Analysis of Airport Taxi Problem based

on M/Ek/1 Queuing Model

Jie Huan

, MengQing Xiao

and YuQian Zhao

Institute of Collaborative Innovation, University of Macau, Macau, China

Institute of Accounting, Tianjin University of Finance and Economics, Tian Jin, China

Institute of mathematics and statistics Hebei University of Economics and Business,Shi Jiazhuang, China

Keywords: Taxi Queuing Theory M/Ek/1 Model, Lingo.

Abstract: The abstract should summarize the contents of the paper and should contain at least 70 and at most 200 words.

It should be set in 9-point font size, justified and should have a hanging indent of 2-centimenter. There should

be a space before of 12-point and after of 30-point. Nowadays, with the growth of the number of private cars

and the number of taxis, the problem of taxi queuing in the airport transportation system needs to be solved

urgently. This paper combines the M/Ek/1 service model in queuing theory (Qie, Wang 2007), effectively

proposed to establish a decision model for the taxi waiting for passengers or no-load return journey, and the

actual airport information verification model is more reasonable.

Scheme A and Scheme B decisions were made by comparing the time costs. Under Scheme A, because the

taxi arrival time follows the parameter of 𝜆 Poisson distribution and the ride service time follows the Erlang

distribution of 𝑘 order, the M/Ek/1 queuing model can be established to list the system equation of state

through the relationship between the total exponential service steps j in the system and the probability

distribution of k customers in the system. Then the parent function is introduced, and finally the average

waiting time of the passenger is 𝑊



()

()

, set the time cost of no-load backhaul in scheme B is Q, when

Wq <Q, scheme A, and when Wq> Q, scheme B.

Shanghai Pudong Airport and Shanghai taxi data were selected for model test until reasonable. The results

were calculated using Lingo to compare the time hours required for schemes A and B. The length of Scheme

B can be calculated from the speed of taxi driving on the Shanghai viaduct and the urban speed of Scheme B

is 1.1917 hours. When the time required of Scheme A is less than 1.1917 hours, Scheme A is selected,

otherwise Scheme B.

The M/Ek/1 queuing model established in this paper avoids the limitations of negative exponential distribution

and can be more applicable to multiple serial processes, or if no memory assumption is not significant;

queuing theory can not only solve the taxi queuing problem, but also has broad applications in medical and

communication fields.

1 INTRODUCTION

A large number of tangible or invisible queuing or

crowded phenomenon as a common life problem,

such as restaurant dining queuing problems, banking

business queuing problems and so on. AS the

economic growth increases and the number of private

cars increases, the road traffic queuing phenomenon

is particularly common. Queuing theory has been

widely used in communication systems, storage

systems, etc.

Production management and other aspects play an

important role, queuing theory for the transportation,

especially the taxi queuing problem research.

Through the statistical study of the arrival time

and the arrival time of taxi drivers, the statistical law

of the waiting time of taxi drivers and passengers and

the peak of taxi passengers are obtained. Then,

according to these laws, we can improve the taxi

queuing problem at the airport terminal or make

decisions for taxi drivers, so that not only taxis can

carry passengers efficiently, but also solve the

problem of passenger retention at the airport.

Huan, J., Xiao, M. and Zhao, Y.

Analysis of Airport Taxi Problem based on M/Ek/1 Queuing Model.

DOI: 10.5220/0011154500003437

In Proceedings of the 1st International Conference on Public Management and Big Data Analysis (PMBDA 2021), pages 191-196

ISBN: 978-989-758-589-0

191

In the problem of queuing in the airport terminal,

the taxi said that the drivers have two schemes: A and

B, to choose. The two schemes compare the waiting

time, and the short time, the time cost is low, which is

selected as the final decision plan. In the case of

scheme A, the taxi is relatively free, can approximate

the Poisson distribution, the time of the taxi is

independent (Wang, Shi, Wang 2015) so

approximately obey the Erlang distribution, from the

traditional M/M/1 waiting system queuing model can

be optimized to the M/Ek/1 queuing model system,

introduce the parent function, using the L'Hopital law

to remove the waiting time. In scenario B, the taxi

driver will immediately return with an empty load,

and the time cost is the return time (or plus the empty

load in the urban area Between). Comparing the two-

time costs gives the selection strategy. As shown in

Figure1:

Figure 1: Scheme selection flow chart.

This question selects Shanghai Pudong airport

taxi data and airport traffic data, using LINGO and

MATLAB software simulation calculation, intercept

a day (with peak and stationary period) for the

research period, bring data into the problem of the

decision results, and analyze the accuracy and

rationality of the results, and discuss the correlation.

2 MODEL ESTABLISHMENT

2.1 Model Hypotheses

1. Suppose that taxis and passengers are generally

unlimited.

2. Suppose that the passenger arrival is

independent of each other (Xue 2004).

3. Taxi and passengers wait (queue without

leaving).

4. Suppose that the number of passengers arriving

is subject to the Poisson distribution (Tang 2017) (the

passengers waiting for the taxi are stable and

ineffective, Ordinary), the service time follows the

Erlang distribution (can represent the time interval of

independent events, good fitting effect).

5. Suppose the driver drives freely, not affected by

the weather, and the traffic flow is smooth.

2.2 Representation of Symbol

Table 1: representation of symbol.

symbol meanings

Team length: the total number

of taxis in the car storage pool

Length: Number of taxis in the

storage pool

Stay time: the time that the taxi

stays in the storage pool

Line up time: the time the taxi

waits in the storage pool

Taxi reach rate per unit of time

Number of taxis completing the

service per unit of time

Average service time per unit of

service desk time

Access capacity of taxi point

(vehicle / hour)

Average taxi stop time

(boarding time and getting in

and out the sum of points)

Number of valid berths

To offset the reduction

coefficient of the docking time

fluctuations, it is usually taken

R = 0.833

2.3 Establishment of Model

The arrival time of the taxi camera follows the

Poisson distribution with the parameter, the number

of stations of the passenger boarding point is 1 (Geng,

Song, Zhao 2013), and the service time is abstract as

the time interval of independent random events, so it

follows the Erlang distribution of order k; the

passenger arrival time can be abstracted as random,

thus approximating the Poisson distribution. Using

the Poisson distribution and the Erlang distribution,

the M/Ek/1 queuing model can also be established to

form the queuing system of the airport and make a

state transfer diagram to list the probability

distribution of the exponential service steps j in the

system and the probability distribution of k customers

in the system. According to the dynamic transfer

diagram, we can list the system transfer equation. It is

then solved by introducing the parent function (Baidu

Encyclopedia 2019).

PMBDA 2021 - International Conference on Public Management and Big Data Analysis

192

3 MODEL SOLUTION

3.1 Model Solution Process

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area.

The form should be completed and signed by one

author on behalf of all the other authors.

Taxi arrival time is assumed to obey the Poisson

distribution of the parameters, and the probability

distribution function is:

()

() ,( 0)

Pt t

−

=≥

Taxi service time follows the Erlang distribution

of k, and its probability distribution function is:

()

() ,( 0)

(1)!

kkt

ft e t

μμ

−

=≥

−

Each taxi in the system leaves the system after the

k-step index service and is set in the time t system

The number of car rental is n, and the number of

service steps at the time t index is j.

Record P

= P {The total number of exponential

service steps in the system is j} ； P

=P {There are

k cars out of the system}, The relation

for these two

quantities is：

(1)1

,( 1,2,3, )

jn k

PPk

=− +

==Λ



The following system equations of state are listed

by the state transfer diagram:

,( 1,2,3, )

()

jjk j

PkP

kP P kP

λμ

λμ λ μ

−+





=Λ



+=+





Because the variables in this model are all non-

negative integers, the concept of the probability

mother function is introduced, let's consider:

()

Pz Pz

∞



Type (1) is available thus:

𝑃(𝑧) =





()





()

(1)

Definition is given by the probabilistic mother

function P (z)= 1，we can get（2）:







(2)

then

11 1P

==− =−

Cut off（1-z）and

Type (3) is available by the partial split method

𝑃(𝑧) =









(



Λ



)



(





)(





)Λ(





)

=(1−𝜌)

∑

𝐴







/(1 −





) (3)

Among

,,,

zz zΛ

are the root of the denominator

polynomial, and have:

1/(1 )

nni

=≠

=−

∏

Expand Type (3) and compare the coefficient to

get that the taxi reaches the car storage pool and find

the car storage pool. The probability of j cars already

inside:

(1 ) ( ) , ( 1, 2, 3, )

jii

PAzj

−

=− = Λ



The average service time of a taxi is the average

pickup time is 1/k

μ

The average taxi pickup time for

j taxis is j/k.

Therefore, the average waiting time required for a

new taxi to receive guests is type (4):

(4)

For z to derive formula (5):

(5)

When z →i ，we can use the Lobida rule to get

the (6) formula:

(6)

From types (4) and (6): the average queuing time

of the taxi in the storage pool

(1)

2(1 )

μρ

−

The average number of taxis queuing in the

storage pool Lq：

(1)

2(1 )

−

Analysis of Airport Taxi Problem based on M/Ek/1 Queuing Model

193

Therefore, the waiting time for Scheme A is W

, if

the wasted time cost of the taxi going directly empty

to the city in Scheme B is Q.

3.2 Results Discussion

Let the waiting time of the taxi in the storage pool be

，t

=Ws；In Scheme B, the taxi to return directly to

the city is t

，the cost of no-load loss is D(D < 0).

When t

< t

，that is, the taxi waiting time in the

storage pool is shorter than the return to the city

directly empty load, at this time, in the min (t

, t

)

period, the income of Scheme A is 0, while Scheme B

income D. At this time, the taxi driver chooses

Scheme A, that is, the taxi queue to the designated car

storage pool and queue into the passengers according

to "come first to arrive".

When t

> t

，that is, the taxi waits longer in the storage

pool than to the city directly on empty load During

the max (t

, t

) time period, the return of Scheme A is

0, while the return of Scheme B is unknown and

requires further discussion:

let t

-t

1.In t

period, if the taxi cannot receive the guest,

scheme B proceeds to D (D < 0) ，At this time,

scheme A income is 0, so scheme A is selected.

2.In t

period, if the taxi receives the customer,

set the taxi manned income is Q, then the scheme B

income is (D+Q). When D+Q > 0, we select Scheme

B, and vice versa, Scheme A.

4 MODELLING VERIFICATION

Select Shanghai Pudong Airport and collect the

relevant data of Shanghai taxi as shown in Table 1.

According to the official website of Shanghai Pudong

Airport, the taxi charging standard is as follows:

Table 2: Taxi Charging Standard in Shanghai.

daytime（05:00-

23:00）

nighttime（

23:00-05:00）

0-3

kilometers

14 CNY 18 CNY

3-10

kilometers

2.5 CNY / km 3.1 CNY / km

than15

kilometers

3.6 CNY/ km 4.7 CNY / km

Referring to the data of the "Third Comprehensive

Traffic Survey Report of Shanghai", we learned that

the empty driving rate of Shanghai district in 2004

was 39%, the total daily mileage was 9.5 million

kilometers, and the service number was 1.493 million

/ day.

It is calculated that the average daily mileage per

taxi is 6.36 kilometers, and the average daily mileage

per hour is 0.265 kilometers, and about 14 CNY and

18 CNY per hour between day and night. Shanghai

Pudong Airport is about 55 kilometers away from the

city center (Wu, Zheng, Deng 2009). Taxi vehicles

use no. 93 gasoline, consuming an average of 8 liters

per 100 km and 7.55 CNY per liter. Therefore, the

one-way oil cost from downtown to the airport is

33.22CNY. It can accommodate 10 taxis at the same

time.

Table 3: Passenger boarding time and frequency table.

Time

interval

（s）

Number of

Vehicles

(Units)

Time

interval

（s）

Number of

Vehicles

(Units)

3.4-13.4 103 43.5-53.4 8

13.5-23.4 62 53.5-63.4 5

23.5-33.4 27 63.5-73.4 2

33.5-43.4 16 73.5-83.4 2

Figure 2: Time and frequency distribution diagram of

passengers' boarding time (Sun, Ding, Chen 2017).

The time frequency distribution of passenger

boarding is shown in Table 2, the data are analyzed

by Excel to approximate the exponential distribution,

and the variance of the above 225 random variables is

0.876, and the passenger boarding time

approximately follows the exponential distribution of

the mean 15, the passenger boarding time frequency

distribution diagram is shown in Figure 2.

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194

Figure 3: Taxi flow distribution map of Shanghai Pudong

International Airport within one day (Yan 2015).

Figure 4: Taxi waiting time in the car storage pool(h).

Figure 5: Number of vehicles in the vehicle storage pool.

The actual data of Shanghai Pudong Airport will

be brought in Fig. 3, Fig.4 and Figure 5 into the

model, with two schemes A and B. The time-

consuming scheme of 1.1917 is the priority decision

model, and according to the Lingo calculation results,

scheme A is the following period:

14:30-15:30, 9:30-10:30, 10:30-11:30, 11:30-12:30,

12:30-13:30, 13:30-14:30, 8:30-9:30, 20:30-21:30,

23:30-00:30, 00:30-01:30, 01:30-02:30, 02:30-03:30,

03:30-04:30, 05:30-06:30, 04:30-05:30, 06:30-07:30,

07:30-08:30.

Select Scheme B for the rest of the period.

5 CONCLUSIONS

The simple negative exponential distribution is

broken through in the model establishment, so that the

service time of the passengers is basically subordinate

to the Erlang Distribution, with a better fit (Lin 2018),

reduces the model error. Select the queuing theory in

operation research, the statistical rules can be

analyzed with the calculated statistical indicators, and

then improve the service system structure according

to these rules, or reorganize the served object, so that

the service system can not only meet the needs of the

service object, but also achieve the optimization of

some indicators of the organization. The M/E

model of queuing theory can not only compute the

statistics, but also optimize the subsequent results (Li

2014), and optimize the overall service level of the

model with the marginal analysis method, dynamic

planning and other methods.

The longitudinal generalization of the model is

extended through the M/E

/1 waiting queuing model,

to the queuing model of multi-service desk, M/M/S/

∞ model (Wu, Li, Liang 2012), with multiple service

desks, and customers can receive service immediately

with free service desk. To continue to improve the

model complexity, we can build the M/M/S/K hybrid

model, which can solve the queuing problem of the

loss system, which is more in line with the actual

situation.

Horizontal promotion: queuing theory can be

widely used in the medical field, banking service

system, airport security check system, campus

express peak service system, college students'

canteen dining problems and so on. For example,

through considering the problem of dining in the

college students' canteen, deduce the peak period of

students during the class, count the number and

location of the canteen Windows, and get the

rationalization suggestions for the canteen service

after the queuing model.

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