A Discrete-Time 𝐺𝑒𝑜/𝐺/1 Retrial Queue with 𝐽 Vacations and Two Types of Breakdowns

Abstract

This paper is concerned with a discrete-time 𝐺𝑒𝑜/𝐺/1 retrial queueing model with 𝐽 vacations and two types of breakdowns. If the orbit is empty, the server takes at most 𝐽 vacations repeatedly until at least one customer appears in the orbit upon returning from a vacation. It is assumed that the server is subject to two types of different breakdowns and is sent immediately for repair. We analyze the Markov chain underlying the considered queueing system and derive the system state distribution as well as the orbit size and the system size distributions in terms of their generating functions. Then, we obtain some performance measures through the generating functions. Moreover, the stochastic decomposition property and the corresponding continuous-time queueing system are investigated. Finally, some numerical examples are provided to illustrate the effect of vacations and breakdowns on several performance measures of the system.

1. Introduction

During the past few decades, retrial queueing systems have been widely studied due to their important applications in many practical systems such as inventory systems, computer systems, and telecommunication networks. Retrial queueing systems are characterized by the fact that arriving customers who find the server busy may join a retrial group to retry their requests after some random time. For excellent bibliographies on retrial queues, the readers are referred to [1–3].

Although many continuous-time retrial queueing models have been studied extensively in the past years, only few researchers studied the discrete-time retrial queues. In fact, the discrete-time retrial queueing systems are suitable for the description of the various phenomena in slotted time computer and communication systems such as broad-band integrated services digital network (B-ISDN) and time-division multiple access (TDMA) systems. Yang and Li [4] firstly studied a discrete-time 𝐺𝑒𝑜/𝐺/1 retrial queue and obtained some performance measures of the systems. This work was generalized to discrete-time 𝐺𝑒𝑜/𝐺/1 retrial queue with general retrial times by Atencia and Moreno [5]. In recent years, there has been a growing interest in the investigation of discrete-time retrial queue with server breakdowns. Such phenomena occur in day-to-day life, for example, the components of the computer and communication systems are subject to random breakdowns. Discrete-time retrial queues that take into account server breakdowns were studied by Atencia and Moreno [6, 7], Moreno [8], Wang and Zhao [9], Wang [10], Wang and Zhang [11], and Atencia et al. [12].

On the other hand, queueing systems with server vacations have been widely used to analyze the performance of some systems such as manufacturing systems and computer systems. In recent years, retrial queues with server vacations have drawn more and more attention. Li and Yang [13] studied a 𝑀/𝐺/1 retrial queue with a finite number of input sources and Bernoulli vacations by the method of supplementary variables. Since the work of Li and Yang, the retrial queues with Bernoulli vacations have been studied by many authors, see Kumar and Arivudainambi [14], Kumar et al. [15], and Choudhury [16]. In addition, the retrial queues with exhaustive service vacations were also studied by several authors. For example, Artalejo [17] analyzed an 𝑀/𝐺/1 retrial queue with multiple vacations and obtained two stochastic decomposition results for the system size. As a generalization of single and multiple vacation policy, Chang and Ke [18] and Ke and Chang [19] introduced the concept of 𝐽 vacations into 𝑀/𝐺/1 retrial queueing systems. Ke and Chang [20] and Wu [21] studied the models concerning both server breakdowns and vacations, simultaneously.

The majority of articles studied the continuous-time retrial queues with server vacations. Most recently, Wang [22] extended the continuous-time 𝑀/𝐺/1 retrial queue with Bernoulli vacations to discrete-time counterpart. Li et al. [23], Liu and Song [24] considered the discrete-time 𝐺𝑒𝑜/𝐺𝑒𝑜/1 retrial queue with working vacation. Zhang et al. [25] studied a discrete-time 𝐺𝑒𝑜/𝐺/1 retrial queue with single vacation and starting failure. To the best of our knowledge, no work appeared till now concerning the discrete-time retrial queue with exhaustive 𝐽 vacations policy and server’s different types of breakdowns. In fact, the discrete-time retrial queues with 𝐽 vacations and server breakdowns are more appropriate to analyze some real situations, such as e-mail systems, IP access networks, and computer systems. This motivates us to analyze the discrete-time retrial queue with 𝐽 vacations and and general retrial times. In this work, we consider a discrete-time 𝐺𝑒𝑜/𝐺/1 retrial queue with 𝐽 vacations where the server may be subjected to two different types of breakdowns.

The rest of the paper is organized as follows. In Section 2, the description of our model is given. In Section 3, We derive the generating functions of the number of customers in the orbit and in the system. The closed-form expressions of some performance measures of the system are also obtained. In Section 4, we give two stochastic decomposition results for the system size. In Section 5, the relationship between our model and the continuous-time counterpart is given. In Section 6, we present some numerical results to show the influence of the parameters on some performance measures of the system.

2. Description of the Mathematical Model

We consider a discrete-time 𝐺𝑒𝑜/𝐺/1 retrial queue with 𝐽 vacations and two types of server breakdowns where the time axis is segmented into slots of equal length and all queueing activities occur at the slot boundaries. Specifically, let the time axis be marked by 0,1,…,𝑚,…. Consider the epoch 𝑚 and assume that the departures, the end of the vacations and the end of repairs occur in the interval (𝑚⁻,𝑚), while arrivals, retrials, the beginning of the vacations, and the beginning of repairs occur in the interval (𝑚,𝑚⁺) in sequence. That is, we consider an early arrival system (EAS) policy in our model.

Customers arrive at the system according to a geometrical arrival process with parameter 𝑝, where 𝑝 (0<𝑝<1) is the probability that an arrival occurs in a slot. If an arriving customer finds that the server is idle, he commences his service immediately. Otherwise, if the server is busy, under repair or on vacation, the arriving customer joins the retrial orbit. It is assumed that only the customer at the head of the orbit is permitted to access to the server. The time between retrials is assumed to follow a general probability distribution {𝑎_𝑖}^∞_𝑖=0 with the generating function 𝐴(𝑥)=∑^∞_𝑖=0𝑎_𝑖𝑥^𝑖.

The service times are independent and identically distributed with arbitrary distribution {𝑠_𝑖}^∞_𝑖=1, generating function 𝑆(𝑥)=∑^∞_𝑖=1𝑠_𝑖𝑥^𝑖 and 𝑛th factorial moment 𝑆_𝑛. It is assumed that the server is subject to two different types of breakdowns. The first type of breakdown is starting failure. In this model, an arriving customer who finds the server idle must turn on the server. If the server is activated successfully (with a probability 𝜃₁=1−𝜃₁), the customer begins his service immediately; otherwise, if the server is started unsuccessfully (with a complementary probability 𝜃₁), the server is sent to repair immediately and the customer being served must join the orbit. The second type of breakdown is normal breakdown, that is, the server may break down during serving customers. The customer just being served before breakdown waits until the server is repaired to complete his remaining service time. The lifetime of the server is assumed to be geometrically distributed with parameter 𝜃₂. The repair times for the server of two types of breakdowns are independent and identically distributed with arbitrary distribution {𝑟_𝑙,𝑖}^∞_𝑖=1, generating functions 𝑅_𝑙(𝑥)=∑^∞_𝑖=0𝑟_𝑙,𝑖𝑥^𝑖, and 𝑛th factorial moment are 𝑅_𝑙,𝑛, 𝑙=1,2, respectively.

As soon as the system is empty, the server takes a vacation immediately. The vacation time is assumed to follow a general probability distribution {𝑣_𝑖}^∞_𝑖=1 with generating function 𝑉(𝑥)=∑^∞_𝑖=1𝑣_𝑖𝑥^𝑖 and 𝑛th factorial moment 𝑉_𝑛. If there is at least one customer in the orbit at the end of a vacation, the server begins to provide service for customers immediately. Otherwise, if no customers appear in the orbit when the server returns from the vacation, it leaves for another vacation. This pattern continues until the server has taken a maximum number 𝐽 of vacations. If the system is empty at the end of the 𝐽th vacation, the server becomes idle and waits for new arriving customers.

Finally, we assume that the interarrival times, the service times, the retrial times, the repair times and the vacation times are mutually independent.

3. The Markov Chain and Some Performance Measures

At time 𝑚⁺, the system can be described by the process {𝑌_𝑚;𝑚≥1} with 𝑌_𝑚=(𝐶_𝑚,𝜉_𝑚,𝜁_𝑚,𝐿_𝑚,𝑁_𝑚), where 𝐶_𝑚 denotes the state of the server and 𝑁_𝑚 be the number of the customers in the orbit. It is assumed that 𝐶_𝑚=0,1,2,3, or 4 according to whether the server is free, busy, under repair for first type of breakdown, under repair for second type of breakdown, or on vacations, respectively. If 𝐶_𝑚=0, 𝜉_𝑚 represents the remaining retrial time. If 𝐶_𝑚=1, 𝜉_𝑚 represents the remaining service time of the customer currently being served. If 𝐶_𝑚=2, 𝜉_𝑚 represents the remaining repair time for the first type of breakdown. If 𝐶_𝑚=3, 𝜉_𝑚 represents the remaining service time of a customer currently being served and 𝜁_𝑚 represents the remaining repair time of the second type of breakdowns. If 𝐶_𝑚=4, let 𝐿_𝑚=𝑛 represent that the server takes the 𝑛th vacation and let 𝜉_𝑚 represent the remaining vacation time, 𝑛=1,2,…,𝐽. It can be shown that {𝑌_𝑚;𝑚≥1} is a Markov chain with the following state space: );()()()()}.Ω={(0,00,𝑖,𝑘∶𝑖≥1,𝑘≥1;1,𝑖,𝑘∶𝑖≥1,𝑘≥0;2,𝑖,𝑘∶𝑖≥1,𝑘≥1;3,𝑖,𝑗,𝑘∶𝑖≥1,𝑘≥04,𝑖,𝑘∶𝑖≥1,𝑘≥0,𝑛=1,2,…,𝐽 (1)

Our object is to find the stationary distribution of the Markov chain which is defined as follows: 𝜋_0,0=lim_𝑚→∞𝑃𝐶_𝑚=0,𝑁_𝑚,𝜋=0_0,𝑖,𝑘=lim_𝑚→∞𝑃𝐶_𝑚=0,𝜉_𝑚=𝑖,𝑁_𝑚𝜋=𝑘;𝑖≥1,𝑘≥1,_1,𝑖,𝑘=lim_𝑚→∞𝑃𝐶_𝑚=1,𝜉_𝑚=𝑖,𝑁_𝑚𝜋=𝑘;𝑖≥1,𝑘≥0,_2,𝑖,𝑘=lim_𝑚→∞𝑃𝐶_𝑚=2,𝜉_𝑚=𝑖,𝑁_𝑚𝜋=𝑘;𝑖≥1,𝑘≥1,_{3,𝑖,𝑗,𝑘}=lim_𝑚→∞𝑃𝐶_𝑚=3,𝜉_𝑚=𝑖,𝜁_𝑚=𝑗,𝑁_𝑚;𝜔=𝑘𝑖≥1,𝑗≥1,𝑘≥0,_{𝑛,𝑖,𝑘}=lim_𝑚→∞𝑃𝐶_𝑚=4,𝐿_𝑚=𝑛,𝜉_𝑚=𝑖,𝑁_𝑚=𝑘;𝑖≥1,𝑘≥0,𝑛=1,2,…,𝐽. (2) The Kolmogorov equations for the stationary distribution of the system are 𝜋_0,0=𝑝𝜋_0,0+𝑝𝜔_𝐽,1,0,𝜋_0,𝑖,𝑘=𝑝𝜋_{0,𝑖+1,𝑘}+𝑝𝑎_𝑖𝜋_1,1,𝑘+𝑝𝑎_𝑖𝜋_2,1,𝑘+𝑝𝑎_𝑖^𝐽_𝑛=1𝜔_𝑛,1,𝑘,𝑖≥1,𝑘≥1,𝜋_1,𝑖,𝑘=𝛿_0,𝑘𝑝𝜃₁𝜃₂𝑠_𝑖𝜋_0,0+𝛿_0,𝑘𝑝𝜃₁𝜃₂𝑠_𝑖^∞_𝑗=1𝜋_0,𝑗,𝑘+𝑝𝜃₁𝜃₂𝑠_𝑖𝜋_0,1,𝑘+1+𝑝𝜃₁𝜃₂𝑠_𝑖𝜔_𝐽,1,𝑘+𝑝𝑎₀𝜃₁𝜃₂𝑠_𝑖𝜋_1,1,𝑘+1+𝛿_0,𝑘𝑝𝜃₁𝜃₂𝑠_𝑖𝜋_1,1,𝑘+𝑝𝜃₂𝜋_{1,𝑖+1,𝑘}+𝛿_0,𝑘𝑝𝜃₂𝜋_{1,𝑖+1,𝑘−1}+𝛿_0,𝑘𝑝𝜃₁𝜃₂𝑠_𝑖𝜋_2,1,𝑘+𝑝𝑎₀𝜃₁𝜃₂𝑠_𝑖𝜋_2,1,𝑘+1+𝑝𝜃₂𝜋_{3,𝑖,1,𝑘}+𝛿_0,𝑘𝑝𝜃₂𝜋_{3,𝑖,1,𝑘−1}+𝑝𝑎₀𝜃₁𝜃₂𝑠_𝑖^𝐽_𝑛=1𝜔_{𝑛,1,𝑘+1}+𝛿_0,𝑘𝑝𝜃₁𝜃₂𝑠_𝑖^𝐽−1_𝑛=1𝜔_𝑛,1,𝑘,𝑖≥1,𝑘≥0,𝜋_2,𝑖,𝑘=𝛿_1,𝑘𝑝𝜃₁𝑟_1,𝑖𝜋_0,0+𝛿_1,𝑘𝑝𝜃₁𝑟^∞_1,𝑖𝑗=1𝜋_{0,𝑗,𝑘−1}+𝑝𝜃₁𝑟_1,𝑖𝜋_0,1,𝑘+𝑝𝑎₀𝜃₁𝑟_1,𝑖𝜋_1,1,𝑘+𝛿_1,𝑘𝑝𝜃₁𝑟_1,𝑖𝜋_1,1,𝑘−1+𝛿_1,𝑘𝑝𝜋_{2,𝑖+1,𝑘−1}+𝑝𝜋_{2,𝑖+1,𝑘}+𝑝𝑎₀𝜃₁𝑟_1,𝑖𝜋_2,1,𝑘+𝛿_1,𝑘𝑝𝜃₁𝑟_1,𝑖𝜋_2,1,𝑘−1+𝑝𝑎₀𝜃₁𝑟^𝐽_1,𝑖𝑛=1𝜔_𝑛,1,𝑘+𝛿_1,𝑘𝑝𝜃₁𝑟_1,𝑖^𝐽−1_𝑛=1𝜔_{𝑛,1,𝑘−1}+𝑝𝜃₁𝑟_1,𝑖𝜔_{𝐽,1,𝑘−1},𝑖≥1,𝑘≥1,𝜋_{3,𝑖,𝑗,𝑘}=𝛿_0,𝑘𝑝𝜃₁𝜃₂𝑠_𝑖𝑟_2,𝑗𝜋_0,0+𝛿_0,𝑘𝑝𝜃₁𝜃₂𝑠_𝑖𝑟^∞_2,𝑗𝑙=1𝜋_0,𝑙,𝑘+𝑝𝜃₁𝜃₂𝑠_𝑖𝑟_2,𝑗𝜋_0,1,𝑘+1+𝑝𝜃₁𝜃₂𝑠_𝑖𝑟_2,𝑗𝜔_𝐽,1,𝑘+𝑝𝑎₀𝜃₁𝜃₂𝑠_𝑖𝑟_2,𝑗𝜋_1,1,𝑘+1+𝛿_0,𝑘𝑝𝜃₁𝜃₂𝑠_𝑖𝑟_2,𝑗𝜋_1,1,𝑘+𝑝𝜃₂𝑟_2,𝑗𝜋_{1,𝑖+1,𝑘}+𝛿_0,𝑘𝑝𝜃₂𝑟_2,𝑗𝜋_{1,𝑖+1,𝑘−1}+𝛿_0,𝑘𝑝𝜃₁𝜃₂𝑠_𝑖𝑟_2,𝑗𝜋_2,1,𝑘+𝑝𝑎₀𝜃₁𝜃₂𝑠_𝑖𝑟_2,𝑗𝜋_2,1,𝑘+1+𝑝𝜋_{3,𝑖,𝑗+1,𝑘}+𝑝𝜃₂𝑟_2,𝑗𝜋_{3,𝑖,1,𝑘}+𝑝𝑎₀𝜃₁𝜃₂𝑠_𝑖𝑟^𝐽_2,𝑗𝑛=1𝜔_{𝑛,1,𝑘+1}+𝛿_0,𝑘𝑝𝜃₁𝜃₂𝑠_𝑖𝑟_2,𝑗^𝐽−1_𝑛=1𝜔_𝑛,1,𝑘+𝛿_0,𝑘𝑝𝜃₂𝑟_2,𝑗𝜋_{3,𝑖,1,𝑘−1}+𝛿_0,𝑘𝑝𝜋_{3,𝑖,𝑗+1,𝑘−1},𝑖≥1,𝑗≥1,𝑘≥0,𝜔_1,𝑖,𝑘=𝑝𝜔_{1,𝑖+1,𝑘}+𝛿_0,𝑘𝑝𝜔_{1,𝑖+1,𝑘−1}+𝛿_0,𝑘𝑝𝑣_𝑖𝜋_1,1,0+𝛿_1,𝑘𝑝𝑣_𝑖𝜋_1,1,0,𝑖≥1,𝑘≥0,𝜔_{𝑛,𝑖,𝑘}=𝑝𝜔_{𝑛,𝑖+1,𝑘}+𝛿_0,𝑘𝑝𝜔_{𝑛,𝑖+1,𝑘−1}+𝛿_0,𝑘𝑝𝑣_𝑖𝜔_𝑛−1,1,0+𝛿_1,𝑘𝑝𝑣_𝑖𝜔_𝑛−1,1,0,𝑖≥1,𝑘≥0,𝑛=2,3,…,𝐽, (3)(4)(5)(6)(7)(8)(9) where 𝛿_𝑖,𝑗 is the Kronecker’s symbol, 𝛿_𝑖,𝑗=1−𝛿_𝑖,𝑗, and the normalizing condition is 𝜋_0,0+^∞∞_𝑖=1𝑘=1𝜋_0,𝑖,𝑘+𝜋_2,𝑖,𝑘+^∞∞_𝑖=1𝑘=0𝜋_1,𝑖,𝑘+^∞∞_𝑖=1^∞_𝑗=1𝑘=0𝜋_{3,𝑖,𝑗,𝑘}+^𝐽∞_𝑛=1^∞_𝑖=1𝑘=0𝜔_{𝑛,𝑖,𝑘}=1. (10) To solve (3)–(10), we introduce the following generating functions: 𝜑₀(𝑥,𝑧)=^∞∞_𝑖=1𝑘=1𝜋_0,𝑖,𝑘𝑥^𝑖𝑧^𝑘,𝜑₁(𝑥,𝑧)=^∞∞_𝑖=1𝑘=0𝜋_1,𝑖,𝑘𝑥^𝑖𝑧^𝑘,𝜑₂(𝑥,𝑧)=^∞∞_𝑖=1𝑘=1𝜋_2,𝑖,𝑘𝑥^𝑖𝑧^𝑘,𝜑₃𝑥,𝑦,𝑧=^∞∞_𝑖=1^∞_𝑗=1𝑘=0𝜋_{3,𝑖,𝑗,𝑘}𝑥^𝑖𝑦^𝑖𝑧^𝑘,𝜓_𝑛(𝑥,𝑧)=^∞∞_𝑖=1𝑘=0𝜔_{𝑛,𝑖,𝑘}𝑥^𝑖𝑧^𝑘𝜑,𝑛=1,2,…,𝐽,₄(𝑥,𝑧)=^𝐽_𝑛=1𝜓_𝑛(𝑥,𝑧), (11) and the following auxiliary generating functions: 𝜑_0,𝑖(𝑧)=^∞_𝑘=1𝜋_0,𝑖,𝑘𝑧^𝑘,𝜑_1,𝑖(𝑧)=^∞_𝑘=0𝜋_1,𝑖,𝑘𝑧^𝑘,𝜑_2,𝑖(𝑧)=^∞_𝑘=1𝜋_2,𝑖,𝑘𝑧^𝑘,𝜓_𝑛,𝑖(𝑧)=^∞_𝑘=1𝜔_{𝑛,𝑖,𝑘}𝑧^𝑘,𝜑_3,𝑖,𝑗(𝑧)=^∞_𝑘=0𝜋_{3,𝑖,𝑗,𝑘}𝑧^𝑘,𝜑_3,𝑖𝑦,𝑧=^∞_𝑘=0𝜋_{3,𝑖,𝑗,𝑘}𝑦^𝑗𝑧^𝑘,𝜙₁(𝑥,𝑧)=^∞∞_𝑖=1𝑘=0𝜋_{3,𝑖,1,𝑘}𝑥^𝑖𝑧^𝑘. (12) Now, we can solve (3)–(10) by using the generating function technique. We first give some lemmas which will be used later on and their proof which can be readily obtained. Thus, they are omitted. The following inequalities hold.

Lemma 1. If  𝜃₁𝜌₁+𝜃₁+𝜃₁𝜌₂<𝑝+𝑝𝐴(𝑝), then the inequality 𝑧+(1−𝑧)𝑝𝐴𝑝Ω(𝑧)−𝑧𝛾(𝑧)>0 (13) holds for 0≤𝑧<1, where 𝛾(𝑧)=𝑝+𝑝𝑧,𝜌₁=𝑝𝑆₁𝜃1+²𝜃₂𝑅_2,1,𝜌₂=𝑝𝑅_1,1,Ω(𝑧)=𝜃₁𝑆𝜏𝛾(𝑧)+𝑧𝜃₁𝑅₁𝛾(𝑧),𝜏(𝑧)=𝜃₂𝑧1−𝜃₂𝑅₂(𝑧). (14)

Lemma 2. The following limits exist, if 𝜃₁𝜌₁+𝜃₁+𝜃₁𝜌₂<𝑝+𝑝𝐴(𝑝)lim_𝑧→1𝛾(𝑧)Γ(𝑧)−𝜂₃Ω(𝑧)Λ(𝑧)=𝑝𝑉₁𝜂₂+𝑝𝜂₃𝜃₁𝑆₁1+𝜃₂/𝜃₂𝑅_2,1−𝜃₁+𝜃₁𝑅_1,1𝑝+𝑝𝐴𝑝−𝜃₁𝜌₁−𝜃₁−𝜃₁𝜌₂,lim_𝑧→1Θ(𝑧)Λ(𝑧)=𝑝𝑉₁𝜂₂+𝑝𝐴𝑝𝜂₃𝑝+𝑝𝐴𝑝−𝜃₁𝜌₁−𝜃₁−𝜃₁𝜌₂, (15) where 𝜉=𝑉𝑝,𝜂₁=1−𝜉^𝐽+1,𝜂₂=1−𝜉^𝐽,𝜂₃=(1−𝜉)𝜉^𝐽,Θ(𝑧)=𝑧𝜂₂1−𝑉𝛾(𝑧)+𝑝𝐴𝑝()Γ(𝑧),Γ(𝑧)1−𝑧=𝜂₁−𝜂₂𝑉𝛾(𝑧),Λ(𝑧)=𝑧+𝑝𝐴𝑝(1−𝑧)Ω(𝑧)−𝑧𝛾(𝑧). (16)

By using Lemmas 1 and 2, we can obtain the generating functions of the stationary distribution of the system which are given by the following theorem.

Theorem 3. If 𝜃₁𝜌₁+𝜃₁+𝜃₁𝜌₂<𝑝+𝑝𝐴(𝑝), the stationary distribution of the Markov chain {𝑌_𝑚,𝑚=0,1,2,…} has the following generating functions: 𝜑₀()=𝐴(𝑥)𝑥,𝑧−𝐴𝑝𝑥−𝑝𝑝𝑥𝑧𝛾(𝑧)Γ(𝑧)−𝜂₃Ω(𝑧)𝜂₃Λ(𝑧)𝜋_0,0,𝜑₁()=𝑆(𝑥)𝜏𝛾(𝑧𝑥,𝑧−𝑆)𝛾(𝑧)Θ(𝑧)𝑥−𝜏𝑝𝑥𝜃₁𝜏𝛾(𝑧)𝑝𝜂₃Λ(𝑧)𝜋_0,0,𝜑₂()=𝑅𝑥,𝑧¹(𝑥)−𝑅₁𝛾(𝑧)(𝑧)Θ(𝑧)𝑥−𝛾𝑝𝑥𝜃₁𝑧𝛾(𝑧)𝑝𝜂₃Λ(𝑧)𝜋_0,0,𝜑₃=𝑆(𝑥)𝜏𝛾(𝑧)𝑥,𝑦,𝑧−𝑆𝛾(𝑧)×𝑅𝑥−𝜏₂𝑦−𝑅₂𝛾(𝑧)(𝑧)Θ(𝑧)𝑦−𝛾𝑝𝑥𝑦𝜃₁𝜏𝛾(𝑧)𝑝𝜂₃Λ(𝑧)𝜃₂𝜃₂𝜋_0,0,𝜑₄(𝑥,𝑧)=𝑝𝜂₂𝑥𝛾(𝑧)𝑉(𝑥)−𝑉𝛾(𝑧)𝑝𝜂₃(𝑧𝜋𝑥−𝛾)^0,0, (17) where 𝜋_0,0=𝑝𝜂₃𝑝+𝑝𝐴𝑝−𝜃₁𝜌₁−𝜃₁−𝜃₁𝜌₂𝜃₁𝑝𝑉₁𝜂₂+𝑝𝐴𝑝𝜂₃. (18)

Proof. Multiplying (3)–(9) by 𝑧^𝑘 and summing over 𝑘, we get the following equations: 𝜑_0,𝑖(𝑧)=𝑝𝜑_0,𝑖+1(𝑧)+𝑝𝑎_𝑖𝜑_1,1(𝑧)+𝜑_2,1(𝑧)+^𝐽_𝑛=1𝜓_𝑛,1(𝑧−)𝑝𝑎_𝑖𝜋_1,1,0−𝑝𝑎_𝑖^𝐽_𝑛=1𝜔_𝑛,1,0,𝑖≥1,𝜑_1,𝑖(𝑧)=𝑝𝜃₁𝜃₂𝑠_𝑖𝜋_0,0+𝑝𝜃₁𝜃₂𝑠_𝑖𝜑₀()+1,𝑧𝑝𝑧𝜃₁𝜃₂𝑠_𝑖𝜑_0,1(𝑧)+𝜃₂𝛾(𝑧)𝜑_1,𝑖+1(𝑧)+𝑝+𝑝𝑧𝑎₀𝜃₁𝜃₂𝑠_𝑖𝜑_1,1(𝑧)+𝜑_2,1(𝑧)+^𝐽_𝑛=1𝜓_𝑛,1(𝑧−)𝑝+𝑝𝑧𝑎₀𝜃₁𝜃₂𝑠_𝑖𝜋_1,1,0,−𝑝𝜃₁𝜃₂𝑠_𝑖^𝐽−1_𝑛=1𝜔_𝑛,1,0−𝑝𝑧𝑎₀𝜃₁𝜃₂𝑠_𝑖^𝐽_𝑛=1𝜔_𝑛,1,0+𝜃₂𝛾(𝑧)𝜑_3,𝑖,1(𝑧),𝑖≥1,𝜑_2,𝑖(𝑧)=𝑝𝜃₁𝑧𝑟_1,𝑖𝜋_0,0+𝑝𝜃₁𝑧𝑟_1,𝑖𝜑₀()+1,𝑧𝑝𝜃₁𝑟_1,𝑖𝜑_0,1(𝑧)+𝛾(𝑧)𝜑_2,𝑖+1(𝑧)+𝑝𝑧+𝑝𝑎₀𝜃₁𝑟_1,𝑖𝜑_1,1(𝑧)+𝜑_2,1(𝑧)+^𝐽_𝑛=1𝜓_𝑛,1(𝑧−)𝑝𝑧+𝑝𝑎₀𝜃₁𝑟_1,𝑖𝜋_1,1,0−𝑝𝑎₀𝜃₁𝑟^𝐽_1,𝑖𝑛=1𝜔_𝑛,1,0−𝑝𝜃₁𝑟_1,𝑖𝑧^𝐽−1_𝑛=1𝜔_𝑛,1,0,𝑖≥1,𝜑_3,𝑖,𝑗(𝑧)=𝑝𝜃₁𝜃₂𝑠_𝑖𝑟_2,𝑗𝜋_0,0+𝑝𝜃₁𝜃₂𝑠_𝑖𝑟_2,𝑗𝜑₀()+1,𝑧𝑝𝑧𝜃₁𝜃₂𝑠_𝑖𝑟_2,𝑗𝜑_0,1(𝑧)+𝜃₂𝛾(𝑧)𝑟_2,𝑗𝜑_1,𝑖+1(𝑧)+𝑝+𝑝𝑧𝑎₀𝜃₁𝜃₂𝑠_𝑖𝑟_2,𝑗𝜑_1,1(𝑧)+𝜑_2,1(𝑧)+^𝐽_𝑛=1𝜓_𝑛,1(𝑧(𝑧)𝜑)+𝛾_{3,𝑖,𝑗+1}(𝑧)−𝑝𝜃₁𝜃₂𝑠_𝑖𝑟_2,𝑗^𝐽−1_𝑛=1𝜔_𝑛,1,0−𝑝𝑧𝑎₀𝜃₁𝜃₂𝑠_𝑖𝑟^𝐽_2,𝑗𝑛=1𝜔_𝑛,1,0+𝜃₂𝛾(𝑧)𝜑_3,𝑖,1(𝑧)−𝑝+𝑝𝑧𝑎₀𝜃₁𝜃₂𝑠_𝑖𝑟_2,𝑗𝜋_1,1,0,𝑖≥1,𝜓_1,𝑖(𝑧)=𝑝𝜓_1,𝑖+1(𝑧)+𝑝𝑧𝜓_1,𝑖+1(𝑧)+𝛾(𝑧)𝑣_𝑖𝜋_1,1,0,𝑖≥1,𝑘≥0,𝜓_𝑛,𝑖(𝑧)=𝑝𝜓_𝑛,𝑖+1(𝑧)+𝑝𝑧𝜓_𝑛,𝑖+1(𝑧)+𝛾(𝑧)𝑣_𝑖𝜔_𝑛−1,1,0,𝑖≥1,𝑘≥0,2≤𝑛≤𝐽. (19)(20)(21)(22)(23)(24) Multiplying (23) and (24) by 𝑥^𝑖, and summing over 𝑖, we get 𝑥−𝛾(𝑧)𝑥𝜓₁(𝑥,𝑧)=−𝛾(𝑧)𝜓_1,1(𝑧)+𝛾(𝑧)𝑉(𝑥)𝜋_1,1,0,𝑥−𝛾(𝑧)𝑥𝜓_𝑛(𝑥,𝑧)=−𝛾(𝑧)𝜓_𝑛,1(𝑧)+𝛾(𝑧)𝑉(𝑥)𝜔_𝑛−1,1,0,2≤𝑛≤𝐽. (25)(26) Setting 𝑥=𝛾(𝑧) in (25) and (26), we get 𝜓_1,1(𝑧)=𝑉𝛾(𝑧)𝜋_1,1,0,𝜓_𝑛,1(𝑧)=𝑉𝛾(𝑧)𝜔_𝑛−1,1,0,2≤𝑛≤𝐽. (27)(28) Substituting (27) and (28) into (25) and (26), respectively, we obtain 𝜓₁(𝑥,𝑧)=𝑝𝑥𝑉(𝑥)−𝑉𝛾(𝑧)(𝑧)𝜋𝑥−𝛾^1,1,0,𝜓_𝑛(𝑥,𝑧)=𝑝𝑥𝑉(𝑥)−𝑉𝛾(𝑧)(𝑧)𝜔𝑥−𝛾^𝑛−1,1,0,2≤𝑛≤𝐽. (29) Taking the derivative of (29) with respective to 𝑥, and letting 𝑥=𝑧=0, we get 𝜔_1,1,0=𝜉𝜋_1,1,0,𝜔_𝑛,1,0=𝜉𝜔_𝑛−1,1,0=𝜉^𝑛𝜋_1,1,0,2≤𝑛≤𝐽. (30)(31) Substituting (31) into (3), we get 𝜋_1,1,0=𝑝𝜋_0,0𝑝𝜉^𝐽. (32) Multiplying (19) by 𝑥^𝑗, summing over 𝑗 and taking into account (30)–(32), we obtain 𝑥−𝑝𝑥𝜑₀()=𝑥,𝑧𝑝𝐴(𝑥)−𝑎₀𝜑_1,1(𝑧)+𝜑_2,1(𝑧)−𝑝𝜑_0,1(𝑧)−𝑝𝐴(𝑥)−𝑎₀𝜂₁−𝜂₂𝑉𝛾(𝑧)𝜂₃𝜋_0,0. (33) Note that we can find 𝜑₀(1,𝑧) by setting 𝑥=1 in (33), then multiplying (20) and (21) by 𝑥^𝑗, summing over 𝑗, and taking into account (30)–(32), we obtain 𝑥−𝛾(𝑧)𝜃₂𝑥𝜑₁()=𝑥,𝑧𝑧+𝑝𝑎₀(1−𝑧)𝑧𝜃₁𝜃₂𝑆(𝑥)𝜑_1,1(𝑧)+𝜑_2,1(𝑧(𝑧))−𝛾𝜃₂𝜑_1,1(𝑧)+𝑝(1−𝑧)𝑧𝜃₁𝜃₂𝑆(𝑥)𝜑_0,1(𝑧)+𝛾(𝑧)𝜃₂𝜙₁()−𝐾(𝑧)𝑥,𝑧𝑧𝑝𝜂₃𝑝𝜃₁𝜃₂𝑆(𝑥)𝜋_0,0, (34)𝑥−𝛾(𝑧)𝑥𝜑₂(𝑥,𝑧)=𝑧+𝑝𝑎₀(1−𝑧)𝜃₁𝑅₁(𝑥)×𝜑_1,1(𝑧)+𝜑_2,1(𝑧)−𝛾(𝑧)𝜑_2,1(𝑧)+𝑝𝜃₁𝑅₁(𝑥)(1−𝑧)𝜑_0,1(𝑧)−𝐾(𝑧)𝑝𝜂₃𝑝𝜃₁𝑅₁(𝑥)𝜋_0,0, (35) where 𝐾(𝑧)=𝑧𝜂₂[1−𝑉(𝛾(𝑧))]+𝑝𝑎₀(1−𝑧)Γ(𝑧).
Multiplying (22) by 𝑦^𝑗 and 𝑥^𝑖, summing over 𝑗 and 𝑖, respectively, and taking into account (30)–(32), we obtain 𝑦−𝛾(𝑧)𝑦𝜑₃𝑥,𝑦,𝑧=𝑧+𝑝𝑎₀(1−𝑧)𝑧𝜃₁𝜃₂𝑆(𝑥)𝑅₂𝑦×𝜑_1,1(𝑧)+𝜑_2,1(𝑧(𝑧)𝜃)−𝛾₂𝑅₂𝑦𝜑_1,1(𝑧)+𝑝(1−𝑧)𝑧𝜃₁𝜃₂𝑆(𝑥)𝑅_2(𝑦)𝜑_0,1(𝑧)+𝛾(𝑧)1−𝜃₂𝑅₂𝑦𝜙₁()−𝐾(𝑧)𝑥,𝑧𝑧𝑝𝜂₃𝑝𝜃₁𝜃₂𝑆(𝑥)𝑅₂𝑦𝜋_0,0+𝛾(𝑧)𝜃₂𝑅₂𝑦𝑥𝜑₁(𝑥,𝑧). (36) Note that we can find 𝜙₁(𝑥,𝑧) by setting 𝑦=𝛾(𝑧) in (36) and substituting the expression of 𝜙₁(𝑥,𝑧) into (34), we get 𝑥−𝜏₁𝛾(𝑧)𝑥1−𝜃₂𝑅₂𝛾(𝑧)𝜑₁()=𝑥,𝑧𝜃₂𝑧+𝑝𝑎₀(1−𝑧)𝑧𝜃₁𝑆(𝑥)𝜑_1,1(𝑧)+𝜑_2,1(𝑧(𝑧)𝜑)−𝛾_1,1(𝑧)+𝑝(1−𝑧)𝑧𝜃₁𝑆(𝑥)𝜑_0,1(𝑧)+𝛾(𝑧)𝜙₁()−𝐾(𝑧)𝑥,𝑧𝑧𝑝𝜂₃𝑝𝜃₁𝑆(𝑥)𝜋_0,0. (37) Setting 𝑥=𝑝 in (33), 𝑥=𝛾(𝑧) in (35), and 𝑥=𝜏(𝛾(𝑧)) in (37), respectively, we can get the equations for 𝜑_0,1(𝑧), 𝜑_1,1(𝑧) and 𝜑_2,1(𝑧). By solving these equations, we get the the generating functions as follows: 𝜑_0,1(𝑧)=𝑝𝑧𝐴𝑝−𝑎₀𝛾(𝑧)Γ(𝑧)−𝜂₃Ω(𝑧)𝑝𝜂₃Λ(𝑧)𝜋_0,0,𝜑_1,1(𝑧)=Θ(𝑧)𝑝𝜃₁𝑆𝜏𝛾(𝑧)𝑝𝜂₃Λ(𝑧)𝜋_0,0,𝜑_2,1(𝑧)=Θ(𝑧)𝑝𝜃₁𝑧𝑅𝛾(𝑧)𝑝𝜂₃Λ(𝑧)𝜋_0,0. (38)
Using Lemmas 1 and 2, it is easy to show that 𝜑_0,1(𝑧), 𝜑_1,1(𝑧), and 𝜑_2,1(𝑧) are defined for 𝑧∈[0,1) and can be extended by continuity in 𝑧=1, if 𝜃₁𝜌₁+𝜃₁+𝜃₁𝜌₂<𝑝+𝑝𝐴(𝑝). Now substituting (30)–(32) into (29), we can get 𝜑₄(𝑥,𝑧). Similarly, substituting (38) into (33), (35)–(37) we derive 𝜑₀(𝑥,𝑧), 𝜑₁(𝑥,𝑧), 𝜑₂(𝑥,𝑧) and 𝜑₃(𝑥,𝑦,𝑧). Using the normalizing condition, we can find the unknown constant 𝜋_0,0. This completes the proof.

Based on Theorem 3, we can easily obtain the marginal generating functions of the number of customers when the server is in various states and some performance measures. They are summarized in the following Corollary. Their proofs are very easy, and thus are omitted. For convenience, we define variable 𝑁 be the orbit size and 𝐿 be the system size.

Corollary 4.  (1) The marginal generating function of the number of customers in the orbit when the server is idle or vacation is given by 𝜋_0,0+𝜑₀()=𝑧1,𝑧1−𝐴𝑝𝛾(𝑧)Γ(𝑧)−𝜂₃Ω(𝑧)+𝜂₃Λ(𝑧)𝜂₃Λ(𝑧)𝜋_0,0. (39)(2) The marginal generating function of the number of customers in the orbit when the server is busy is given by 𝜑₁()=𝜏𝛾(𝑧1,𝑧1−𝑆)𝛾(𝑧)Θ(𝑧)1−𝜏𝜃₁𝜏𝛾(𝑧)𝑝𝜂₃Λ(𝑧)𝜋_0,0. (40)(3) The marginal generating function of the number of customers in the orbit when the server is under the first type of repair 𝜑₂(1,𝑧)=1−𝑅₁𝛾(𝑧)Θ(𝑧)𝜃1−𝑧¹𝑧𝛾(𝑧)𝑝𝜂₃Λ(𝑧)𝜋_0,0. (41)(4) The marginal generating function of the number of customers in the orbit when the server is under the second type of repair 𝜑₃()=𝜏𝛾(𝑧)1,1,𝑧1−𝑆1−𝜏𝛾(𝑧)1−𝑅₂𝛾(𝑧)×Θ(𝑧)1−𝑧𝜃₁𝜏𝛾(𝑧)𝑝𝜂₃Λ(𝑧)𝜋_0,0. (42)(5) The marginal generating function of the number of customers in the orbit when the server is on vacation is given by 𝜑₄()=𝜂1,𝑧²𝛾(𝑧)1−𝑉𝛾(𝑧)𝑝𝜂₃()𝜋1−𝑧^0,0. (43)(6) The generating function of the number of customers in the orbit is given by Ψ(𝑧)=𝜋_0,0+𝜑₀(1,𝑧)+𝜑₁(1,𝑧)+𝜑₂(1,𝑧)+𝜑₃(1,1,𝑧)+𝜑₄()=Θ(𝑧)1,𝑧𝜃₁𝛾(𝑧)𝑝𝜂₃Λ(𝑧)𝜋_0,0. (44)(7) The probability generating function of the number of customers in the system is given by Φ(𝑧)=𝜋_0,0+𝜑₀(1,𝑧)+𝑧𝜑₁(1,𝑧)+𝜑₂(1,𝑧)+𝑧𝜑₃(1,1,𝑧)+𝜑₄()=Θ(𝑧)1,𝑧𝜃₁𝛾(𝑧)𝑆𝜏𝛾(𝑧)𝑝𝜂₃Λ(𝑧)𝜋_0,0. (45)

Corollary 5.  (1) The system is idle with probability 𝜋_0,0=𝑝𝜂₃𝑝+𝑝𝐴𝑝−𝜃₁𝜌₁−𝜃₁−𝜃₁𝜌₂𝜃₁𝑝𝑉₁𝜂₂+𝑝𝐴𝑝𝜂₃. (46)(2) The probability that the server is free is 𝜋_0,0+𝜑₀(1,1)+𝜑₄(1,1)=1−𝜌₁−𝜃₁𝜃₁𝜌₂. (47)(3) The probability that the server is busy is 𝜑₁(1,1)=𝑝𝑆₁. (48)(4) The probability that the server is under the first type of repair is 𝜑₂(1,1)=𝑝𝑅_1,1𝜃₁𝜃₁. (49)(5) The probability that the server is under the second type of repair is 𝜑₃(1,1,1)=𝑝𝑆₁𝑅_2,1𝜃₂𝜃₂. (50)(6) The probability that the server is on vacation is 𝜑₄(1,1)=𝑝𝜂₂𝑉₁𝑝+𝑝𝐴𝑝−𝜃₁𝜌₁−𝜃₁−𝜃₁𝜌₂𝜃₁𝑝𝑉₁𝜂₂+𝑝𝐴𝑝𝜂₃. (51)(7) The mean number of customers in the orbit is 𝐸(𝑁)=Ψ^′(𝑧)_𝑧=1=Ω^′′(1)+2𝑝1−𝐴𝑝Ω^′(1)−𝑝2𝑝+𝑝𝐴𝑝−Ω^′(1+𝜂₂)2𝑝𝑉₁1−𝑝𝐴𝑝+𝑝²𝑉₂2𝑝𝑉₁𝜂₂+𝑝𝐴𝑝𝜂₃, (52) where Ω^′(1)=𝜃₁𝜌₁+𝜃₁+𝜃₁𝜌₂,Ω^′′(1)=𝜃₁𝑝²𝑆₂𝜃1+²𝜃₂²+𝑝²𝑆₁×𝜃₂𝜃₂𝑅_2,2𝜃+2²𝜃₂𝑅_2,1𝜃1+²𝜃₂+2𝜃₁𝑝𝑅_1,1+𝜃₁𝑝²𝑅_1,2. (53)(54)(8) The mean number of customers in the system is 𝐸(𝐿)=Φ^′(𝑧)_𝑧=1=𝐸(𝑁)+𝜌₁. (55)

Remark 6. Consider some special cases as follows.(i) When 𝐽=0, 𝜃₁=𝜃₂=0, our model becomes a discrete-time 𝐺𝑒𝑜/𝐺/1 queue with general retrial times. In this case, Φ(𝑧) reduces to Φ(𝑧)=(1−𝑧)𝛾(𝑧)𝑝+𝑝𝐴𝑝−𝑝𝑆₁𝑆𝛾(𝑧)𝑧+𝑝𝐴𝑝(𝑆𝛾(𝑧)(𝑧),1−𝑧)−𝑧𝛾 (56) which coincides with results of the model studied by Atencia and Moreno [5].(ii) When 𝐽=0, 𝜃₂=0, the present model becomes a discrete-time 𝐺𝑒𝑜/𝐺/1 queue with general retrial times and starting failure. In this case Φ(𝑧) reduces to Φ(𝑧)=()𝛾(𝑧)𝑆𝛾(𝑧×1−𝑧)𝑝+𝑝𝐴𝑝−𝜃₁𝑝𝑆₁−𝜃₁−𝜃₁𝑝𝑅_2,1×𝑧+𝑝𝐴𝑝(×1−𝑧)𝜃₁𝑆𝛾(𝑧)+𝑧𝜃₁𝑅₁𝛾(𝑧)−𝑧𝛾(𝑧)⁻¹. (57)(iii) When 𝜃₁=𝜃₂=0, our model becomes a discrete-time 𝐺𝑒𝑜/𝐺/1 queue with 𝐽 vacations. In this case, Φ(𝑧) reduces to Φ(𝑧)=𝑧𝜂₂1−𝑉𝛾(𝑧)+𝑝𝐴𝑝(1−𝑧)Γ(𝑧)𝑧+𝑝𝐴𝑝(𝑆𝛾(𝑧)(𝑧)𝑆𝛾(𝑧𝜋1−𝑧)−𝑧𝛾)_0,0, (58) where 𝜋_0,0=𝑝𝜂₃𝑝+𝑝𝐴𝑝−𝑝𝑆₁𝑝𝑉₁𝜂₂+𝑝𝐴𝑝𝜂₃. (59)

4. Stochastic Decomposition

The property of stochastic decomposition was proposed firstly in queueing system with vacations. This property has also been studied for retrial queues with vacations by Artalejo [17]. The property of stochastic decomposition shows that the steady-state system size at an arbitrary point can be represented as the sum of two independent random variable, one of which is the system size of the corresponding queueing system without server vacations and the other is the orbit size given that the server is on vacations. In this section, we present two different stochastic decompositions of the system size distribution in our model.

Theorem 7. The total number of customers (𝐿) in the system can be represented as the sum of two independent random variables 𝐿=𝐿₁+𝑀₁. 𝐿₁ is the number of customers in the 𝐺𝑒𝑜/𝐺/1/ queue with server breakdowns and 𝑀₁ is the number of repeated customers given that the server is idle, under the first type of repair or on vacation.

Proof. After some algebra operation, Φ(𝑧) can be expressed by Φ(𝑧)=Φ₁(𝑧)Φ₂(𝑧), (60) where Φ₁(𝑧)=1−𝜌₁𝑆𝜏𝛾(𝑧)(1−𝑧)𝑆𝜏𝛾(𝑧)−𝑧 (61) is the generating function of the 𝐺𝑒𝑜/𝐺/1 queue server with breakdowns, and Φ₂(𝑧)=𝑆𝜏𝛾(𝑧)−𝑧×1−𝑧𝑧𝜂₂1−𝑉𝛾(𝑧)+𝑝𝐴𝑝(1−𝑧)Γ(𝑧)1−𝜌₁Λ(𝑧)×𝑝+𝑝𝐴𝑝−𝜃₁𝜌₁−𝜃₁−𝜃₁𝜌₂𝑝𝑉₁𝜂₂+𝑝𝐴𝑝𝜂₃=𝜋_0,0+𝜑₀(1,𝑧)+𝜑₂(1,𝑧)+𝜑₄(1,𝑧)𝜋_0,0+𝜑₀(1,1)+𝜑₂(1,1)+𝜑₄(1,1) (62) is the generating function given that the server is idle, under first type of repair or on vacations. This completes the proof.

Theorem 8. The total number of customers (𝐿) in the system can be expressed as the sum of two independent random variables 𝐿=𝐿₂+𝑀₂. 𝐿₂ is the number of customers in the 𝐺𝑒𝑜/𝐺/1/ queue with two types of breakdowns and 𝑀₂ is the number of repeated customers given that the server is idle or on vacation.

Proof. After some algebra operation, Φ(𝑧) can be expressed by Φ(𝑧)=Φ₁(𝑧)Φ₂(𝑧), (63) where Φ₁(𝑧)=1−𝜃₁𝜌₁−𝜃₁−𝜃₁𝜌₂𝑆𝜏𝛾(𝑧)(1−𝑧)𝜃₁𝑆𝜏𝛾(𝑧)+𝑧𝜃₁𝑅₁𝛾(𝑧)−𝑧 (64) is the generating function of the unreliable 𝐺𝑒𝑜/𝐺/1 queue, and Φ₂(𝑧)=𝜃₁𝑆𝜏𝛾(𝑧)+𝑧𝜃₁𝑅₁𝛾(𝑧)−𝑧×1−𝑧𝑧𝜂₂1−𝑉𝛾(𝑧)+𝑝𝐴𝑝(1−𝑧)Γ(𝑧)1−𝜃₁𝜌₁−𝜃₁−𝜃₁𝜌₂Λ(𝑧)×𝑝+𝑝𝐴𝑝−𝜃₁𝜌₁−𝜃₁−𝜃₁𝜌₂𝑝𝑉₁𝜂₂+𝑝𝐴𝑝𝜂₃=𝜋_0,0+𝜑₀(1,𝑧)+𝜑₄(1,𝑧)𝜋_0,0+𝜑₀(1,1)+𝜑₄(1,1) (65) is the generating function given that the server is idle or on vacations. This completes the proof.

5. Relationship to the Corresponding Continuous-Time Model

In this section, we investigate the relationship between our discrete-time model and the corresponding continuous-time counterpart. We consider a continuous-time 𝑀/𝐺/1 general retrial queue with 𝐽 vacations and two types of breakdowns. The customers arrive according to a Poisson process with rate 𝜆. If an arriving customer finds that the server is busy, under repair or on vacations, the customer joins the orbit and only the first customer in the orbit is permitted to retry for service. The server is subject to two types of different breakdowns. As soon as the server fails, it is repaired immediately. The first type of breakdown is starting failure, that is, an arriving customer must start the server, which takes negligible time and the server is started successfully with a probability 𝜃₁. In addition to starting failure, the server may break down during serving an customer and the lifetime of the server follows exponential distribution with rate 𝜃₂.

Once the system becomes empty, the server takes vacations immediately according to 𝐽 vacation policy. The retail times, the service times, the repair times for the two types of breakdowns and the vacation times are all assumed to follow general continuous-time distributions denoted by 𝐴(𝑥), 𝑆(𝑥), 𝑅₁(𝑥), 𝑅₂(𝑥), and 𝑉(𝑥), respectively, and their Laplace-Stieltjes transforms denoted by 𝐴(𝑠), 𝑆(𝑠), 𝑅₁(𝑠), 𝑅₂(𝑠), and 𝑉(𝑠), respectively, and their finite means are denoted by 𝐴₁, 𝑆₁ and 𝑅_1,1, 𝑅_2,1, and 𝑉₁ respectively.

We suppose that the time is divided into sufficiently small intervals of equal length Δ. Then, the continuous-time 𝑀/𝐺/1 general retrial queue with 𝐽 vacations and two types of breakdowns can be approximated by the following: 𝑝=𝜆Δ,𝑎_𝑖=^(𝑖+1)Δ_𝑖Δ𝑑𝐴(𝑥),𝑠_𝑖=_()Δ^𝑖Δ_𝑖−1(𝑥),𝑅𝑑𝑆_𝑙,𝑖=_()Δ^𝑖Δ_𝑖−1𝑑𝑅_𝑙(𝑥),𝑙=1,2,𝑣_𝑖=_()Δ^𝑖Δ_𝑖−1𝑑𝑉(𝑥). (66) By using the definition of Lebesgue integration, we can get the following equalities: lim_Δ→0𝜌₂=𝜆𝑅_2,1=𝜌₂,lim_Δ→0𝑝𝑉_𝑖=𝜆𝑉₁,lim_Δ→0𝜌₁=𝜆𝑆₁𝜃1+²𝜃₂𝑅_2,1=𝜌₁lim_Δ→0𝑆𝜏𝛾(𝑧)=𝑆𝜆(1−𝑧)+𝜃₂𝑅₂(𝜆(=𝑆𝜏𝛾(𝑧),1−𝑧))lim_Δ→0𝑉₂𝑝=𝑉₂(𝜆),lim_Δ→0𝑅₁𝛾(𝑧)=𝑅₁[𝜆(1−𝑧)],lim_Δ→0𝐴𝑝=𝐴(𝜆),lim_Δ→0𝑉𝛾(𝑧)=𝑉[𝜆(1−𝑧)],lim_Δ→0𝜂_𝑖=𝜂_𝑖,𝑖=1,2,3. (67) From the above relations, we obtain the generating function of the system size for the corresponding continuous-time model by lim_Δ→0Φ(𝑧)=𝑧𝜂₂𝑉[𝜆(+𝐴(𝜆)𝜂1−1−𝑧)])(1−𝑧₁−𝜂₂𝑉[𝜆(×1−𝑧)]𝜃₁𝑆𝜏𝛾(𝑧)+𝑧𝜃₁𝑅₁[𝜆(×𝐴(𝜆1−𝑧)]−𝑧𝑧+)(1−𝑧)−𝑧⁻¹×𝐴(𝜆)−𝜃₁𝜌₁−𝜃₁−𝜃₁𝜌₂𝜆𝑉₁𝜂₂+𝐴(𝜆)𝜂₃𝑆𝜏𝛾(𝑧). (68)

6. Numerical Examples

In this section, we present some numerical examples to illustrate the effect of the parameters on the main performance measures of the system. We consider two important performance measures: the probability that server is on vacation 𝑃_𝑉 and the mean orbit size 𝐸(𝑁). It is assumed that the retrial time is geometric distribution with generating function 𝐴(𝑥)=(1−𝑟)/(1−𝑟𝑥). The service time, the repair times of the server for the two types of breakdowns, and the vacation time are also assumed to follow geometric distribution with generating functions 𝑆(𝑥)=(1−𝜇)𝑥/(1−𝜇𝑥), 𝑅_𝑙(𝑥)=(1−𝑟_𝑙)𝑥/(1−𝑟_𝑙𝑥), 𝑙=1,2, and 𝑉(𝑥)=(1−𝑣)𝑥/(1−𝑣𝑥), respectively. In general, we have chosen the arrival rate 𝑝=0.1, the retrial rate 𝑟=0.25, the service rate 𝜇=0.5, and the vacation rate 𝑣=0.5, and we have presented three curves which correspond to 𝜃₁=0.1, 0.3 and 0.5, respectively, in Figures 1–4, where 𝜃₁ is the probability that the server experience the first type of breakdown.

Figure 1 

The curve of 𝑃𝑉 with 𝐽.

Figure 2 

The curve of 𝑃𝑉 with 𝜃2.

Figure 3 

The curve of 𝐸(𝑁) with 𝐽.

Figure 4 

The curve of 𝐸(𝑁) with 𝜃2.

Figure 1 describes 𝑃_𝑉 with varying values of the maximum number of the server's vacations 𝐽. As illustrated in Figure 1, 𝑃_𝑉 is increasing as a function of 𝐽. For different values of 𝜃₁, we observe that 𝑃_𝑉 decreases with increasing values of 𝜃₁. That is because the mean service time of a customer is getting longer with increasing values of 𝜃₁ and 𝑃_𝑉 decreases accordingly. Figure 2 describes the effect of 𝜃₂ on 𝑃_𝑉. The curve in Figure 2 shows that 𝑃_𝑉 is decreasing as a function of 𝜃₂ which agrees with intuitive expectations.

Figures 3 and 4 describe the influence of the parameter 𝐽 and 𝜃₂ on the mean orbit size 𝐸(𝑁), respectively. As to be expected, 𝐸(𝑁) increases with increasing value of 𝐽 and 𝜃₂, that is, there are more customers in the orbit as 𝐽 and 𝜃₂ increase.

As we can observe in Figures 3 and 4, the mean orbit size 𝐸(𝑁) increases with increasing values of 𝜃₁. As the the value of 𝜃₁ increases, the mean sojourn time time for an customer increases, and therefore the number of customers in the system increases. Moreover, the parameter 𝜃₁ affects 𝐸(𝑁) more apparently, when the value of the parameter 𝜃₁ is getting greater.

Acknowledgments

This research is partially supported by the Youth Science Foundation of Shanxi Province (No. 2012021001-1).