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International Journal of Computer Networks & Communications (IJCNC) Vol.13, No.4, July 2021
DOI: 10.5121/ijcnc.2021.13406 85
PERFORMANCE ANALYSIS IN CELLULAR
NETWORKS CONSIDERING THE QOS BY RETRIAL
QUEUEING MODEL WITH THE FRACTIONAL GUARD
CHANNELS POLICIES
Dang ThanhChuong1
, Hoa Ly Cuong1
,
Pham Trung Duc1
and Duong Duc Hung2
1
Faculty of Information Technology, University of Sciences, Hue University, Viet Nam
2
Hue University, Viet Nam
ABSTRACT
In this article, a retrial queueing model will be considered with persevering customers for wireless cellular
networks which can be frequently applied in the Fractional Guard Channel (FGC) policies, including
Limited FGC (LFGC), Uniform FGC (UFGC), Limited Average FGC (LAFGC) and Quasi Uniform FGC
(QUFGC). In this model, the examination on the retrial phenomena permits the analyses of important
effectiveness measures pertained to the standard of services undergone by users with the probability that a
fresh call first arrives the system and find all busy channels at the time, the probability that a fresh call
arrives the system from the orbit and find all busy channels at the time and the probability that a handover
call arrives the system and find all busy channels at the time. Comparison between four types of the FGC
policy can befound to evaluate the performance of the system.
KEYWORDS
Cellular Mobile Networks, Retrial, FGC, QoS.
1. INTRODUCTION
In the present, with the aid of the rapid development of wireless and mobile networks, many
queueing models, especially retrial queues, have been proposed to evaluate the actual
performances of these network systems. In cellular networks, it is important to design network
models that the handover calls are more prior than the fresh calls properly. The concerns are
limited resources, for example, a number of channels, sharings, and competitions in a certain cell
(the collisions between the handover calls and the fresh calls), which can lead to the call
interruptions when a user frequently moves among cells in the network as cell edges. Therefore,
it is necessary that the appropriate policy admitting and processing the arrival calls minimize
congestion risks of the system, including the handover and fresh calls. The call admission
controls have been proposed to fulfillQoS requirements [1].
The research [1] and [2] first introduced the application of the retrial queueing models on
evaluating cellular mobile networks involving the fractional guard channel policy to minimize the
handover probability. Then [3] constructed the novel model and obtained some important results
but that model bases only one probability ๐œƒ. In later years [4] and [5] reapplied and renovated [3]
by considering the additional probability of the fresh calls ๐œƒโ€ฒ. [6] used the fractional guard
channel without retrial customers. [7] improved [3] by using the fractional guard channel, but it
considered only one probability ๐œƒ. The fractional guard channel policy allocates resources in the
International Journal of Computer Networks & Communications (IJCNC) Vol.13, No.4, July 2021
86
cellular mobile network to reserve for the handover calls. This policy is considered the general
form of the guard channel ones. When the system has available resources, the fresh call
requirements are accepted by a certain probability, depending on the cell states (a number of the
busy channels).
Ramjee [8] first proposed the fractional guard channel policy with the accepted probability of the
fresh calls ๐›ฝ๐‘– (0 โ‰ค ๐›ฝ๐‘– โ‰ค 1, ๐‘– = 0, ๐‘, where ๐‘ is the number of the channels), and it depends on the
number of the operating channels. The problem is that the optimal probability ๐›ฝ๐‘– is chosen by the
fractional guard channel policy. The limited fractional guard channel LFGC, the uniform
fractional guard channel UFGC, the limited average fractional guard channel LAFGC and the
quasi-uniform fractional guard channel QUFGC policies have been initiated based on the features
of different network systems.
To generalize the above results, the research can be examined the retrial queueing model whose
form is ๐‘€/๐‘€/๐‘/(๐‘ + ๐ฟ) in combination with the fractional guard channel policies and the
characteristic parameters, such as the probability that a fresh call first arrives and then it enters
the orbit and the impatient customer probability that a call resumes to enter the servers. In
contrast, it is unsuccessful and eternally departs and the additional features are the probability ๐œƒ
(0 < ๐œƒ โ‰ค 1) that a call first enters the orbit when failing and the probability ๐œƒโ€ฒ (0 < ๐œƒโ€ฒ โ‰ค 1)
which can be considered as a call next enters the orbit when failing, whereas [7] has only one
fixed probability ๐œƒ for all calls entering the orbit. The main aim of our model is to lessen
possibilities of being blocked with retrial and handover calls, significant factors to accomplish
QoS, with the additional measure which is a novel attribute in comparison with the
aforementioned works.
The organization of the article is as follows, In Section 2, the detailed problems with the
parameters and the models are described. The analysis results will be presented evaluated the
model performances in Section 3.
2. ANALYSIS MODEL
2.1. Performance Analysis of Fractional Guard Channel
The model that we use in this study is some what like the models in [3], [4], [5] and [7] by
considering the retrial customers for the fresh and handover calls. The salient point is that we
implement the fractional guard channel with the two probabilities of the fresh calls (Figure 1). In
the guard channel model, the guard channels reserve for the handover calls. It means that a fresh
call arrives in the system and is accepted with the probability ๐›ฝ๐‘– (0 โ‰ค ๐›ฝ๐‘– โ‰ค 1, ๐‘– = 0, ๐‘, where ๐‘ is
the number of the channels) and the system has at least a vacant channel, otherwise it will be
rejected. A handover call is denied when all channels of the system are busy. Also in this model,
the fairness is considered in the resource stores for all calls, including the fresh and handover
calls. The innovations are that implement the fractional guard channels (Definition 2.1) with the
limited capacity of the orbit.
International Journal of Computer Networks & Communications (IJCNC) Vol.13, No.4, July 2021
87
F
โ‹ฎ
Call admission
control
i=๐›ฝi F+ H
H
Orbit
1-๐œƒ
๐œƒ
1-๐œƒโ€™
๐œƒโ€™
โ‹ฎ
c channels
Figure 1.The retrial queueing model in the cellular mobile network.
Definition 2.1 [7]: A fresh or retrial call is admitted by the system in accordance with the
fractional guard channel policy. The handover calls are accepted if the system exists a certainly
available channel, and the fresh calls only are served with the probabilities ๐›ฝ๐‘–(0 โ‰ค ๐›ฝ๐‘– โ‰ค 1)
depending on the system state.
As the aforementioned issue, we consider the fractional guard channel policies, such as LFGC,
UFGC, LAFGC, and QUFGC. In addition, we assume that the system consists of ๐‘ the operation
channels, ๐‘” the permission channels, the accepted probability of the fresh calls ๐‘(0 โ‰ค ๐‘ โ‰ค 1),
and the average value ๐‘Ÿ(๐‘Ÿ โˆˆ โ„, 0 โ‰ค ๐‘Ÿ โ‰ค ๐‘).
๏‚ท LFGC: ๐›ฝ๐‘– = 1 (0 โ‰ค ๐‘– โ‰ค ๐‘” โˆ’ 1), ๐›ฝ๐‘” = ๐‘, ๐›ฝ๐‘– = 0 (๐‘” + 1 โ‰ค ๐‘– < ๐‘).
๏‚ท LAFGC: ๐›ฝ๐‘– = 1 (0 โ‰ค ๐‘– โ‰ค ๐‘ โˆ’ โŒŠ๐‘ŸโŒ‹ โˆ’ 2), ๐›ฝ๐‘โˆ’โŒŠ๐‘ŸโŒ‹โˆ’1 = 1 โˆ’ ๐‘Ÿ + โŒŠ๐‘ŸโŒ‹, ๐›ฝ๐‘– = 0 (๐‘ โˆ’ โŒŠ๐‘ŸโŒ‹ โ‰ค ๐‘– < ๐‘).
๏‚ท UFGC: ๐›ฝ๐‘– = ๐‘ (0 โ‰ค ๐‘– < ๐‘).
๏‚ท QUFGC: ๐›ฝ๐‘– = 1 (0 โ‰ค ๐‘– โ‰ค ๐‘” โˆ’ 1), ๐›ฝ๐‘– = ๐‘ (๐‘” โ‰ค ๐‘– < ๐‘).
Obviously, we have๐›ฝ๐‘ = 0.
2.2. The Parameters
Considering a certain cell in the cellular mobile network, the system is modeled in Figure 1. The
arrival calls will be admitted by the call admission control with the fractional guard channel
policies. The interarrival times for fresh and handover calls follow the exponential distribution
with the rate ๐น and ๐ป, respectively. Let is defined as the interarrival time between
consecutively incoming calls regardless of fresh and handover calls, thus we have = ๐น + ๐ป.
Also, the served time follows the exponential distribution with the rate .
A call first arriving the system is blocked due to lacking or allocating the resources. Now it is:
๏‚ท Permanently depart the system with the probability1 โˆ’ ๐œƒ(0 < ๐œƒ โ‰ค 1).
๏‚ท or enter the orbit with the probability ๐œƒ.
In this model, the orbit is considered a queue (capacity๐ฟ) to store the blocked calls (or the fresh
calls).Similarly, when the retrial calls are unsuccessful, they will:
๏‚ท Permanently depart the system with the probability1 โˆ’ ๐œƒโ€ฒ (0 < ๐œƒโ€ฒ โ‰ค 1).
๏‚ท or continue entering the orbit with the probability ๐œƒโ€ฒ.
International Journal of Computer Networks & Communications (IJCNC) Vol.13, No.4, July 2021
88
In addition, the retrial rate for the fresh calls from the orbit to the system is .
The fresh or retrial calls enter the system with the probability ๐›ฝ๐‘– in accordance with the state
system. A handover call is served immediately if the system exists at least one vacant channel. In
contrast, it will permanently depart and does not enter orbit.
2.3. Modelling
The system is illustrated by the two-dimensional continuous-time Markov chains ๐‘‹ =
{(๐ผ(๐‘ก), ๐ฝ(๐‘ก)), ๐‘ก โ‰ฅ 0}, where ๐ผ(๐‘ก)(0 โ‰ค ๐ผ(๐‘ก) โ‰ค ๐‘)are the numbers ofthecustomers being served by
the system, and ๐ฝ(๐‘ก)(0 โ‰ค ๐ฝ(๐‘ก) โ‰ค ๐ฟ)are the numbers of the customerswaiting to enter the system
at the time ๐‘ก. The state space of the above chains is depicted by Figure 2, where (๐‘–, ๐‘—)are the
states that the numbers of the occupied channels at the time ๐‘กare๐‘–(๐‘– = 0, 1, 2, โ€ฆ , ๐‘ โˆ’ ๐‘”, ๐‘ โˆ’ ๐‘” +
1, โ€ฆ , ๐‘), and the numbers of the customers in the orbit at the time ๐‘กare๐‘—(๐‘— = 0, 1, 2, โ€ฆ , ๐ฟ).
1,0
0,0
1,1
0,1 ...
... 1,L-1
0,L-1
1,L
0,L
l0 m
l1 2m
b0a
c,0 c,1 c,L-1 c,L
...
lc-1 cm
(1-qโ€™)a
lFq
cm cm cm
l1 2m l1 2m l1 2m
l0 m l0 m l0 m
L(1-qโ€™)a
lc-1 lc-1 lc-1
c-1,0 c-1,1 ... c-1,
L-1
c-1,L
lc-2 (c-1)m
lF(1- b0)q
lF(1- b1)q
lF(1- bc-1)q
lc-2 lc-2 lc-2
lF(1- b0)q
lF(1- b1)q
lF(1- bc-1)q
lFq
b1a
bc-2a
bc-1a
Lb0a
Lb1a
Lbc-2a
Lbc-1a
(1- bc-1)(1-qโ€™)a
(1- b1)(1-qโ€™)a
(1- b0)(1-qโ€™)a L(1- b0)(1-qโ€™)a
L(1- b1)(1-qโ€™)a
L(1- bc-1)(1-qโ€™)a
2b0a
2(1-qโ€™)a
lFq
lF(1- b0)q
lF(1- b1)q
lF(1- bc-1)q
2b1a
2bc-2a
2bc-1a
2(1- bc-1)(1-qโ€™)a
2(1- b1)(1-qโ€™)a
2(1- b0)(1-qโ€™)a
(L-1)b0a
(L-1)(1-qโ€™)a
lFq
lF(1- b0)q
lF(1- b1)q
lF(1- bc-1)q
(L-1)b1a
(L-1)bc-2a
(L-1)bc-1a
(L-1)(1- bc-1)(1-qโ€™)a
(L-1)(1- b1)(1-qโ€™)a
(L-1)(1- b0)(1-qโ€™)a
(c-1)m (c-1)m (c-1)m
Figure 2. State transition diagram of the model
Let ๐œ‹๐‘–,๐‘— = lim
๐‘กโ†’โˆž
๐‘ƒ(๐ผ(๐‘ก) = ๐‘–, ๐ฝ(๐‘ก) = ๐‘—), where ๐œ‹๐‘–,๐‘— are the steady-state probabilities for the state
(๐‘–, ๐‘—). When the chains ๐‘‹ = {๐ผ(๐‘ก), ๐ฝ(๐‘ก); ๐‘ก โ‰ฅ 0}are represented through the state transition
matrixes ๐ด๐‘—, ๐ต๐‘—, ๐ถ๐‘— (with theirdimension(๐‘ + 1) ร— (๐‘ + 1)) [9-13]:
๏‚ท ๐ด๐‘—(๐‘–, ๐‘˜) are the state transition matrixes for the states from (๐‘–, ๐‘—) to (๐‘˜, ๐‘—)(0 โ‰ค ๐‘–, ๐‘˜ โ‰ค
๐‘, 0 โ‰ค ๐‘— โ‰ค ๐ฟ). They express an successful call, which is a fresh or handover call, or a call
having been completed and departing. ๐ด๐‘— has the entries ๐ด๐‘—(๐‘–, ๐‘– โˆ’ 1) = ๐‘– (๐‘– = 1, ๐‘) and
๐ด๐‘—(๐‘–, ๐‘– + 1) = ๐‘– = ๐›ฝ๐‘– ๐น + ๐ป(๐‘– = 0, ๐‘ โˆ’ 1) and the rest 0.
International Journal of Computer Networks & Communications (IJCNC) Vol.13, No.4, July 2021
89
๐ด๐‘— = ๐ด =
(
0 0 0 โ‹ฏ 0 0 0
0 1 โ‹ฏ 0 0 0
โ‹ฎ โ‹ฎ โ‹ฎ โ‹ฎ โ‹ฎ โ‹ฎ โ‹ฎ
0 0 0 โ‹ฏ (๐‘ โˆ’ 1) 0 ๐‘โˆ’1
0 0 0 โ‹ฏ 0 ๐‘ 0 )
, (๐‘— โ‰ฅ 0) (1)
๏‚ท ๐ต๐‘—(๐‘–, ๐‘˜)are the state transition matrixes for the states from (๐‘–, ๐‘—) to (๐‘˜, ๐‘— + 1)(0 โ‰ค ๐‘–, ๐‘˜ โ‰ค
๐‘, 0 โ‰ค ๐‘— โ‰ค ๐ฟ โˆ’ 1). They express a fresh call first arriving the system and entering the
orbit with the probability ๐œƒ. ๐ต๐‘— has the entries ๐ต๐‘—(๐‘–, ๐‘–) = (1 โˆ’ ๐›ฝ๐‘–)๐œƒ ๐น(๐‘– = 0, ๐‘) (note
๐ต๐‘—(๐‘, ๐‘) = ๐œƒ ๐น due to ๐›ฝ๐‘ = 0) and the rest 0.
๐ต๐‘— = ๐ต =
(
๐œŽ0 โ‹ฏ 0 0
0 โ‹ฏ 0 0
โ‹ฎ โ‹ฎ โ‹ฎ โ‹ฎ
0 โ‹ฏ ๐œŽ๐‘โˆ’1 0
0 โ‹ฏ 0 ๐œŽ๐‘)
, (๐‘— โ‰ฅ 0) (2)
Where ๐œŽ๐‘– = (1 โˆ’ ๐›ฝ๐‘–)๐œƒ ๐น, (0 โ‰ค ๐‘– โ‰ค ๐‘).
๏‚ท ๐ถ๐‘—(๐‘–, ๐‘˜)are the state transition matrixes for the states from (๐‘–, ๐‘—) to (๐‘˜, ๐‘— โˆ’ 1)(0 โ‰ค
๐‘–, ๐‘˜ โ‰ค ๐‘, 1 โ‰ค ๐‘— โ‰ค ๐ฟ). They express a retrial call returning the system. Then it is
served or permanently departs due to its impatience and all channels that are busy.
๐ถ๐‘— has the entries ๐ถ๐‘—(๐‘–, ๐‘– + 1) = ๐›ฝ๐‘–๐‘— (๐‘– = 0, ๐‘ โˆ’ 1) and ๐ถ๐‘—(๐‘–, ๐‘–) = (1 โˆ’ ๐›ฝ๐‘–)(1 โˆ’
๐œƒโ€ฒ)๐‘— (๐‘– = 0, ๐‘) (note ๐ถ๐‘—(๐‘, ๐‘) = (1 โˆ’ ๐œƒโ€ฒ)๐‘— due to ๐›ฝ๐‘ = 0) and the rest 0.
๐ถ๐‘— =
(
๐œŽ0,๐‘— ๐›ฝ0๐‘— 0 โ‹ฏ 0 0
0 ๐œŽ1,๐‘— ๐›ฝ1๐‘— โ‹ฏ 0 0
โ‹ฎ โ‹ฎ โ‹ฎ โ‹ฎ โ‹ฎ โ‹ฎ
0 0 0 โ‹ฏ ๐œŽ๐‘โˆ’1,๐‘— ๐›ฝ๐‘โˆ’1๐‘—
0 0 0 โ‹ฏ 0 ๐œŽ๐‘,๐‘— )
, (๐‘— โ‰ฅ 1) (3)
Where ๐œŽ๐‘–,๐‘— = (1 โˆ’ ๐›ฝ๐‘–)(1 โˆ’ ๐œƒโ€ฒ)๐‘— (0 โ‰ค ๐‘– โ‰ค ๐‘, 1 โ‰ค ๐‘— โ‰ค ๐ฟ).
We have the generator matrix ๐‘„:
๐‘„ =
(
๐‘„1
(0)
๐‘„2
(0)
๐ŸŽ ๐ŸŽ ๐ŸŽ
๐‘„0
(1)
๐‘„1
(1)
โ‹ฑ ๐ŸŽ ๐ŸŽ
๐ŸŽ โ‹ฑ โ‹ฑ ๐‘„2
(๐ฟโˆ’2)
๐ŸŽ
๐ŸŽ ๐ŸŽ ๐‘„0
(๐ฟโˆ’1)
๐‘„1
(๐ฟโˆ’1)
๐‘„2
(๐ฟโˆ’1)
๐ŸŽ ๐ŸŽ ๐ŸŽ ๐‘„0
(๐ฟ)
๐‘„1
(๐ฟ)
)
(4)
International Journal of Computer Networks & Communications (IJCNC) Vol.13, No.4, July 2021
90
Where:
{
๐‘„1
(0)
= ๐ด โˆ’ ๐ท๐ด
โˆ’ ๐ท๐ต
๐‘„1
(๐‘—)
= ๐ด โˆ’ ๐ท๐ด
โˆ’ ๐ท๐ต
โˆ’ ๐ท๐ถ๐‘—(1 โ‰ค ๐‘— โ‰ค ๐ฟ โˆ’ 1)
๐‘„๐ฟ
(0)
= ๐ด โˆ’ ๐ท๐ด
โˆ’ ๐ท๐ถ๐ฟ
๐‘„0
(๐‘—)
= ๐ถ๐‘—(1 โ‰ค ๐‘— โ‰ค ๐ฟ)
๐‘„2
(๐‘—)
= ๐ต (0 โ‰ค ๐‘— โ‰ค ๐ฟ โˆ’ 1)
Where ๐ท๐‘‹(๐‘‹ = ๐ด, ๐ต, ๐ถ๐‘—) are the diagonal matrixes with the entries on the main diagonal that are
the sum of the entries on respective row of the matrix ๐‘‹. We have:
๐‘ฃ๐‘„ = (0,0, โ€ฆ ,0)
โŸ
(๐‘+1)ร—(๐‘+1)
(5)
Where ๐‘ฃ = (๐‘ฃ0,๐‘ฃ1, โ€ฆ , ๐‘ฃ๐ฟ).
Let the probabilities (๐‘–, ๐‘—) are defined by ๐‘๐‘–,๐‘— = lim
๐‘กโ†’+โˆž
๐‘ƒ(๐ผ(๐‘ก) = ๐‘–, ๐ฝ(๐‘ก) = ๐‘—), and the level
probabilities vectors ๐‘ฃ๐‘— = (๐‘0,๐‘—, ๐‘1,๐‘—, โ€ฆ , ๐‘๐‘,๐‘—)(1 โ‰ค ๐‘— โ‰ค ๐ฟ). The balance equations of the system
are:
๐‘ฃ0๐‘„1
(0)
+ ๐‘ฃ1๐‘„0
(1)
= (0,0, โ€ฆ ,0)
โŸ
๐‘+1
(6)
๐‘ฃ๐‘—โˆ’1๐‘„2
(๐‘—โˆ’1)
+ ๐‘ฃ๐‘—๐‘„1
(๐‘—)
+ ๐‘ฃ๐‘—+1๐‘„1
(๐‘—+1)
= (0,0, โ€ฆ ,0)
โŸ
๐‘+1
(1 โ‰ค ๐‘— โ‰ค ๐ฟ โˆ’ 1) (7)
๐‘ฃ๐ฟโˆ’1๐‘„2
(๐ฟโˆ’1)
+ ๐‘ฃ๐ฟ๐‘„1
(๐ฟ)
= (0,0, โ€ฆ ,0)
โŸ
๐‘+1
(8)
โˆ‘ ๐‘ฃ๐‘—
๐ฟ
๐‘—=0
ร— (
1
1
โ‹ฎ
1
)
โŸ
๐‘+1
= 1 (9)
From (5) and (9), we calculate ๐‘ฃ:
๐‘ฃ = ๐‘’(๐‘„ + ๐ธ)โˆ’1 (10)
Where ๐‘’ is the row vector (1 ร— (๐‘ + 1)) and ๐ธ is the matrix ((๐‘ + 1) ร— (๐‘ + 1)) all of whose
elements are the constants 1.
2.4. The Performance Measures of the System
The probability of the fresh calls ๐‘ƒ๐ต๐น firstly arrives the system and find all busy channels at the
time or is rejected with the probability (1 โˆ’ ๐›ฝ๐‘–).
International Journal of Computer Networks & Communications (IJCNC) Vol.13, No.4, July 2021
91
๐‘ƒ๐ต๐น = โˆ‘ โˆ‘ ๐‘๐‘–,๐‘—(1 โˆ’ ๐›ฝ๐‘–)
๐‘
๐‘–=0
๐ฟ
๐‘—=0
(11)
The probability of the retrial calls ๐‘ƒ๐ต๐น,๐‘Ÿ๐‘’๐‘ก๐‘Ÿ๐‘–๐‘Ž๐‘™ then arrives the system from the orbit and find all
busy channels at the time is rejected with the probability (1 โˆ’ ๐œƒโ€ฒ).
๐‘ƒ๐ต๐น,๐‘Ÿ๐‘’๐‘ก๐‘Ÿ๐‘–๐‘Ž๐‘™ =
(1 โˆ’ ๐œƒโ€ฒ)
๐œƒ ๐น
โˆ‘ โˆ‘ ๐‘—๐‘๐‘–,๐‘—
๐‘
๐‘–=0
๐ฟ
๐‘—=1
(12)
The probability of the handover calls ๐‘ƒ๐ต๐น,๐‘Ÿ๐‘’๐‘ก๐‘Ÿ๐‘–๐‘Ž๐‘™ clearly arrives the system and find all busy
channels at the time.
๐‘ƒ๐ต๐ป = โˆ‘ ๐‘๐‘,๐‘—
๐ฟ
๐‘—=0
(13)
3. RESULTS
The efficiency of the performance as the change of the parameters of the system is firstly
considered with the fresh and handover calls. When the blocks occur, the fresh calls reattempt to
connect in the intervals of the stochastic distribution. We assume the base station of the cell that
can process ๐‘ the connection simultaneously. Table 1 enumerates the parameters to analyze
results. The Mathematica program of Wolfram Research denoted [14] is a power tool to compute
and simulate for network models and is utilized in the proposed model of the paper.
Table 1. The parameters of the model.
The parameters Values Describe the parameters
๐‘ 10 Number of the channels in the cell
๐‘” 1 to 7 Number of the accepted channels
๐‘ 0.1 to 1 Probability that a fresh call is served
๐ฟ 1 to 10 Number of the customers in the orbit
๐ป 0.002 to 2 Arrival rate for the handover calls
๐น 0.018 to 0.2 Arrival rate for the fresh calls
0.2 Service rate
0.4 Retrial rate
๐œƒ 0.1 to 1 Probability that a fresh call first enters the orbit
๐œƒโ€ฒ 0.1 to 1 Retrial probability that a fresh call re-enters the orbit
3.1. Analysing and Comparing the Blocking Probabilities with Values ๐†
First, we compare the blocking probabilities ๐‘ƒ๐ต (๐‘ƒ๐ต๐น, ๐‘ƒ๐ต๐น,๐‘Ÿ๐‘’๐‘ก๐‘Ÿ๐‘–๐‘Ž๐‘™ or ๐‘ƒ๐ต๐‘…, ๐‘ƒ๐ต๐ป) according to a
variety of the fractional guard channel policies with the fluctuations in values ๐œŒ. It can be seen in
Figure 3, the value ๐œŒ, the traffic load, is definited by the expression ๐œŒ = ๐‘
โ„ . We have ๐‘” = 7,
๐‘ = 0.5, ๐‘Ÿ = 2.3 (for the LAFGC policy), ๐ฟ = 8, ๐œƒ = 0.5, ๐œƒโ€ฒ = 0.5, ๐น = 2 ๐ป, ๐œŒ ranging from
0.3 to 1.9. We find that the blocking probabilities for the LFGC and LAFGC policies are better
than the blocking probabilities for the UFGC and QUFGC policies.
International Journal of Computer Networks & Communications (IJCNC) Vol.13, No.4, July 2021
92
Figure 3. The blocking probability of ๐‘ƒ๐ต๐น, ๐‘ƒ๐ต๐น,๐‘Ÿ๐‘’๐‘ก๐‘Ÿ๐‘–๐‘Ž๐‘™ and ๐‘ƒ๐ต๐ปby ๐œŒ.
This may partly explain that the UFGC and QUFGC policies permit the fresh calls with ๐›ฝ๐‘– = 1 or
๐›ฝ๐‘– = ๐‘(0 โ‰ค ๐‘– < ๐‘). As a result, the fresh calls are easier to enter the system, and the handover
calls are significantly restricted. In the meanwhile, the LFGC and LAFGC policies control the
fresh calls with ๐›ฝ๐‘– = 0 (0 < ๐ฟ โ‰ค ๐‘– < ๐‘, ๐ฟ: a particular parameter) at some of the last states with
๐‘– โ‰ฅ ๐ฟ.
3.2. Analysing and comparing the blocking probabilities with values๐’ˆ
With ๐œŒ = 0.7, ๐‘ = 0.5, ๐ฟ = 8, ๐œƒ = 0.5, ๐œƒโ€ฒ
= 0.5, ๐น = 2 ๐ป, ๐‘” ranging from 1 to 7, we obtain
the results as presented in Figure 4.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9
PB
r
UFGC
PB_F
PB_R
PB_H
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9
PB
r
LFGC
PB_F
PB_R
PB_H
0
0.1
0.2
0.3
0.4
0.5
0.6
1 2 3 4 5 6 7
PB
g
UFGC
PB_F
PB_R
PB_H
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9
PB
r
QUFGC
PB_F
PB_R
PB_H
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9
PB
r
LAFGC
PB_F
PB_R
PB_H
0
0.1
0.2
0.3
0.4
0.5
0.6
1 2 3 4 5 6 7
PB
g
QUFGC
PB_F
PB_R
PB_H
International Journal of Computer Networks & Communications (IJCNC) Vol.13, No.4, July 2021
93
Figure 4. The blocking probability of ๐‘ƒ๐ต๐น, ๐‘ƒ๐ต๐น,๐‘Ÿ๐‘’๐‘ก๐‘Ÿ๐‘–๐‘Ž๐‘™ , and ๐‘ƒ๐ต๐ปby ๐‘”.
When we change values ๐‘”, the probabilities ๐‘ƒ๐ต๐น and ๐‘ƒ๐ต๐น,๐‘Ÿ๐‘’๐‘ก๐‘Ÿ๐‘–๐‘Ž๐‘™ for the QUFGC policy decrease
inversely proportional to values ๐‘”, and the probabilities ๐‘ƒ๐ต๐ป for the QUFGC, UFGC, LFGC and
LAFGC policies remain steady and thus we need to choose an appropriate value ๐‘” for the low
blocking probabilities ๐‘ƒ๐ต๐ป. The UFGC policy has the probabilities ๐‘ƒ๐ต๐น and ๐‘ƒ๐ต๐น,๐‘Ÿ๐‘’๐‘ก๐‘Ÿ๐‘–๐‘Ž๐‘™ which
maintain stable due to ๐›ฝ๐‘– = ๐‘(0 โ‰ค ๐‘– < ๐‘). Similar to the QUFGC policy, the probabilities ๐‘ƒ๐ต๐น
and ๐‘ƒ๐ต๐น,๐‘Ÿ๐‘’๐‘ก๐‘Ÿ๐‘–๐‘Ž๐‘™ for the LFGC and LAFGC policies significantly decline because the numbers of
the accepted channels for the fresh and retrial calls gradually increase.
3.2. Analysing and Comparing the Blocking Probabilities with Values ๐‘ณ
With ๐œŒ = 0.7, ๐‘” = 7, ๐‘ = 0.5, ๐‘Ÿ = 2.3 (for the LAFGC policy), ๐œƒ = 0.5, ๐œƒโ€ฒ = 0.5, ๐น = 2 ๐ป, ๐ฟ
ranging from 1 to 8, we obtain the results presented in Figure 5.
Figure 5. The blocking probability of ๐‘ƒ๐ต๐น, ๐‘ƒ๐ต๐น,๐‘Ÿ๐‘’๐‘ก๐‘Ÿ๐‘–๐‘Ž๐‘™ , and ๐‘ƒ๐ต๐ปby ๐ฟ.
The blocking probabilities ๐‘ƒ๐ต๐น and ๐‘ƒ๐ต๐น,๐‘Ÿ๐‘’๐‘ก๐‘Ÿ๐‘–๐‘Ž๐‘™ are influenced by the number of the maximum
customers in the orbit ๐ฟ when ๐ฟ < 4 as Figure 5. When ๐ฟ โ‰ฅ 4 these probalities are almost
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1 2 3 4 5 6 7
PB
g
LFGC
PB_F
PB_R
PB_H
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
1 2 3 4 5 6 7 8
PB
L
LFGC
PB_F
PB_R
PB_H
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1 2 3 4 5 6 7
PB
g
LAFGC
PB_F
PB_R
PB_H
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
1 2 3 4 5 6 7 8
PB
L
LAFGC
PB_F
PB_R
PB_H
International Journal of Computer Networks & Communications (IJCNC) Vol.13, No.4, July 2021
94
steady. While the blocking probabilities ๐‘ƒ๐ต๐ป are not affected regardless of the value ๐ฟ. This can
be explained that the number of the servers is fixed, the more the customers in the orbit are, the
more blocked the system is. For this reason, we can approximate to the infinite model with ๐ฟ โ†’
+โˆž.
3.3. Analysing and comparing the blocking probabilities with values ๐’‘
With ๐œŒ = 0.7, ๐‘” = 7, ๐‘ = 0.5, ๐ฟ = 8, ๐‘Ÿ = 2.3 (for the LAFGC policy), ๐œƒ = 0.5, ๐œƒโ€ฒ = 0.5, ๐น =
2 ๐ป, ๐‘ ranging from 0.1 to 1, we obtain the results as listed in Figure 6.
Figure 6. The blocking probability of ๐‘ƒ๐ต๐น, ๐‘ƒ๐ต๐น,๐‘Ÿ๐‘’๐‘ก๐‘Ÿ๐‘–๐‘Ž๐‘™ , and ๐‘ƒ๐ต๐ปby ๐‘.
We found/find that the accepted probabilities of the fresh calls ๐‘ influence on how the
congestions of the fresh and retrial calls are. In these analyses, the blocking probabilities of the
fresh calls for the UFGC and QUFGC policies are better than the ones for the LFGC and LAFGC
policies due to the fact that the the probabilities ๐›ฝ๐‘–(0 โ‰ค ๐‘– โ‰ค ๐‘) for the LFGC and LAFGC
policies have more the values ๐‘ than the ones for the UFGC and QUFGC policies. As a result, the
fresh and retrial calls ease to enters the system for the LFGC and LAFGC policies.
3.4. Analysing and comparing the blocking probabilities with values ๐€๐‘ฏ ๐€๐‘ญ
โ„
With ๐œŒ = 0.7, ๐‘” = 7, ๐ฟ = 8, ๐‘ = 0.5, ๐œƒ = 0.5, ๐œƒโ€ฒ = 0.5, ๐ป ๐น
โ„ ranging from 1 5
โ„ to 5, we
obtain the results as shown in Figure 7.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
PB
p
UFGC
PB_F
PB_R
PB_H
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
PB
p
LFGC
PB_F
PB_R
PB_H
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
PB
p
QUFGC
PB_F
PB_R
PB_H
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
PB
p
LAFGC
PB_F
PB_R
PB_H
International Journal of Computer Networks & Communications (IJCNC) Vol.13, No.4, July 2021
95
Figure 7. The blocking probability of ๐‘ƒ๐ต๐น, ๐‘ƒ๐ต๐น,๐‘Ÿ๐‘’๐‘ก๐‘Ÿ๐‘–๐‘Ž๐‘™ , and ๐‘ƒ๐ต๐ปby ๐ป ๐น
โ„ .
We find that if the ratio ๐ป ๐น
โ„ < 1/3, the probabilities ๐‘ƒ๐ต๐น, ๐‘ƒ๐ต๐น,๐‘Ÿ and ๐‘ƒ๐ต๐ป are changed, but if
the ratio ๐ป ๐น
โ„ โ‰ฅ 1/3, the probabilities ๐‘ƒ๐ต๐น, ๐‘ƒ๐ต๐น,๐‘Ÿ and ๐‘ƒ๐ต๐ป remain almost unchanged.
Therefore, we choose the ratio ๐ป ๐น
โ„ โ‰ฅ 1/3 to maintain the stability of the system.
3.5. Analysing and comparing the blocking probabilities with values ๐œฝ
With ๐œŒ = 0.7, ๐‘” = 7, ๐ฟ = 8, ๐‘ = 0.5, ๐‘Ÿ = 2.3 (for the LAFGC policy), ๐œƒโ€ฒ = 0.5, ๐น = 2 ๐ป, ๐œƒ
ranging from 0.1 to 1, we gain the results as presented in Figure 8.
0
0.1
0.2
0.3
0.4
0.5
0.6
1/5 1/4 1/3 1/2 1 2 3 4 5
PB
H/ F
UFGC
PB_F
PB_R
PB_H
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
1/5 1/4 1/3 1/2 1 2 3 4 5
PB
H/ F
LFGC
PB_F
PB_R
PB_H
0
0.1
0.2
0.3
0.4
0.5
0.6
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
PB
q
UFGC
PB_F
PB_R
PB_H
0
0.05
0.1
0.15
0.2
0.25
0.3
1/5 1/4 1/3 1/2 1 2 3 4 5
PB
H/ F
QUFGC
PB_F
PB_R
PB_H
0
0.05
0.1
0.15
0.2
0.25
0.3
1/5 1/4 1/3 1/2 1 2 3 4 5
PB
H/ F
LAFGC
PB_F
PB_R
PB_H
0
0.05
0.1
0.15
0.2
0.25
0.3
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
PB
q
QUFGC
PB_F
PB_R
PB_H
International Journal of Computer Networks & Communications (IJCNC) Vol.13, No.4, July 2021
96
Figure 8. The blocking probability of ๐‘ƒ๐ต๐น, ๐‘ƒ๐ต๐น,๐‘Ÿ๐‘’๐‘ก๐‘Ÿ๐‘–๐‘Ž๐‘™ , and ๐‘ƒ๐ต๐ปby ๐œƒ.
The parameter ๐œƒ gives the influence on the blocking probabilities of the retrial calls rather than
the ones of the fresh calls. This can be explained that the probabilities that a fresh call first enters
the orbit increase resulting in a numerous number of the customers in the the orbit, and the
blocking probabilities of the retrial calls will raise. In the meantime, the probabilities of the
handover calls remain almost unchanged. In these analyses, we find that the parameters โˆˆ
(0.2 โˆ’ ๐œ€, 0.2 + ๐œ€), with ๐œ€ โ‰ฅ 0 is a arbitrarily small real. The system performances are optimized
by the given parameters.
3.6. Analysing and comparing the blocking probabilities with values๐œฝโ€ฒ
With ๐œŒ = 0.7, ๐‘” = 7, ๐ฟ = 8, ๐‘ = 0.5, ๐‘Ÿ = 2.3 (for the LAFGC policy), ๐œƒ = 0.5, ๐น = 2 ๐ป, ๐œƒโ€ฒ
ranging from 0.1 to 1, we obtain the results as presented Figure 9.
Figure 9. The blocking probability of ๐‘ƒ๐ต๐น, ๐‘ƒ๐ต๐น,๐‘Ÿ๐‘’๐‘ก๐‘Ÿ๐‘–๐‘Ž๐‘™ , and ๐‘ƒ๐ต๐ปby ๐œƒโ€ฒ.
When taking into account the increase in the retrial probabilities ๐œƒโ€ฒ(0.1 โ‰ค ๐œƒโ€ฒ โ‰ค 0.9), the
blocking probabilities ๐‘ƒ๐ต๐น,๐‘Ÿ are declined. Then the retrial calls re-enter the orbit to attempt the
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
PB
q
LFGC
PB_F
PB_R
PB_H
0
0.1
0.2
0.3
0.4
0.5
0.6
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
PB
q'
UFGC
PB_F
PB_R
PB_H
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
PB
q'
LFGC
PB_F
PB_R
PB_H
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
PB
q
LAFGC
PB_F
PB_R
PB_H
0
0.05
0.1
0.15
0.2
0.25
0.3
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
PB
q'
QUFGC
PB_F
PB_R
PB_H
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
PB
q'
LAFGC
PB_F
PB_R
PB_H
International Journal of Computer Networks & Communications (IJCNC) Vol.13, No.4, July 2021
97
next occasions. On the contrary, the blocking probabilities ๐‘ƒ๐ต๐น and ๐‘ƒ๐ต๐ป are slightly affected. As
the results ofinFigure 9, when the parameter ๐œƒโ€ฒ tends to the asymptotic value of 1, the blocking
probabilities ๐‘ƒ๐ต๐น,๐‘Ÿ almost optimize the certain performances.
(a)
(b)
Figure 10. The blocking probability of ๐‘ƒ๐ต๐น,๐‘Ÿ๐‘’๐‘ก๐‘Ÿ๐‘–๐‘Ž๐‘™ (a) and ๐‘ƒ๐ต๐ป(b) among LAFGC, LFGC, QUFGC and
UFGC
We find that the blocking probabilities ๐‘ƒ๐ต๐น,๐‘Ÿ๐‘’๐‘ก๐‘Ÿ๐‘–๐‘Ž๐‘™ and ๐‘ƒ๐ต๐ป of LAFGC and LFGC are almost
equivalent (Figure 10). Also, the blocking probabilities ๐‘ƒ๐ต๐ป of QUFGC and UFGC are
approximately equal, yet the blocking probabilities ๐‘ƒ๐ต๐น,๐‘Ÿ of QUFGC are better than those of
UFGC. This is explained that the accepted probabilities ๐›ฝ๐‘– of QUFGC are 1 at some initial states.
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
0,3 0,5 0,7 0,9 1,1 1,3 1,5 1,7 1,9
PB
F,retrial
r
LAFGC
LFGC
QUFGC
UFGC
0
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0,4
0,45
0,3 0,5 0,7 0,9 1,1 1,3 1,5 1,7 1,9
PB
H
r
LAFGC
LFGC
QUFGC
UFGC
International Journal of Computer Networks & Communications (IJCNC) Vol.13, No.4, July 2021
98
Figure 11. The blocking probability of ๐‘ƒ๐ต๐น between in [7] and the article.
According to figures extracted from [7] (of Do Van Tien) with ๐ฟ = ๐‘ โˆ’ 1, ๐›ฝ = 0, ๐‘ = 15, =
1
120
, = 20 , ๐น = 24 ๐ป, we ascertain that the probabilties of ๐‘ƒ๐ต๐น in the article are superior to
those of [7]. The disparity is that the further value ๐œƒโ€ฒ
can be varied, depending on usersโ€™s
adjustments and playing a signal role in reducing the possibilities of obstructing the fresh calls.
4. CONCLUSIONS
We have introduced the retrial queueing model in the cellular mobile network using the fractional
guard channel policy with impatient customers according to the probabilities ฮธ and ฮธ'. The
advantage of the model is the significant decrease of the blocking probabilities for the fresh and
retrial calls as well as the support of the network operation to maintain the stability of QoS under
the influences on the network traffic and additional factors. The generality in the model is
weighed by choosing the retrial queueing model in combination with the fractional guard channel
policies. The analysis results are simple and guarantee the low blocking probabilities of the
handover calls for the fractional guard channel policy in compassion to the blocking probabilities
of the guard channel policy. In addition, the retrial queueing model with the fractional guard
channel policy in the cellular mobile network can be used for the criteria of the different network
traffics to measure the performances. The corollaries elucidate that the model is appropriate and
precise. Besides, the research considers a variety of the fractional guard channel policies,
including the UFGC, QUFGC, LFGC, and LAFGC policies, to fall the blocking probabilities of
the fresh calls, but they still guarantee the handover calls protected with the call admission
control.
CONFLICTS OF INTEREST
The authors declare no conflict of interest.
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0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
PB
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LFGC
Do V Tien
This article
International Journal of Computer Networks & Communications (IJCNC) Vol.13, No.4, July 2021
99
REFERENCES
[1] Mandles M. and Tutschku K., (1996). Efficient Call Handling Procedures in Cellular Mobile
Networks.Forschungsbericht, Preprint-Reihe 144, Institut fur Informatik, Universitat Wurzburg,
Germany.
[2] Tran-Gia P. and Mandles M., (1997). Modeling of Customer Retrial Phenomenon in Cellular Mobile
Networks.IEEE Journal on Selected Areas in Communications, 15(8), pp. 1406-1414.
[3] Do T. V. (2010). A new computational algorithm for retrial queues to cellular mobile systems with
guard channels. Computers & Industrial Engineering, 59(4), pp. 865โ€“872.
[4] Linda A. C. and Ani C. I. (2014). A Reservation and Retrial Queue Analysis for Cellular
Networks.International Journal of Engineering Research & Technology (IJERT), ISSN: 2278-0181,
Vol. 3 Issue 2, February โ€“ 2014
[5] Bhargava A., Bhagat A. and Singh J., (2016). Modelling and Performance Analysis of Cellular
Mobile Networks.Indian Journal of Science and Technology, Vol 9 (48), December 2016.
[6] Vazquez-Avila J., Cruz-Pรฉrez F.A. and Ortigoza-Guerrero L., (2006). Performance Analysis of
Fractional Guard Channel Policies in Mobile Cellular Networks.IEEE Transactions on Wireless
Communications, 5 (2006), pp. 301-305.
[7] Do T. V. (2011). Solution for A Retrial Queueing Problem in Cellular Networks with the Fractional
Guard Channel Policy.Mathematical and Computer Modelling, 53(11โ€“12), pp. 2058-2065.
[8] Ramjee R., Towsley D. and Nagarajan R., (1997). On Optimal Call Admission Control in Cellular
Networks.Wireless Network, 3(1), March 1997.
[9] Neuts M.F. (1981). Matrix Geometric Solutions in Stochastic Model, Johns Hopkins University
Press, Baltimore.
[10] Tien Van Do, Ram Chakka, An efficient method to compute the rate matrix for retrial queues with
large number of servers, Applied Mathematics Letters 23, (2010) 638-643.
[11] Tien, D.V. (2013). An Enhanced Algorithm to Solve Multiserver Retrial Queueing Systems with
Impatient Customer. Computers & Industrial Engineering, Vol 65, No 4, pp. 719-728.
[12] Kulshrestha R, Agarwal A, Shruti (2019). An adaptive fractional guard channel based CAC scheme
for heterogeneous traf๏ฌc in wireless cellular networks. In: Proceedings of IEEE 6th international
conference in computing for sustainable global development, Delhi, India, pp 1066โ€“1070.
[13] Do T.V. (2010). An Efficient Computation Algorithm for A Multiserver Feedback retrial Queue With
A Large Queueing Capacity, Applied Mathematical Modelling, 34(8), pp. 2272-2278.
[14] https://www.wolfram.com/mathematica/
International Journal of Computer Networks & Communications (IJCNC) Vol.13, No.4, July 2021
100
AUTHORS
Dang Thanh Chuong obtained his doctorate in Mathematical Foundation for
Computers and Computing Systems in 2014 from the Institute of Information
Technology, Vietnam Academy of Science and Technology (VAST). He has published
over 20 research papers. His research interests are in the fields of all-optical networks
with emphasis on packet/burst-based switching, Contention Resolution, and Quality of
Service; Queueing Theory and Retrial Queue. Email: dtchuong@hueuni.edu.vn.
Hoa Ly Cuong procuring MSc in Computer Science in 2017 from the Hue University
of Science, Hue University. The areas he has engaged in comprise Queueing Theory
and Wireless Networks.
Email: hlcuong90@gmail.com.
Pham TrungDuc In 2010, he graduated with a Bachelor of Information Technology
from University of Sciences, Hue University. In 2012, he received a Master's degree in
Computer Science from Hue University of Sciences. Currently, he is a PhD student at
Hue University of Sciences, Hue University (from December 2016 to the present).
Research fields: OBS network, QoS differentiation, scheduling admission control, QoS
improvement, QoS provisioning. Email: phamtrungduc@hueuni.edu.vn
Duong Duc Hung is a Technical Editor at the HU Journal of Science, Hue University,
Vietnam. His main research topics are Computer Networks and Communications; Text
Mining.
Email: ddhung@hueuni.edu.vn

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  • 1. International Journal of Computer Networks & Communications (IJCNC) Vol.13, No.4, July 2021 DOI: 10.5121/ijcnc.2021.13406 85 PERFORMANCE ANALYSIS IN CELLULAR NETWORKS CONSIDERING THE QOS BY RETRIAL QUEUEING MODEL WITH THE FRACTIONAL GUARD CHANNELS POLICIES Dang ThanhChuong1 , Hoa Ly Cuong1 , Pham Trung Duc1 and Duong Duc Hung2 1 Faculty of Information Technology, University of Sciences, Hue University, Viet Nam 2 Hue University, Viet Nam ABSTRACT In this article, a retrial queueing model will be considered with persevering customers for wireless cellular networks which can be frequently applied in the Fractional Guard Channel (FGC) policies, including Limited FGC (LFGC), Uniform FGC (UFGC), Limited Average FGC (LAFGC) and Quasi Uniform FGC (QUFGC). In this model, the examination on the retrial phenomena permits the analyses of important effectiveness measures pertained to the standard of services undergone by users with the probability that a fresh call first arrives the system and find all busy channels at the time, the probability that a fresh call arrives the system from the orbit and find all busy channels at the time and the probability that a handover call arrives the system and find all busy channels at the time. Comparison between four types of the FGC policy can befound to evaluate the performance of the system. KEYWORDS Cellular Mobile Networks, Retrial, FGC, QoS. 1. INTRODUCTION In the present, with the aid of the rapid development of wireless and mobile networks, many queueing models, especially retrial queues, have been proposed to evaluate the actual performances of these network systems. In cellular networks, it is important to design network models that the handover calls are more prior than the fresh calls properly. The concerns are limited resources, for example, a number of channels, sharings, and competitions in a certain cell (the collisions between the handover calls and the fresh calls), which can lead to the call interruptions when a user frequently moves among cells in the network as cell edges. Therefore, it is necessary that the appropriate policy admitting and processing the arrival calls minimize congestion risks of the system, including the handover and fresh calls. The call admission controls have been proposed to fulfillQoS requirements [1]. The research [1] and [2] first introduced the application of the retrial queueing models on evaluating cellular mobile networks involving the fractional guard channel policy to minimize the handover probability. Then [3] constructed the novel model and obtained some important results but that model bases only one probability ๐œƒ. In later years [4] and [5] reapplied and renovated [3] by considering the additional probability of the fresh calls ๐œƒโ€ฒ. [6] used the fractional guard channel without retrial customers. [7] improved [3] by using the fractional guard channel, but it considered only one probability ๐œƒ. The fractional guard channel policy allocates resources in the
  • 2. International Journal of Computer Networks & Communications (IJCNC) Vol.13, No.4, July 2021 86 cellular mobile network to reserve for the handover calls. This policy is considered the general form of the guard channel ones. When the system has available resources, the fresh call requirements are accepted by a certain probability, depending on the cell states (a number of the busy channels). Ramjee [8] first proposed the fractional guard channel policy with the accepted probability of the fresh calls ๐›ฝ๐‘– (0 โ‰ค ๐›ฝ๐‘– โ‰ค 1, ๐‘– = 0, ๐‘, where ๐‘ is the number of the channels), and it depends on the number of the operating channels. The problem is that the optimal probability ๐›ฝ๐‘– is chosen by the fractional guard channel policy. The limited fractional guard channel LFGC, the uniform fractional guard channel UFGC, the limited average fractional guard channel LAFGC and the quasi-uniform fractional guard channel QUFGC policies have been initiated based on the features of different network systems. To generalize the above results, the research can be examined the retrial queueing model whose form is ๐‘€/๐‘€/๐‘/(๐‘ + ๐ฟ) in combination with the fractional guard channel policies and the characteristic parameters, such as the probability that a fresh call first arrives and then it enters the orbit and the impatient customer probability that a call resumes to enter the servers. In contrast, it is unsuccessful and eternally departs and the additional features are the probability ๐œƒ (0 < ๐œƒ โ‰ค 1) that a call first enters the orbit when failing and the probability ๐œƒโ€ฒ (0 < ๐œƒโ€ฒ โ‰ค 1) which can be considered as a call next enters the orbit when failing, whereas [7] has only one fixed probability ๐œƒ for all calls entering the orbit. The main aim of our model is to lessen possibilities of being blocked with retrial and handover calls, significant factors to accomplish QoS, with the additional measure which is a novel attribute in comparison with the aforementioned works. The organization of the article is as follows, In Section 2, the detailed problems with the parameters and the models are described. The analysis results will be presented evaluated the model performances in Section 3. 2. ANALYSIS MODEL 2.1. Performance Analysis of Fractional Guard Channel The model that we use in this study is some what like the models in [3], [4], [5] and [7] by considering the retrial customers for the fresh and handover calls. The salient point is that we implement the fractional guard channel with the two probabilities of the fresh calls (Figure 1). In the guard channel model, the guard channels reserve for the handover calls. It means that a fresh call arrives in the system and is accepted with the probability ๐›ฝ๐‘– (0 โ‰ค ๐›ฝ๐‘– โ‰ค 1, ๐‘– = 0, ๐‘, where ๐‘ is the number of the channels) and the system has at least a vacant channel, otherwise it will be rejected. A handover call is denied when all channels of the system are busy. Also in this model, the fairness is considered in the resource stores for all calls, including the fresh and handover calls. The innovations are that implement the fractional guard channels (Definition 2.1) with the limited capacity of the orbit.
  • 3. International Journal of Computer Networks & Communications (IJCNC) Vol.13, No.4, July 2021 87 F โ‹ฎ Call admission control i=๐›ฝi F+ H H Orbit 1-๐œƒ ๐œƒ 1-๐œƒโ€™ ๐œƒโ€™ โ‹ฎ c channels Figure 1.The retrial queueing model in the cellular mobile network. Definition 2.1 [7]: A fresh or retrial call is admitted by the system in accordance with the fractional guard channel policy. The handover calls are accepted if the system exists a certainly available channel, and the fresh calls only are served with the probabilities ๐›ฝ๐‘–(0 โ‰ค ๐›ฝ๐‘– โ‰ค 1) depending on the system state. As the aforementioned issue, we consider the fractional guard channel policies, such as LFGC, UFGC, LAFGC, and QUFGC. In addition, we assume that the system consists of ๐‘ the operation channels, ๐‘” the permission channels, the accepted probability of the fresh calls ๐‘(0 โ‰ค ๐‘ โ‰ค 1), and the average value ๐‘Ÿ(๐‘Ÿ โˆˆ โ„, 0 โ‰ค ๐‘Ÿ โ‰ค ๐‘). ๏‚ท LFGC: ๐›ฝ๐‘– = 1 (0 โ‰ค ๐‘– โ‰ค ๐‘” โˆ’ 1), ๐›ฝ๐‘” = ๐‘, ๐›ฝ๐‘– = 0 (๐‘” + 1 โ‰ค ๐‘– < ๐‘). ๏‚ท LAFGC: ๐›ฝ๐‘– = 1 (0 โ‰ค ๐‘– โ‰ค ๐‘ โˆ’ โŒŠ๐‘ŸโŒ‹ โˆ’ 2), ๐›ฝ๐‘โˆ’โŒŠ๐‘ŸโŒ‹โˆ’1 = 1 โˆ’ ๐‘Ÿ + โŒŠ๐‘ŸโŒ‹, ๐›ฝ๐‘– = 0 (๐‘ โˆ’ โŒŠ๐‘ŸโŒ‹ โ‰ค ๐‘– < ๐‘). ๏‚ท UFGC: ๐›ฝ๐‘– = ๐‘ (0 โ‰ค ๐‘– < ๐‘). ๏‚ท QUFGC: ๐›ฝ๐‘– = 1 (0 โ‰ค ๐‘– โ‰ค ๐‘” โˆ’ 1), ๐›ฝ๐‘– = ๐‘ (๐‘” โ‰ค ๐‘– < ๐‘). Obviously, we have๐›ฝ๐‘ = 0. 2.2. The Parameters Considering a certain cell in the cellular mobile network, the system is modeled in Figure 1. The arrival calls will be admitted by the call admission control with the fractional guard channel policies. The interarrival times for fresh and handover calls follow the exponential distribution with the rate ๐น and ๐ป, respectively. Let is defined as the interarrival time between consecutively incoming calls regardless of fresh and handover calls, thus we have = ๐น + ๐ป. Also, the served time follows the exponential distribution with the rate . A call first arriving the system is blocked due to lacking or allocating the resources. Now it is: ๏‚ท Permanently depart the system with the probability1 โˆ’ ๐œƒ(0 < ๐œƒ โ‰ค 1). ๏‚ท or enter the orbit with the probability ๐œƒ. In this model, the orbit is considered a queue (capacity๐ฟ) to store the blocked calls (or the fresh calls).Similarly, when the retrial calls are unsuccessful, they will: ๏‚ท Permanently depart the system with the probability1 โˆ’ ๐œƒโ€ฒ (0 < ๐œƒโ€ฒ โ‰ค 1). ๏‚ท or continue entering the orbit with the probability ๐œƒโ€ฒ.
  • 4. International Journal of Computer Networks & Communications (IJCNC) Vol.13, No.4, July 2021 88 In addition, the retrial rate for the fresh calls from the orbit to the system is . The fresh or retrial calls enter the system with the probability ๐›ฝ๐‘– in accordance with the state system. A handover call is served immediately if the system exists at least one vacant channel. In contrast, it will permanently depart and does not enter orbit. 2.3. Modelling The system is illustrated by the two-dimensional continuous-time Markov chains ๐‘‹ = {(๐ผ(๐‘ก), ๐ฝ(๐‘ก)), ๐‘ก โ‰ฅ 0}, where ๐ผ(๐‘ก)(0 โ‰ค ๐ผ(๐‘ก) โ‰ค ๐‘)are the numbers ofthecustomers being served by the system, and ๐ฝ(๐‘ก)(0 โ‰ค ๐ฝ(๐‘ก) โ‰ค ๐ฟ)are the numbers of the customerswaiting to enter the system at the time ๐‘ก. The state space of the above chains is depicted by Figure 2, where (๐‘–, ๐‘—)are the states that the numbers of the occupied channels at the time ๐‘กare๐‘–(๐‘– = 0, 1, 2, โ€ฆ , ๐‘ โˆ’ ๐‘”, ๐‘ โˆ’ ๐‘” + 1, โ€ฆ , ๐‘), and the numbers of the customers in the orbit at the time ๐‘กare๐‘—(๐‘— = 0, 1, 2, โ€ฆ , ๐ฟ). 1,0 0,0 1,1 0,1 ... ... 1,L-1 0,L-1 1,L 0,L l0 m l1 2m b0a c,0 c,1 c,L-1 c,L ... lc-1 cm (1-qโ€™)a lFq cm cm cm l1 2m l1 2m l1 2m l0 m l0 m l0 m L(1-qโ€™)a lc-1 lc-1 lc-1 c-1,0 c-1,1 ... c-1, L-1 c-1,L lc-2 (c-1)m lF(1- b0)q lF(1- b1)q lF(1- bc-1)q lc-2 lc-2 lc-2 lF(1- b0)q lF(1- b1)q lF(1- bc-1)q lFq b1a bc-2a bc-1a Lb0a Lb1a Lbc-2a Lbc-1a (1- bc-1)(1-qโ€™)a (1- b1)(1-qโ€™)a (1- b0)(1-qโ€™)a L(1- b0)(1-qโ€™)a L(1- b1)(1-qโ€™)a L(1- bc-1)(1-qโ€™)a 2b0a 2(1-qโ€™)a lFq lF(1- b0)q lF(1- b1)q lF(1- bc-1)q 2b1a 2bc-2a 2bc-1a 2(1- bc-1)(1-qโ€™)a 2(1- b1)(1-qโ€™)a 2(1- b0)(1-qโ€™)a (L-1)b0a (L-1)(1-qโ€™)a lFq lF(1- b0)q lF(1- b1)q lF(1- bc-1)q (L-1)b1a (L-1)bc-2a (L-1)bc-1a (L-1)(1- bc-1)(1-qโ€™)a (L-1)(1- b1)(1-qโ€™)a (L-1)(1- b0)(1-qโ€™)a (c-1)m (c-1)m (c-1)m Figure 2. State transition diagram of the model Let ๐œ‹๐‘–,๐‘— = lim ๐‘กโ†’โˆž ๐‘ƒ(๐ผ(๐‘ก) = ๐‘–, ๐ฝ(๐‘ก) = ๐‘—), where ๐œ‹๐‘–,๐‘— are the steady-state probabilities for the state (๐‘–, ๐‘—). When the chains ๐‘‹ = {๐ผ(๐‘ก), ๐ฝ(๐‘ก); ๐‘ก โ‰ฅ 0}are represented through the state transition matrixes ๐ด๐‘—, ๐ต๐‘—, ๐ถ๐‘— (with theirdimension(๐‘ + 1) ร— (๐‘ + 1)) [9-13]: ๏‚ท ๐ด๐‘—(๐‘–, ๐‘˜) are the state transition matrixes for the states from (๐‘–, ๐‘—) to (๐‘˜, ๐‘—)(0 โ‰ค ๐‘–, ๐‘˜ โ‰ค ๐‘, 0 โ‰ค ๐‘— โ‰ค ๐ฟ). They express an successful call, which is a fresh or handover call, or a call having been completed and departing. ๐ด๐‘— has the entries ๐ด๐‘—(๐‘–, ๐‘– โˆ’ 1) = ๐‘– (๐‘– = 1, ๐‘) and ๐ด๐‘—(๐‘–, ๐‘– + 1) = ๐‘– = ๐›ฝ๐‘– ๐น + ๐ป(๐‘– = 0, ๐‘ โˆ’ 1) and the rest 0.
  • 5. International Journal of Computer Networks & Communications (IJCNC) Vol.13, No.4, July 2021 89 ๐ด๐‘— = ๐ด = ( 0 0 0 โ‹ฏ 0 0 0 0 1 โ‹ฏ 0 0 0 โ‹ฎ โ‹ฎ โ‹ฎ โ‹ฎ โ‹ฎ โ‹ฎ โ‹ฎ 0 0 0 โ‹ฏ (๐‘ โˆ’ 1) 0 ๐‘โˆ’1 0 0 0 โ‹ฏ 0 ๐‘ 0 ) , (๐‘— โ‰ฅ 0) (1) ๏‚ท ๐ต๐‘—(๐‘–, ๐‘˜)are the state transition matrixes for the states from (๐‘–, ๐‘—) to (๐‘˜, ๐‘— + 1)(0 โ‰ค ๐‘–, ๐‘˜ โ‰ค ๐‘, 0 โ‰ค ๐‘— โ‰ค ๐ฟ โˆ’ 1). They express a fresh call first arriving the system and entering the orbit with the probability ๐œƒ. ๐ต๐‘— has the entries ๐ต๐‘—(๐‘–, ๐‘–) = (1 โˆ’ ๐›ฝ๐‘–)๐œƒ ๐น(๐‘– = 0, ๐‘) (note ๐ต๐‘—(๐‘, ๐‘) = ๐œƒ ๐น due to ๐›ฝ๐‘ = 0) and the rest 0. ๐ต๐‘— = ๐ต = ( ๐œŽ0 โ‹ฏ 0 0 0 โ‹ฏ 0 0 โ‹ฎ โ‹ฎ โ‹ฎ โ‹ฎ 0 โ‹ฏ ๐œŽ๐‘โˆ’1 0 0 โ‹ฏ 0 ๐œŽ๐‘) , (๐‘— โ‰ฅ 0) (2) Where ๐œŽ๐‘– = (1 โˆ’ ๐›ฝ๐‘–)๐œƒ ๐น, (0 โ‰ค ๐‘– โ‰ค ๐‘). ๏‚ท ๐ถ๐‘—(๐‘–, ๐‘˜)are the state transition matrixes for the states from (๐‘–, ๐‘—) to (๐‘˜, ๐‘— โˆ’ 1)(0 โ‰ค ๐‘–, ๐‘˜ โ‰ค ๐‘, 1 โ‰ค ๐‘— โ‰ค ๐ฟ). They express a retrial call returning the system. Then it is served or permanently departs due to its impatience and all channels that are busy. ๐ถ๐‘— has the entries ๐ถ๐‘—(๐‘–, ๐‘– + 1) = ๐›ฝ๐‘–๐‘— (๐‘– = 0, ๐‘ โˆ’ 1) and ๐ถ๐‘—(๐‘–, ๐‘–) = (1 โˆ’ ๐›ฝ๐‘–)(1 โˆ’ ๐œƒโ€ฒ)๐‘— (๐‘– = 0, ๐‘) (note ๐ถ๐‘—(๐‘, ๐‘) = (1 โˆ’ ๐œƒโ€ฒ)๐‘— due to ๐›ฝ๐‘ = 0) and the rest 0. ๐ถ๐‘— = ( ๐œŽ0,๐‘— ๐›ฝ0๐‘— 0 โ‹ฏ 0 0 0 ๐œŽ1,๐‘— ๐›ฝ1๐‘— โ‹ฏ 0 0 โ‹ฎ โ‹ฎ โ‹ฎ โ‹ฎ โ‹ฎ โ‹ฎ 0 0 0 โ‹ฏ ๐œŽ๐‘โˆ’1,๐‘— ๐›ฝ๐‘โˆ’1๐‘— 0 0 0 โ‹ฏ 0 ๐œŽ๐‘,๐‘— ) , (๐‘— โ‰ฅ 1) (3) Where ๐œŽ๐‘–,๐‘— = (1 โˆ’ ๐›ฝ๐‘–)(1 โˆ’ ๐œƒโ€ฒ)๐‘— (0 โ‰ค ๐‘– โ‰ค ๐‘, 1 โ‰ค ๐‘— โ‰ค ๐ฟ). We have the generator matrix ๐‘„: ๐‘„ = ( ๐‘„1 (0) ๐‘„2 (0) ๐ŸŽ ๐ŸŽ ๐ŸŽ ๐‘„0 (1) ๐‘„1 (1) โ‹ฑ ๐ŸŽ ๐ŸŽ ๐ŸŽ โ‹ฑ โ‹ฑ ๐‘„2 (๐ฟโˆ’2) ๐ŸŽ ๐ŸŽ ๐ŸŽ ๐‘„0 (๐ฟโˆ’1) ๐‘„1 (๐ฟโˆ’1) ๐‘„2 (๐ฟโˆ’1) ๐ŸŽ ๐ŸŽ ๐ŸŽ ๐‘„0 (๐ฟ) ๐‘„1 (๐ฟ) ) (4)
  • 6. International Journal of Computer Networks & Communications (IJCNC) Vol.13, No.4, July 2021 90 Where: { ๐‘„1 (0) = ๐ด โˆ’ ๐ท๐ด โˆ’ ๐ท๐ต ๐‘„1 (๐‘—) = ๐ด โˆ’ ๐ท๐ด โˆ’ ๐ท๐ต โˆ’ ๐ท๐ถ๐‘—(1 โ‰ค ๐‘— โ‰ค ๐ฟ โˆ’ 1) ๐‘„๐ฟ (0) = ๐ด โˆ’ ๐ท๐ด โˆ’ ๐ท๐ถ๐ฟ ๐‘„0 (๐‘—) = ๐ถ๐‘—(1 โ‰ค ๐‘— โ‰ค ๐ฟ) ๐‘„2 (๐‘—) = ๐ต (0 โ‰ค ๐‘— โ‰ค ๐ฟ โˆ’ 1) Where ๐ท๐‘‹(๐‘‹ = ๐ด, ๐ต, ๐ถ๐‘—) are the diagonal matrixes with the entries on the main diagonal that are the sum of the entries on respective row of the matrix ๐‘‹. We have: ๐‘ฃ๐‘„ = (0,0, โ€ฆ ,0) โŸ (๐‘+1)ร—(๐‘+1) (5) Where ๐‘ฃ = (๐‘ฃ0,๐‘ฃ1, โ€ฆ , ๐‘ฃ๐ฟ). Let the probabilities (๐‘–, ๐‘—) are defined by ๐‘๐‘–,๐‘— = lim ๐‘กโ†’+โˆž ๐‘ƒ(๐ผ(๐‘ก) = ๐‘–, ๐ฝ(๐‘ก) = ๐‘—), and the level probabilities vectors ๐‘ฃ๐‘— = (๐‘0,๐‘—, ๐‘1,๐‘—, โ€ฆ , ๐‘๐‘,๐‘—)(1 โ‰ค ๐‘— โ‰ค ๐ฟ). The balance equations of the system are: ๐‘ฃ0๐‘„1 (0) + ๐‘ฃ1๐‘„0 (1) = (0,0, โ€ฆ ,0) โŸ ๐‘+1 (6) ๐‘ฃ๐‘—โˆ’1๐‘„2 (๐‘—โˆ’1) + ๐‘ฃ๐‘—๐‘„1 (๐‘—) + ๐‘ฃ๐‘—+1๐‘„1 (๐‘—+1) = (0,0, โ€ฆ ,0) โŸ ๐‘+1 (1 โ‰ค ๐‘— โ‰ค ๐ฟ โˆ’ 1) (7) ๐‘ฃ๐ฟโˆ’1๐‘„2 (๐ฟโˆ’1) + ๐‘ฃ๐ฟ๐‘„1 (๐ฟ) = (0,0, โ€ฆ ,0) โŸ ๐‘+1 (8) โˆ‘ ๐‘ฃ๐‘— ๐ฟ ๐‘—=0 ร— ( 1 1 โ‹ฎ 1 ) โŸ ๐‘+1 = 1 (9) From (5) and (9), we calculate ๐‘ฃ: ๐‘ฃ = ๐‘’(๐‘„ + ๐ธ)โˆ’1 (10) Where ๐‘’ is the row vector (1 ร— (๐‘ + 1)) and ๐ธ is the matrix ((๐‘ + 1) ร— (๐‘ + 1)) all of whose elements are the constants 1. 2.4. The Performance Measures of the System The probability of the fresh calls ๐‘ƒ๐ต๐น firstly arrives the system and find all busy channels at the time or is rejected with the probability (1 โˆ’ ๐›ฝ๐‘–).
  • 7. International Journal of Computer Networks & Communications (IJCNC) Vol.13, No.4, July 2021 91 ๐‘ƒ๐ต๐น = โˆ‘ โˆ‘ ๐‘๐‘–,๐‘—(1 โˆ’ ๐›ฝ๐‘–) ๐‘ ๐‘–=0 ๐ฟ ๐‘—=0 (11) The probability of the retrial calls ๐‘ƒ๐ต๐น,๐‘Ÿ๐‘’๐‘ก๐‘Ÿ๐‘–๐‘Ž๐‘™ then arrives the system from the orbit and find all busy channels at the time is rejected with the probability (1 โˆ’ ๐œƒโ€ฒ). ๐‘ƒ๐ต๐น,๐‘Ÿ๐‘’๐‘ก๐‘Ÿ๐‘–๐‘Ž๐‘™ = (1 โˆ’ ๐œƒโ€ฒ) ๐œƒ ๐น โˆ‘ โˆ‘ ๐‘—๐‘๐‘–,๐‘— ๐‘ ๐‘–=0 ๐ฟ ๐‘—=1 (12) The probability of the handover calls ๐‘ƒ๐ต๐น,๐‘Ÿ๐‘’๐‘ก๐‘Ÿ๐‘–๐‘Ž๐‘™ clearly arrives the system and find all busy channels at the time. ๐‘ƒ๐ต๐ป = โˆ‘ ๐‘๐‘,๐‘— ๐ฟ ๐‘—=0 (13) 3. RESULTS The efficiency of the performance as the change of the parameters of the system is firstly considered with the fresh and handover calls. When the blocks occur, the fresh calls reattempt to connect in the intervals of the stochastic distribution. We assume the base station of the cell that can process ๐‘ the connection simultaneously. Table 1 enumerates the parameters to analyze results. The Mathematica program of Wolfram Research denoted [14] is a power tool to compute and simulate for network models and is utilized in the proposed model of the paper. Table 1. The parameters of the model. The parameters Values Describe the parameters ๐‘ 10 Number of the channels in the cell ๐‘” 1 to 7 Number of the accepted channels ๐‘ 0.1 to 1 Probability that a fresh call is served ๐ฟ 1 to 10 Number of the customers in the orbit ๐ป 0.002 to 2 Arrival rate for the handover calls ๐น 0.018 to 0.2 Arrival rate for the fresh calls 0.2 Service rate 0.4 Retrial rate ๐œƒ 0.1 to 1 Probability that a fresh call first enters the orbit ๐œƒโ€ฒ 0.1 to 1 Retrial probability that a fresh call re-enters the orbit 3.1. Analysing and Comparing the Blocking Probabilities with Values ๐† First, we compare the blocking probabilities ๐‘ƒ๐ต (๐‘ƒ๐ต๐น, ๐‘ƒ๐ต๐น,๐‘Ÿ๐‘’๐‘ก๐‘Ÿ๐‘–๐‘Ž๐‘™ or ๐‘ƒ๐ต๐‘…, ๐‘ƒ๐ต๐ป) according to a variety of the fractional guard channel policies with the fluctuations in values ๐œŒ. It can be seen in Figure 3, the value ๐œŒ, the traffic load, is definited by the expression ๐œŒ = ๐‘ โ„ . We have ๐‘” = 7, ๐‘ = 0.5, ๐‘Ÿ = 2.3 (for the LAFGC policy), ๐ฟ = 8, ๐œƒ = 0.5, ๐œƒโ€ฒ = 0.5, ๐น = 2 ๐ป, ๐œŒ ranging from 0.3 to 1.9. We find that the blocking probabilities for the LFGC and LAFGC policies are better than the blocking probabilities for the UFGC and QUFGC policies.
  • 8. International Journal of Computer Networks & Communications (IJCNC) Vol.13, No.4, July 2021 92 Figure 3. The blocking probability of ๐‘ƒ๐ต๐น, ๐‘ƒ๐ต๐น,๐‘Ÿ๐‘’๐‘ก๐‘Ÿ๐‘–๐‘Ž๐‘™ and ๐‘ƒ๐ต๐ปby ๐œŒ. This may partly explain that the UFGC and QUFGC policies permit the fresh calls with ๐›ฝ๐‘– = 1 or ๐›ฝ๐‘– = ๐‘(0 โ‰ค ๐‘– < ๐‘). As a result, the fresh calls are easier to enter the system, and the handover calls are significantly restricted. In the meanwhile, the LFGC and LAFGC policies control the fresh calls with ๐›ฝ๐‘– = 0 (0 < ๐ฟ โ‰ค ๐‘– < ๐‘, ๐ฟ: a particular parameter) at some of the last states with ๐‘– โ‰ฅ ๐ฟ. 3.2. Analysing and comparing the blocking probabilities with values๐’ˆ With ๐œŒ = 0.7, ๐‘ = 0.5, ๐ฟ = 8, ๐œƒ = 0.5, ๐œƒโ€ฒ = 0.5, ๐น = 2 ๐ป, ๐‘” ranging from 1 to 7, we obtain the results as presented in Figure 4. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 PB r UFGC PB_F PB_R PB_H 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 PB r LFGC PB_F PB_R PB_H 0 0.1 0.2 0.3 0.4 0.5 0.6 1 2 3 4 5 6 7 PB g UFGC PB_F PB_R PB_H 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 PB r QUFGC PB_F PB_R PB_H 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 PB r LAFGC PB_F PB_R PB_H 0 0.1 0.2 0.3 0.4 0.5 0.6 1 2 3 4 5 6 7 PB g QUFGC PB_F PB_R PB_H
  • 9. International Journal of Computer Networks & Communications (IJCNC) Vol.13, No.4, July 2021 93 Figure 4. The blocking probability of ๐‘ƒ๐ต๐น, ๐‘ƒ๐ต๐น,๐‘Ÿ๐‘’๐‘ก๐‘Ÿ๐‘–๐‘Ž๐‘™ , and ๐‘ƒ๐ต๐ปby ๐‘”. When we change values ๐‘”, the probabilities ๐‘ƒ๐ต๐น and ๐‘ƒ๐ต๐น,๐‘Ÿ๐‘’๐‘ก๐‘Ÿ๐‘–๐‘Ž๐‘™ for the QUFGC policy decrease inversely proportional to values ๐‘”, and the probabilities ๐‘ƒ๐ต๐ป for the QUFGC, UFGC, LFGC and LAFGC policies remain steady and thus we need to choose an appropriate value ๐‘” for the low blocking probabilities ๐‘ƒ๐ต๐ป. The UFGC policy has the probabilities ๐‘ƒ๐ต๐น and ๐‘ƒ๐ต๐น,๐‘Ÿ๐‘’๐‘ก๐‘Ÿ๐‘–๐‘Ž๐‘™ which maintain stable due to ๐›ฝ๐‘– = ๐‘(0 โ‰ค ๐‘– < ๐‘). Similar to the QUFGC policy, the probabilities ๐‘ƒ๐ต๐น and ๐‘ƒ๐ต๐น,๐‘Ÿ๐‘’๐‘ก๐‘Ÿ๐‘–๐‘Ž๐‘™ for the LFGC and LAFGC policies significantly decline because the numbers of the accepted channels for the fresh and retrial calls gradually increase. 3.2. Analysing and Comparing the Blocking Probabilities with Values ๐‘ณ With ๐œŒ = 0.7, ๐‘” = 7, ๐‘ = 0.5, ๐‘Ÿ = 2.3 (for the LAFGC policy), ๐œƒ = 0.5, ๐œƒโ€ฒ = 0.5, ๐น = 2 ๐ป, ๐ฟ ranging from 1 to 8, we obtain the results presented in Figure 5. Figure 5. The blocking probability of ๐‘ƒ๐ต๐น, ๐‘ƒ๐ต๐น,๐‘Ÿ๐‘’๐‘ก๐‘Ÿ๐‘–๐‘Ž๐‘™ , and ๐‘ƒ๐ต๐ปby ๐ฟ. The blocking probabilities ๐‘ƒ๐ต๐น and ๐‘ƒ๐ต๐น,๐‘Ÿ๐‘’๐‘ก๐‘Ÿ๐‘–๐‘Ž๐‘™ are influenced by the number of the maximum customers in the orbit ๐ฟ when ๐ฟ < 4 as Figure 5. When ๐ฟ โ‰ฅ 4 these probalities are almost 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5 6 7 PB g LFGC PB_F PB_R PB_H 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 1 2 3 4 5 6 7 8 PB L LFGC PB_F PB_R PB_H 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5 6 7 PB g LAFGC PB_F PB_R PB_H 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 1 2 3 4 5 6 7 8 PB L LAFGC PB_F PB_R PB_H
  • 10. International Journal of Computer Networks & Communications (IJCNC) Vol.13, No.4, July 2021 94 steady. While the blocking probabilities ๐‘ƒ๐ต๐ป are not affected regardless of the value ๐ฟ. This can be explained that the number of the servers is fixed, the more the customers in the orbit are, the more blocked the system is. For this reason, we can approximate to the infinite model with ๐ฟ โ†’ +โˆž. 3.3. Analysing and comparing the blocking probabilities with values ๐’‘ With ๐œŒ = 0.7, ๐‘” = 7, ๐‘ = 0.5, ๐ฟ = 8, ๐‘Ÿ = 2.3 (for the LAFGC policy), ๐œƒ = 0.5, ๐œƒโ€ฒ = 0.5, ๐น = 2 ๐ป, ๐‘ ranging from 0.1 to 1, we obtain the results as listed in Figure 6. Figure 6. The blocking probability of ๐‘ƒ๐ต๐น, ๐‘ƒ๐ต๐น,๐‘Ÿ๐‘’๐‘ก๐‘Ÿ๐‘–๐‘Ž๐‘™ , and ๐‘ƒ๐ต๐ปby ๐‘. We found/find that the accepted probabilities of the fresh calls ๐‘ influence on how the congestions of the fresh and retrial calls are. In these analyses, the blocking probabilities of the fresh calls for the UFGC and QUFGC policies are better than the ones for the LFGC and LAFGC policies due to the fact that the the probabilities ๐›ฝ๐‘–(0 โ‰ค ๐‘– โ‰ค ๐‘) for the LFGC and LAFGC policies have more the values ๐‘ than the ones for the UFGC and QUFGC policies. As a result, the fresh and retrial calls ease to enters the system for the LFGC and LAFGC policies. 3.4. Analysing and comparing the blocking probabilities with values ๐€๐‘ฏ ๐€๐‘ญ โ„ With ๐œŒ = 0.7, ๐‘” = 7, ๐ฟ = 8, ๐‘ = 0.5, ๐œƒ = 0.5, ๐œƒโ€ฒ = 0.5, ๐ป ๐น โ„ ranging from 1 5 โ„ to 5, we obtain the results as shown in Figure 7. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 PB p UFGC PB_F PB_R PB_H 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 PB p LFGC PB_F PB_R PB_H 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 PB p QUFGC PB_F PB_R PB_H 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 PB p LAFGC PB_F PB_R PB_H
  • 11. International Journal of Computer Networks & Communications (IJCNC) Vol.13, No.4, July 2021 95 Figure 7. The blocking probability of ๐‘ƒ๐ต๐น, ๐‘ƒ๐ต๐น,๐‘Ÿ๐‘’๐‘ก๐‘Ÿ๐‘–๐‘Ž๐‘™ , and ๐‘ƒ๐ต๐ปby ๐ป ๐น โ„ . We find that if the ratio ๐ป ๐น โ„ < 1/3, the probabilities ๐‘ƒ๐ต๐น, ๐‘ƒ๐ต๐น,๐‘Ÿ and ๐‘ƒ๐ต๐ป are changed, but if the ratio ๐ป ๐น โ„ โ‰ฅ 1/3, the probabilities ๐‘ƒ๐ต๐น, ๐‘ƒ๐ต๐น,๐‘Ÿ and ๐‘ƒ๐ต๐ป remain almost unchanged. Therefore, we choose the ratio ๐ป ๐น โ„ โ‰ฅ 1/3 to maintain the stability of the system. 3.5. Analysing and comparing the blocking probabilities with values ๐œฝ With ๐œŒ = 0.7, ๐‘” = 7, ๐ฟ = 8, ๐‘ = 0.5, ๐‘Ÿ = 2.3 (for the LAFGC policy), ๐œƒโ€ฒ = 0.5, ๐น = 2 ๐ป, ๐œƒ ranging from 0.1 to 1, we gain the results as presented in Figure 8. 0 0.1 0.2 0.3 0.4 0.5 0.6 1/5 1/4 1/3 1/2 1 2 3 4 5 PB H/ F UFGC PB_F PB_R PB_H 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 1/5 1/4 1/3 1/2 1 2 3 4 5 PB H/ F LFGC PB_F PB_R PB_H 0 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 PB q UFGC PB_F PB_R PB_H 0 0.05 0.1 0.15 0.2 0.25 0.3 1/5 1/4 1/3 1/2 1 2 3 4 5 PB H/ F QUFGC PB_F PB_R PB_H 0 0.05 0.1 0.15 0.2 0.25 0.3 1/5 1/4 1/3 1/2 1 2 3 4 5 PB H/ F LAFGC PB_F PB_R PB_H 0 0.05 0.1 0.15 0.2 0.25 0.3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 PB q QUFGC PB_F PB_R PB_H
  • 12. International Journal of Computer Networks & Communications (IJCNC) Vol.13, No.4, July 2021 96 Figure 8. The blocking probability of ๐‘ƒ๐ต๐น, ๐‘ƒ๐ต๐น,๐‘Ÿ๐‘’๐‘ก๐‘Ÿ๐‘–๐‘Ž๐‘™ , and ๐‘ƒ๐ต๐ปby ๐œƒ. The parameter ๐œƒ gives the influence on the blocking probabilities of the retrial calls rather than the ones of the fresh calls. This can be explained that the probabilities that a fresh call first enters the orbit increase resulting in a numerous number of the customers in the the orbit, and the blocking probabilities of the retrial calls will raise. In the meantime, the probabilities of the handover calls remain almost unchanged. In these analyses, we find that the parameters โˆˆ (0.2 โˆ’ ๐œ€, 0.2 + ๐œ€), with ๐œ€ โ‰ฅ 0 is a arbitrarily small real. The system performances are optimized by the given parameters. 3.6. Analysing and comparing the blocking probabilities with values๐œฝโ€ฒ With ๐œŒ = 0.7, ๐‘” = 7, ๐ฟ = 8, ๐‘ = 0.5, ๐‘Ÿ = 2.3 (for the LAFGC policy), ๐œƒ = 0.5, ๐น = 2 ๐ป, ๐œƒโ€ฒ ranging from 0.1 to 1, we obtain the results as presented Figure 9. Figure 9. The blocking probability of ๐‘ƒ๐ต๐น, ๐‘ƒ๐ต๐น,๐‘Ÿ๐‘’๐‘ก๐‘Ÿ๐‘–๐‘Ž๐‘™ , and ๐‘ƒ๐ต๐ปby ๐œƒโ€ฒ. When taking into account the increase in the retrial probabilities ๐œƒโ€ฒ(0.1 โ‰ค ๐œƒโ€ฒ โ‰ค 0.9), the blocking probabilities ๐‘ƒ๐ต๐น,๐‘Ÿ are declined. Then the retrial calls re-enter the orbit to attempt the 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 PB q LFGC PB_F PB_R PB_H 0 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 PB q' UFGC PB_F PB_R PB_H 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 PB q' LFGC PB_F PB_R PB_H 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 PB q LAFGC PB_F PB_R PB_H 0 0.05 0.1 0.15 0.2 0.25 0.3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 PB q' QUFGC PB_F PB_R PB_H 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 PB q' LAFGC PB_F PB_R PB_H
  • 13. International Journal of Computer Networks & Communications (IJCNC) Vol.13, No.4, July 2021 97 next occasions. On the contrary, the blocking probabilities ๐‘ƒ๐ต๐น and ๐‘ƒ๐ต๐ป are slightly affected. As the results ofinFigure 9, when the parameter ๐œƒโ€ฒ tends to the asymptotic value of 1, the blocking probabilities ๐‘ƒ๐ต๐น,๐‘Ÿ almost optimize the certain performances. (a) (b) Figure 10. The blocking probability of ๐‘ƒ๐ต๐น,๐‘Ÿ๐‘’๐‘ก๐‘Ÿ๐‘–๐‘Ž๐‘™ (a) and ๐‘ƒ๐ต๐ป(b) among LAFGC, LFGC, QUFGC and UFGC We find that the blocking probabilities ๐‘ƒ๐ต๐น,๐‘Ÿ๐‘’๐‘ก๐‘Ÿ๐‘–๐‘Ž๐‘™ and ๐‘ƒ๐ต๐ป of LAFGC and LFGC are almost equivalent (Figure 10). Also, the blocking probabilities ๐‘ƒ๐ต๐ป of QUFGC and UFGC are approximately equal, yet the blocking probabilities ๐‘ƒ๐ต๐น,๐‘Ÿ of QUFGC are better than those of UFGC. This is explained that the accepted probabilities ๐›ฝ๐‘– of QUFGC are 1 at some initial states. 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0,3 0,5 0,7 0,9 1,1 1,3 1,5 1,7 1,9 PB F,retrial r LAFGC LFGC QUFGC UFGC 0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0,3 0,5 0,7 0,9 1,1 1,3 1,5 1,7 1,9 PB H r LAFGC LFGC QUFGC UFGC
  • 14. International Journal of Computer Networks & Communications (IJCNC) Vol.13, No.4, July 2021 98 Figure 11. The blocking probability of ๐‘ƒ๐ต๐น between in [7] and the article. According to figures extracted from [7] (of Do Van Tien) with ๐ฟ = ๐‘ โˆ’ 1, ๐›ฝ = 0, ๐‘ = 15, = 1 120 , = 20 , ๐น = 24 ๐ป, we ascertain that the probabilties of ๐‘ƒ๐ต๐น in the article are superior to those of [7]. The disparity is that the further value ๐œƒโ€ฒ can be varied, depending on usersโ€™s adjustments and playing a signal role in reducing the possibilities of obstructing the fresh calls. 4. CONCLUSIONS We have introduced the retrial queueing model in the cellular mobile network using the fractional guard channel policy with impatient customers according to the probabilities ฮธ and ฮธ'. The advantage of the model is the significant decrease of the blocking probabilities for the fresh and retrial calls as well as the support of the network operation to maintain the stability of QoS under the influences on the network traffic and additional factors. The generality in the model is weighed by choosing the retrial queueing model in combination with the fractional guard channel policies. The analysis results are simple and guarantee the low blocking probabilities of the handover calls for the fractional guard channel policy in compassion to the blocking probabilities of the guard channel policy. In addition, the retrial queueing model with the fractional guard channel policy in the cellular mobile network can be used for the criteria of the different network traffics to measure the performances. The corollaries elucidate that the model is appropriate and precise. Besides, the research considers a variety of the fractional guard channel policies, including the UFGC, QUFGC, LFGC, and LAFGC policies, to fall the blocking probabilities of the fresh calls, but they still guarantee the handover calls protected with the call admission control. CONFLICTS OF INTEREST The authors declare no conflict of interest. 0 0,02 0,04 0,06 0,08 0,1 0,12 0,14 0,16 0,18 0,2 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 PB F r LFGC Do V Tien This article
  • 15. International Journal of Computer Networks & Communications (IJCNC) Vol.13, No.4, July 2021 99 REFERENCES [1] Mandles M. and Tutschku K., (1996). Efficient Call Handling Procedures in Cellular Mobile Networks.Forschungsbericht, Preprint-Reihe 144, Institut fur Informatik, Universitat Wurzburg, Germany. [2] Tran-Gia P. and Mandles M., (1997). Modeling of Customer Retrial Phenomenon in Cellular Mobile Networks.IEEE Journal on Selected Areas in Communications, 15(8), pp. 1406-1414. [3] Do T. V. (2010). A new computational algorithm for retrial queues to cellular mobile systems with guard channels. Computers & Industrial Engineering, 59(4), pp. 865โ€“872. [4] Linda A. C. and Ani C. I. (2014). A Reservation and Retrial Queue Analysis for Cellular Networks.International Journal of Engineering Research & Technology (IJERT), ISSN: 2278-0181, Vol. 3 Issue 2, February โ€“ 2014 [5] Bhargava A., Bhagat A. and Singh J., (2016). Modelling and Performance Analysis of Cellular Mobile Networks.Indian Journal of Science and Technology, Vol 9 (48), December 2016. [6] Vazquez-Avila J., Cruz-Pรฉrez F.A. and Ortigoza-Guerrero L., (2006). Performance Analysis of Fractional Guard Channel Policies in Mobile Cellular Networks.IEEE Transactions on Wireless Communications, 5 (2006), pp. 301-305. [7] Do T. V. (2011). Solution for A Retrial Queueing Problem in Cellular Networks with the Fractional Guard Channel Policy.Mathematical and Computer Modelling, 53(11โ€“12), pp. 2058-2065. [8] Ramjee R., Towsley D. and Nagarajan R., (1997). On Optimal Call Admission Control in Cellular Networks.Wireless Network, 3(1), March 1997. [9] Neuts M.F. (1981). Matrix Geometric Solutions in Stochastic Model, Johns Hopkins University Press, Baltimore. [10] Tien Van Do, Ram Chakka, An efficient method to compute the rate matrix for retrial queues with large number of servers, Applied Mathematics Letters 23, (2010) 638-643. [11] Tien, D.V. (2013). An Enhanced Algorithm to Solve Multiserver Retrial Queueing Systems with Impatient Customer. Computers & Industrial Engineering, Vol 65, No 4, pp. 719-728. [12] Kulshrestha R, Agarwal A, Shruti (2019). An adaptive fractional guard channel based CAC scheme for heterogeneous traf๏ฌc in wireless cellular networks. In: Proceedings of IEEE 6th international conference in computing for sustainable global development, Delhi, India, pp 1066โ€“1070. [13] Do T.V. (2010). An Efficient Computation Algorithm for A Multiserver Feedback retrial Queue With A Large Queueing Capacity, Applied Mathematical Modelling, 34(8), pp. 2272-2278. [14] https://www.wolfram.com/mathematica/
  • 16. International Journal of Computer Networks & Communications (IJCNC) Vol.13, No.4, July 2021 100 AUTHORS Dang Thanh Chuong obtained his doctorate in Mathematical Foundation for Computers and Computing Systems in 2014 from the Institute of Information Technology, Vietnam Academy of Science and Technology (VAST). He has published over 20 research papers. His research interests are in the fields of all-optical networks with emphasis on packet/burst-based switching, Contention Resolution, and Quality of Service; Queueing Theory and Retrial Queue. Email: dtchuong@hueuni.edu.vn. Hoa Ly Cuong procuring MSc in Computer Science in 2017 from the Hue University of Science, Hue University. The areas he has engaged in comprise Queueing Theory and Wireless Networks. Email: hlcuong90@gmail.com. Pham TrungDuc In 2010, he graduated with a Bachelor of Information Technology from University of Sciences, Hue University. In 2012, he received a Master's degree in Computer Science from Hue University of Sciences. Currently, he is a PhD student at Hue University of Sciences, Hue University (from December 2016 to the present). Research fields: OBS network, QoS differentiation, scheduling admission control, QoS improvement, QoS provisioning. Email: phamtrungduc@hueuni.edu.vn Duong Duc Hung is a Technical Editor at the HU Journal of Science, Hue University, Vietnam. His main research topics are Computer Networks and Communications; Text Mining. Email: ddhung@hueuni.edu.vn