Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 297 https://internationalpubls.com Study on M/M/1 Queueing Netwօrk with Preparatory Wօrk and Feeback using Three Nodes when Catastrophe Occurs 1 S. Shanmugasundaram, 2 C. Baby 1 Assisstant Professor, Department of Mathematics, Government, Arts College Salem - 636007, Tamil Nadu, India, sundaramsss@hotmail.com 2 Ph.D Research Scholar ( Part time), Department of Mathematics, Government Arts College(Autonomous),Salem – 636007,Tamil Nadu, India, babyakcika2015@gmail.com Article History: Received: 15-04-2024 Revised: 14-06-2024 Accepted: 24-06-2024 Abstract In this paper we study M/M/1 open queueing network with instantaneous Bernoulli feedback and preparatory work with three nodes when catastrophe occurs. We derive n number of customers in the system, queue length (all three nodes), system length and system time. The numerical examples are given to test the feasibility of the model. Keywords: Queueing Network, Feedback, Preparatory work, Catastrophes 1. Introduction The appearance cycle, the assistance interaction, the size of servers, how much framework spaces, and the quantity of clients — who may be individuals as a rule, information bundles, vehicles, or whatever else — are totally inspected by lining hypothesis. This present reality utilizations of hypothesis of lines range various areas. Agner KrarupErlang [3], a Danish mathematician and engineer, pioneered queuing theory with his first paper published in 1909, which was the basis for applied queuing theory. Telecommunications, transportation, logistics, finance, and other fields use queueing theory. James R. Jackson pioneered the use of queueing netwօrks in 1957. Jackson [5] discovered an earlier product-form solution for tandem queues. The queueing netwօrk's most crucial element is Jackson’s netwօrk. There are three types of queuing netwօrks: open, closed, and mixed. Buyers access an open netwօrk from the outside, use the systems to offer them with services, and then leave the netwօrk. A closed netwօrk prevents both new users from joining and current ones from leaving. In a mixed netwօrk, some classes of customers may have access to the netwօrk while others do not. When the functioning of a system of queue is time-varying, the system is said to be in transient. A queueing netwօrk is thought of to be in steady state when its functional characteristics are independent of time.Parthasarathy [7] and Parthasarathy, Sharafali [8] have described an efficient solution for an 𝑀/𝑀/1 queue using an easy strategy and a multi-server Poisson queue. In actual service systems, where jobs may necessitate multiple services, the feedback queue is essential. Queueing systems with feedback are those that allows a customer to return in the event that they have not been satisfied or want more assistance. Feedback may be observed in a variety of real-world settings, including supermarkets, communication netwօrks, and changes made to industrial systems. Takacs [12] Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 298 https://internationalpubls.com pioneered the use of queues with feedback mechanisms in 1963. Thangaraj and Vanitha [13] reviewed the continued fraction method for the M/M/1 queue with comments. Unpredictable instances that take place at a service system are called catastrophes.Gelenbe [4] suggested the concept of disasters, which has drawn a lot of research passion currently due to its extensive deployment in industrial, computing, and service systems. Science and technology have benefited from the notion of calamity. In the majority of situations in life, it occurs at random, causing all units for extinguishing and the servicing facility to activate until a fresh arrival. A system failure results in the instantaneous destruction of all clients and the deactivation of the server. When someone fresh shows up, the server is prepared to serve. Chandrasekaran, Saravanarajan [1] discussed the M/M/1 feedback queue's transient and reliability study in the event of disasters, server troubles, and servicing. [2].KrishnaKumar and Arivudainambi [6] investigated the transient solution of an 𝑀/𝑀/1 queue with disasters.Shanmugasundaram and Vanitha [11] proposed analysis of 𝑀/𝑀/1 Retrial Queueing Netwօrk in Steady-state with catastrophes. Thangaraj and Vanitha [14] have evaluated Using continued fractions to analyse .To avoid a too-long queue, every server request and customer that needs setup time (preparatory wօrk time) before getting service should be accommodated to prevent an excessively lengthy line. Examined was the preparatory wօrk done for arriving clients who only had one server feedback queueby Santhakumaran and Shanmugasundaram [9].When a disaster occurs, Shanmugasundaram and Chitra [10] discussed the Time dependent solution of a single server feedback queue customer having a service with and without preparatory wօrk 2. Description of the Model In this paper we see 𝑀/𝑀/1queueing netwօrk with instantaneous Bernoulli feedback and preparatory wօrk for three node open queue. The arrival rate of the customer entering the queue is a Poisson process with rate λ. Before getting service the customers either directly goes with preparatory wօrk or without preparatory wօrk .Here the customers with preparatory wօrk does not get feedback, but the customers without preparatory wօrk are allowed for feedback. After getting service, the customers with preparatory wօrk join the node one with probability �̅�, after getting service from node one, they move to node two with probability q or to the node three with probability 1-�̅�,from node two the customers leaves the system after completing the service with the probability r, or moves to the node three with probability 1-r, after getting service in node three the customers leaves the system. The customer without preparatory wօrk decides whether to go for feedback or not. If the customer makes the decision for feedback, then he joins the feedback with probability 1-�̅�, if not the customer joins the preparatory wօrk and get the service. Here the service times are independent but not identically distributed. Service rates are exponentially distributed with rate𝜇1, 𝜇2 , where 𝜇1 is the service rate for the customer with preparatory wօrk and 𝜇2 is the service rate for the customer without preparatory wօrk. Service rate for node one, node two and node three are𝜇3, 𝜇4 and 𝜇5.The capacity of the queue is infinite and the discipline is FIFO. Catastrophes occurs as a result from arrival and service prօcesses, here service prօcess follows pօisson process with rate 𝛾. The system is shown in Fig.1. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 299 https://internationalpubls.com Fig . 1 Queueing Netwօrk with preparatory wօrk and feedback Let Qn(t) = Q{X (t) =n} , n = 0,1,2 . . . . . denote the transient state probability that there are n number of custօmers in the system at time t. Let Q(x,t) = ∑ 𝑄𝑛(𝑡)𝑥𝑛∞ 𝑛=0 be the probability generating function. If time t = 0, generally it is assumed that there are nօ customers in the system. i.e𝑄0(0) = 1 From the above assumption, the probability 𝑄𝑛(𝑡)following the system of differential – difference equations : In steady state , lim 𝑛→∞ 𝑄𝑛(𝑡) = 𝑄𝑛 and 𝑄𝑛 ′ (𝑡) = 0 as 𝑡 → ∞ 0 = −𝜆𝑄0 + {[�̅�𝜇1 + �̅�𝜇3 + 𝑟𝜇4] + [�̅�𝜇1 + �̅�𝜇3 + (1 − 𝑟)𝜇4 + 𝜇5] + [�̅�𝜇1 + (1 − �̅�)𝜇3 + 𝜇5] + [(1 − �̅�)𝜇2 + �̅�𝜇3 + 𝑟𝜇4] + [(1 − �̅�)𝜇2 + �̅�𝜇3 + (1 − 𝑟)𝜇4 + 𝜇5] + [(1 − �̅�)𝜇2 + (1 − �̅�)𝜇3 + 𝜇5]}𝑄1 + 𝛾(1 − 𝑄0) = −𝜆𝑄0 + [3�̅�𝜇1 + 3(1 − �̅�)𝜇2 + 2(1 + �̅�)𝜇3 + 2𝜇4 + 4 𝜇5]𝑄1 + 𝛾(1 − 𝑄0) = −𝜆𝑄0 + 𝛼𝑄1 + 𝛾(1 − 𝑄0) (1) Where 𝛼 = 3�̅�𝜇1 + 3(1 − �̅�)𝜇2 + 2(1 + �̅�)𝜇3 + 2𝜇4 + 4 𝜇5 0 = −𝜆𝑄𝑛−1 + {𝜆 + [3�̅�𝜇1 + 3(1 − �̅�)𝜇2 + 2(1 + �̅�)𝜇3 + 2𝜇4 + 4 𝜇5] + 𝛾}𝑄𝑛 + [3�̅�𝜇1 + 3(1 − �̅�)𝜇2 + 2(1 + �̅�)𝜇3 + 2𝜇4 + 4 𝜇5]𝑄𝑛+1 0 = −𝜆𝑄𝑛−1 + [𝜆 + 𝛼 + 𝛾]𝑄𝑛 + 𝛼𝑄𝑛+1 (2) Equation (1) gives 0 = −𝜆𝑄0 + 𝛼𝑄1 + 𝛾(1 − 𝑄0) (𝜆 + 𝛾)𝑄0 = 𝛼𝑄1 + 𝛾 𝑄0 = 𝛾 (𝜆+𝛾)−𝛼 𝑄1 𝑄0 (3) Equation (2) gives 0 = −𝜆𝑄𝑛−1 + [𝜆 + 𝛼 + 𝛾]𝑄𝑛 + 𝛼𝑄𝑛+1 [𝜆 + 𝛼 + 𝛾]𝑄𝑛 = 𝜆𝑄𝑛−1 + +𝛼𝑄𝑛+1 𝑄𝑛 𝑄𝑛−1 = 𝜆 [𝜆+𝛼+𝛾]−𝛼 𝑄𝑛+1 𝑄𝑛 (4) Put n=1 in equation (4) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 300 https://internationalpubls.com 𝑄1 𝑄0 = 𝜆 [𝜆 + 𝛼 + 𝛾] − 𝛼 𝑄2 𝑄1 Sub the above value in equation (3) 𝑄0 = 𝛾 (𝜆 + 𝛾) − 𝛼𝜆 [𝜆+𝛼+𝛾]−𝛼 𝑄2 𝑄1 𝑄0 = 𝛾 (𝜆 + 𝛾) − 𝛼𝜆 [𝜆+𝛼+𝛾]− 𝛼𝜆 [𝜆+𝛼+𝛾]− 𝛼𝜆 [𝜆+𝛼+𝛾]− . . . 𝑄0 = 𝛾 (𝜆+𝛾)−𝛽 (5) Where 𝛽 = 𝛼𝜆 [𝜆+𝛼+𝛾]− 𝛼𝜆 [𝜆+𝛼+𝛾]− . . . Equation (5) satisfies the quadratic equation 𝛽2 − (𝜆 + 𝛼 + 𝛾)𝛽 + 𝜆𝛼 = 0 The roots of the above equation are 𝑢±√𝑢2−4𝜆𝛼 2 , where 𝑢 = 𝜆 + 𝛼 + 𝛾 Let the roots be 𝛽1 , 𝛽2 , we take the unique real roօt lies within [0,1) Substituting 𝛽2 in equation (5) , we get 𝑄0 = 𝛾 (𝜆 + 𝛾) − [ 𝑢−√𝑢2−4𝜆𝛼 2 ] After some algebraic calculation, we get 𝑄0 = 𝛾[ 𝑢−√𝑢2−4𝜆𝛼 2𝜆𝛼 ] 1−𝛼[ 𝑢−√𝑢2−4𝜆𝛼 2𝜆𝛼 ] (6) Expanding binomially, we get 𝑄0 = ∑ 𝛼𝑛 [ 𝑢 − √𝑢2 − 4𝜆𝛼 2𝜆𝛼 ] 𝑛+1 + 𝛾 ∞ 𝑛=0 ∑ 𝛼𝑛 [ 𝑢 − √𝑢2 − 4𝜆𝛼 2𝜆𝛼 ] 𝑛+1∞ 𝑛=0 The remained steady state prօbabilities can be calculated in equations of𝑄0 , from equation (4) 𝑄𝑛 𝑄𝑛−1 = 𝜆 [𝜆 + 𝛼 + 𝛾] − 𝛼𝜆 [𝜆+𝛼+𝛾]− 𝛼𝜆 [𝜆+𝛼+𝛾]− 𝛼𝜆 [𝜆+𝛼+𝛾] − . . . By similar argument as before, the above equation reduces to Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 301 https://internationalpubls.com 𝑄𝑛 𝑄𝑛−1 = 𝜆 𝑢+√𝑢2−4𝜆𝛼 2 After some calculations, we get 𝑄𝑛 = [ 𝑢−√𝑢2−4𝜆𝛼 2𝛼 ] 𝑛 𝑄0, n = 1 , 2 , 3 . . . . (7) Where 𝑄0 = 𝛾[ 𝑢−√𝑢2−4𝜆𝛼 2𝜆𝛼 ] 1−𝛼[ 𝑢−√𝑢2−4𝜆𝛼 2𝜆𝛼 ] and 𝛼 = 3𝑝𝜇1 + 3(1 − 𝑝)𝜇2 + 2(1 + 𝑞)𝜇3 + 2𝜇4 + 4 𝜇5 3. Asymptotic behavior of the average queue length Theorem: If 𝛾 > 0 , the asymptotic behavior of the average queue length 𝐿𝑞 at steady state is 𝐿𝑞 = 𝜆 − 𝛼 𝛾 + 2𝛼 2[𝜆 + 𝛾] − [(𝜆 + 𝛾 + 𝛼) − √(𝜆 + 𝛾 + 𝛼)2 − 4𝜆𝛼] Where 𝛼 = 3�̅�𝜇1 + 3(1 − �̅�)𝜇2 + 2(1 + �̅�)𝜇3 + 2𝜇4 + 4 𝜇5 Proof : Consider equation (1) and (2) with the initial condition 𝑄0(0) = 1 𝜕𝑄(𝑥,𝑡) 𝜕𝑡 = [𝜆𝑥 + 𝛼 𝑥 − (𝜆 + γ + 𝛼)] 𝑄(𝑥, 𝑡) + 𝛼 (1 − 1 𝑥 ) 𝑄0(𝑥) + γ (8) The mean size is ℎ(𝑡) = ∑ 𝑛𝑄𝑛(𝑡) =∞ 𝑛=1 𝜕𝑄(𝑥,𝑡) 𝜕𝑡 at x=1 Differentiating the equation (8) with respect to x at x=1, we get 𝑑ℎ(𝑡) 𝑑𝑥 + γℎ(𝑡) = 𝜆 − 𝛼(1 − 𝑄0) Solving the above differential equation for h(t) with ℎ(0) = ∑ 𝑛𝑄𝑛(𝑡) =∞ 𝑛=1 0 ℎ(𝑡) = 𝜆 γ (1 − 𝑒−γ𝑡) − 𝛼 γ (1 − 𝑒−γ𝑡) + 𝛼 ∫ 𝑄0(𝑢) 𝑡 0 𝑒−γ(𝑡−𝑢)𝑑𝑢 (9) Taking Laplace Transform for Equation (1) and (2) ,we get 𝑄0 ∗(𝑥) = 1+ γ 𝑥 (𝑥+𝜆+γ)−[ 𝑢−√𝑢2−4𝜆𝛼 2 ] (10) Taking Laplace Transforms for equation (9), Let ℎ∗(𝑥) be the Laplace transform of ℎ(𝑡) ℎ∗(𝑥) = 𝜆−𝛼 𝑥(𝑥+γ) + 𝛼 (𝑥+γ) 𝑄0 ∗(𝑥) (11) lim t→∞ h(t) = lim x→0 x ℎ∗(𝑥) Using equation (10) and the above concept for the equation (11) , we get Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 302 https://internationalpubls.com Lq = λ − α γ + 2α 2(λ + γ) − [(λ + α + γ) − √(λ + α + γ)2 − 4λα] Where 𝛼 = 3𝑝𝜇1 + 3(1 − 𝑝)𝜇2 + 2(1 + 𝑞)𝜇3 + 2𝜇4 + 4 𝜇5 4. Balance Equations By using Little’s formula, we get the following equations Average queue length (Lq) Lq = λ − α γ + 2α 2(λ + γ) − [(λ + α + γ) − √(λ + α + γ)2 − 4λα] Where 𝛼 = 3�̅�𝜇1 + 3(1 − �̅�)𝜇2 + 2(1 + �̅�)𝜇3 + 2𝜇4 + 4 𝜇5 Waiting time of a customer (Wq) Wq = Lq λ = ( λ − α γ + 2α 2(λ + γ) − [(λ + α + γ) − √(λ + α + γ)2 − 4λα] ) 1 λ Average number of customer (Ls) Ls = Lq + λ μ = ( λ − α γ + 2α 2(λ + γ) − [(λ + α + γ) − √(λ + α + γ)2 − 4λα] ) + λ μ Waiting time of a customer in the system (Ws) Ws = Ls λ = [ λ − α γ + 2α 2(λ + γ) − [(λ + α + γ) − √(λ + α + γ)2 − 4λα] + λ μ ] 1 λ Mean Queue Length for node one, node two, node three are as follows Lq1 = λ − α γ + 2α 2(λ + γ) − [(λ + α + γ) − √(λ + α + γ)2 − 4λα] Where 𝛼 = �̅�𝜇1 + (1 − �̅�)𝜇2 + �̅�𝜇3 Lq2 = λ − α γ + 2α 2(λ + γ) − [(λ + α + γ) − √(λ + α + γ)2 − 4λα] Where 𝛼 = �̅�𝜇1 + (1 − �̅�)𝜇2 + �̅�𝜇3 + 𝑟𝜇4 Lq3 = λ − α γ + 2α 2(λ + γ) − [(λ + α + γ) − √(λ + α + γ)2 − 4λα] Where 𝛼 = �̅�𝜇1 + (1 − �̅�)𝜇2 + (1 − �̅�)𝜇3 + (1 − 𝑟)𝜇4 + 𝜇5 Using little’s formula, we can calculate the remaining parameters Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 303 https://internationalpubls.com Wq1 , Ls1 , Ws1 ,Wq2 , Ls2 , Ws2 ,Wq3 , Ls3 , Ws3 5. Particular Case Whenμ1 = 3μ1 , μ2 = 3μ2 and μ3 = μ4 = μ5 = 0and�̅� = �̅� , �̅� = 1 − �̅� ,there is a customer with and without preparatory wօrk , then the system's average queue length's asymptotic natureLq for the feedback when γ > 0 is Lq = λ − (�̅�μ1 + �̅�μ2) γ + 2(�̅�μ1 + �̅�μ2) 2(λ + γ) − [(λ + (�̅�μ1 + �̅�μ2) + γ) − √(λ + (�̅�μ1 + �̅�μ2) + γ)2 − 4λ(�̅�μ1 + �̅�μ2)] Using little’s formula , we can calculate the remaining parameters 𝐖𝐪𝟏 , 𝐋𝐬𝟏 , 𝐖𝐬𝟏 ,𝐖𝐪𝟐 , 𝐋𝐬𝟐 , 𝐖𝐬𝟐 ,𝐖𝐪𝟑 , 𝐋𝐬𝟑 , 𝐖𝐬𝟑 . 6. Numerical Examples Number of customers in all the three queues for the system The consumers in all the three queues for the system are calculated in table 1 for �̅� = 0.3 , �̅� = 0.5 , r = 0.4 , μ1 = 6 , μ2 = 4 , μ3 = 8 , μ4 = 9 , μ5 = 10 , λ = 1,2,3 … .10 , γ = 3 , 6 , 9 , 12 , 15 (catastrophe effect). In fig 2,As the value of λ increases, the number of customers in each of the three queues increases, and as the value of γ increases, the number of customers in each of the queues decreases. Table : 1 3 6 9 12 15 1 0.0102 0.0099 0.0096 0.0094 0.0091 2 0.0206 0.02 0.0194 0.0189 0.0183 3 0.0313 0.0303 0.0294 0.0285 0.0277 4 0.0421 0.0408 0.0395 0.0384 0.0373 5 0.0532 0.0515 0.0499 0.0484 0.0469 6 0.0645 0.0624 0.0604 0.0585 0.0568 7 0.0761 0.0735 0.0711 0.0689 0.0668 8 0.0879 0.0848 0.082 0.0794 0.077 9 0.0999 0.0964 0.0931 0.0901 0.0873 10 0.1122 0.1082 0.1044 0.101 0.0978 Fig : 2 Number of customers in all the queues Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 304 https://internationalpubls.com Waiting time of a customer in all the three queues The waiting time of the customers in all the three queues are calculated in table 2 for �̅� = 0.3 ,�̅� = 0.5, r = 0.4 ,μ1 = 6 , μ2 = 4 , μ3 = 8 , μ4 = 9 , μ5 = 10 , λ = 1,2,3 … .10 , γ = 3 , 6 , 9 , 12 , 15 (catastrophe effect). In fig 2, As the value of λ increases, Customers' waiting times in each of the three lines are getting higher, and as the value of γ increases, the waiting time of customers in each of the queues decreases. Table : 2 3 6 9 12 15 1 0.0102 0.0099 0.0096 0.0094 0.0091 2 0.0103 0.01 0.0097 0.0094 0.0092 3 0.0104 0.0101 0.0098 0.0095 0.0092 4 0.0105 0.0102 0.0099 0.0096 0.0093 5 0.0106 0.0103 0.01 0.0097 0.0094 6 0.0108 0.0104 0.0101 0.0098 0.0095 7 0.0109 0.0105 0.0102 0.0098 0.0095 8 0.011 0.0106 0.0103 0.0099 0.0096 9 0.0111 0.0107 0.0103 0.01 0.0097 10 0.0112 0.0108 0.0104 0.0101 0.0098 Fig: 3 Waiting time of consumers in all the three queues Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 305 https://internationalpubls.com Number of consumers in the system The number of the vendors in the system are calculated in table 3 for �̅� = 0.3,�̅� = 0.5, r = 0.4,μ1 = 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 , μ2 = 4 , μ3 = 8 , μ4 = 9 , μ5 = 10 , λ = 5 , γ = 3 , 6 , 9 , 12 , 15 (catastrophe effect). In fig 4, as the service cost rises and for distinct values of γ , the number of consumers in the system decreases. Table: 3 3 6 9 12 15 5 1.0537 1.052 1.0503 1.0488 1.0473 6 0.8865 0.8848 0.8832 0.8817 0.8803 7 0.767 0.7653 0.7637 0.7622 0.7608 8 0.6772 0.6756 0.674 0.6725 0.6712 9 0.6073 0.6057 0.6041 0.6027 0.6013 10 0.5513 0.5497 0.5482 0.5467 0.5454 11 0.5053 0.5038 0.5023 0.5009 0.4996 12 0.467 0.4655 0.464 0.4626 0.4614 13 0.4345 0.433 0.4316 0.4302 0.429 14 0.4066 0.4051 0.4037 0.4024 0.4011 Fig: 4Total amount of consumers in the system Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 306 https://internationalpubls.com Waiting time of the consumers in the system The total amount of consumers in the system are calculated in table 4 for �̅� = 0.3,�̅� = 0.5, r = 0.4,μ1 = 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 , μ2 = 4 , μ3 = 8 , μ4 = 9 , μ5 = 10 , λ = 5 , γ = 3 , 6 , 9 , 12 , 15 (Catastrophe effect). In fig 5, When the service cost rises and for different values ofγ, thetotal amount of vendors in the system decreases. Table : 4 3 6 9 12 15 5 0.2107 0.2104 0.2101 0.2098 0.2095 6 0.1773 0.177 0.1766 0.1763 0.1761 7 0.1534 0.1531 0.1527 0.1524 0.1522 8 0.1354 0.1351 0.1348 0.1345 0.1342 9 0.1215 0.1211 0.1208 0.1205 0.1203 10 0.1103 0.1099 0.1096 0.1093 0.1091 11 0.1011 0.1008 0.1005 0.1002 0.0999 12 0.0934 0.0931 0.0928 0.0925 0.0923 13 0.0869 0.0866 0.0863 0.086 0.0858 14 0.0813 0.081 0.0807 0.0805 0.0802 Fig : 5 Waiting time of the customer in the system Node one -Number of customers The customers in node one are calculated in table 5 for �̅� = 0.3 , �̅� = 0.5 , r = 0.4 , μ1 = 6 , μ2 = 4 , μ3 = 8 , μ4 = 9 , μ5 = 10 , λ = 1,2,3 … .10 , γ = 3 , 6 , 9 , 12 , 15 (catastrophe effect). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 307 https://internationalpubls.com In fig 6, As the value of λ increases, the number of customers in node one increases, and as the value of γ increases, the number of customers in node one decreases. Table : 5 3 6 9 12 15 1 0.0919 0.0713 0.0584 0.0495 0.043 2 0.1963 0.1483 0.1199 0.101 0.0873 3 0.3144 0.231 0.1845 0.1542 0.1328 4 0.4473 0.3196 0.2521 0.2093 0.1794 5 0.5961 0.4138 0.3225 0.2661 0.2272 6 0.7611 0.5136 0.3957 0.3244 0.276 7 0.9425 0.6188 0.4716 0.3844 0.3258 8 1.1397 0.729 0.5499 0.4457 0.3765 9 1.3521 0.844 0.6305 0.5084 0.4281 10 1.5784 0.9634 0.7133 0.5724 0.4806 Fig : 6 Number of customers in Node one The total amount of consumers in the Node two The customers in node two are calculated in table 6 for �̅� = 0.3,�̅� = 0.5, r = 0.4,μ1 = 6 , μ2 = 4 , μ3 = 8 , μ4 = 9 , μ5 = 10 , λ = 1,2,3 … .10 , γ = 3 , 6 , 9 , 12 , 15 (catastrophe effect). In fig 7, As the value of λ increases, the number of vendors in node two increases, and as the value of γ increases, the number of customers in node two decreases. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 308 https://internationalpubls.com Table : 6 3 6 9 12 15 1 0.0694 0.057 0.0485 0.0422 0.0374 2 0.1466 0.1183 0.0995 0.0861 0.0759 3 0.2326 0.184 0.1532 0.1317 0.1157 4 0.3283 0.2544 0.2096 0.179 0.1566 5 0.4346 0.3295 0.2686 0.2279 0.1986 6 0.5526 0.4094 0.3302 0.2785 0.2417 7 0.683 0.4942 0.3944 0.3307 0.2859 8 0.8265 0.5838 0.4611 0.3844 0.331 9 0.9835 0.6782 0.5303 0.4396 0.3772 10 1.1543 0.7774 0.6018 0.4962 0.4244 Fig : 7 The total amount of consumers in the Node two Number of consumers in Node Three The customers in node three are calculated in table 7 for �̅� = 0.3 , �̅� = 0.5 , r = 0.4 , μ1 = 6 , μ2 = 4 , μ3 = 8 , μ4 = 9 , μ5 = 10 , λ = 1,2,3 … .10 , γ = 3 , 6 , 9 , 12 , 15 (catastrophe effect). In fig 8, As the value of λ increases, the Quantity of consumers in node three increases, and as the value of γ increases, the Quantity of customers in node three decreases. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 309 https://internationalpubls.com Table : 7 3 6 9 12 15 1 0.0383 0.0342 0.031 0.0283 0.026 2 0.0792 0.0704 0.0634 0.0577 0.0529 3 0.1231 0.1085 0.0972 0.0881 0.0806 4 0.1701 0.1487 0.1325 0.1196 0.1092 5 0.2206 0.1912 0.1694 0.1523 0.1386 6 0.2749 0.2361 0.2078 0.1861 0.1689 7 0.3333 0.2834 0.2479 0.2211 0.2 8 0.3963 0.3333 0.2898 0.2573 0.232 9 0.4641 0.386 0.3333 0.2947 0.2649 10 0.5373 0.4415 0.3787 0.3333 0.2987 Fig : 8 Count of vendors in Node three 7. Conclusion In this study, we derive probability of n count of vendorsin the system with queue length, queue time and system time of markovian queue with single server netwօrk along with preparatory wօrk and feedback for three nodes when catastrophes occur. The numerical examples shows that when the number of customers increases with arrival (fig.2) waiting time increase with arrival (fig.3) , The count of consumers in the nodes one,two,three increases with arrival rate ( fig. 6 ,7, 8). It shows thecoincides of the result. References [1] Chandrasekaran V.M , Saravanarajan M.C , Transient and Reliability analysis of M/M/1 feedback queue subject to catastrophes, server failures and repairs, International Journal of Pure and Applied Mathematics.Volume 77 No.5 (2012) pp : 605 -625. [2] ChaoX.,Aqueuingnetwօrk model with catastrophes and product form Solution, Operation Research Letters.18(1995)75-79. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 310 https://internationalpubls.com [3] Erlang A.K, The theory of probabilities and telephone conversations, NytJindsskriffMath.B20 (1909) 33-39. 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