appropriate, the
overall project plan should include
- A statement of the alternative systems
- A method for evaluating the effectiveness of these alternatives
- Plans for the study in terms of the number of people involved
- Cost of the study
- The number of days required to accomplish each phase of the work
with the anticipated
results.
3. Model conceptualization
The construction of a model of
a system is probably as much art as science. The art of
modeling is enhanced by an
ability
- To abstract the essential features of a problem
- To select and modify basic assumptions that characterize the system
- To enrich and elaborate the
model until a useful approximation results
Thus, it is best to start with
a simple model and build toward greater complexity. Model
conceptualization enhance the
quality of the resulting model and increase the confidence of the
model user in the application
of the model.
4. Data collection
There is a constant interplay
between the construction of model and the collection of
needed input data. Done in the
early stages.
Objective kind of data are to
be collected.
5. Model translation
Real-world systems result in
models that require a great deal of information storage and
computation. It can be
programmed by using simulation languages or special purpose simulation
software.
Simulation languages are
powerful and flexible. Simulation software models
development time can be
reduced.
6. Verified
It pertains to the computer
program and checking the performance. If the input parameters
and logical structure and
correctly represented, verification is completed.
7. Validated
It is the determination that a
model is an accurate representation of the real system.
Achieved through calibration of
the model, an iterative process of comparing the model to actual
system behavior and the
discrepancies between the two.
8. Experimental Design
The alternatives that are to be
simulated must be determined. Which alternatives to
simulate may be a function of
runs. For each system design, decisions need to be made
concerning
- Length of the initialization period
- Length of simulation runs
- Number of replication to be made of each run
9. Production runs and analysis
They are used to estimate
measures of performance for the system designs that are being
simulated.
10. More runs
Based on the analysis of runs
that have been completed. The analyst determines if
additional runs are needed and
what design those additional experiments should follow.
11. Documentation and reporting
Two types of documentation.
- Program documentation
Program documentation
Can be used again by the same
or different analysts to understand how the program
operates. Further modification
will be easier. Model users can change the input parameters for
better performance.
- Process documentation
Gives the history of a simulation
project. The result of all analysis should be reported
clearly and concisely in a
final report. This enables to review the final formulation and
alternatives, results of the
experiments and the recommended solution to the problem. The final
report provides a vehicle of
certification.
12. Implementation
Success depends on the previous
steps. If the model user has been thoroughly involved
and understands the nature of
the model and its outputs, likelihood of a vigorous implementation
is enhanced.
