Application of Queueing theory to Salem railway ticket counters

Sam Dickson; Jithin Ravi; Jayarama Mohan; H Sabareesh; G.S. Kumar

Research article · 2018 · Full text

International Journal of Statistics and Applied Mathematics

Application of Queueing theory to Salem railway ticket counters

Sam DicksonJithin RaviJayarama MohanH SabareeshG.S. Kumar

Volume 3(3) · Pages 203–207

Research areasAdvanced Queuing Theory Analysis Urban Transport Systems Analysis IoT and GPS-based Vehicle Safety Systems

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Abstract

Author-written

In this paper the Queueing model (M/M/C: /FIFO) is applied to the ticket counters of the Salem railway station. In this research the various characters of that queueing model have been analyzed and concluded.

Summary

Source-checked

Plain-language reading of the full paper, prepared with AI and checked against the text. Each point links to the authors' own words.

Why it matters
- Queues form at service facilities when the system lacks the capacity to serve all arriving customers simultaneously. View evidence ↗
  Why it matters
  Queues form at service facilities when the system lacks the capacity to serve all arriving customers simultaneously.
  In the paper's own words
  “A flow of customers from infinite or finite population towards the service facility forms a queue on account of lack of capability to serve them all at a time.”
  Page 1
  Found in: Introduction
  Read it in the full text Open PDF at page 1 ↗
The question
- The study aims to apply a specific queueing model to the Salem railway station reservation centre in Tamil Nadu, India, to determine its operational characteristics. View evidence ↗
  The question
  The study aims to apply a specific queueing model to the Salem railway station reservation centre in Tamil Nadu, India, to determine its operational characteristics.
  In the paper's own words
  “To apply the basic concepts of the selected queueing model to the Salem railway station, Tamil Nadu, India.”
  Page 1
  “To determine the various characteristics of the selected queueing model to the queueing system in the railway reservation centre at Salem Railway station.”
  Page 1
  Found in: Introduction
  Read it in the full text Open PDF at page 1 ↗
What they did
- The study utilized a multi-server queueing model (M/M/C: ∞/FIFO), assuming Poisson arrivals, exponential service rates, three servers, and a first-in-first-out discipline. View evidence ↗
  What they did
  The study utilized a multi-server queueing model (M/M/C: ∞/FIFO), assuming Poisson arrivals, exponential service rates, three servers, and a first-in-first-out discipline.
  In the paper's own words
  “The customers arrive at the railway reservation counters under poisson process and the service is done to the customers in the exponential rate. This queueing model involves three servers. The customers are given service under the FIFO (FIRST IN FIRST OUT) discipline. Thus the queueing model which will fit to our problem is (M/M/C: ∞/FIFO) which is also known as the multi server queueing model.”
  Page 1
  Found in: Data Collection
  Read it in the full text Open PDF at page 1 ↗
What they found
- The average arrival rate of customers to the system was approximately 24 per hour. View evidence ↗
  What they found
  The average arrival rate of customers to the system was approximately 24 per hour.
  In the paper's own words
  “Average number of arrivals to the system per hour (𝜆) = 12 per hour = 24.33 per hour = 24 per hour (appr.)”
  Page 3
  “Average number of arriving rate, 𝜆 = 24 per hour.”
  Page 5
  Found in: Evaluation of the Characteristics
  Read it in the full text Open PDF at page 3 ↗
- The average departure rate from the system was approximately 26 per hour, with an individual service rate of 6.5000 per hour. View evidence ↗
  What they found
  The average departure rate from the system was approximately 26 per hour, with an individual service rate of 6.5000 per hour.
  In the paper's own words
  “Average number of departure from the system (µ) per hour = 12 per hour = 25.67 per hour µ = 26 (appr.)”
  Page 3
  “Average number of service rate, µ = 6.5000 per hour.”
  Page 5
  Found in: Evaluation of the Characteristics
  Read it in the full text Open PDF at page 3 ↗
- The probability that a customer enters service immediately without waiting is 0.1644. View evidence ↗
  What they found
  The probability that a customer enters service immediately without waiting is 0.1644.
  In the paper's own words
  “Probability that an arrival enters the service without wait = 1 −P(n > C) = 1 −0.8356 = 0.1644”
  Page 4
  “Probability that an arrival enters the service without wait 1-p (n≥c) = 0.1644”
  Page 5
  Found in: Observations, Summary 5.1 Observations
  Read it in the full text Open PDF at page 4 ↗
What it means
- The authors conclude that employing four reservation clerks is optimal; adding another would significantly increase idle time, while removing one would lead to rapid queue growth and chaos. View evidence ↗
  What it means
  The authors conclude that employing four reservation clerks is optimal; adding another would significantly increase idle time, while removing one would lead to rapid queue growth and chaos.
  In the paper's own words
  “In our study we have found that it is optional that there are four reservation clerks employed in the salem railway reservation. It means that if we add one more counter to the existing set up, the chance for a reservation clerk to be idle would increase enormously. And also, if one counter is removed from the existing set up, the number of customers waiting in the queue would increase rapidly, thus it would end up in chaos.”
  Page 5
  Found in: Conclusion
  Read it in the full text Open PDF at page 5 ↗
- The system is considered stable because the traffic intensity remains less than one. View evidence ↗
  What it means
  The system is considered stable because the traffic intensity remains less than one.
  In the paper's own words
  “Also we have analyzed that the traffic intensity of the queueing system prevailing in salem railway station always remains less then unity and therefore the system attains stability.”
  Page 5
  Found in: Conclusion
  Read it in the full text Open PDF at page 5 ↗

Visual summary

2 visuals

Illustrations explain the paper; charts are replotted from the paper's own table values.

Queueing System Flow
AI-generated conceptual illustration (Railway ticket counters). It explains the idea and does not show study data.
View source evidence ↗

No. of arrivals Per Hour and No. of Departure Per Hour: 8.00-9.00 vs 9.00-10.00 vs 10.00-11.00 vs 11.00-12.00 vs 12.00-13.00
Replotted from Table 2 (page 3) of the original paper. Every bar is the paper's own table value.
View source evidence ↗

Figures & tables

From the original paper

Table 2Page 2

Table 2Page 3

Table 3Page 3

FULL TEXT Original article · reflowed for reading

Introduction

A flow of customers from infinite or finite population towards the service facility forms a queue on account of lack of capability to serve them all at a time. Our main objectives of our research are i. To apply the basic concepts of the selected queueing model to the Salem railway station, Tamil Nadu, India. ii. To determine the various characteristics of the selected queueing model to the queueing system in the railway reservation centre at Salem Railway station.

Data Collection

We have visited the railway reservation counters at Salem railway Station on various days during 8.00am to 8pm and conducted a survey about the arrivals of customers to the reservation counters and about the service rendered to the customers. The data pertaining to the arrival and Departures of customers to the reservation counters of Salem railway station are given in this section.

The customers arrive at the railway reservation counters under poisson process and the service is done to the customers in the exponential rate. This queueing model involves three servers.

Evaluation of the Characteristics

From the above table, 1. Average number of arrivals to the system per hour (𝜆) =

Observations, Summary 5.1 Observations

In our study we have made the following observations the following results. i. Traffic intensity, 𝜌 = 0.9231 ii. Probability that there are no customers in the system, 𝑃 0 = 0.0083 iii. Probability that an arrival has to wait, p (n≥c) = 0.8356 iv. Probability that an arrival enters the service without wait 1-p (n≥c) = 0.1644 v.

Summary

After conducting the survey and the analysis of data obtained from the reservation centre of salem railway station the following predictions are made regarding the characteristics of the queueing made 1 (M/M/C: ∞/FIFO):

i. An average of 24 customers are arriving to the reservation counters in an hour.

ii. An average of 26 customers are being served in the counters in an hour.

iii. The traffic intensity p= 𝜆 𝑐µ =3.6923<1 and therefore the queueing system followed in the salem railway reservation centre attains stability. iv. The probability that all the servers bring idle is nearly 2 percent. v. The probability that an arrival has to wait before getting served is 86 percent. vi. The probability that an arrival enters the service straight away without having to wait is 14 percent. vii. At any time an average of 10 customers are found waiting in the queue. viii. At any time an average of 13 customers are found waiting in the system. ix. On arrival, a customer has to wait in the queue for 25 minutes before getting served.

x. On arrival, a customer has to wait in the system for 34minutes before getting served. xi. A reservation clerk takes an average of 7 minutes to reserve a ticket.

Conclusion

In our study we have found that it is optional that there are four reservation clerks employed in the salem railway reservation. It means that if we add one more counter to the existing set up, the chance for a reservation clerk to be idle would increase enormously. And also, if one counter is removed from the existing set up, the number of customers waiting in the queue would increase rapidly, thus it would end up in chaos. Also we have analyzed that the traffic intensity of the queueing system prevailing in salem railway station always remains less then unity and therefore the system attains stability.

References

1. Kanti Swarup PK, Gupta manmohan. Operations research Sultan & Sons, New Delhi, 1995. 2. Gupta Manmohan PK. Operations Research and Statistical Analysis, Sultan & Son, New Delhi, 1991. 3. Hamdy Taha A. Operations research. Macmillan publishing Company, New York, 1987.

Figures and tables from the paper

References

0 in the original paper

Cited by 3

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Time	Time-intervals	Number of Arrivals	Number of Departure
8.00-9.00	0-10	//	-
	10-20	/	/
	20-30	///	//
	30-40	//	///
	40-50	///	//
	50-60	//	///
9.00-10.00	0-10	///	////
	10-20	///// //	///// /
	20-30	/	///// /
	30-40	/////	/////
	40-50	///// /	/////
	50-60	///// //	///
10.00-11.00	0-10	////	//////
	10-20	///// /	/////
	20-30	////	///// /
	30-40	///// /	/////
	40-50	///// /	///// /
	50-60	///// ///	/////
11.00-12.00	0-10	///// /////	///// /////
	10-20	///// //	///// //
	20-30	///// /////	///// //
	30-40	///// ///	///// ///
	40-50	///// /	///
	50-60	///// ///// //	///// ///// /
12.00-13.00	0-10	///// /////	///// //
	10-20	///// ///	///// ///
	20-30	/////	////// ////
	30-40	///// /	///// /////
	40-50	///// /////	/////
	50-60	/////	/////
13.00-14.00	0-10	///	///// /
	10-20	///	///// //
	20-30	///	///// /
	30-40	//	/////
	40-50	/	///
	50-60	/	///
14.00-15.00	0-10	-	-
	10-20	/	//
	20-30	///	///
	30-40	///	/////
	40-50	/////	////
	50-60	///// /	///// /
15.00-16.00	0-10	////	//////
	10-20	////	///
	20-30	//	///
	30-40	///	///
	40-50	//	//
	50-60	/	///
16.00-17.00	0-10	///	//
	10-20	////	////
	20-30	/////	///
	30-40	/	//
	40-50	///	////
	50-60	/	//
17.00-18.00	0-10	///	///
	10-20	/////	////
	20-30	////	//
	30-40	//	/////
	40-50	///// /	///
	50-60	/	//
18.00-19.00	0-10	///// //	///
	10-20	///	/////
	20-30	////	///
	30-40	///	///// /
	40-50	////	////
	50-60	////	//

0-10	//	///
10-20	//	/
20-30	///	////
30-40	/	///
40-50	//	//
50-60	/	///
0-10	///	///
10-20	////	/////
20-30	///	///
30-40	////	///// /
40-50	////	//
50-60	-	/

Abstract

Summary

1Why it matters

2The question

3What they did

4What they found

5What it means

Visual summary

Queueing System Flow

No. of arrivals Per Hour and No. of Departure Per Hour: 8.00-9.00 vs 9.00-10.00 vs 10.00-11.00 vs 11.00-12.00 vs 12.00-13.00

Figures & tables

Introduction

Data Collection

Evaluation of the Characteristics

Observations, Summary 5.1 Observations

Summary

Conclusion

References

Figures and tables from the paper

References

Cited by 3

Why it matters

The question

What they did

What they found

What it means