IMT CAFETERIA -STUDY ON QUEUING SYSTEM

BY

J.SHIVALKAR

HARSHITA CHINTAM

POORNACHAND KALYAMPUDI

BACKGROUND STUDY OF THE SYSTEM:

Institute of Management Technology Cafeteria employs two staffs and they sell items

such as Biscuits, Chocolates, Cakes, Puffs, Sandwiches, Omlette and Noodles. One

of the staffs(Staff 1) takes care by serving the students only Sandwiches, Omlettes

and Noodles and the other staff (Staff 2) takes care of Biscuits, Chocolates and

microwave oven related food. IMT Cafeteria imitates a system with two servers and

two queue.

OBJECTIVE OF THE PROJECT:

Around 410 students study in IMT and the average waiting time of the students in the

queue, average waiting time of students in the system, average staff service time

,probability of Staff being idle, average time between arrivals and the probability of

the student waiting in the queue for both servers Staff 1 and Staff 2 are studied.

DATA FINDINGS:

The data found below has been collected using stopwatch timer on September

26th,2013 form evening 6:00 p.m to 8:00 p.m.

Staff 1 service time(seconds)

Staff 2 service time(seconds)

Arrival time Seconds

Interarrival time seconds

1 396 6 206 0

2 630 9 343 137

3 417 72 389 46

4 569 87 562 173

5 556 17 724 162

6 603 48 1032 308

7 623 31 1166 134

8 604 94 1388 222

9 483 37 1459 71

10 279 132 1602 143

11 388 9 1631 29

12 562 56 1836 205

13 630 108 1902 66

STAFF 1 STAFF 2

14 503 47 1959 57

15 370 102 2005 46

16 276 33 2218 213

17 616 96 2346 128

18 498 63 2579 233

19 530 53 2755 176

20 554 104 2822 67

21 559 56 2888 66

22 527 68 3184 296

23 494 23 3258 74

24 505 40 3435 177

25 513 159 3451 16

The numbers are in seconds which are continuous variables which would help to

determine the distribution of arrival time and service time distribution well.

The Student arrival time follows a Uniform Distribution which is show in below Fig 1.

Fig 2. Arrival time of Students

The Student inter-arrival time follows Normal, Uniform and Weibull distribution is

shown below in Fig 2. Since all have same chi-square statistics. Uniform distribution

was preferred to be taken.

Fig 2. Inter Arrival time of Students

Serivice time of Staffs:

Staff 1 service time follows a triangular distribution shown in Fig 3 and Staff 2’s

service time too follows a triangular distribution shown in Fig 4.

Fig 3. Staff 1 service time

Fig 4. Staff 2 service time

METHODOLOGY:

As the data of Interarrival time follows an Uniform distribution,Riskfunction of

RiskUniform(60,1200) was kept. As the Service time followed Triangular distribution

minimum,most likely and maximum seconds were kept for Service time of both the

staffs and Risktriangular function was used.

Staff 1 Service Time seconds Item Code Item

Minimum 276 1 Maggi

most likely 507 2 Sandwich

Maximum 630 3 Omlette

Staff 2 Service Time Seconds Item code Item

Minimum 6 1 Chocolates

Most Likely 62 2 Cool Drinks

Maximum 159 3 Puff

Risk outptut function was added to these fields:

Average Waiting Time for a student in system

Average Waiting Time in Queue

Average Staff Service Time

Prob of idle server

Avg time between arrivals

Prob that a student waits in the queue

And the simulation was run for 1000 iterations with first 100 students.

OUTPUT:

The final output which was computed is shown in the below table

Staff 1 Minutes Staff 2 Minutes

Average Waiting Time for a student in system

939 15:39 98.199 01:38

Average Waiting Time in Queue 445 07:25 8.87 00:09

Average Staff Service Time 494 08:14 89 01:29

Prob of idle server 22.33% 57.43%

Avg time between arrivals 629 10:29 190.9088 03:11

Prob that a student waits in the queue 0.5725 0.1245

VALIDATION:

Validation for Staff 1 model

6 Replication scenarios were considered and t-statistic was computed.

Mean(mu) 456

Expected Mean 311.3333333

Standard deviation 241.4635928

t -1.467548428

t(critical) 2.571

Absolute value of t is lesser that t(critical) hence the Staff 1 model is valid. Validation for Staff 2 model

Mean(mu) 4.8767

Expected Mean Rep 1 3.846153846

Standard deviation 2.556779443

t -0.987301521

t(critical) 2.571

Absolute value of t is lesser that t(critical) hence the Staff 2 model is valid. CONCLUSION: The average waiting time of a student in queue in Staff 1 model is comparatively greater than Staff 2 model, hence the Staff 1 model should additionally be employed by one more staff so that the average waiting of student in queue would drastically reduce to half or the Staff 2’s probability of being idle is more than 50% so he should also take the work of Staff 1 to reduce the average waiting time of a student in queue and this would help the students community to get benefitted at greater level.

ANNEXURE:

STAFF 1 MODEL

STAFF 2 MODEL