Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 3s (2024) 

 

44 
https://internationalpubls.com 

Cost Analysis Approach-the 𝑴𝒙/𝑴/𝟏 Queues with Working 

Breakdowns and Interruptions 
 

B. Deepa1*, K. Kalidass2 and K. Vijaya3* 

1Department of Mathematics, Faculty of Engineering, Karpagam Academy of Higher Education, Coimbatore-21, 

Tamil Nadu, India.email:  Deepa09pragu@gmail.com 

2Department of Mathematics, Chikkanna Government Arts college,Thirupur. 

3Research Scholar, Karpagam Academy of Higher Education.email: vijayakirubai@gmail.com 

 

Article History: 

Received: 10-04-2024 

Revised: 19-05-2024 

Accepted: 05-06-2024 

Abstract 

In this paper, we have studied the steady-state analysis of single-server Markovian bulk 

arrival queueing systems with working breakdowns and interruptions. For this model, the 

probability-generating functions of the system size distributions in steady state are 

obtained. Various important characteristics of the model, like the expected number of 

customers in the system, the expected waiting time of a customer in the system, etc., are 

derived. In steady state, we have discussed some important performance measures of the 

system and provided some numerical examples to show that our results are computationally 

tractable. The suitability of the model is tested in a cost optimization issue where the mean 

number of customers is optimized by reducing the average cost per unit hour using a direct 

search technique. 

Keywords: Poisson arrival, Markovian Queue, vacation, single server, working 

breakdown, maintenance. 

  1 INTRODUCTION 

     Markovian queues are a major subfield of applied probability. In fact, Markovian queues are crucial 

to the theory and practice of continuous-time Markov chains as well as to the creation of broader 

queueing models. The current analysis considers a Markovian queue in which the server experiences 

failures while serving customers. Instead of stopping all service delivery during a breakdown phase, 

the server reduces the service pace. 

Good references are just one of several that [1] V. Jacob, S.R. Chakravarthy, and A. Krishnamoorthy 

discussed the interpretation of interruptions caused by customers and the retrying of those consumers. 

N.K. Jaiswal conducted research on the preemptive resume priority queue in [3]. [4] Jau-Chuan Ke 

used server breakdown and startup/closedown times to model the batch arrival queues under vacation 

policies. [5] Kim BK and Lee DH investigated and examined the M/G/1 queue that had malfunctions 

and catastrophic events. The M/G/1/1 queue modeled by B. Krishna Kumar, D. Arivuadainambi, and 

A. Vijayakumar has an unreliable server and no waiting capacity. [6]. [7] A study conducted by 

Krishnamoorthy A, Pramod PK, and Chakravarthy examined interrupted lines. [8] In stochastic 

models, Keilson, J., and Servi, L.D. investigated the queue as a distributional form of matrix-geometric 

solutions for Little's law. [9] Khalaf R. F., Madan K. C., and Lukas C. A. were taken into consideration 

as a queue model that included Bernoulli schedules for random breakdowns, general repair times, 

general vacation periods, and general prolonged vacation times. The cost optimization of a single-

server queue with working breakdowns under the N policy was examined by Chen, J.-Y., Yen, T.-C., 

and Wang, K.-H. [10]. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 3s (2024) 

 

45 
https://internationalpubls.com 

The article has a structure as follows: The model is illustrated in Section 2. In Section 3, we derive the 

stability condition through the matrix geometric method. For the model, the probability-generating 

functions of the system size distributions in steady state are obtained in Section 4. Stochastic 

decomposition is provided in Section 5. Various important characteristics for the model, like the 

expected number of customers in the system, the expected waiting time of a customer in the system, 

etc., are derived from Section 6. Section 7 discusses the reliability measures of the model. Practical 

applications is provided in section 8. Numerical examples are exhibited to study the parameters and 

their effects on the model in Section 9. In Section 10, we have discussed the cost model and its 

optimization through a direct search method. 

2 THE MODEL 

The inter-arrival stream of customers constitutes a compound Poisson process with rate 𝜆 .Services are 

provided to the customer who is at the head of the queue and the service is single at a time according 

to an exponential distribution with rate 𝜇 .The arrival stream of customers joins the system in 

batches/bulks. The size of batches is a random variable X of nonnegative values with a probability 

distribution 𝑏𝑘  =  𝑃𝑟 {𝑋 =  𝑘}, 𝑘 ≥ 1. The server is possible to partial failures while providing 

service to the customer. At the time of failure, the server continues the service process with slower 

service rate 𝜇𝑏(𝜇𝑏 <  𝜇). The time between failures and the time between repairs is assumed to be an 

exponential random variable with rate 𝛼 and 𝛾 respectively. After service completion, the server is 

sent to repair. Once the repair process completed, the server starts to provide service in regular service 

rate 𝜇.The arrival process, service, breakdown and maintenance times are assumed to be mutually 

independent. 

Let C(t) be the state of the server at time t. Then 

𝐶(𝑡) = {
0, 𝑡ℎ𝑒 𝑠𝑒𝑟𝑣𝑒𝑟 𝑖𝑠 𝑖𝑛 𝑚𝑎𝑖𝑛𝑡𝑒𝑛𝑎𝑛𝑐𝑒 𝑠𝑡𝑎𝑡𝑒,

1, 𝑡ℎ𝑒 𝑠𝑒𝑟𝑣𝑒𝑟 𝑖𝑠 𝑖𝑛 𝑛𝑜𝑟𝑚𝑎𝑙 𝑠𝑡𝑎𝑡𝑒,
2, 𝑡ℎ𝑒 𝑠𝑒𝑟𝑣𝑒𝑟 𝑖𝑠 𝑖𝑛 𝑤𝑜𝑟𝑘𝑖𝑛𝑔 𝑏𝑟𝑒𝑎𝑘𝑑𝑤𝑛 𝑠𝑡𝑎𝑡𝑒

 

Let X (t) be the number of customers present in the system at time t. Then {(𝐶(𝑡), 𝑋(𝑡)), 𝑡 ≥ 0} is a 

CTMC. 

𝑃𝑖,𝑛(𝑡) = 𝑃𝑟𝑜𝑏{𝐶(𝑡) = 𝑖, 𝑋(𝑡) = 𝑛}, 𝑖 = 0,1,2, 𝑎𝑛𝑑 𝑛 ≥ 0. 

3 STABILITY CONDITION 

          A matrix geometric method is applied to analyze the problem and to derive the stability 

condition of the model. Using the lexicographical order for the states, the infinitesimal generator of 

the process {(𝑋(𝑡), 𝐶(𝑡)), 𝑡 ≥ 0} can be written as a block Jacobi matrix and is denoted as 

𝑄 =

[
 
 
 
 
𝐴00

𝐵00

⬚
⬚
⋮

𝐴1

𝐵0

𝐶0

⬚
⋮

𝐴2

𝐵1

𝐵0

𝐶0

⋮

𝐴3

𝐵2

𝐵1

𝐵0

⋮

𝐴4

𝐵3

𝐵2

𝐵1

⋮

𝐴5 …
𝐵4 …
𝐵3 …
𝐵2 …

⋮ ⋮

⋯

]
 
 
 
 

 ------(1) 

Where 𝐶0, 𝐵0 and 𝐵𝑖(𝑖 = 1,2, . . . ) are matrices of order m (the number of environmental states, 

that is m=4 in this study). 𝐵0 is the generator of the environmental process, 𝐵𝑖(𝑖 = 1,2, . . . ) are 

diagonal matrices. Each element of the matrix Q is listed in the following. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 3s (2024) 

 

46 
https://internationalpubls.com 

𝐴00 = [
−𝜆 + 𝛾 𝛾

0 −𝜆
], 

𝐴𝑖 = [
𝜆𝑔𝑖

0 0

0 𝜆𝑔𝑖
0
], 

𝐵00 = [
0 0
0 𝜇
𝜇𝑏 0

 ], 

, 

𝐵0 = [

−𝜆 + 𝛾 𝛾 0
0 −𝜆 + 𝛼 + 𝜇 𝛾
0 0 −𝜆 + 𝜇𝑏

] 

𝐵𝑖 = [

𝜆𝑔𝑖
0 0

0 𝜆𝑔𝑖
0

0 0 𝜆𝑔𝑖

], 

𝐶0 = [
0 0 0
0 𝜇 0
𝜇𝑏 0 0

], 

Consider  

𝑏 = ∑𝑖𝐵𝑖 = [

𝜆𝑔 0 0

0 𝜆𝑔 0

0 0 𝜆𝑔

]

∞

𝑖=1

 

where g is the expectation. 

From the Theorem of Neuts (1981), we know that the steady-state probability vector exists if and 

only    

if  

𝑋𝑏𝑒 < 𝑋𝐶0𝑒 ------(2) 

X is the invariant probability of the matrix 𝐷 = 𝐶0 + 𝐵0 + 𝐵1 + ⋯ 

𝐷 = [
−𝛾 𝛾 0
0 −𝛼 𝛼
𝜇𝑏 0 −𝜇𝑏

], 

The vector X satisfies XD=0 and Xe=1, where 𝑒 = [
1
1
1
]

3𝑥1

.After some calculations, the vector  

𝑋 =
1

𝜇𝑏𝛼+𝜇𝑏𝛾+𝛼𝛾
[𝜇𝑏𝛼 𝜇𝑏𝛾 𝜇𝑏𝛾]1𝑥3  ----------------(3) 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 3s (2024) 

 

47 
https://internationalpubls.com 

Can be obtained and substituting in the condition (2) and doing some algebraic manipulation leads 

to 

𝜌 =
𝜆𝑔[𝜇𝑏(𝛼+𝛾)+𝛼𝛾]

𝜇𝑏𝛾[𝜇+𝛼]
< 1     --------(4) 

4 STEADY STATE ANALYSIS 

The steady state probabilities equations governing the model are derived as follows: 

(𝜆 + 𝛾)𝑃0,0 = 𝜇𝑏𝑃2,1   , 𝑛 = 0                                         ---------------(5) 

(𝜆 + 𝛾)𝑃0,𝑛 = 𝜆 ∑ 𝑔𝑖𝑃0,𝑛−𝑖 + 𝜇𝑏𝑃2,𝑛+1     , 𝑛 ≥ 1𝑛
𝑖=1                 

                                                                                           --------------(6) 

𝜆𝑃1,0 = 𝜇𝑃1,1  + 𝛾𝑃0,0   , 𝑛 = 0                                      -----------------(7) 

(𝜆 + 𝜇 + 𝛼)𝑃1,𝑛 = 𝜇𝑃1,𝑛+1 + 𝛾𝑃0,𝑛 + 𝜆 ∑ 𝑔𝑖𝑃1,𝑛−𝑖     , 𝑛 ≥ 1𝑛
𝑖=1           (8) 

(𝜆 + 𝜇𝑏)𝑃2,1 = 𝛼𝑃1,1  ,     𝑛 = 1                                     ---------------(9) 

(𝜆 + 𝜇𝑏)𝑃2,𝑛 = 𝛼𝑃1,𝑛 + 𝜆 ∑ 𝑔𝑖𝑃2,𝑛−𝑖     , 𝑛 ≥ 2𝑛−1
𝑖=1         --------------(10) 

Define the following partial probability generating functions: 

𝑃0(𝑍) = ∑ 𝑃0,𝑛𝑍𝑛∞
𝑛=0  ,𝑃1(𝑍) = ∑ 𝑃1,𝑛𝑍𝑛∞

𝑛=1   ,  

𝑃2(𝑍) = ∑ 𝑃2,𝑛𝑍𝑛∞
𝑛=1 , 𝐺(𝑧) = ∑ 𝑔𝑖𝑧

𝑛, 𝑖 ≥ 1∞
𝑛=1   

Solving the above equations along with the corresponding probability generating functions we get: 

[𝜆𝑍(1 − 𝐺(𝑍)) + 𝛾𝑍]𝑃0(𝑍) − 𝜇𝑏𝑃2(𝑍) = 0 -----------(11) 

−𝛾𝑃0(𝑍) + [𝜆(1 − 𝐺(𝑍)) + 𝛼 + 𝜇(1 − 1/𝑍)]𝑃1(𝑍) = [𝛼+𝜇(1 − 1/𝑍)]𝑃1,0 ---------(12) 

[𝜆𝑍(1 − 𝐺(𝑍)) + 𝜇𝑏]𝑃2(𝑍) − 𝛼𝑃1(𝑍) = −𝛼𝑃1,0 -------------(13) 

Which is turn yields 

𝑃0(𝑍) =
𝐴0(𝑍)

Δ′(𝑍)
                                                                        -------------(14) 

𝑃1(𝑍) =
𝐴1(𝑍)

Δ′(𝑍)
                                                                        ------------(15) 

𝑃2(𝑍) =
𝐴2(𝑍)

Δ′(𝑍)
                                                                       -------------(16) 

𝑃(𝑍) = 𝑃0(𝑍) + 𝑃1(𝑍) + 𝑃2(𝑍) + 𝑃3(𝑍)                               ----------(17) 

Where, 𝐴0(𝑍) = 𝜆𝜇𝑏𝛼𝐺′(𝑍)𝑃1,0 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 3s (2024) 

 

48 
https://internationalpubls.com 

𝐴1(𝑍) = [𝜆2𝛼[−2𝑍(1 − 𝐺(𝑍))𝐺′(𝑍) + (1 − 𝐺(𝑍)))2] + [𝜆𝛼𝜇𝑏 + 𝛾𝛼𝜆][−𝑍𝐺′(𝑍) + (1 − 𝐺(𝑍))]

+ [𝜆𝜇𝜇𝑏 + 𝜆𝜇𝛾]𝑥[(1 − 𝑍)𝐺′(𝑍) + (1 − 𝐺(𝑍))] + 𝜆2𝜇(1 − 𝑍)2(1 − 𝐺(𝑍))𝐺′(𝑍)

+ (1 − 𝐺(𝑍)))2] + 𝛾𝛼𝜇𝑏 + 𝛾𝜇𝜇𝑏]𝑃1,0 

𝐴2(𝑍) = [2𝜆2𝛼𝑍(1 − 𝐺(𝑍))𝐺′(𝑍) − 𝛼𝜆2(1 − 𝐺(𝑍)))2 − 𝛾𝜆𝛼[(1 − 𝐺(𝑍) − 𝑍𝐺′(𝑍)]]𝑃1,0 

 

Δ′(𝑍) = 𝜆3(1 − 𝐺(𝑍)))3 + (1 − 𝐺(𝑍)))2[−3𝜆3𝑍𝐺′(𝑍) + 𝜆2(𝜇𝑏 + 𝛼 + 𝜇 + 𝛾) + 𝜆(𝛼𝜇𝑏 + 𝜇𝜇𝑏

+ 𝛾𝜇𝑏 + 𝛾𝛼 + 𝛾𝜇)] + 𝐺′(𝑍)𝜆𝑍[−𝛾𝛼 − 𝛾𝜇 +
𝛾𝜇

𝑍
− 𝛾𝜇𝑏 − 𝛼𝜇𝑏 − 𝜇𝜇𝑏 +

𝜇𝜇𝑏

𝑍
]

+ 𝛾𝜇𝑏[𝛼 + 𝜇] 

Now letting Z=1 then the above equations can be rewritten as follows: 

𝑃0(1) =
𝐺′(1)𝜇𝑏𝜆𝛼𝑃1,0

−𝜆𝐺′′(1)[𝛾𝛼+𝛾𝜇𝑏+𝛼𝜇𝑏]+𝛾𝜇𝑏[𝛼+𝜇]
                            ---------(18) 

𝑃1(1) =
−𝜆𝛼𝐺′(1)[𝜇𝑏+𝛾]+𝛾𝜇𝑏[𝜇+𝛼]𝑃1,0

−𝜆𝐺′(1)[𝜇𝑏𝛾+𝛼𝛾+𝛼𝜇𝑏]+𝜇𝑏𝛾[𝜇+𝛼]
                              ------------(19) 

𝑃2(1) =
𝜆𝛼𝛾𝐺′(1)𝑃1,0

−𝜆𝐺′(1)[𝜇𝑏𝛾+𝛼𝛾+𝛼𝜇𝑏]+𝜇𝑏𝛾[𝜇+𝛼]
                         ------------(20) 

 

Using the normalization condition and solving the above equations: 

𝑃0(1) + 𝑃1(1) + 𝑃2(1) = 1                                

We obtain  

𝑃1,0 =
−𝜆𝐺′(1)[𝜇𝑏𝛾+𝛼𝛾+𝛼𝜇𝑏]+𝜇𝑏𝛾[𝜇+𝛼]

𝛾𝜇𝑏[𝜇+𝛼]
 --------------(21) 

From equation (5),(7) and (9) the other probabilities are obtained as follows: 

𝑃0,0 =
𝜆𝛼𝜇𝑏𝑃1,0

[𝜇(𝜆+𝛾)(𝜆+𝜇𝑏)+𝛾𝛼𝜇𝑏]
     -------------(22) 

𝑃2,1 =
𝜆(𝜆+𝛾)(𝜇𝑏𝛼)𝑃1,0

[𝜇(𝜆+𝛾)(𝜆+𝜇𝑏)+𝛾𝛼𝜇𝑏]
 ------(23) 

5 STOCHASTIC DECOMPOSITION APPROACH 

This section concerns the analysis of the stochastic decomposition property of the system size 

distribution. The literature on vacations models recognizes this property as one of the most interesting 

features on this matter. Yang and Templeton [11] studied the stochastic decomposition of retrial 

queues, whose applications were discussed later by Artalejo and Falin [12] and Yang [13].The classical 

interpretation of the stochastic decomposition property shows that the system size distribution 

decomposes into two random variables one of which corresponds to the system size of the ordinary 

queue without working breakdowns and the other random variable gives the increase in the system 

size due to the presence of working breakdowns in the system. 

In particular, in the context of our system, we observe that the probability generating function of the 

system size P(z) can be written as 

𝑃(𝑍) = 𝑃𝑐(𝑍) 𝑥 𝑃𝑤𝑏(𝑍) 

Where 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 3s (2024) 

 

49 
https://internationalpubls.com 

𝑃𝑤𝑏(𝑍) =
𝑁(𝑍) 𝑥 𝑈(𝑍)

Δ′(𝑍)𝛾𝜇𝑏[𝜇 + 𝛼][𝜆𝑔 − 𝜇][1 − 𝑍]
 

𝑃𝑐(𝑍) =
(𝜆𝑔 − 𝜇)(1 − 𝑍)

(𝜆 + 𝜇)𝑍 − 𝜇 − 𝜆𝑍𝐺(𝑍)
 

Where, 

𝑁(𝑍) = 𝜆2𝜇(1 − 𝐺(𝑍))2 + 𝜆(1 − 𝐺(𝑍))[𝛼𝜇𝑏 + 𝜇𝜇𝑏 + 𝜇𝛾 + 2𝜆𝜇(1 − 𝑍)𝐺′(𝑍)] + 𝐺′(𝑍)𝜆(1
− 𝑍)[𝜇𝑏𝛼 + 𝜇𝜇𝑏 + 𝜇𝛾] + 𝛾𝜇𝑏[𝛼 + 𝜇] 

𝑈(𝑍) = −𝜇𝛾𝜇𝑏[𝛼 + 𝜇] + 𝜆2𝑍𝐺′(1)𝐺(𝑍)[𝛼𝛾 + 𝛾𝜇𝑏 + 𝛼𝜇𝑏] − 𝜆𝑍𝐺(𝑍)𝛾𝜇𝑏[𝛼 + 𝜇] − 𝜆𝐺′(1)𝑍(𝜆
+ 𝜇)[𝛼𝛾 + 𝛾𝜇𝑏 + 𝛼𝜇𝑏] + (𝜆 + 𝜇)𝑍𝛾𝜇𝑏[𝛼 + 𝜇] + 𝜆𝐺′(1)𝜇)[𝛼𝛾 + 𝛾𝜇𝑏 + 𝛼𝜇𝑏] 

𝑃𝑐(𝑍) is the probability generating function of the random variable N which is the number in the system 

in the classical 𝑀𝑥/𝑀/1 queueing model. 𝑃𝑤𝑏(𝑍) is the probability generating function of the extra 

load in the system due to the presence of working break downs. 

6 PERFORMANCE MEASURES 

According to the distribution of the steady state, various system performance measures can be 

developed. 

1.Expected number of customers in the system: 𝐸(𝑋) = 𝑃0′(1) + 𝑃1′(1) + 𝑃2′(1) 

[
𝜃1𝜃4 − 𝜃3𝜃2

2θ1
2 +

𝜃1𝜃6 − 𝜃5𝜃2

2θ1
2 +

𝜃1𝜃8 − 𝜃7𝜃2

2θ1
2 ] 

Where, 

𝜃1 = −𝜆𝐺′(1)[μbα + γα + γμb] + μb𝛾[𝜇 + 𝛼]  

𝜃2 = −𝜆𝐺′′(1)[μbα + γα + γμb] + 𝐺′(1)22𝜆2[𝛾 + μb + 𝛼] − 2𝜆𝐺′(1)[𝛾𝜇 + 𝛾𝛼 + μb𝜇 + 𝛾μb +

𝛼μb]  

𝜃3 = μb𝜆𝛼𝐺′(1)𝑃1,0  

𝜃4 = μb𝜆𝛼𝐺′′(1)𝑃1,0  

𝜃5 = [−𝜆𝐺′(1)[𝛼μb + 𝛼𝛾] + 𝛾μb[𝜇 + 𝛼]]𝑃1,0  

𝜃6 = 𝑃1,0[−2𝜆𝐺′(1)[𝛼μb + 𝛼𝛾 + 𝜇μb + 𝜇𝛾] + 2𝜆2𝛼(𝐺′(1))2 − 𝜆𝐺′′(1)[𝛾μb + 𝛾𝛼]]  

𝜃7 = 𝛾𝜆𝛼𝐺′(1)𝑃1,0  

𝜃8 = [−2𝛼𝜆2(𝐺′(1))2 + 2𝛾𝜆𝛼𝐺′(1) + 𝛾𝜆𝛼𝐺′′(1)]𝑃1,0  

2.Expected number of customer customers in the queue 

  𝐸(𝑄) = ∑ (𝑛 − 1)𝑃𝑛
∞
𝑛=1  

3.Expected waiting time in the system  𝐸(𝑊𝑠) =
𝐸(𝑋)

𝜆𝑔
 

4.Expected waiting time in the queue 𝐸(𝑊𝑞) =
𝐸(𝑄)

𝜆𝑔
 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 3s (2024) 

 

50 
https://internationalpubls.com 

5.The proportion of time, the server is busy=𝑃1(1) − 𝑃1,0 

6. The proportion of time, the server being in maintenance 𝑄0 = 𝑃0(1) 

7. The proportion of time, the server being normal 

  𝑄1 = 𝑃1(1) 

8. The proportion of time, the server being in non-reliable state  𝑄2 = 𝑃2(1) 

7 RELIABILITY MEASURES 

In queuing theory literature, some studies are devoted to reliability measures under different 

queuing situations because of its importance. Availability is defined as the probability that the system 

is operating properly when it is requested for use. On the other hand, failure frequency of the server is 

the probability when the system is not operating in normal condition. Thus, we obtain the following 

two important reliability measures, 

(i)The availability of the server is  

𝑃1(1) =
−𝜆𝛼𝐺′(1)[μb + 𝛾] + 𝛾μb[𝜇 + 𝛼]

−𝜆𝐺′(1)[𝛾𝛼 + 𝛾μb + 𝛼μb] + 𝛾μb[𝛼 + 𝜇]
 

(ii)The failure frequency of the server is 𝑃0(1) + 𝑃2(1) = 𝑃1,0
𝐺′(1)μb𝜆𝛼+𝜆𝛼𝛾𝐺′(1)

−𝜆𝐺′(1)[𝛾𝛼+𝛾μb+𝛼μb]+𝛾μb[𝛼+𝜇]
 

8 PRACTICAL APPLICATION 

In queueing theory, a breakdown time refers to a period when a system, such as a server or a machine, 

becomes unavailable due to maintenance, repair, or unexpected failure. During breakdown times, the 

service provided to customers may be affected, leading to delays and disruptions in the queueing 

system. One strategy for managing breakdown times in queueing systems is to continue providing 

service at a reduced rate rather than stopping work immediately. This approach can help in several 

ways: In a restaurant setting, if a piece of equipment breaks down, like a dishwasher or oven, the staff 

might continue serving customers but at a slower pace, using alternative methods or equipment until 

the issue is resolved. This way, they can still provide some level of service to customers, rather than 

abruptly stopping operations. 

Customer Service Centers: In call centers or customer service departments, agents may continue to 

work but at a reduced capacity during breakdowns, such as when their computer systems experience 

technical issues or when they face disruptions in internet connectivity. This can lead to longer call 

handling times and increased queue lengths, affecting customer satisfaction levels. 

Public Transportation: During breakdowns or delays in public transportation systems like buses or 

trains, operators may still provide service, but at a slower pace. For example, a bus experiencing 

mechanical problems might continue its route but at reduced speed, leading to longer waiting times at 

bus stops and increased passenger frustration. 

Health care Facilities: In hospitals or clinics, medical staff may continue to provide care during 

equipment breakdowns or technical issues, but at a slower pace or with limited functionality. For 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 3s (2024) 

 

51 
https://internationalpubls.com 

instance, a medical imaging machine experiencing technical difficulties may still be operational but 

produce scans at a slower rate, affecting patient throughput and waiting times. 

Manufacturing Plants: In manufacturing settings, breakdowns of machinery or equipment can result 

in slower production rates. Workers may attempt to continue production using alternative methods or 

manual processes, but output levels are typically reduced, leading to bottlenecks in the production line 

and delays in fulfilling orders. 

Information Technology Services: In IT departments or data centers, slow work during breakdown 

times can occur when servers or network infrastructure experience performance degradation or 

outages. IT staff may implement temporary fixes or workarounds to maintain essential services, but at 

a reduced capacity, resulting in slower response times for user requests and increased downtime. 

However, it's important to communicate with customers about the situation transparently. Letting them 

know about the issue, the steps being taken to resolve it, and any potential delays can help manage 

expectations and maintain customer satisfaction. 

9 NUMERICAL RESULTS 

In this section we discuss a few illustrative examples to show the qualitative aspects of the queuing 

model under study. We consider rated fifteen examples including 4 on reliability measures. The 

purpose of the following four examples are to study the reliability measures, varying some selected 

parameters (the failure rate and repair rate). We fix 𝜆 = 5, 𝜇 = 15, 𝜇𝑏 = 10 . 

Example 1 Figure 1 illustrate the availability of the server against the failure rate 𝛼. We observe 

that the availability of the server under study state decreases with increase in the failure rate 𝛼 for the 

different repair rates 𝛾 = 2,2.5 𝑎𝑛𝑑 3 as expected. 

 

Figure 1: The effect of 𝛼 on the server availability 

Example 2 Figure 2 tells the effect of the repair rate 𝛾 for three different values of the failure rate 

𝛼 = 1.2, 1.4 𝑎𝑛𝑑 1.6 on the availability of the server. It is seen that the availability of the server 

increases with increasing 𝛾 .This is because the increasing repair rate brings the server to available. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 3s (2024) 

 

52 
https://internationalpubls.com 

 

Figure 2: The effect of 𝛾 on the server availability 

Example 3 Figure 3. depicts variation of the failure frequency of the server against the failure rate 

𝛼 for three different values of the arrival rate 𝜆 = 4,5 𝑎𝑛𝑑 6. As we expect, the failure frequency of 

the server increases with increasing in the failure rate 𝛼. 

 

Figure 3: The effect of 𝛼 on the server failure 

Example 4 Figure 4 displays the effect of the repair rate 𝛾 on the failure frequency of the server 

for three different values of 𝜆 .The failure frequency of the server decreases with increase in the repair 

rate 𝛾 as it is quite natural. 

 

Figure 4: The effect of 𝛾 on the server failure 

Example 5 In this example, we study the effect of varying 𝜆  and 𝜇  on the probability 𝑃1,0   . We 

fix 𝛼 = 2 𝜇𝑏  = 50, 𝛾 = 3, 𝐺′(1) = 0.4 𝑎𝑛𝑑 𝐺′′(1) = 0.03 . As we expected, in figure 5 we notice that 

the probability 𝑃1,0  appears to decrease as a function 𝜆 for fixed 𝜇 and for fixed 𝜆 the probability 

measure 𝑃1,0   increases as a function of the service rate 𝜇. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 3s (2024) 

 

53 
https://internationalpubls.com 

 

Figure 5: The effect of 𝜆 𝑎𝑛𝑑 𝜇 on 𝑃1,0 

Example 6 In this example, we study the effect of varying 𝜆 and 𝜇  on the probability 𝑃0,0 As we 

expected, figure 6 shows that the probability measure 𝑃0,0 decreases as a function of 𝜇 for fixed 𝜆  .For 

fixed 𝜇 the probability measure 𝑃0,0 decreases as function of the arrival rate 𝜆. 

 

Figure 6: The effect of 𝜆 𝑎𝑛𝑑 𝜇 on 𝑃0,0 

Example 7 In this example, we study the effect of service rate 𝜇  and the arrival rate 𝜆 on the 

probability measure 𝑃0(1) and the results are plotted in the figure 7. It is seen that the probability 

measure 𝑃0(1)  increases as a function of the arrival rate 𝜆 for fixed 𝜇 .Also, we observe that the 

probability measure 𝑃0(1) decreases as a function of the service rate 𝜇 for fixed 𝜆 . 

 

Figure 7: The effect of 𝜆 𝑎𝑛𝑑 𝜇 on 𝑃0(1) 

Example 8 This example, clearly shows the effect of varying arrival rate 𝜆 and the service rate 𝜇 

on the probability measure  𝑃1(1)Here figure 8 shows that the probability measure 𝑃1(1) decreases as 

a function of the arrival rate 𝜆 for fixed 𝜇 and for fixed 𝜆 we observe that the probability measure  

𝑃1(1)decreases as a function of the service rate 𝜇 for fixed 𝜆 . 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 3s (2024) 

 

54 
https://internationalpubls.com 

 

Figure 8: The effect of 𝜆 𝑎𝑛𝑑 𝜇 on 𝑃1(1) 

Example 9 In this example, we study the effect of varying 𝜆 and 𝜇 on the probability measure 

𝑃2(1) .As we expected, in figure 9 we notice that the probability measure 𝑃2(1)appears to increase as 

a function of 𝜆 for fixed 𝜇 and it decreases as a function of the service rate 𝜇 for fixed 𝜆 . 

 

Figure 9: The effect of 𝜆 𝑎𝑛𝑑 𝜇 on 𝑃2(1) 

Example 10 In this example, we investigate the impact of the normal service rate 𝜇 and partial 

service rate 𝜇𝑏 on the study state mean number of customers in the system under the ergodicity 

condition. In figure 10, we notice that the study state mean number of customer E(X) increases as the 

partial service rate 𝜇𝑏for fixed normal service rate 𝜇 and it also increases for the normal service rate 𝜇 

for the fixed partial service rate 𝜇𝑏. 

 

Figure 10: The effect of  𝜇 𝑎𝑛𝑑 𝜇𝑏 on E(X) 

Example 11 we study the effect of the varying arrival rate 𝜆 and the repair rate 𝛾 on the probability 

measure 𝑃0(1)and the results are plotted in the figure 11. It is seen that the probability measure 

𝑃0(1) increases as the function of the arrival rate 𝜆 for fixed repair rate 𝛾 and we also observe that the 

probability measure 𝑃0(1) decreases as the function of the repair rate 𝛾 for fixed arrival rate 𝜆. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 3s (2024) 

 

55 
https://internationalpubls.com 

 

Figure 11: The effect of 𝜆 𝑎𝑛𝑑 𝛾 on 𝑃0(1) 

Example 12 This example, clearly shows the effect of varying 𝜆 and the failure rate 𝛼 on the 

probability measure 𝑃1(1)  . As we expected, in figure 12 we notice that the probability measure 𝑃1(1)  

appears to decrease as a function of arrival rate 𝜆 for fixed failure rate 𝛼 For fixed 𝜆 the probability 

measure 𝑃1(1)  decreases as a function of the failure rate 𝛼 . 

 

Figure 12: The effect of 𝜆 𝑎𝑛𝑑 𝛼 on 𝑃1(1) 

Example 13 In this example, we study the effect of the varying service rate 𝜇 and the failure rate 

𝛼 on the probability measure 𝑃1(1). It is clearly noticed in figure 13 that the probability measure 

𝑃1(1)increases as a function 𝜇  for fixed failure rate 𝛼 and the same probability measure𝑃1(1) 

decreases as a function of the failure rate 𝛼 for fixed service rate 𝜇. 

 

Figure 13: The effect of 𝜇 𝑎𝑛𝑑 𝛼 on 𝑃1(1) 

Example 14 we study the effect of varying arrival rate 𝜆 and the failure rate 𝛼 on the probability 

measure 𝑃2(1)  and the results are plotted in figure 14. It is seen that the probability measure 𝑃2(1) 

increases as the function of the arrival rate 𝜆 for fixed failure rate 𝛼. Also observed that the probability 

measure 𝑃2(1) increases as the function of the failure rate 𝛼 for fixed 𝜆 . 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 3s (2024) 

 

56 
https://internationalpubls.com 

 

Figure 14: The effect of 𝜆 𝑎𝑛𝑑 𝛼 on 𝑃2(1) 

10 THE COST MODEL AND ITS OPTIMIZATION 

In this section, we study the total expected cost function with decision variable 𝜇 and 𝜇𝑏 for the 

above discussed model. Our main aim is to obtain the optimal values for the continuous random 

variables say (𝜇∗, 𝜇𝑏
∗) in order to minimize the cost.  

Cost Function: Let us define the following cost elements: 

𝐶0: cost per unit time when the server is in maintenance state. 

𝐶1: cost per unit time for when the server is busy during the normal   

      service period. 

𝐶2: break down cost per unit time for a broken server. 

𝐶3: fixed cost for fast service rate and  

𝐶4: fixed cost for slow service rate. 

Using the definition of these cost elements listed above, the expected cost function per unit time is 

given by 

𝐹(𝜇, 𝜇𝑏) = 𝐶0𝑃0′(1) + 𝐶1𝑃1′(1) + 𝐶2𝑃2′(1) + 𝐶3𝜇 + 𝐶4𝜇𝑏      -------(24) 

The cost minimization problem can be formulated as 𝐹(𝜇∗, 𝜇𝑏
∗) = 𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒𝐹(𝜇, 𝜇𝑏),Subject to 

𝜇 > 𝜇𝑏 and 𝜌 < 1   ---------------------(25) 

The complexity of expressions 𝑃1′(1), 𝑃2′(1) and 𝑃0′(1) complicate the cost function in Equation. 

Unfortunately, it is impossible to derive the analytic solutions for the optimal service rates at the 

minimum expected cost. Thus, we sought to compute the numerical results of the optimal service rates 

𝜇∗ and 𝜇𝑏
∗  using Direct search method. 

Direct Search Method: 

We assume the following cost parameters as 𝐶0 =  200, 𝐶1 = 500, 𝐶2 = 300, 𝐶3 =  20, 𝐶4 = 10 

Numerical examples are presented to determine the optimal value 𝜇 by means of the direct search 

method. Also, we fix 𝜆=2, 𝛼 =0.02, 𝛾 =1, G’(1)=0.4, G’’(1)=0.03 Varying 𝜇 from 4 to 14, and choosing 

different values of 𝜇𝑏 ,the minimum expected cost F(𝜇, 𝜇𝑏) are shown in Table 1 and figure 15 for 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 3s (2024) 

 

57 
https://internationalpubls.com 

𝜇𝑏=2.5,3,3.5 the minimum expected cost Rs.222.3553 is achieved at 𝜇 =5 for 𝜇𝑏=2.5, Rs. 227.1716 is 

achieved at 𝜇 =5 for 𝜇𝑏=3 and Rs. Rs.232.0458 is achieved at 𝜇 =5 for 𝜇𝑏=3.5. 

It is seen that initially the total cost decreases and starts increasing with the grant of 𝜇 for fixed 

values of 𝜇𝑏. The convex nature of the cost function with respect to 𝜇 show the trend for the optimum 

cost by increasing the normal service domain of the customers. From Table2, we concluded that the 

minimum expected cost Rs.204.1258 is attained at 𝜇 =5 for 𝜆 =1.5, Rs.232.0458 is attained at 𝜇 =5, 

for 𝜆 = 2 and Rs.2517.3021 is attained at 𝜇 =6 for 𝜆 =4.2 for fixed 𝜆 =2.5, 𝛼 =0.002, 𝜇𝑏=3.5, 𝛾 =1, G’ 

(1) =0.4, G’’ (1) =0.03. 

From figure16 it is evident that total cost function decreases first and then increases consequently 

the convex nature arises in the total cost function. This confirms the possibility of obtaining the 

optimum service rates. From Table 3, we concluded that the minimum expected cost Rs.236.9736 is 

attained at 𝜇 =5 for 𝛼 =0.1, Rs.242.3192 is attained at 𝜇 =6 for 𝛼 =0.2 and Rs.247.2811 is attained at 

𝜇 =6 for 𝛼 =0.3 for fixed 𝜆 =2, 𝛾 =1,  𝜇𝑏=3.5, G’ (1) =0.4, G’’ (1) =0.03. As we hoped, the figure 17 

shows the convexity in the total cost function. From Table 4, we conclude that the minimum expected 

cost Rs.367.8189 is attained at 𝜇 =9 for 𝛾 =0.2 Rs.275.5773 is attained at 𝜇 =7 for 𝛾 =0.45 and 

Rs.251.891 is attained at 𝜇 =6 for 𝛾 =0.6 for fixed 𝜆 =2, 𝜇𝑏=3, 𝛾 =0.6, 𝛼 =0.2, G’ (1) =0.4, G’’ (1) 

=0.03. As we hoped, the figure 18 shows the convexity in the total cost function. 

 

Figure 15: The effect of 𝜇 on the total cost function 

µ 
𝜇𝑏 = 2.5 𝜇𝑏 = 3 𝜇𝑏 = 3.5 

𝑃1,0 cost 𝑃1,0 cost 𝑃1,0 cost 

4 0.7954 232.9611 0.7957 237.7114 0.7959 242.5405 

5 0.8362 222.3553 0.8364 227.1716 0.8365 232.0458 

6 0.8634 223.5591 0.8636 228.4138 0.8637 233.3144 

7 0.8828 230.8441 0.883 235.724 0.8831 240.6418 

8 0.8975 241.6706 0.8976 246.5683 0.8977 251.4983 

Table 1: The effect of µ on the cost function for various values of 𝜇𝑏 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 3s (2024) 

 

58 
https://internationalpubls.com 

 

Figure 16: The effect of 𝜇 on the total cost function 

µ 
𝜆 = 1.5 𝜆 = 2 𝜆 = 2.5 

𝑃1,0 cost 𝑃1,0 cost 𝑃1,0 cost 

4 0.8469 204.5071 0.7959 242.5405 0.7448 286.1305 

5 0.8774 204.1258 0.8365 232.0458 0.7957 263.0633 

6 0.8978 211.303 0.8637 233.3144 0.8296 257.3021 

7 0.9123 222.4923 0.8831 240.6418 0.8539 20.1645 

8 0.9233 236.0656 0.8977 251.4983 0.8721 267.9425 

Table 2: The effect of µ on the cost function for various values of 𝜆 

 

Figure 17: The effect of 𝜇 on the total cost function 

µ 
𝛼 = 0.1 𝛼 = 0.2 𝛼 = 0.3 

𝑃1,0 cost 𝑃1,0 cost 𝑃1,0 cost 

4 0.7798 248.9268 0.7605 256.8546 0.7422 264.7262 

5 0.823 236.9736 0.8066 243.0957 0.7908 249.1787 

6 0.852 237.3286 0.8378 242.3192 0.824 247.2811 

7 0.8728 244.0292 0.8603 248.243 0.8481 252.435 

8 0.8885 254.4287 0.8774 258.0757 0.8664 261.7056 

9 0.9008 267.1723 0.8907 270.3873 0.8808 273.5886 

Table 3: The effect of µ on the cost function for various values of 𝛼 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 3s (2024) 

 

59 
https://internationalpubls.com 

 

Figure 18: The effect of 𝜇 on the total cost function 

µ 
𝛾 = 0.2 𝛾 = 0.45 𝛾 = 0.6 

𝑃1,0 cost 𝑃1,0 cost 𝑃1,0 cost 

4 0.6063 608.7468 0.7016 326.7336 0.7333 278.7352 

5 0.6821 480.2275 0.759 290.554 0.7846 257.072 

6 0.7333 419.026 0.7978 277.4138 0.8194 251.891 

7 0.7704 388.0538 0.8259 275.5773 0.8444 255.0319 

8 0.7984 373.1639 0.8472 280.1133 0.8634 262.9564 

9 0.8203 367.8189 0.8638 288.5918 0.8783 273.8824 

Table 4: The effect of µ on the cost function for various values of 𝛾 

Conclusion 

We have studied the steady-state analysis of single-server Markovian bulk arrival queueing systems 

with working breakdowns and interruptions. For this model, the probability-generating functions of 

the system size distributions in steady state are obtained. Various important characteristics of the 

model, like the expected number of customers in the system, the expected waiting time of a customer 

in the system, etc., are derived. Numerical examples are presented to study the parameters and their 

effects on the model. Optimizing the cost function is analyzed using MATLAB. Mathematica is used 

for presenting three-dimensional figures. 

References 

[1] V.Jacob, S.R.Chakravarthy, A.Krishnamoorthy, (2012). On a customer-induced interruption in a service 

system.Stochastic Analysis and Applications 30(6),949-962. 

[2] V.Jacob, A.Krishnamoorthy, (2015). Analysis of customer-Induced interruption and retrial of interrupted 

customers.American Journal of Mathematical and Management Sciences 34(4),34-. 

[3] Jaiswal, N.K. (1961). Preemptive resume priority queue. Operations Research, 732-770. 

[4] Jau-Chuan Ke, (2007) Batch arrival queues under vacation policies with server breakdown and startup/closedown 

times, Applied Mathematical modeling, Vol.31, 1282-1292. 

[5] Kim BK. Lee DH. (2014) The M/G/1 queue with disasters and working breakdowns, Applied Mathematical 

Modelling 38, 1788-1798.   

[6] B.Krishna Kumar.,D.Arivuadainambi and A.Vijayakumar,(2002). An M/G/1/1 queue with unreliable server and no 

waiting capacity,Information and Management Science, 13(2) 35-50. 

[7] Krishnamoorthy A, Pramod PK, Chakravarthy SR.9(2014) Queues with interruptions: a survey. Top,22:290-320. 

[8] Keilson, J., and Servi, L.D. (1988): A Distributional form of Little’s law. M.F. (1994): Matrix-Geometric Solutions 

in Stochastic Models - An Algorithmic Approach, 2nd ed., Dover Publications, Inc., NewYork.    



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 3s (2024) 

 

60 
https://internationalpubls.com 

[9] Khalaf R. F., Madan K. C. and Lukas C. A. (2011). An 𝑀𝑥/𝐺/1 Queue with Bernoulli Schedule General Vacation 

Times, General Extended Vacations, Random Breakdowns, General Delay Times for Repairs to Start and General 

Repair Times.Journal of Mathematics Research, 3, 8 -20.  

[10]  Chen,J.-Y., Yen, T.-C., and Wang, K.-H., (2016). “Cost Optimization of a Single- Server Queue with Working 

Breakdowns Under the N Policy,” Journal of Testing and Evaluation, Vol 44, No.5, 2059-2067. 

[11]  Neuts,M. (1981).Matrix geometric solutions in stochastic models, Baltimore: John Hopkins University Press.