J. Nig. Soc. Phys. Sci. 5 (2023) 1148 Journal of the Nigerian Society of Physical Sciences An Alleviation of Cloud Congestion Analysis of Fluid Retrial User on Matrix Analytic Method in IoT-based Application K. Nandhini, V. Vidhya∗ Division of Mathematics, Vellore Institute of Technology, Chennai, Tamilnadu, India Abstract Cloud Computing (CC) and Internet of Things (IoT) are upgrowing human intervention to enhance the daily lifestyle. Currently, the heavy loaded traffic congestion is a very big challenge over IoT-based applications. For that purpose, the researchers approached various ways to overcome the congestion mechanism in recent years. Even though, they have futile to acheive the best resource storage accessing capacity expectation other than, Cloud Computing. Data sharing is a key impediment of Cloud Computing as well as Internet of Things. These are the constituent that give rise to the combination of the IoT and cloud computing paradigm as IoT Cloud. Though, preserving the missed data during the execution time is a key factor to indulge the Retrial Queueing Theory (RQT), who is facing issue upon accessing Cloud Service Provider (CSP) enter into virtual pool to preserve the data for reuse. The paper imposes Markov Fluid analysis with Matrix Analytic Method (MAM) allows the data as continuous length of data rather than individual data to avoid the congestion. The virtual orbit queue follow constant retrial rate discipline, that is, head of the orbital users makes attempt to occupy the server are assumed to be independent and identically distributed (i.i.d). Steady-state expression presented to study the behaviour of congestion. An illustrative analysis is produced to gain deep perception into the system model. DOI:10.46481/jnsps.2023.1148 Keywords: Retrial queue, Stationary distribution, Congestion, Markov Fluid queue, Matrix analytic Method, IoT cloud computing Article History : Received: 26 October 2022 Received in revised form: 13 January 2023 Accepted for publication: 18 January 2023 Published: 04 April 2023 c© 2023 The Author(s). Published by the Nigerian Society of Physical Sciences under the terms of the Creative Commons Attribution 4.0 International license (https://creativecommons.org/licenses/by/4.0). Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Communicated by: S. Fadugba 1. Introduction The rising prevalence of cloud users and their anticipations has resulted in an incredible growth in the state-of-the-art of cloud computing approaches in recent years. A key primary characteristics of cloud computing and the IoT environment is better communication. The cloud is a stand-alone platform that enables users to access cloud data and resources via the Inter- net from any location or device. Similarly, IoT devices may ∗Corresponding author tel. no: +91 9791020444 Email address: vidhya.v@vit.ac.in (V. Vidhya) access data and resources from any location. The necessary components of IoT technology include cloud properties like re- source pooling, flexibility, and ubiquitous network connectivity. The on-demand and elastic nature of the exertion service en- hance service reliability, which is further enhanced by massive resource pooled storage. All of these strong arguments sup- port the necessity to combine IoT with Cloud Computing stan- dards. Researchers have worked on numerous projects in this field. The potential and stigma associated with the bifurcation of the IoT and cloud were extensively investigated by [15] and [16]. In order to govern and monitor the complex cloud infras- tructure, [25] conducted a survey on the cloud integrated IoT 1 Nandhini & Vidhya / J. Nig. Soc. Phys. Sci. 5 (2023) 1148 2 dependent supervisory control and data collection security sys- tems. The premium approach was created by [26] and is based on the amount of time the user data should wait whilst being uploaded to the cloud. Cloud storage is a critical aspect of cloud-based technolo- gies. As both the user’s data volume and the network bandwidth are increasing at an exponential rate periodically, the device’s capacity cannot keep up with the user’s requirements. People were seeking a brand-new alternative to save their data, and they uncovered cloud storage, which has more powerful floor space. The practice of using cloud storage becomes increas- ingly sophisticated, and in the foreseeable years ahead, users’ information will be held there instead. Cloud storage merely refers to a computing environment where processing and infor- mation information is made available. The cloud storage con- sists of a cluster of distributed file systems, applications, and cooperating network technology. The firms that provide cloud storage services include iCloud, Dropbox, Baidu Cloud, Ama- zon Web Services S3, Azure Cloud, Google Drive, and so forth. All such enterprises are successful in persuading an immense number of users by offering sustainable throughputs and many administrations pertaining to well-known applications. The un- derlying storage in this instance ensures resource pooling for different kinds of data sources. Contributions to [17],[19] as well as several sources of literature where numerous sorts of cloud storage and forthcoming concerns have been emphasised and reviewed. Traffic congestion is the main concern among these prob- lems. When significant amounts of heterogeneous data are trans- ferred to sensor nodes concurrently in a cloud environment, net- work bandwidth congestion ensues, which in turn affects cloud application service or a contravene of the service level agree- ment (SLA). Few background studies on network congestion were previously known; see[20]. As in meantime, there is in- deed a considerable amount of traffic whenever the utilization of cloud data has increased substantially. Now, the usage of data of cloud uses have been raised significantly leads to heavy traffic. The user has access to store data directly in the cloud, and the cloud service provider (CSP) administers the data there. The partitioning, positioning, and management of information do occurs without the user presiding over the actual storage of their data. Due to the heterogeneous dataset with diverse inter- face and various geographical paths to store the data, the CSP independently accesses the data in the cloud, and there is a sig- nificant risk that the user will receive inaccurate transmission. As previously stated, data losses will result from the instances taken. Information is repeated as a consequence of it though. Accessing data from the virtual pool and waiting for CSP for web-based services are crucial components of cloud data repetition. Due to massive infrastructure and increasing num- ber of cloud primary user and repeated user, collision among the user have been increased. New collision prevention tech- niques are developing in response to environmental changes. Fluid Cloud model is the ongoing model used to freely opt the CSP providing accessible data into clusters omitting the indi- vidual performance and reduce the blocking of Network band- width congestion, see [21]. Therefore, we proposed the model of Markov fluid queue (MFQ) approach to simplify the Matrix Analytic Method based on Retrial queueing theory to validate the result. There is some hope for effective background traffic produc- tion in mathematical representations of traffic congestion. The idea behind abstraction is that it aggregates activity requiring less computational work. [22] established the practise of fluid queue analysis in Markovian environments. This research fo- cuses on the distinctive elements of network behaviour, like the buffer. Use of important sampling is an additional intrigu- ing method for quick stochastic modelling and simulation. e.g. [23]; nevertheless, given that it focuses on its quick estimation of statistics like Probability of packet loss, this is unlikely to suit our goals of effectively creating realistic background traffic. The phenomenon of fluid queues has garnered a lot of attention over the last two decades. Whilst also modelling ”packet trains” as instead of individual packets, networks can be simulated at a lower computing cost. Such fluid queues have been widely regarded as valuable mathematical tools for modelling, for in- stance, packet speech, video systems with or without back- ground data, computer networks, including call admission con- trol, traffic shaping and modelling of Traffic Control Protocol (TCP), and production and inventory control. Refer to [7],[8] . The research presented in this article effectively employs the traffic model used in the previous two articles, where a traffic flow is defined by a piecewise-constant rate function. According to [24], an effort with a specific intent analogous to our own has been made. In order to demonstrate how to make packets and the fluid model work together, a simulation model that is packet-oriented and is based on SDE was incorporated. As a packet travels through a region characterised by the fluid approach, the fluid model was utilised to estimate the overall queueing delay. Although packet flow data was used to feed the fluid model’s input for the time step, neither the packets themselves nor the fluid in the fluid network directly interacted with the packet streams that passed the fluid networks at the same time. As indicated earlier, fluid models enabled by traditional queu- ing systems make some progress. There is, however, no article on a fluid queue system that is driven over retrial queue back- ground. The analytical theory of fluid models can be provided by studies on the fluid model over various queueing models. We can offer the fluid model more diversity and flexibility in the design and control of input and output rate by putting forth sev- eral retry strategies. For instance, user’s requests are organized into packets and relayed via a network of routers. Requests are blocked unless the router is accessible to send packets to the server for decoding the data. In order to minimize the packet latency, the user request flow must therefore be continuous to neglect the individual effect on system performance over the Retrial Queueing paradigm. The network bandwidth conges- tion describes with repeated transmission using MAM to obtain system stability. The work [1], [2] the author instigated the homogeneity of the fluid queue by explicitly defined the closed form solution. [3], have studied the buffer content distribution by expressing closed form solution of fluid over two distict queues. The study 2 Nandhini & Vidhya / J. Nig. Soc. Phys. Sci. 5 (2023) 1148 3 [4] approaches a catastrophic queueing model in random en- vironment over fluid queue investigates the stability of the sys- tem. Mao et al. [5] and [6] have studied the fluid vacation model expressing Laplacian of the stationary distribution. Kapoor, S., and Dharmaraja, S. [9] have obtained the transient analysis of fluid queue by exploiting the explicit solution of eigenvalue de- composition and subsequently computing the eigenvector in- terms of bisection algorithm. N.J Starreveld, R. Bekker and M. Mandjes [10] approaches an fluid queue with finite buffer along with workload process by martingale method. In [11] encour- aged Markov Modulated arrival process by obtaining steady state probabiliy distribution with phase type service and impa- tient customers. Reader can refer from [12],[13],[14] for further reference of fluid queues This paper makes two significant contributions. First, (i) To present a Matrix analytic method (MAM) based on data repeti- tion phenomena to acquire the data from cloud data warehouse. (ii) to introduce a Markov fluid queue (MFQ) approach-based Retrial queueing mechanism which is used to access the hetero- geneous data from various interfaces. The proposed model (iii) developed modified Bessel function of first kind which is used to show linearly independent solution leads to gradual stability and (iv) Using the fluid queue technique, we show a method- ological benefit that might be used to intuitively get the station- ary probability of the system states using the matrix-analytic method. Finally, by conducting extensive numerical trials, we significantly broaden the stationary performance analyses’ ap- plicability. Following of how the remaining of this work is organized: The more pertinent and related research in the field of fluid flow approach and cloud storage has been discussed. System de- sign is discussed with real-world instance in Section 1.1 The conceptual mathematical framework is thoroughly explained in Section 2 and Section 3. Section 4 presents the experimental setup and provides justifications with practical augmentation. The conclusion of the suggested work is provided in Section 5 with potential improvement. 1.1. System Architecture A real-world instance of an online education cloud system is provided for illustration in Fig 1. The major components of the system including the user interface module (UI), a cloud database contains Virtual Machine Manager (VMM), a cloud broker, a Cloud Information Service (CIS), a cloud server. Here cloud user send and receive the data via User interface mod- ule. The data storage and retrieval process of the data in the cloud server are the primary responsibility of the user interface module. The cloud database contains the volume of informa- tion of cloud users. VMM is responsible for processing the as- signed task, and a cloud broker partition the task into cloudlets but in our case, it is evenly spaced packet trunks where each trunk represents multiple packets, that is, ”fluid packet trunks”, a cloud information service (CIS) corresponds to obtain the list of resources for the assigned task to provide service and vir- tual server to fulfill the request of cloud users in connection for online-learning. Each Virtual Machine receives one of the 16 CPU cores that are available. A 1 TB Hadoop distributed file system storage will be deployed in the interim to increase the I/O effectiveness of the online learning system. Each VM has a preinstalled Linux operating system and ”Moodle” online learning platform with a root file system size of 50GB. Each VM has a 300 GB additional root filesystem mounted in order to retain any lost data and provide an orbital storage pool in case of transmission errors. Thus, a maximum of 8 missing in- formation in VMs can be transmitted to the server to wait for service. To guarantee the quality of service, unsuccessful task are transmitted to a VM server, which can inspect and make the corrupt root filesystem accessible again. If the server is accessi- ble, a failing virtual machine will proceed right away to access the service in accordance with its system log file. In the event that this does not occur, the failing VM is sent to the server’s storage pool to wait for a filesystem check and reconfiguration. When a failed VM does not access the server, the cloud- stack has a 50% (treated as q1) chance to inhibit any failed in- formation due to non-availability of server. The cloudstack may failed to prevent a impatient re-accessed information with prob- ability 50% (treated as p̄1) and cause an error due to excessive congestion of cloud user. In the beginning, all the VM are bootstrapped to speed up the process. Let the arrival rate of the failed VM be stepwise function with parameter λ = 1.2/hr. The failed VM can be re- accessed at an instant if the server is accessible. The service time is exponentially distributed with service rate µ = 2.0/hr if the server is in busy period. On the contrary, the failed infor- mation in VM is sent to storage pool if the server is busy and repeat its service with exponential amount of retrial time with rate θ = 0.9/hr. 2. A framework of Mathematical structure As a way to examine the behaviour of congestion, we take into account a single standby server Markovian queueing mech- anism with retrial users. Figure 2 represents the state transition diagram. The system consist of single server Markov Fluid queue model with constant retrial rate policy. The customer arrive according to the piecewise-constant process and the inter-arrival time dis- tribution follows an i.i.d exponential distribution with parame- ter λ. The service time of a user are i.i.d follows an exponential distribution with parameter µ. If a user encounters a free server upon arrival, this user receives service immediately. If a server are busy during an arbitary user arrival epoch, the user join in virtual buffer and wait until a server becomes available with probability q1 and if the server is not accessible, they again en- ter the so-called orbit after an arbitary amount of time. A user who arrive to the system from orbit is called a retrial user or exit the sytem with complementary probability p̄1, 0 ≤ p̄1 ≤ 1, p̄1 = 1 − p1. 2.1. The process of background queueing model The behaviour of the process under examination can be de- rived as the conventional irreducible continuous-time Markov Chain (CTMC) ζt = {(kt, it)}, t ≥ 0, where kt ∈ Kt signify the 3 Nandhini & Vidhya / J. Nig. Soc. Phys. Sci. 5 (2023) 1148 4 architecture.png Figure 1. Overview of system design of cloud data storage transition diagram 2.png Figure 2. Transition state diagram ζt = {(kt, it )}, t ≥ 0 amount of data packets accumulated in the orbit, Kt ≥ 0; it ∈ It designate the state of the underlying process of QBD process, it ∈ {0, 1}, where, it = 0, the state is in operational mode; it = 1, the state is in non-operational mode. Theorem 1. The infinitesimal generator Q structured as block- tridiagonal matrices of the Markov chain ζt, t ≥ 0 Q =  A0 C . A B C . A B C . A B C . . . . . . .  4 Nandhini & Vidhya / J. Nig. Soc. Phys. Sci. 5 (2023) 1148 5 where A0 = ( −λ λ µ −(λq1 + µ) ) C = ( 0 0 0 λq1 ) A = ( 0 θ 0 θp̄1 ) B = ( −(λ + θ) λ µ −(λq1 + µ + θp̄1) ) Proof. Theorem 2.1 is demonstrated by analysing each transi- tion in the Markov chain ζt, t ≥ 0 during the interval of infin- tesimal length, and by reconfiguring the intensities into block matrices. Here the intensities of the transition assumed to be continuous entity, termed as fluid. Individual users impact will be negli- gible since the flow will be continuous in the system during an interval of an infintesimal length, the non-diagonal blocks are zero matrix, thus the generator Q has the block-tridiagonal structure. The diagonal entries of the 2×2 square block matrix A0, define the intensities of the transition of the chain ζt, t ≥ 0 that does not affect the number of users in the orbit. The intensities of an event are given by −(λq1 + µ) defines the user enter orbit with probability q1 and ends with customers service. The intensities of the former event are defined by the entries of the event that insist arrival of users in the orbit while the latter event defines the service of the user µ. The super-diagonal 2×2 matrix C, defines the intensities of the event are given by λq1, that is, the probability of joining the so- called orbit who finds server busy. The sub-diagonal 2 × 2 ma- trix A, defines the intensities of the event are given by θ, enter into a retrial orbit with the transition probability (1, 0) → (0, 1) and leave with impatience with θp̄1 The diagonal entries of the matrix B, defines the intensities of the event are given by −(λ+θ) that the arrival of user encounters an busy server, enter into a retrial orbit with probability θ and probability of joining the so-called orbit who finds server busy λq1. The intensity of the former event is arrival of users in the orbit while the latter event describes zero Hence the proof 3. Markov Fluid Analysis 1 This section includes the set of governing equation driven by QBD process over fluid retrial queue with non-persistent users. Let x be the buffer occupancy at time ′t′. Let B(t) de- notes the amount of fluid data packets in the buffer at time ′t′. 1The efficacious action of the buffer is defined as η(B(t), K(t), I(t)) = d(B(t)) dt =  r0, (K(t), I(t)) = (k, 0), k ≥ 0 r, (K(t), I(t)) = (k, 1), k ≥ 0 0, B(t) = 0 We know that, r0 ≥ r ≥ 0. The occupancy of the buffer when the server is in non-operational (idle) period is much greater than the buffer occupancy in the operational (busy) period. Clearly, buffer occupancy is linear increasing at the rate of r0, as soon as users begin the driving process and the process is still in progress; buffer occupancy is linear increasing at the rate of r according to the driving process stays in operational period. Also, the buffer occupancy level cannot depleted until the buffer is empty. As soon as the buffer occupancy level varies, the ef- fective request-service rate of fluid remains unstable. In order to ensure the stability condition, define the average drift condi- tion should satisfy d < 0 d = r0 ∞∑ k=1 Fk0 + r ∞∑ k=1 Fk1 < 0 The following set of differential equations that satisfy a quasi- birth and death process for a stationary joint distribution Fki(t) can easily be solved using the conventional approach. Define Fki(t) is the stationary joint distribution when kt ∈ Kt users in the retrial orbit at time ′t′ and it ∈ It represents the affiction of the server r0 dF00(t) dt = −λF00(t) + µF01(t) (1) r0 dFk0(t) dt = −λFk0(t) + µFk1(t) − θFk+1,0(t) (2) r dF01(t) dt =λF00(t) − (λq1 + µ)F01(t) − θF1,0(t) + θp̄1 F11(t) (3) r dF11(t) dt =λq1F01(t) + λF10(t) + θF2,0(t) − (λp1 + µ + θp̄1)F11(t) + θp̄1F21(t) (4) with boundary conditions F00(0) = a1 F01(0) = a2 Fkt,it (0) = (kt, it), (kt, it) ∈ Ω{(kt, 0) ∪ (kt, 1)} where 0 < a1, a2 < 1 The boundary conditions are feasible to the solution since there is linear growth of customers in the background driving process and it has some non-negative probability distributed around the boundary region. Also the buffer gets empty when the buffer occupancy level decreases at the rate of r0. Thus the boundary conditions are satisfied. 3.1. Markov Decision Process In this theoretical evaluation, the steady state analysis of the fluid retrial queue is presented with the detailed study of aver- age buffer occupancy expressed in an closed form solution of stationary distribution and buffer occupancy is determined in the form of modified Bessel function of first kind. The set of differential equations are governed by Quasi Birth-Death pro- cess evaluated interms of infintesimal generator matrix. The 5 Nandhini & Vidhya / J. Nig. Soc. Phys. Sci. 5 (2023) 1148 6 set of matrix quadratic equations defined to provide an explicit solution for joint steady state analytic distribution. Now define F0(t) = {F00(t),F01(t)} Fk(t) = {Fk,0(t),Fk,1(t)}, j = 1, 2, ... Therefore, F (t) = (F0(t),F1(t),F2(t), ...) The set of differential equation can be rewritten with the joint probability density sequence in the matrix form as dF (t) dt Λ = F (t)Q (5) where Λ =  Λ0 . . Λ1 . Λ1 . Λ1 . . . .  Here Λ0 = ( r0 0 0 r0 ) , Λ1 = ( r 0 0 r ) To depict the Laplace transform of the system of matrix so- lutions to provide an analytical interpretation to the stationary analysis of the buffer capacity distribution. F̂ (s) = (F̂0(s),F̂1(s),F̂2(s), . . . ) then the above matrix form is given as F̂ (s)(Q− sΛ) = (−āΛ, 0, 0...), ā = (r0a1, r0a2) The set of differential equations is given by F̂0(s)(A − sΛ0) −F̂1(s)A = −āΛ (6) F̂0(s)C + F̂1(s)(B − sΛ1) + F̂2(s)A = 0 (7) F̂k−1(s)C + F̂k(s)(B − sΛ1) + F̂k+1(s)A = 0 (8) Theorem 2. A necessary sufficient condition for the driving queueing process {Kt = kt, It = it; t ≥ 0} is given by ρ = λq1 (λ+θ) θµp where µp = µ(λ + θ) p̄1 Proof. From the framework of the generator matrix, we know that {Kt = kt, It = it; t ≥ 0} is a Quasi Birth-Death process. Let us define L = A + B + C. L = ( 0 0 0 λq1 ) + ( 0 θ 0 θp̄1 ) + ( −(λ + θ) λ 0 λq1 ) = ( −(λ + θ) (λ + θ) µ −µ ) Clearly, L is a finite integrable matrix. Its steady-state probabil- ity vector defined as π = (π0,π1) satisfies (π0,π1)L = 0 where π0 + π1 = 1 The obtained solution is π0 = µ λ+θ and π1 = λ+θ µ A necessary sufficient condition for the driving queueing process {Kt = kt, It = it; t ≥ 0} is πCe < πAe, where e is a two dimen- sional column vector whose elements are all equal to one. Then using simple expression, we obtained the result ρ = λq1 (λ+θ) θµp where µp = µ(λ + θ) p̄1 Theorem 3. The evaluation of rate matrix is determined in the form of matrix quadratic equation is given by R2(s)A + R(s)(B− sΛ1) + C = 0 yields minimal non-negative solution such as R(s) = ( χ(s) β(s) 0 r(s) ) Proof. We make an assumption for the rate matrix R(s) has the structure is of the 2 × 2 matrix form such that R = ( r11 r12 0 r22 ) From the equation above, we obtain,( r211 r12(r11 + r22) 0 r222 ) ( 0 θ 0 θp̄1 ) + ( r11 r12 0 r22 ) ( −(λ + θ + sr) λ µ −(λq1 + µ + θp̄1) + sr ) + ( 0 0 0 λq1 ) = 0 The set of differential equations are determined as follows: r11(λ + θ + sr) + r12µ = 0 (9) r211θ + r12(r11 + r22)θp̄1 + r11(λq1 + µ + θp̄1 + sr) = 0 (10) r22µ = 0 (11) r222θp̄1 − r22(λq1 + µ + θp̄1 + sr) + λq1 = 0 (12) The solution from Equation 7 expressed as r22 = r(s) = (λq1 + µ + θp̄1 + sr) ± √ (λq1 + µ + θp̄1 + sr)2 − 4θp̄1λq1 2θp̄1 (13) Let the above equation has positive and negative eigenvalues namely,r(s) and r1(s) .Since the root that lies inside the unit disc, we consider r(s) for further reference. Therefore it satis- fies the following recurrsion relation as follows: (λq1 + µ + sr + θp̄1(1 − r)) = µ 1 − r + θp̄1 + sr 1 − r = λq1 r (14) From the equation above, we get, χ(s) = ρ + sr r(s) − λµ θµp (15) where ρ = λq1 (λ+θ+sr) θµp and µp = µ + (λ + θ + sr)p̄1. Using the expression given in Equation 8, the solution yields, β(s) = (λ + θ + sr)ρ µr(s) −λµ λ + θ + sr µθµp (16) 6 Nandhini & Vidhya / J. Nig. Soc. Phys. Sci. 5 (2023) 1148 7 Hence, the rate matrix is given by R(s) =  ρ r(s) − λµ θµp (λ+θ+sr)ρ µr(s) −λµ λ+θ+sr µθµp 0 (λq1 + µ + θp̄1 + sr) ± √ (λq1 +µ+θp̄1 +sr)2−4θp̄1λq1 2θp̄1  = ( χ(s) β(s) 0 r(s) ) ... Rn−1 = ( χ(s)n−1 β(s)(χ(s)n−2 + r(s)n−2 + ∑n−3 i=1 χ(s) ir(s)n−2−i) 0 r(s)n−1 ) In general, Rn = ( χ(s)n β(s)(χ(s)n−1 + r(s)n−1 + ∑n−2 i=1 χ(s) ir(s)n−1−i) 0 r(s)n ) Hence the result is proved Theorem 4. The steady state probability distribution of the buffer occupancy in the driving queueing model gives an an- alytical solution in an Laplace domain determined as F̂k0(s) = F̂00(s)χ(s) n F̂k1(s) = F̂00(s)β(s)(χ(s) n−1 + r(s)n−1 n−2∑ i=0 χ(s)ir(s)n−1−i) + F̂01(s)r(s) n Proof. Without loss of generality, let us assume Fk(s) = Fk−1(s)R(s) can be rewritten as Fk(s) = F0(s)Rk(s) From Equation 6,7, we have =F̂k−1(s)C + F̂k(s)(B − sR) + F̂k+1(s)A =F̂k−1(s)C + F̂k−1(s)R(s)(B − sR) + F̂k−1R 2(s)A =F̂k−1(s){C + R(s)(B − sR) + R 2(s)A} = 0 =F̂0(s)C + F̂1(s)(B − sR) + F̂2(s)A =F̂0(s)C + F̂0(s)R(s)(B − sR) + F̂0(s)R 2(s)A =F̂0(s){C + R(s)(B − sR) + R 2(s)A} = 0 From Equation 6, we get F̂0(s)(A − sR1) −F̂1(s)A = −ā F̂0(s) = ā sR1 − A0 − R(s)A (17) The proof is provided in the Appendix with all sufficient condi- tions. 3.2. Stationary Analysis of Buffer Content Distribution Theorem 5. The stationary distribution of the buffer occupancy level can be expressed analytically to provide explicit solution to the joint steady probability vector[1] F̂ (s) = P(X ≤ x) = ∞∑ k=0 {F̂k0(s) + F̂k1(s)} (18) buffer occupancy.png Figure 3. A Stationary buffer occupancy distribution F(x) against x Proof. Taking Laplace transform on the above equation, we get F̂ (s) = F̂0(s)e1 + F̂0(s) ∞∑ k=1 Rn(s)ek (19) = F̂0(s)e1 + F̂0(s)e(I − R(s)) −1e2 where (I − R(s))−1 = 1 (I −χ(s))(I − r(s)) ( Ir (s) β(s) 0 I −χ(s) ) = ( (Iχ(s))−1 β(s)(I −χ(s))(I − r(s)) −1 0 (I − r(s))−1 ) F̂ (s) = F̂00(s) n−1∑ j=0 χ(s) j + F̂00(s) n−1∑ j=0 χ(s) jβ(s) ∞∑ k=0 r(s)n + F̂01(s)r(s) n = ∞∑ k=0 ( ρe− (λq1 +µ+θp̄1 ) r x jI j(βx)β j x ( r 2θp̄1 ) j − λµ θµp )∗ kF00(s) + β(x) ∗ ∞∑ k=0 ( r 2θp̄1 )k kIk(βx)β j x e− (λq+µ+θp̄1 ) r x ∗ ∞∑ k=0 ( ρe− (λq1 +µ+θp̄1 ) r x jI j(βx)β j x ( r 2θp̄1 ) j − λµ θµp )∗ kF00(s) + F̂01(s) ∗ ∞∑ k=0 ( r 2θp̄1 )k kIk(βx)β j x e− (λq1 +µ+θp̄1 ) r x (20) Hence, evaluation of buffer content distribution are expressed in terms of F̂00(s) and F̂01(s). Therefore the measure of perfor- mance can be given using this explicit analytical solution given in the form of modified Bessel function of first kind. 4. Analysis of Statistical illustration To acquire a holistic view of the model stability and per- formance, this section describe the findings of certain numeri- cal experiments in the aspect of system stability with intrinsic of Markov fluid analysis. We examine the steady-state con- vergence of several measures to their exact theoretical values 7 Nandhini & Vidhya / J. Nig. Soc. Phys. Sci. 5 (2023) 1148 8 of buffer occupancy distribution.png Figure 4. Dependency of buffer occupancy distribution against the variation of parameters where analytical expressions are given in the depiction for the purpose of model validation. Following the model’s validation, we use empirical correlation to conduct a numerical investiga- tion of a few auxiliary measures for the dependence of orbits, arrivals, and service patterns. The simulation is carried out us- ing MATLAB and is based on the continuous event procedure. As a desired set approach, the following preceding param- eters are chosen for the illustrated example of an online edu- cation cloud system shown in Section 1.1: λ = 1.2/hr, µ = 2.0/hr, θ = 0.9/hr, q1 = 0.5, p̄1 = 0.5. The values are se- lected in such a way to guarantee sufficient stability condition Equation 20 holds true. The graph displayed in Figure 3 displays the difference of buffer capacity distribution against time t with retrial rate θ us- ing the appropriate value of θ considered for the purpose of comparison. It is to be noted that the retrial rate θ value in- creases with increases of buffer capacity distribution, particu- larly at the values of buffer capacity t more than 0.4. It is strictly convex montonically increasing function proves stability in re- lation with the dependent variable of λ, µ and θ. In Figure 4 depicts the behaviour of buffer content distribu- tion in three dimensions with variation of the system estimation parameters dependence of λ, µ and θ. In particular, Figure 4(a) explains the variation of buffer content distribution along with service rate µ and retrial rate θ has validate the result. Fig- ure 4(b) depicts the evolution of stationary distribution in three dimension with variation of the arrival λ and retrial rate θ. This figure explains the variation of buffer content distribution in- crease along with increment of the arrival rate λ and retrial rate θ. It is concluded that the numerical illustration provided has validate the result. Figure 4(c) depicts the behaviour of buffer content distribution in three dimensions with variation of the ar- rival λ and service µ. This figure explains the evolution of buffer content distribution is strictly increasing along with increment of the arrival rate λ and service rate µ. It is to be concluded that the numerical illustration provided has validate the result. To illustrate our result, we simulate M/M/1 type queueing system. The main motivation of our simulation study is to learn the behaviour of the performance measure which are associated with the buffer content distribution against the variation of some system parameters. 8 Nandhini & Vidhya / J. Nig. Soc. Phys. Sci. 5 (2023) 1148 9 5. Conclusion In this research, steady-state evaluation of the Markovian queueing model with constant retrial rate with non-persistent users is demonstrated. It is displayed that the service and inter- retrial times are linearly independent of one another. A Markov fluid queue model serves as a foundation for analytic viewpoint. Besides, we demonstrate how an interaction of retrial correla- tion with matrix analytic method allows to optimize the per- formance efficiency by reducing the packet service delay and congestion avoidance. In future, the extended result will con- template to control the congestion using time-inhomogeneous in a finite source retrial queue. 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The proof is built on the analysis of buffer occupancy distribution given by F̂0(s) = a1r0, a2r0 s ( r0 0 0 r0 ) − ( −λ λ µ −(λq + µ) ) − ( χ(s) β(s) 0 r(s) ) ( 0 θ 0 θp̄ ) = (a1r0, a2r0) ( sr0 + λ −(λθχ(s) + θp̄β(s)) −µ sr0 + (λq + µ)θp̄r(s) )−1 F̂00(s) = a1r0(sr0 + λq1 + µ− θp̄1r(s)) + a2r0µ (sr0 + λ)(sr0 + λq1 + µ− θp̄1r(s)) −µ(λ + θχ(s) + θp̄1β(s)) F̂01(s) = a1r0(λ + θχ(s) + θp̄1β(s)) + a2r0(sr0 + λ) (sr0 + λ)(sr0 + λq + µ− θp̄1r(s)) −µ(λ + θχ(s) + θp̄1β(s)) 9 Nandhini & Vidhya / J. Nig. Soc. Phys. Sci. 5 (2023) 1148 10 Upon factorization which leads to F̂00(s) = a1r0(sr0 + λq1 + µ− θp̄1r(s)) + a2r0µ (sr0 + λ)(sr0 + λq1 + µ− θp̄1r(s)) −µ(λ + θχ(s) + θp̄1β(s)) = ∞∑ k=0 ω(s)k{a2σ0µD(s) − a1σ0 sσ0 + λ } (21) Similarly for F̂01(s) F̂01(s) = a1r0(λ + θχ(s) + θp̄β(s)) + a2r0(sr0 + λ) (sr0 + λ)(sr0 + λq + µ− θp̄1r(s)) −µ(λ + θχ(s) + θp̄1β(s)) = (a1r0(λ + θχ(s) + θp̄1β(s))D(s)) + a2r0 sr0 + λq1 + µ− θp̄1r(s) ∞∑ k=0 ω(s)k (22) where ω(s) = µ(λ + θχ(s) + θp̄1β(s)) (sr0 + λ)(sr0 + λq1 + µ− θp̄1r(s)) D(s) = 1 (sr0 + λ)(sr0 + λq1 + µ− θp̄1r(s)) From inversion of (12),(14),(15) with β = 2 √ λθp̄q σ , we have, β(s) = (λ + θ + sr)ρ µr(s) −λµ λ + θ + sr µθµp = e− (λ+θ) r x ( r µ )k xk k! ∗χ(x)∗i ω(s) = µ(λ + θχ(s) + θp̄1β(s)) (sr0 + λ)(sr0 + λq1 + µ− θp̄1r(s)) = µ(λ + θχ(x) + θp̄1β(x)) ∗ ∞∑ i=0 ∞∑ j=0 θp̄i1 ri+2 e− λ r0 xe− (λq1 +µ) r0 x xi i! iIi(βx)βi x ( r 2θp̄1 )ie−( λq1 +µ+θp̄1 r )x χ(s) = ρ + sr r(s) − λµ θµp = 1 θp1 ∞∑ k=0 ( µ p̄1r )k xk−1 (k − 1)! e− (λ+θ) r x ( λµ r ) e− (λ+θ) r x − 1 θp1 ∗ kIk(βx)βk x e− (λq1 +µ+θp1 ) r x ( r 2θp̄1 )k D(s) = 1 (sr0 + λ)(sr0 + λq1 + µ− θp̄1r(s)) = ∞∑ i=0 ∞∑ j=0 θp̄i1 ri+1 e− (λq1 +µ) r0 x xi+ j (i + j)!  ∗ iI j(βx)βi x ( r 2θp̄1 )ie− (λq1 +µ+θp̄1 ) r x From Equation 17, we arrive at the solution in terms of the modified Bessel function of the first kind, which signifies ex- plicitly the exponentially growing function achieved the char- acteristic of the proposed work by significantly increasing the function of the number of packets/signals in the virtual buffer (Retrial) F̂00(s) = ( −a1r0 (sr0 + λ) + a2r0µD(s) ) ∞∑ k=0 ω(s)k = ∞∑ k=0 (ω(s))∗k ∗ ( a2 ∞∑ v=0 ( θp̄1 r0 )ve− (λq1 +µ) r0 x xv v! ∗ vIv(βx)βv x ( r 2θp̄1 )ve− (λq1 +µ+θp̄1 ) r x ) + a1r0(λ + θχ(x) + θp̄1β(x)) ∗ D(x) F̂01(s) =(a1r0(λ + θχ(s) + θp̄1β(s))D(s)) + a2r0 sr0 + λq1 + µ− θp̄1r(s) ∞∑ k=0 ω(s)k = ∞∑ k=0 (ω(s))∗k ∗ ( a2r0µ∗ D(x) − a1e − λ r0 x ) (F̂k0(s),F̂k1(s)) = (F̂00(s),F̂01(s))R n(s) = (F̂00(s),F̂01(s)) ( χ(s)n ∑n−1 i=0 χ(s) iβ(s)r(s)n−1−i 0 r(s)n ) = ( F̂00(s)χ(s)n F̂00(s) ∑n−1 i=0 χ(s) iβ(s)r(s)n−1−i +F̂01(s)r(s)n ) which is simplified in the form of F̂k0(s) = F̂00(s)χ(s) n (23) F̂k1(s) = F̂00(s) n−1∑ i=0 χ(s)iβ(s)r(s)n−1−i + F̂01(s)r(s) n (24) Thus all steady state probabilities are obtained interms of mod- ified Bessel function of first kind to determine the flow of fluid in retrial queue along with impatient customers are observed. 10