1-s2.0-S187705092101139X-main

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Procedia Computer Science 00 (2019) 000–000
Available online at www.sciencedirect.com

Procedia Computer Science 00 (2019) 000–000

ScienceDirect

Procedia Computer Science 188 (2021) 69–77

CQVIP Conference on Data Driven Intelligence and Innovation
CQVIP Conference on Data Driven Intelligence and Innovation

Application of Queuing Model in Library Service
Application of Queuing Model in Library Service

Xia Lyua, Fang Xiaoa , Xin Fana,*
aHuazhong UnivXerisaityLofyScuiean,ceFaandnTgecXhnoilaoogyaL,ibXrariyn, WFuhaann aa,n*d 430074, China

aHuazhong University of Science and Technology Library, Wuhan and 430074, China

Abstract
Abstract
Determining the number of service facilities to be purchased or allocated, in order to relieve the contradiction between reader
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determine the number of access control channels in our library, thus promoting the effective allocation of library facilities.
© 2021 The Authors. Published by ELSEVIER B.V.
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Innovation (CCDDII2021)
Keywords: Library; queuing; resource allocation; service desk

Keywords: Library; queuing; resource allocation; service desk

1. Introduction
1. Introduction

Queuing is a common phenomenon encountered in daily life. For example, people often have to queue when buying
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And in a broad sense, even each book can be seen as a “service desk” for readers. These desks can only serve one

* Corresponding author. Tel: +86-155 2778 0640;
* CE-omrraeislpaodnddriensgs:aufathnoxrin. @Tehl:u+st8.e6d-u1.5c5n2778 0640;

E-mail address: [email protected]
1877-0509 © 2021 The Authors. Published by ELSEVIER B.V.
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(PCeeCrD-rDevIiIe2w02u1n)der responsibility of the scientific committee of the CQVIP Conference on Data Driven Intelligence and Innovation
(CCDDII2021)

1877-0509 © 2021 The Authors. Published by Elsevier B.V.
This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0)
Peer-review under responsibility of the scientific committee of the CQVIP Conference on Data Driven Intelligence and Innovation
(CCDDII2021)
10.1016/j.procs.2021.05.054

70 Xia LyXuiaetLayul. /ePt arol.c/edPiraocCeodmiapuCtoemr SpcuiteenrcSec0ie0n(c2e011898) (020002–10)0609–77

reader at a time. For example, access channels can only be passed by one reader at a time, self-service loan machine
can only allow one reader at a time to borrow, and a book can only be borrowed by one reader at a time [1]. Therefore,
we can use the queuing model to quantitatively analyze the library's services and put forward quantifiable suggestions
on the allocation of various resources in the library.

Queuing Theory is a mathematical theory and method for studying the phenomenon of random dispersion of
systems and the working process of random service systems. It is a branch of operations research. Through the
statistical research on the arrival and service time of the service objects, we can obtain the statistical laws of these
quantitative indicators (waiting time, queue length, busy period, etc.) [2]. And then the structure of the service system
is improved or the object is reorganized according to these laws, so that the service system can not only meet the needs
of service objects, but also make the cost of the organization the most economical or some indicators optimal.

Queuing Theory, also known as random service system theory, is a discipline developed to solve these above
problems. Its studies include the following three parts:

 Behavioral problem, that is, to study the probabilistic regularity of various queuing systems, mainly to study of
queue length distribution, waiting time distribution and busy period distribution, including transient and steady
states;

 The optimization problem. It is divided into static optimization and dynamic optimization. The former refers to
the optimal design, and the latter refers to the optimal operation of the queuing system.

 The statistical inference of the queuing system. It is to judge which kind of queuing system is in conformity with
the models for analysis and research based on queuing theory.

Taking the library as an example, each reader randomly comes to the service organization (access control, self-
service loan machine, books, information desk, etc.) according to certain queuing rules and waits for service. They
leave the queuing system after receiving the service according to certain service rules (entry, loan operation, reading
books, consulting guide, etc.). In the queuing system, all readers who require service referred to as customers, and the
person or thing for the customer service are called service desks. And the service system is composed of customers
and service desks. A queuing service system can be denoted as: X/Y/Z/A/B/C, where X represents the distribution of
customer arrival time intervals, Y represents the distribution of service time, Z represents the number of service desks,
A represents the system capacity, B represents the number of customer sources, and C represents service rules. When
A and B are unlimited and C is first-come, first-served, it can be simplified to X/Y/Z.

For a service system, if the service organization is too small to meet the needs of so many customers who require
services, then crowding will occur and the service quality will be reduced. Therefore, customers always hope that the
larger the service organization is, the better. However, if the service organization is too large, the manpower and
material costs will increase accordingly, which will cause waste. Therefore, the purpose of studying the queuing model
of the library is to make a trade-off decision between the service time needs of readers and the scale of service facilities
to achieve a reasonable balance, such as the scale of access control, the number of self-service loan machines, the
number of book copies, and so on [3-6].

However, most of the current researches focus on the setting of the number of inquiries at the service desk, the
setting of the number of labor at the circulation desk, and other studies. And there is no statistical verification of reader
arrival rate. Based on this, this paper analyzes the log data generated by the reader's use of the library process, fits it
into analysis and determines its laws, and then analyzes the library's service facilities through the queuing model, so
as to provide a quantitative library facility setting [7].

2. Establishment of service queuing model

As a service organization, the library’s queuing model shows in Fig.1. The equipment, facilities and books that
provide services to readers can be regarded as “service desks”, through which the library provides services to readers.
In figure 1, the “service desk” is providing services for the readers [8-10].

Xia Lyu et al. / Procedia Computer Science 188 (2021) 69–77 71
Xia Lyu et al. / Procedia Computer Science 00 (2019) 000–000

Reader arrival Reader queue Reader i Service desk 1
…… Reader j

Service desk 2

……

Reader k Service desk n

Fig.1. Queuing model of library service

For the study of the queuing model in Fig.1. three elements need to be determined: the input process, that is, the
rules in which customers arrive at the system; the queuing rules, that is, the rules of customer waiting, including three
types and they are waiting system, loss system and mixed system; service organization, that is, the number of service
desks and service methods, and it can be one or more service desks which can be arranged in parallel or in series [11-
13].

In terms of queuing rules, readers have a certain purpose in coming to the library and will not leave at will, so the
model in Figure 1 should be a waiting system; in terms of service organizations, whether it is access control channels,
self-service loan machines, information desks or copies of books are all parallel to provide services to readers, so they
belong to the parallel mode of multiple service desks. For the input process, the fit can be determined through the
statistical analysis of the reader's arrival.

2.1. Statistical analysis of reader arrival

In order to fit the distribution probability of readers’ arrival, we collected statistics on the access control records
and borrowing records of readers every day throughout the year 2019. The statistics are based on 10 minutes as the
unit of measurement, counting the number of readers entering the library every 10 minutes, and readers using self-
service loan and return every 10 minutes.

72 Xia LyXuiaetLayul. /ePt arol.c/edPiraocCeodmiapuCtoemr SpcuitenrcSec0ie0n(c2e011898) (020002–10)0609–77

Fig.2. (a) Time-by-time statistics of access control in the library;
(b) Time-by-time statistics of access control in Hall C of the main library;
(c) Time-by-time statistics of self-service loan and return in the library.

As the chart showing in Fig.2., the K-S test was used as a non-linear test statistic for the statistical data to test the
goodness of fit, and the critical value of significance 0.05 was selected. It can be found that the reader arrival rates in
Fig.2. are all subject to the Poisson distribution, where in Fig.2(a).and Fig.2(b). the Poisson distribution follows the
parameter = 1.625, and in Fig.2(c). the Poisson distribution of the parameter = 1.76.

2.2. Birth and death process of library services

Suppose there is a service desk for a certain service in the system; readers arrive at the service desk according to
the Poisson flow, and the arrival intensity is . The service time of the service desk is negative exponentially
distributed, and the average service rate is , so the average service rate of the entire system is . At this point, the
system is an M/M/k queue. Then a reasonable number of service desks needs to get the value of the service window
in a stable state, that is, the number of service desks on the basis of service quality (reader waiting time) and reader
arrival intensity.

Observing the queuing model of the library, the following characteristics can be seen: the time interval of reader
arrival obeys the negative exponential distribution of parameter ; the time interval of reader departure obeys the
negative exponential distribution of parameter ; for each service desk, readers arrive successively and only one reader
arrives or leaves at a time.

It can be seen that the queuing model is a birth and death process, which is shown in Fig.3. For a certain service
ServiceA provided by the library, the state k indicates that there are access requests from k readers in the queuing
system. For the number of service desk n, there are service desks in the system to provide services for readers where
> , and the other − readers are waiting in line; there are k service desks in the system to provide services for
readers where ≤ .

λ λ λ λ λ λλ λ λ

… … …0 1
k k+1 n-1 n n+1

μ 2μ kμ (k+1)μ (k+2)μ (n-1)μ nμ (n+1)μ

Fig.3. The birth and death process of library services

According to the queuing theory, the important indicator symbols in the birth and death process of ServiceA in
Fig.3. are defined as follows, see table1.

Table 1. Definition of important indicator symbols in the birth and death process of ServiceA
Symbol Definition

Xia Lyu et al. / Procedia Computer Science 188 (2021) 69–77 73
Xia Lyu et al. / Procedia Computer Science 00 (2019) 000–000

,1/ average number of arrivals per unit of time, 1/ represents average time between arrivals
µ,1/µ average number of units processed in a unit of time for a continuously busy service facility, 1/µ represents
average time to service a unit
ρ system service intensity, namely /nµ, and set ρ1=/µ
Lq average queue length, the average number of readers waiting to be served in the queue
Lb
Ls average queue length, the average number of readers receiving service in the queue
Wq
Ws average number of requests, the average number of all readers in the queue, that is Lb+Lq
n average time of readers waiting, that is, the average response time
P0 average time the reader stays in the system, that is, the average completion time
Pk number of service desks of ServiceA

probability that ServiceA has 0 readers

the probability that ServiceA has k readers

3. Analysis of service model

In order to ensure the service quality of ServiceA, the reader's stay time has to be less than a certain threshold. At
this point, the birth and death process established in 2.2 can be used to solve the problem: find the number of service
desks of the ServiceA, so that the reader’s stay time of ServiceA is less than the threshold.

While the birth and death process in Figure 3 is in a balanced state, it satisfies the “inflow=outflow” principle in
statistical equilibrium, that is, the expected value of the transfer-in rate in each state and the expected value of the
transfer-out rate are equal. In Figure 3, when the birth and death process reaches equilibrium, if the number of service
desks of ServiceA is n, the expected value of the transfer-out rate or the state k in the birth and death process is:

{ − 1 +−1(+ + 1) +1, < , , ∈
+1, ≥

And the expected value of the transfer-in rate is:

{ + , ≤ , , ∈
+ , >

Therefore, for state 0, the expected value of the transfer-out rate is 0, and the expected value of the transfer-in
rate is 1, namely, 0 = 1

For state 1, the expected value of the transfer-out rate is 1 + 1, and the expected value of the transfer-in rate
is 0 + 2 2, namely, 1 + 1 = 0 + 2 2

Because of 0 = 1, so for state 1, 1 = 2 2

For state 2, the expected value of the transfer-out rate is 2 + 2 2, and the expected value of the transfer-in rate
is 1 + 3 3, namely, 2 + 2 2 = 1 + 3 3

Because of 1 = 2 2, so for state 2, 2 = 3 3

By analogy, for state n-1, the expected value of the transfer-out rate is −1 + ( − 1) −1, and the expected
value of the transfer-in rate is −2 + , that is,

−1 + ( − 1) −1 = −2 +

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Similarly, we can also get −1 =
For state n, the expected value of the transfer-out rate is + , but for the expected value of the transfer-in
rate, since ServiceA has n service desks, there are still only n service desks providing services in the state n+1, that is,
for state n+1, the expected value of transition to state n is +1, then the expected value of transition-in rate of state
n is −1 + +1, namely, + = −1 + +1
The same can be obtained, = +1
By analogy, for any state r(r>n), + = + +1

Therefore, when there are n service desks in ServiceA, the equilibrium equations in any state can be known.

0: 0 = 1
1: 1 = 2 2

⋮
: = +1

⋮
{ + − 1: + −1 = +

Combining the definitions in Table 1, according to the above balance equations, it can be calculated:

0: 1 = 1 × 0 = 0
12 2
1: 2 = 2! 0 = 2! 2 0
⋯
1 ! + 1⋯ 0
n: +1 = = ! +1 0

{n+r-1: + = 1 + 0 = + 0
! !

Therefore, the equilibrium equation for any state k can be obtained:

= { 1 !! 1 − 0 = 0 = ! ! 0 , 0 0≤, ≥< (1)

At the same time, the sum of the probabilities of each state in the queuing model is 1. Therefore, combining formula
1, we can get

∞

∑ = 1

=0

Namely,

1 = −1 1 + ∞ ! 1 − ) 0
!
(∑ ∑

=0 =

And so,

Xia LXyuiaeLt yaul. /ePt arol.c/edPiraocCeodmiapCutoemr SpcuiteenrcSec0ie0nc(2e011898) (020002–10)0609–77 75

(2)

0 = −1 1 + 1 1 1 −1
! ! −
(∑ )

=0

Because the queuing model in ServiceA is a first-come, first-served model, readers in the waiting state in the queue
need to wait until the previous request is completed before they can get the service. Combining the definition of the
average queue length in Table 1, we can get,

∞ (3)

= ∑ +

=1

Substituting + in formula 1 into formula 3 can be obtained:

= 1 0 (4)
(1− )2⋅ !

Readers using ServiceA not only include the ones in the waiting state in the waiting queue, but also the ones who
are using the service. The sum of the two is all readers in the system. Therefore, combining the definition in Table 1,
is the number of readers that the system is using ServiceA, and is the number of all readers in the system, we
can get: = +

Combining the definitions in Table 1, we can substitute formula 3 into the above formula,

= 1 0 + 1 (5)
(1− )2 !

According to the definition in Table 1, is the average waiting time of readers in the queue and is the average
staying time of readers in the system, which is regarded as the average completion time of reader service in the system.
According to Little's law, we can get,

= = 1 0 (6)
(1− )2⋅ ! (7)

= = + = + 1


According to formula 2 and formula 7, the average response time (Wq) and average completion time (Ws) of the

system to readers depends on the reader arrival intensity , service intensity u and the number of service desks .

For the queuing system, after the system is completed, if no upgrade operations such as expansion are performed, its

reader processing capacity has been finalized. At this time, the average service intensity of readers u in the system

is determined by the file size. Therefore, the average response time (Wq) and average completion time (Ws) of readers
in ServiceA depend on the intensity of reader arrival and the number of service desks .

After obtaining the reader's arrival intensity, the number of service desks can be changed according to the needs
of Wq and Ws (that is, the needs of service quality). According to formula 6, while the number of service desks is
larger, the values of Wq and Ws are smaller. Therefore, there is a critical value for the number of service desks ′.
When ≥ ′, the average response time (Wq) and average completion time (Ws) of readers in the system both meet

the needs of library service quality.

76 Xia LXyuiaeLt yaul. /etParlo.c/ePdiraocCeodmiapCutoemr pSuctieerncSeci0e0nc(2e011898) (020002–10)0609–77

4. Application model

Taking a practical example to calculate, suppose that in the access control channel system in Hall B of the main
library, the arrival of readers obeys the characteristics of Poisson distribution, and the arrival rate is 48 person-times
per minute, that is, the reader's arrival intensity is = 48. At the same time, the service time of the access control
system for readers obeys the exponential distribution and the service time is 25 people per minute, that is = 25. We

want to solve the relationship between the number of access control channels n and the number of readers at the

channel and the average stay time of readers .

Solution: Because of = 48, = 25

So, 1 = = 1.92


According to formula 2, we can get:

0 = −1 (1.92) + ( − (1.92) − −1
! 1.92) ⋅ (
[∑ 1)!]

=0

According to formula 5, we can get:

= ( (1.92) +1 ⋅ 0 1)! + 1.92
− 1.92)2 ⋅ ( −

Substitute 0 into to get:

= (∑ =−01 (1.92) ) ⋅ ( − (1.92) +1 + (1.92) ⋅ ( − 1.92) + 1.92
! 1.92)2 ⋅ ( − 1)!

According to formula 6, we can get: =
48

Since the number of passages in Hall B is a discrete natural number, the relationship between the number of
passages in Hall B and the number of readers at the gate can be calculated. The specific values are shown in
Table 2. It should be noted that the arrival intensity and service intensity in this example are based on the time in
minutes. After the calculation of = 4 8 , the result needs to be converted into a value in seconds.

Table 2. The relationship between the number of different channels

(second)

2 24.96 31.2
3 2.65 3.36
4 2.06 2.58

According to Table 2, for the access control channel system in Hall B of the main library of our library, if the
reader’s stay time requirement is less than 5 seconds, the critical value of the number of access control channels ′ is
3; if the reader’s stay time requirement is less than 3 Second, the critical value of the number of access control channels
′ is 4.

Xia LXyuiaeLt yaul. /ePt arol.c/edPiraocCeodmiapCutoemr SpcuiteenrcSec0ie0nc(2e011898) (020002–10)0609–77 77

5. Conclusion

The resource allocation of the library has always been a hot research topic. This paper uses the queuing model to
determine the library resource allocation model on the basis of the reader service quality. The model determines the
quantitative measurement of the library resource allocation through the birth and death process of the number of
readers. Finally, it uses the model to determine the number of access control channels as a practical verification of the
model.

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