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• Level: GCSE
• Subject: Maths
• Word count: 2957

# Decision and Discrete.

Extracts from this document...

Introduction

Name: Minesh Mepa                Centre Number: - 20780

Decision and Discrete 1 Coursework: - Queuing Problem

DECISION AND DISCRETE 1 COURSEWORKBY MINESH MEPA

What is simulation?

I am running a simulation by generating random numbers on my calculator and by doing this I can work out the solution to my problem.

## The problem

For this piece of coursework, we had to go into a place (possibly a store), which had a queuing problem in there. This was so that we could then simulate the times in, which a customer entered the shop, and simulate the time in which, the person gets served.

For my piece of coursework, I decided to go into Greggs; I decided to go at lunchtime. This is because the shop would mostly get crowded then, so it would be a good idea to do it at lunchtime and simulate the problem.

When I went in the shop to collect the data, there wasn’t much of a queue, but then it gradually began to grow, and it grew to upto the entrance.

There were only three servers in the shop, but only one server was open at the time, although there was a lot of staff serving behind the one server.

I noted the arrival and serving times of 50 people.

Middle

0.01

1.54.58

1.55.12

0.14

After I had collected the above data, I had to put the data into groups so that it would be easier for simulation. Before I could put the data into groups, I tallied the different seconds up into a table. This was so that I could know the size to put the groups at. For example, if there were a lot of values between 0 and 10, I would size the groups like 0-2, 2-5, and 5-10, but if there weren’t any values for a certain length such as from 20 to 40, I would size the groups into bigger numbers such as 20-30 and 31-40, etc.

The following are the tables for the IATs and the serving times. The first table is for the IATs and the second table is for the serving times: -

 IAT (SECS) FREQUENCY 1 5 2 10 3 1 4 1 5 3 6 0 7 1 8 0 9 0 10 1 11 3 12 2 13 0 14 1 15 1 16 0 17 0 18 0 19 0 20 1 21 0 22 1 23 0 24 0 25 2 26 1 27 1 28 0 29 0 30 0 31 0 32 1 33 1 34 0 35 0 36 1 37 2 38 1 39 0 40 0 41 0 42 0 43 0 44 0 45 0 46 1 47 1 48 1 49 0 50 1 51 1 52 1 SERVING TIMES FREQUENCY 1 0 2 0 3 0 4 0 5 0 6 0 7 1 8 0 9 3 10 2 11 5 12 5 13 3 14 2 15 3 16 4 17 2 18 3 19 3 20 0 21 4 22 3 23 0 24 2 25 1 26 2 27 0 28 0 29 0 30 0 31 0 32 0 33 0 34 0 35 0 36 0 37 0 38 0 39 0 40 1

From the above tables, I can make the groups and extend my coursework. I am now going to group the IATs and the service times, so that I can work the percentage and the cumulative percentage as well.

## IATs

FREQUENCY

Percentage

RN #

MID POINT

1-3

16

32

00 - 31

2

4-6

5

10

32 - 41

5

7-10

2

4

42 - 45

9

11-13

5

10

46 - 54

12

14-19

3

6

55 - 60

17

20-24

2

4

61 - 64

22

25-30

5

10

65 - 74

28

31-38

6

12

75 - 86

35

39-52

6

12

87 - 100

46

Conclusion

There was a major change to the length of the queue when the second sever was introduced. However, there was a lot of idle time.

To see what the effect would be on the serving time, idle time, and the q length, I decided to introduce a third server.

By introducing the third server, I got the following results: -

• Average Q Time
• Idle server time 1
• Idle server time 2
• Idle server time 3
• Average Service Time
 Average Q Time      = Total Q Time = 23 0.46 Total Number of customers 50
 Average Idle            = Total Idle Server Time 1 = 552 11.04 Server Time 1 Total Number of customers 50
 Average Idle            = Total Idle Server Time 2 = 568 11.36 Server Time 2 Total Number of customers 50
 Average Idle            = Total Idle Server Time 3 = 559 11.18 Server Time 3 Total Number of customers 50
 Average                   = Total Service Time = 821 16.42 Service Time Total Number of customers 50

As you can see by introducing the third sever into the shop, there hardly is a queue or the waiting time. You can also identify that the average service time has also been decreased as well.

Solution

However, introducing a third server into the shop would cause some servers such as server 2 and 3 to have a lot of idle time. Because of this, I have changed the actual rule, the new rule for the solution is: -

Have two servers open, if both are free go to server 1, if not, go the server, which has the least queue. If the queue lengthens up to the entrance of the shop, then introduce the third server.

The reason why I have changed the rule so that they only open the third server when it gets busy is so that it will reduce the actual idle time, which is given.

Candidate Number: - 2273                Page  of

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