Communications on Applied Nonlinear Analysis 

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A Study on Catastrophic Retrial Feedback Queueing Network Model 

with Three Nodes in Steady State 
 

S. Shanmugasundaram1, C. Baby2 
1Associate Professor of Mathematics, Government Arts College (Autonomous), Salem – 636007, Tamil Nadu, India. 

Email: sundaramsss@hotmail.com 
2Ph.D Research Scholar (Part time), Department of Mathematics, Government Arts College (Autonomous), Salem – 

636007, Tamil Nadu, India. Email: babyakcika2015@gmail.com 

 

Article History: 

Received: 19-07-2024 

Revised: 09-09-2024 

Accepted: 29-09-2024 

Abstract 

Queueing networks, which are effectively used for performance analysis of various 

systems, including computer, communications, transportation networks, and 

manufacturing, are discussed in this article. Under the catastrophic condition, we evaluate 

a queueing network with feedback and retrial for single server. In the third node the 

customer can decide whether feedback or leave the system with probabilities t, u and 1-t-

u. Also using Little's formula, we are determining the likelihood of having idle customers 

and determining n number of customers in the system. We are deriving 𝐿𝑞 ,𝐿𝑠, 𝑊𝑞 𝑎𝑛𝑑 𝑊𝑠 

for the above model. The purpose of the numerical example gives the richness of our 

result and the feasibility of this concept. 

Keywords: Queueing Network, Feedback, Retrial, Catastrophes. 

 

1. Introduction 

Queues form when there is a current demand for a certain service that exceeds supply. If the 

customer's requested service is not immediately available, a queue will emerge. Our everyday lives 

involve waiting in line for services; examples include waiting at hotels, at train reservation counters, 

and at banks. Other scenarios involve waiting, such as tasks that require a machine to complete, 

aircraft that must circle in the air before landing and vehicles that must stop at traffic signals, among 

others. Erlang [4], a Danish mathematician and engineer, initially contributed in queueing theory, 

which formed the foundation of the system. When the behaviour of the system depends on time, it is 

said to be in a transient state. When the operational characteristics do not depent on time, they are 

said to be in a steady state. 

1.1 Queueing Network  

A collection of nodes, with each one representing a different service facility, forms a queuing 

network. In 1957, Jackson [7] first proposed the application of queueing networks. For the queueing 

network, Jackson's network [8] has contributed the initial model which is still considering till date. 

Queuing network models have been utilized in a wide range of businesses and environments, 

including airport terminals, healthcare systems, flexible manufacturing and production systems, 

computer networks, and networks for communication. Queuing networks fall into three categories: 

mixed, closed, and open networks. An arrival enters the system get service and leaves the system in 

the open network. In closed networks, the number of users is fixed, and they never leave the system, 

they just switch between queues. Mixed networks allow certain kinds of users in but prohibit others. 

1.2 Retrial Queue 

Retrial queueing models are predicated on the idea that arriving customers do not queue up or move 

the system instantly when the server is crowded. Instead, they visit a virtual space called orbit and 

return later to try their luck. Many computer systems and communication networks may be 

mailto:sundaramsss@hotmail.com
mailto:babyakcika2015@gmail.com


Communications on Applied Nonlinear Analysis 

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mathematically modeled using retrial queues, which are employed with great effectiveness in these 

applications. Kosten [9] initially presents the Retrial queue in 1947. Falin [5] examined the key 

results and methods of retrial queues theory in a short discussion. Shanmugasundaram et al. [12] 

examined a feedback retrial queueing network consisting of three nodes under catastrophic 

circumstances. 

1.3 Feedback Queue 

In modern service systems, where specific activities may necessitate repeated services, the feedback 

queue is crucial. Feedback queues are queuing systems that allow customers to return for extra 

services or because they are dissatisfied with the service they received. Numerous real-world 

scenarios, including the redesign of manufacturing systems, communication networks, and 

supermarkets, demonstrate the feedback phenomenon. Takacs introduced the feedback mechanism in 

1963 [15]. Working vacations, server breakdowns, and single-server queues with instant feedback 

have all been taken into consideration by Varalakshmi et al. [17]. 

1.4 Catastrophe 

The evolution of networks and communication systems has resulted in increasing attention to 

queueing systems with disasters. The subject of disaster queues has been extensively researched by 

numerous researchers over the past 20 years. Every customer (including the one in service) must exit 

the system in the case of a complete disaster. There are numerous applications for these kinds of 

queueing systems. For instance, every customer are required to vacate the premises when an ATM 

defects in a bank. Moreover, viruses have the ability to erase all stored data from computer networks 

during a clearing process. 

Gelenbe [6] introduced the concept of catastrophes, Furthermore, Chao [2] is motivated to assess the 

migration process in the context of disasters and virus-infected computer networks. Krishna Kumar 

et al. [10] considered the transient solution of the system size in the queuing model with the 

possibility of disasters in a service station. Shanmugasundaram et al. [13] focused on a time-

dependent solution of an arrival of customer with retrial customers and providing feedback to non-

retrial customers with disasters.  

Thangaraj et al [16] studied the analytically obtained transient solution using continuous fractions of 

the feedback queue under catastrophic condition. Shanmugasundaram et al. [14] have studied a 

steady-state single server retrial queueing network with catastrophes. Using the continuous fractions 

approach, Chandrasekaran et al. [3] have examined the transient and reliable nature of the M/M/1 

feedback queue in the face of disasters, server outages, and repairs. The past few years, investigation 

have been done in disaster-prone queueing systems by Chowdhury et al. [1] and Kumar et al. [11] 

2. Description of The Model 

An investigation is conducted on a queueing network consisting of three nodes, incorporating retrial, 

feedback, and catastrophe. External customers access the system through a Poisson process with a 

rate of arrival 𝜆. If the server is not busy, there is a high likelihood that incoming customers will 

enter the queue with probability 𝑠. Customers have a likelihood of 1 − 𝑠 to join the orbit when the 

server is occupied. It is considered that the capacity is infinite, and that the only person allowed to 

get service on a first-come, first-served basis is the one at the front of the line.  

The customer returns to the orbit, if the server is busy. This procedure persists until the customer 

observes that the server on node one is idle. During the retrial, both requested and required services 

are provided. If the server is busy, customers are required to wait in an endless circle and repeat their 



Communications on Applied Nonlinear Analysis 

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request after an exponential rate. After receiving service at node one, they proceed to node two and 

then to node three. Here, the customer chooses whether or not to request feedback.  

After receiving service in node three with probability1 − 𝑡 − 𝑢, if the customer is happy with the 

service, they exit the system. If the customer is dissatisfied in node three, they will either travel to 

node two for instant feedback with a probability of t or to node one for delayed feedback with a 

probability of u. Every node runs on an M/M/1 basis. The service rate for all the three nodes are 

𝜇1 ,𝜇2 𝑎𝑛𝑑 𝜇3 and are dispersed exponentially. The arrival and service processes, which are followed 

by the poisson process with rate γ, are the sources of catastrophes.  

The server is deactivated, and all available customers are immediately destroyed in the event of a 

catastrophic system catastrophe. When someone arrives, the server is ready to serve. 

In Fig. 1, this mechanism is displayed.  

Fig. 1 

 

The transient state probability of the system having n customers at time t is represented as 

𝑄𝑛(𝑡) = 𝑄{𝑋(𝑡) = 𝑛}, 𝑛 = 0,1,2,3 …  

Consider Q(x,t) as the probability generating function,  

𝑄(𝑥, 𝑡) = ∑ 𝑄𝑛(𝑡)𝑥𝑛∞
𝑛=0                                                                                                                       

In most cases, when time t = 0 it is assumed that there are no customers in the system, denoted as 

𝑄0(0) = 1  

The probability of 𝑄𝑛(𝑡)  as per the system of differential – difference equations.  

𝑄0
′(𝑡) =  −𝜆𝑄0(𝑡)   

+ {[𝑠𝜇1 + 𝜇2 + (1 − 𝑡 − 𝑢)𝜇3] + [𝑠𝜇1 + 𝜇2 + 𝜇3 + 𝑡𝜇2 + (1 − 𝑡 − 𝑢)𝜇3]
+ [𝑠𝜇1 + 𝜇2 + 𝜇3 + 𝑢𝜇1 + 𝜇2 + (1 − 𝑡 − 𝑢)𝜇3] + [(1 − 𝑠)𝜇1 + 𝜇2 + (1 − 𝑡 − 𝑢)𝜇3)]
+ [(1 − 𝑠)𝑠1 + 𝑠2 + 𝑠3 + 𝑠𝑠2 + (1 − 𝑠 − 𝑠)𝑠3]
+ [(1 − 𝑠)𝜇1 + 𝜇2 + 𝜇3 + 𝑢𝜇1 + 𝜇2 + (1 − 𝑡 − 𝑢)𝜇3]}𝑄1(𝑡) + 𝛾[1 − 𝑄0(𝑡)] 

            =  −𝜆𝑄0(𝑡) + [(3 + 2𝑢)𝜇1 + (9 + 𝑡)𝜇2 + (10 − 6𝑡 − 6𝑢)𝜇3]𝑄1(𝑡) + 𝛾[1 − 𝑄0(𝑡)]         (1) 

n = 1,2,3, . . . 

𝑄𝑛
′( 𝑡) =  𝜆 𝑄𝑛−1 (𝑡) − [ 𝜆 + 𝛾 + [(3 + 2𝑢)𝜇1 + ( 9 + 𝑡)𝜇2 + (10 − 6𝑡 − 6𝑢)𝜇3]]𝑄𝑛(𝑡) +

[3𝜇1 + 6𝜇2 + 2(3 + 2𝑢)𝜇1 + (9 + 𝑡)𝜇2 + (10 − 6𝑡 − 6𝑢)𝜇3𝜇3]𝑄𝑛+1(𝑡)                                      (2)   

In Steady state, 

𝑙𝑖𝑚
𝑛→∞

𝑄𝑛(𝑡) = 𝑄𝑛 and  𝑄𝑛
′(𝑡) = 0  𝑎𝑠𝑡 → ∞ 



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These are the steady-state equations that correspond with equations (1) and (2). 

0 =  −𝜆𝑄0 + [3𝜇1 + 6𝜇2 + 2𝜇3]𝑄1 + 𝛾[1 − 𝑄0] 

(𝜆 + 𝛾)𝑄0 = 𝛾 + [(3 + 2𝑢)𝜇1 + (9 + 𝑡)𝜇2 + (10 − 6𝑡 − 6𝑢)𝜇3]𝑄1                                               (3)                

0 =  𝜆𝑄𝑛−1 − [𝜆 + 𝛾 + [(3 + 2𝑢)𝜇1 + (9 + 𝑡)𝜇2 + (10 − 6𝑡 − 6𝑢)𝜇3]]𝑄𝑛

+ [(3 + 2𝑢)𝜇1 + (9 + 𝑡)𝜇2 + (10 − 6𝑡 − 6𝑢)𝜇3]𝑄𝑛+1 

[𝜆 + 𝛾 + [(3 + 2𝑢)𝜇1 + (9 + 𝑡)𝜇2 + (10 − 6𝑡 − 6𝑢)𝜇3]]𝑄𝑛 =  𝜆𝑄𝑛−1 + [(3 + 2𝑢)𝜇1 + (9 + 𝑡)𝜇2 +
(10 − 6𝑡 − 6𝑢)𝜇3]𝑄𝑛+1                                                                                                 (4) 

Equation (3) ⟹ 

𝛾 = (𝜆 + 𝛾)𝑄0 − [(3 + 2𝑢)𝜇1 + (9 + 𝑡)𝜇2 + (10 − 6𝑡 − 6𝑢)𝜇3]𝑄1 

=  [(𝜆 + 𝛾) − [(3 + 2𝑢)𝜇1 + (9 + 𝑡)𝜇2 + (10 − 6𝑡 − 6𝑢)𝜇3]
𝑄1

𝑄0

] 𝑄0 

𝑄0 =
𝛾

(𝜆+𝛾)−[(3+2𝑢)𝜇1+(9+𝑡)𝜇2+(10−6𝑡−6𝑢)𝜇3]
𝑄1
𝑄0

                       (5) 

𝑄𝑛

𝑄𝑛−1
=

𝜆

(𝜆+𝛾+(3+2𝑢)𝜇1+(9+𝑡)𝜇2+(10−6𝑡−6𝑢)𝜇3)−[(3+2𝑢)𝜇1+(9+𝑡)𝜇2+(10−6𝑡−6𝑢)𝜇3]
𝑄𝑛+1

𝑄𝑛

                              (6) 

With the continuing fraction and the two equations mentioned previously, 

𝑄0 =
𝛾

(𝜆 + 𝛾) −
𝜂𝜆

(𝜆 + 𝜂 + 𝛾) − 𝜂 
𝑄2

𝑄1

 

Where  𝜂 = (3 + 2𝑢)𝜇1 + (9 + 𝑡)𝜇2 + (10 − 6𝑡 − 6𝑢)𝜇3 

𝑄0 =
𝛾

(𝜆 + 𝛾) −
𝜂 𝜆

( 𝜆 + 𝜂 + 𝛾 )–
𝜆 𝜂

𝜆 + 𝜂 + 𝛾
− ⋯ .

 

𝑄0 =
𝛾

(  𝜆+ 𝛾 )−𝛼
 , 𝛼 =

𝜂 𝜆

( 𝜆+ 𝜂 +𝛾 ) −
𝜆 𝜂

  𝜆+ 𝜂+ 𝛾
−⋯.

                 (7) 

The quadratic equation is satisfied by equation (7), 

𝛼2 − (𝜆 + 𝜂 + 𝛾)𝛼 + 𝜆𝜂 = 0 

Therefore, the roots are  
𝛽±√𝛽2−4𝜆𝛿

2
  , where 𝛽 = 𝜆 + η + γ 

Assuming 𝛼1 and 𝛼2as the roots, we extract the only real root that falls inside [0, 1). 

Substituting 𝛼2 in equation (7), we get 

𝑄0 =
𝛾

(𝜆+𝛾)−
𝛽−√𝛽2−4𝜆𝜂

2

                 (8) 

𝑄0 =
𝛾

(𝛽 − 𝜂) −
𝛽 − √𝛽2 − 4𝜆𝜂

2

 



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Multiply and divide by  
𝛽−√𝛽2−4𝜆η

2𝜆η
 

After some algebraic calculation, we get 

𝑄0 =

𝛾 (
𝛽−√𝛽2−4𝜆𝜂

2𝜆𝜂
)

1−𝛿 (
𝛽−√𝛽2−4𝜆𝜂

2𝜆𝜂
)

                            (9)  

After expanding binomially, 

𝑄0 = ∑ 𝜂𝑛 (
𝛽−√𝛽2−4𝜆𝜂

2𝜆𝜂
)

𝑛+1
∞
𝑛=0 + 𝛾 ∑ 𝜂𝑛 (

𝛽−√𝛽2−4𝜆𝜂

2𝜆𝜂
)

𝑛+1
∞
𝑛=0                           (10) 

Using equation (6), we can now calculate the remaining steady state probabilities in terms of Q
0
, 

𝑄𝑛

𝑄𝑛−1
=

𝛾

( 𝜆+𝛾+𝜂 )−
𝜆 𝜂

( 𝜆+𝜂+𝛾 ) − 
𝜆 𝜂

𝜆+𝜂+𝛾
−⋯.

  

Using the same reasoning as before, the equation above reduces to 

𝑄𝑛

𝑄𝑛−1
=

𝜆

𝛽+√𝛽2−4𝜆𝜂

2

 ,   n = 1 , 2 . . . .  .                                                     (11) 

Equations (11) can be multiplied and divided by  𝛽 − √𝛽2 − 4𝜆η 

𝑄𝑛 = (
𝛽−√𝛽2−4𝜆𝜂

2𝜂
)

𝑛

𝑄0   , n = 1, 2, . . .,                                             (12)

  

As we know 𝛽 = 𝜆 + γ + η 

Equations (10) and (11) gives the probabilities for Q
0
 and Q

n
 under steady state condition for n = 1, 2 

, . . . , 

2.1 Theorem  

The average queue length asymptotic behavior in steady state, if  γ > 0, is 

𝐿𝑞 =
𝜆−𝜂

𝛾
+

2 𝜂

2 (𝜆+ 𝛾) − [ (𝜆+𝜂+𝛾) − √ (𝜆+𝜂+𝛾)2 – 4 𝜆 𝜂]
 ,  𝜂 = (3 + 2𝑢)𝜇1 + (9 + 𝑡)𝜇2 + (10 − 6𝑡 − 6𝑢)𝜇3   

Proof: 

Examine equations (1) and (2) in relation to the initial condition Q
0
(0) = 1. 

𝜕𝑄(𝑥,𝑡)

𝜕𝑡
= [𝜆𝑥 +

𝜂

𝑥
− (𝜆 + 𝛾 + 𝜂)] 𝑄(𝑥, 𝑡) + 𝜂 (1 −

1

𝑥
) 𝑄0(𝑥) + γ                          (13) 

The mean size is 𝑚(𝑡) = ∑ 𝑛𝑄𝑛(𝑡) =∞
𝑛=1

𝜕𝑄(𝑥,𝑡)

𝜕𝑡
 at x=1 

By taking the derivative of equation (13) with respect to x and evaluating it at x = 1, we obtain 

𝑑𝑚(𝑡)

𝑑𝑥
+ γ𝑚(𝑡) = 𝜆 − (1 − 𝑄0)η 

Finding the solution for m(t) in the differential equation above, where 



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𝑚(0) = ∑ 𝑛𝑄𝑛(𝑡) =

∞

𝑛=1

0 

𝑚(𝑡) =
𝜆

γ
(1 − 𝑒−γ𝑡) −

η

γ
(1 − 𝑒−γ𝑡) + η ∫ 𝑄0(𝑢)

𝑡

0
𝑒−γ(𝑡−𝑢)𝑑𝑢                          (14) 

We get the below equation by applying laplace transform for Equation (1) and (2)  

𝑄0
∗(𝑥) =

1 + 
γ

𝑥

(𝑥+𝜆+γ)−[
𝛽−√ 𝛽2− 4 𝜆 η

2
]

                             (15) 

Taking Laplace Transforms for equation (14), 

 Let 𝑚∗(𝑥) be the laplace transform of  𝑚(𝑡) 

𝑚∗(𝑥) =
𝜆−η

𝑥(𝑥+γ)
+

η

(𝑥+γ)
𝑄0

∗(𝑥)                             (16) 

𝑙𝑖𝑚
𝑡→∞

𝑚(𝑡) = 𝑙𝑖𝑚
𝑥→0

𝑥 𝑚∗(𝑥) 

Using the Laplace Transform to solve Equations (1) and (2), we obtain 

𝐿𝑞 = 𝑚(𝑡) =
𝜆 − 𝜂

𝛾
+

2𝜂

2(𝜆 + 𝛾) − [(𝜆 + 𝜂 + 𝛾) − √(𝜆 + 𝜂 + 𝛾)2 − 4𝜆𝜂]
 

3. Balance Equations 

Using Little's formula, the following equations are obtained. 

Average queue length of the system  

𝐿𝑞 =
𝜆−𝜂

𝛾
+

2𝜂

2(𝜆+𝛾)−[(𝜆+𝜂+𝛾)−√(𝜆+𝜂+𝛾)2−4𝜆𝜂]
 , 

𝜂 = (3 + 2𝑢)𝜇1 + (9 + 𝑡)𝜇2 + (10 − 6𝑡 − 6𝑢) 

Customer wait time in each of the three lines 

𝑊𝑞 =
𝐿𝑞

𝜆
= (

𝜆 − 𝜂

𝛾
+

2𝜂

2(𝜆 + 𝛾) − [(𝜆 + 𝜂 + 𝛾) − √(𝜆 + 𝜂 + 𝛾)2 − 4𝜆𝜂]
)

1

𝜆
 

The mean quantity of customers within the system 

𝐿𝑠 = 𝐿𝑞 +
𝜆

𝜇
= (

𝜆 − 𝜂

𝛾
+

2𝜂

2(𝜆 + 𝛾) − [(𝜆 + 𝜂 + 𝛾) − √(𝜆 + 𝜂 + 𝛾)2 − 4𝜆𝜂]
) +

𝜆

𝜇
 

Customer's wait time within the system 

𝑊𝑠 =
𝐿𝑠

𝜆
= [

𝜆 − 𝜂

𝛾
+

2𝜂

2(𝜆 + 𝛾) − [(𝜆 + 𝜂 + 𝛾) − √(𝜆 + 𝜂 + 𝛾)2 − 4𝜆𝜂]
+

𝜆

𝜇
]

1

𝜆
 

For Node one, the average queue length is 



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𝐿𝑞1
=

𝜆−𝜂

𝛾
+

2𝜂

2(𝜆+𝛾)−[(𝜆+𝜂+𝛾)−√(𝜆+𝜂+𝛾)2−4𝜆𝜂]
 , 

 𝜂 = 𝜇1 + 2𝜇2 + 2𝑢𝜇3 

For Node two, the average queue length is 

𝐿𝑞2
=

𝜆−𝜂

𝛾
+

2𝜂

2(𝜆+𝛾)−[(𝜆+𝜂+𝛾)−√(𝜆+𝜂+𝛾)2−4𝜆𝜂]
 , 

𝜂 = 𝜇1 + 2𝜇2 + 2𝑡𝜇3 

For Node three, the average queue length is 

𝐿𝑞3
=

𝜆−𝜂

𝛾
+

2𝜂

2(𝜆+𝛾)−[(𝜆+𝜂+𝛾)−√(𝜆+𝜂+𝛾)2−4𝜆𝜂]
 , 

𝜂 = 2(1 + 𝑢)𝜇1 + 8𝜇2 + 2𝑡𝜇3 

Using Little’s formula the remaining parametersWq1
,Wq2

,Wq3
,Ls1

,Ls2
,Ls3

,Ws1
,Ws2

,Ws3
 are 

calculated. 

4. PARTICULAR CASE: 

When μ
1

= μ and  μ
2

= μ
3

= 0 , t = s = 0 , there will only be a single node without feedback. When 

s = 0, there are no retrials leading to η=μ. 

Asymptotic behavior of the system Lq when γ > 0 is 

𝐿𝑞 =
𝜆−𝜇

𝛾
+

2𝜇

2(𝜆+𝛾)−[(𝜆+𝜇+𝛾)−√(𝜆+𝜇+𝛾)2−4𝜆𝜇]
                             (17) 

Case 1:  

When γ > 0, the average queue length of the system Lq behaves asymptotically 

𝐿𝑞 =
𝜆−𝜇

𝛾
+

2𝜇

2(𝜆+𝛾)−[(𝜆+𝜇+𝛾)−√(𝜆+𝜇+𝛾)2−𝛼2]
 as  t → ∞, 𝛼 = 2√λμ 

When γ > 0, equation (17) agrees with Krishnakumar et al. [8]. 

Case 2: 

The mean queue length m(t) behaves asymptotically changes when γ is greater than 0. 

𝐿𝑞 =
𝜆−𝜇𝑞

𝛾
+

2𝜇𝑞

2(𝜆+𝛾)−[(𝜆+𝜇𝑞+𝛾)−√(𝜆+𝜇𝑞+𝛾)2−4𝜆𝜇𝑞]
 as  t → ∞ 

By assuming q = 1, the equation (17) agrees with Thangaraj et al. [13]. 

Case 3: 

The mean queue length m(t) exhibits asymptotic behavior when γ>0. 

𝐿𝑞 =
𝜆−𝛽

𝛾
+

2𝛽

2(𝜆+𝛾)−[(𝜆+𝛽+𝛾)−√(𝜆+𝛽+𝛾)2−4𝜆𝛽]
 as  t → ∞ 

Where  𝛽 = (5 + 2𝑝)𝜇1 + 4𝜇2 + 2𝜇3 

Equation (17) agrees with Shanmugasundaram et al. [11] when p = 0, 𝜇1 = 0 , 𝜇2 = 0, 𝑎𝑛𝑑 𝛽 = 𝜇 are 

taken. 

 



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5.  NUMERICAL EXAMPLES  

Calculation for the system 

For t=0.6, u=0.4 ,𝜇1 = 7 , 𝜇2 = 8 , 𝜇3 = 9, 𝜆 = 1,2,3,4,5,6,7,8,9,10 , 𝛾 = 1,3,5,7,9 

Total number of customers in all the three queues: 

Table I shows the number of customers in each of the three nodes for arrival rates λ, which ranges 

from 1 to 10 and for the disaster impact γ, varying from 1 to 9. 

TABLE I 

  1 3 5 7 9 

1 0.007173 0.007071 0.006972 0.006875 0.006781 

2 0.014449 0.014241 0.014038 0.013841 0.01365 

3 0.021831 0.021511 0.021201 0.020899 0.020606 

4 0.029319 0.028884 0.028461 0.028051 0.027653 

5 0.036918 0.036361 0.035822 0.035299 0.034792 

6 0.044628 0.043946 0.043285 0.042644 0.042023 

7 0.052453 0.05164 0.050852 0.050089 0.04935 

8 0.060395 0.059445 0.058525 0.057635 0.056773 

9 0.068457 0.067364 0.066307 0.065285 0.064296 

10 0.076642 0.0754 0.0742 0.07304 0.071918 

 
Fig. 2 The number of customers in the three queues varies depending on the arrival rate under 

calamity impact 

It is evident from Fig. 2 that when the arrival rate rises, there are more customers in the three queues, 

and when γ value rises, there are less customers in three queues. 

Customer's waiting time in the three queues: 

In Table II, The customer's waiting time in the three queues is determined for arrival rates λ ranging 

from 1 to 10, and for γ (Catastrophe effect) ranging from 1 to 9. 

TABLE II 

 1 3 5 7 9 

1 0.007173 0.007071 0.006972 0.006875 0.006781 

2 0.007225 0.00712 0.007019 0.006921 0.006825 

3 0.007277 0.00717 0.007067 0.006966 0.006869 

4 0.00733 0.007221 0.007115 0.007013 0.006913 



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5 0.007384 0.007272 0.007164 0.00706 0.006958 

6 0.007438 0.007324 0.007214 0.007107 0.007004 

7 0.007493 0.007377 0.007265 0.007156 0.00705 

8 0.007549 0.007431 0.007316 0.007204 0.007097 

9 0.007606 0.007485 0.007367 0.007254 0.007144 

10 0.007664 0.00754 0.00742 0.007304 0.007192 

 
Fig 3. For different arrival rate the waiting time of a customer in all the three queues under 

catastrophe effect 

It is evident from Fig. 3 that when the arrival rate rises, customer waiting times in all three queues 

increase, and when γ value rises, customers are waiting time in all three queues decrease. 

For t = 0.6 , u = 0.4, 𝜇1 = 5,5.2,5.4,5.6,5.8,6.0,6.2,6.4,6.6,6.8 , 𝜇2 = 7 , 𝜇3 = 8, 𝜆 = 6 , 𝛾 = 1,3,5,7,9 

Number of customers in the system: 

Table III shows the total customers in the system with service rate 𝜇1 varying from 5 to 6.8 and 

catastrophe effect γ ranging from 1 to 9. 

TABLE III 

 1 3 5 7 9 

5 0.952979 0.952013 0.951083 0.950188 0.949324 

5.2 0.943715 0.942762 0.941845 0.940961 0.940108 

5.4 0.93463 0.93369 0.932785 0.931912 0.931071 

5.6 0.92572 0.924792 0.923899 0.923037 0.922206 

5.8 0.916979 0.916064 0.915182 0.914331 0.91351 

6 0.908402 0.907499 0.906628 0.905789 0.904978 

6.2 0.899986 0.899094 0.898235 0.897405 0.896605 

6.4 0.891724 0.890844 0.889996 0.889177 0.888386 

6.6 0.883614 0.882745 0.881907 0.881099 0.880317 

6.8 0.875651 0.874793 0.873966 0.873167 0.872395 

 

 

 



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Fig 4. For different service rate number of customers in the system under catastrophe effect  

It is evident from figure 4 that when the service rate rises, the number of customers in the system 

falls for different values of γ. 

Customer's waiting time within the system: 

Table IV determines the customer's waiting time based on two factors: the service rate (𝜇1) which 

ranges from 5 to 6.8 and the catastrophe effect (γ) which ranges from 1 to 9. 

TABLE IV 

  1 3 5 7 9 

5 0.15883 0.158669 0.158514 0.158365 0.158221 

5.2 0.157286 0.157127 0.156974 0.156827 0.156685 

5.4 0.155772 0.155615 0.155464 0.155319 0.155178 

5.6 0.154287 0.154132 0.153983 0.15384 0.153701 

5.8 0.15283 0.152677 0.15253 0.152389 0.152252 

6 0.1514 0.15125 0.151105 0.150965 0.15083 

6.2 0.149998 0.149849 0.149706 0.149568 0.149434 

6.4 0.148621 0.148474 0.148333 0.148196 0.148064 

6.6 0.147269 0.147124 0.146985 0.14685 0.14672 

6.8 0.145942 0.145799 0.145661 0.145528 0.145399 

 
Fig 5. For different service rate the waiting time of the customer in the system under catastrophe 

effect 

It is evident from Fig. 5 that when the service rate rises, customer waiting times in the system get 

shorter for different values of γ. 

 

 



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6. Conclusion 

Here, we examine the feedback retrial queueing network's steady state probability in the event of a 

catastrophe and calculate𝑄0(𝑡) or the number of customers in the system at time t, and 𝑄𝑛(𝑡). We 

also determine𝐿𝑞, 𝑊𝑞, 𝐿𝑠, and 𝑊𝑠for each of the three nodes. The numerical example displays the 

number of customers in each of the three system nodes, the number of customers in the system, and 

the amount of time a client has to wait in each of the three system queues. It shows the correctness of 

the results. 

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