Solution of Unit Test I Computer Simulation and Modeling

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

1. Avg. waiting time for a customer= 3/5 = 0.6 min
2. Prob. That customer has to wait in queue= 1/5 = 0.2
3. Prob. Of idle server= 11/25 = 0.44
4. Avg. service time= 14/5 = 2.8 min
5. Avg. time between arrival= 23/4 = 5.75 min
6. Avg. waiting time for customers who wait= 3/1 = 3 min
7. 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

1. A statement of the alternative systems
2. A method for evaluating the effectiveness of these alternatives
3. Plans for the study in terms of the number of people involved
4. Cost of the study
5. 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

1. To abstract the essential features of a problem
2. To select and modify basic assumptions that characterize the system
3. 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

1. Length of the initialization period
2. Length of simulation runs
3. 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.

1. 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.

2. 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.