A Discrete-Time Unreliable Geo/G/1 Retrial Queue with Balking Customers, Second Optional Service, and General Retrial Times

Abstract

This paper deals with the steady-state behavior of a discrete-time unreliable Geo/G/1 retrial queueing system with balking customers and second optional service. The server may break down randomly while serving the customers. If the server breaks down, the server is sent to be repaired immediately. We analyze the Markov chain underlying the considered system and its ergodicity condition. Then, we obtain some performance measures based on the generating functions. Moreover, a stochastic decomposition result of the system size is investigated. Finally, some numerical examples are provided to illustrate the effect of some parameters on main performance measures of the system.

1. Introduction

During the past few decades, there has been increasing interest in studying retrial queueing systems because they are widely used in performance analysis of many practical systems such as call center, computer systems, and telecommunication networks. Retrial queueing systems are characterized by the feature that a blocked customer (a customer who finds the server unavailable) may leave the service area temporarily and join a retrial group in order to retry his request after some random time. For more detailed review of the main results and references on the retrial queueing systems, the readers are referred to [1–3].

In the past, the research of retrial queueing systems focused mainly on the continuous-time case [4, 5]. Compared with the continuous-time counterparts, the discrete-time retrial queueing systems received less attention in the literature. One of the difficulties in developing the discrete-time retrial queueing systems lied in the fact that the complexity of the discrete-time retrial queueing models increases. The reason is that in a discrete-time retrial queueing model, arrivals, departures, and retrials can occur simultaneously; that is, the probability that two or more events occur at every slot is positive. In fact; the research on discrete-time retrial queueing systems is very important due to the fact that they are more appropriate than their continuous-time counterparts for modeling slotted digital computer and communication systems such as broadband integrated services digital network (B-ISDN) and time-division multiple access (TDMA) systems. The first work on discrete-time retrial queueing system was discussed by Yang and Li [6]. They extended the continuous-time M/G/1 retrial queueing system to the discrete-time case and save the relationship between the Geo/G/1 retrial queues and the corresponding continuous-time counterpart. Atencia and Moreno [7] generalized the model of Yang and Li to the discrete-time Geo/G/1 retrial queue with general retrial times. For related literature on discrete-time retrial queueing systems, the reader may refer to [8, 9].

Most of the retrial queueing models in the literature assumed that customers received service eventually. However, in practice, customers may balk or abandon the system before receiving service. Queueing models with balking or impatience customers arise in many diverse applications. Although the incorporation of balking or impatience customers in retrial queueing systems is very important, so far there is only little work on retrial queueing systems with balking or impatience customers in the literature [10–12]. On the other hand, queueing models in which the server may provide a second phase service have been proved very useful to analyze many practical situations, arising in manufacturing systems, call center, and computer systems. The retrial queueing systems with second phase service were studied by Artalejo and Choudhury [13], Atencia and Moreno [14], and Choudhury and Deka [15].

In many practical situations, the server may be subjected to unpredictable breakdowns. The breakdowns of the server may have a great influence on performance measures of some practical systems. So, the research of retrial queues with unreliable server is very important. There have been significant contributions to the retrial queueing systems with unreliable server. For example, the continuous-time retrial queueing systems which take into account servers breakdowns and repairs were discussed by Kulkarni and Choi [16], Wang et al. [17, 18], and Li et al. [19]. Parallel to the continuous-time unreliable retrial queues, the discrete-time retrial queues with starting failure were studied by Atencia and Moreno [20], Wang and Zhao [21], and Atencia et al. [22]. The discrete-time unreliable retrial queueing systems caused by negative customers were discussed by Wang and Zhang [23]. In contrast to the retrial queueing system with starting failure, the analysis of discrete-time retrial queueing system is more complex. Though the discrete-time retrial queueing systems with unreliable server have been studied by some authors, there is little work about the discrete-time unreliable retrial queueing systems which assumed that the server’s life is random time [24].

To the best of our knowledge, there is no published work on discrete-time Geo/G/1 retrial queueing system with balking customer, second optional service, and unreliable server. The main objective in this paper is to extend the analysis of continuous-time M/G/1 retrial queueing system with second optional service [13], exponential server lifetime [17], and general retrial times to discrete-time case. Moreover, our model generalized the discrete-time retrial queueing system studied by Atencia and Moreno [7] and Aboul-Hassan et al. [11] directly. Our queueing system can be used to model many practical situations. For example, in the Voice-over-IP (VoIP) system, the analogue voice information is digitized, sent to the destination, and converted to the original analogue voice signal. In the existing VoIP system, the admission control is proposed for improving the quality of service (QoS) and the VoIP system may reject some data packets request due to network congestion. The admitted packets can be placed in the buffer and then can be sent to another end user for further process. The VoIP systems may break down during establishing connection, digitizing, and transferring information. As soon as the VoIP system breaks down, it is repaired immediately.

The rest of the paper is organized as follows. In Section 2, we give a detailed description of the mathematical model. In Section 3, the Markov chain underlying the queueing system is analyzed and some performance measures of the system are also obtained. In Section 4, we derive a stochastic decomposition result. Finally, some numerical results are provided to show the impact of the some parameters on the performance measures of the system in Section 5.

2. The Mathematical Model

We consider a discrete-time single server retrial queue where the time axis is segmented into slots of equal length and all queueing activities occur at the slot boundaries. Let the time axis be marked by 0,1,2,…,𝑚,…. Specially, we assume that the departures and the end of the repairs occur in the interval (𝑚⁻,𝑚), and the arrivals, the retrials, and the beginning of the repairs occur in the interval (𝑚,𝑚⁺). That is, we consider an early arrival system (EAS) policy.

Customers arrive at the system according to a geometric arrival process with parameter 𝑝, where 𝑝 is the probability that an arrival occurs in a slot. If an arriving customer finds that the server is idle, he begins his service immediately and leaves the system forever after service completion. Otherwise, if an arriving customer finds that the server is busy or under repair, he joins the orbit with probability 𝛼 or leaves the system completely with probability 1−𝛼. We assumed that only the first customer in the orbit is permitted to access to the server. Successive interretrial times of any customer follow an arbitrary distribution {𝑎_𝑖}^∞_𝑖=0 with generating function 𝐴(𝑥)=∑^∞_𝑖=0𝑎_𝑖𝑥^𝑖.

The server provides two phases of different services for customers in succession. As soon as the first-phase service (FPS) of a customer is completed, he may receive the second-phase service (SPS) with probability 𝑟 or leaves the system with probability 𝑟=1−𝑟. The service times of the FPS and SPS are independent and arbitrarily distributed with distributions {𝑠_1,𝑖}^∞_𝑖=0, {𝑠_2,𝑖}^∞_𝑖=0, and generating functions 𝑆₁(𝑥)=∑^∞_𝑖=0𝑠_1,𝑖𝑥^𝑖, 𝑆₂(𝑥)=∑^∞_𝑖=0𝑠_2,𝑖𝑥^𝑖, respectively. The corresponding 𝑛th factorial moments are 𝛽_1,𝑛 and 𝛽_2,𝑛, respectively.

The server may break down during serving customers. If the server breaks down, it is sent to repair immediately; the customer just being served before the server breakdown waits in the service station until the server is repaired to complete his remaining service. It is assumed that the lifetimes of the server for the FPS and SPS are geometrically distributed with parameters 𝜃₁=1−𝜃₁ and 𝜃₂=1−𝜃₂, respectively, where 𝜃₁ and 𝜃₂ are the probability that the failures do not occur in a slot, respectively.

The repair times for the FPS and SPS are independent and identically distributed with arbitrary distributions {𝑠_3,𝑖}^∞_𝑖=0 and {𝑠_4,𝑖}^∞_𝑖=0, respectively. The corresponding generating functions and 𝑛th factorial moments are 𝑆₃(𝑥)=∑^∞_𝑖=0𝑠_3,𝑖𝑥^𝑖, 𝑆₄(𝑥)=∑^∞_𝑖=0𝑠_4,𝑖𝑥^𝑖, 𝛽_3,𝑛 and 𝛽_4,𝑛, respectively.

Finally, we suppose that various stochastic processes involved in the system are independent of each other. In order to avoid trivial cases, it is also supposed that 0<𝑝<1, 0≤𝑟<1, 0<𝜃₁≤1, 0<𝜃₂≤1.

3. The Markov Chain

At time 𝑚⁺, the system can be described by the process 𝑌_𝑚=(𝐶_𝑚,𝜉_0,𝑚,𝜉_1,𝑚,𝜉_2,𝑚,𝜉_3,𝑚,𝜉_4,𝑚,𝑁_𝑚), where 𝐶_𝑚 denotes the state of the server, 0, 1, 2, 3, or 4, according to whether the server is free, busy with FPS, busy with SPS, under repair during FPS, or under repair during SPS, and 𝑁_𝑚 is the number of the customers in the orbit. If 𝐶_𝑚=0, then 𝜉_0,𝑚 represents the remaining retrial time. If 𝐶_𝑚=1,2, then 𝜉_1,𝑚, 𝜉_2,𝑚 represent the customers’ remaining service time for FPS and SPS, respectively. If 𝐶_𝑚=3,4, then 𝜉_1,𝑚, 𝜉_2,𝑚 represent the remaining service time of the customer who was being served just before the server breakdown and 𝜉_3,𝑚, 𝜉_4,𝑚 represent the remaining repair time of the server for the FPS and SPS, respectively. It can be shown that {𝑌_𝑚,𝑚≥1} is a Markov chain with the following state space: ∪)}∪)}∪)}∪∪.Ω={(0,0)}{(0,𝑖,𝑘∶𝑖≥1,𝑘≥1{(1,𝑖,𝑘∶𝑖≥1,𝑘≥0{(2,𝑖,𝑘∶𝑖≥1,𝑘≥03,𝑖,𝑗,𝑘∶𝑖≥1,𝑗≥1,𝑘≥04,𝑖,𝑗,𝑘∶𝑖≥1,𝑗≥1,𝑘≥0 (1) Define the stationary probabilities of the Markov chain as follows: 𝜋_0,0=lim_𝑚→∞𝑃𝐶_𝑚=0,𝑁_𝑚,𝜋=0_0,𝑖,𝑘=lim_𝑚→∞𝑃𝐶_𝑚=0,𝜉_0,𝑚=𝑖,𝑁_𝑚𝜋=𝑘;𝑖≥1,𝑘≥1,_{𝑛,𝑖,𝑘}=lim_𝑚→∞𝑃𝐶_𝑚=𝑛,𝜉_1,𝑚=𝑖,𝑁_𝑚;𝜋=𝑘𝑛=1,2,𝑖≥1,𝑘≥0,_{𝑛,𝑖,𝑗,𝑘}=lim_𝑚→∞𝑃𝐶_𝑚=𝑛,𝜉_1,𝑚=𝑖,𝜉_3,𝑚=𝑗,𝑁_𝑚=𝑘;𝑛=3,4,𝑖≥1,𝑗≥1,𝑘≥0. (2)

Then, the Kolmogorov equations for the stationary distribution are 𝜋_0,0=𝑝𝜋_0,0+𝑝𝑟𝜋_1,1,0+𝑝𝜋_2,1,0,𝜋_0,𝑖,𝑘=𝑝𝜋_{0,𝑖+1,𝑘}+𝑝𝑎_𝑖𝑟𝜋_1,1,𝑘+𝑝𝑎_𝑖𝜋_2,1,𝑘𝜋,𝑖≥1,𝑘≥1,_1,𝑖,𝑘=𝛿_0,𝑘𝑝𝜃₁𝑠_1,𝑖𝜋_0,0+1−𝛿_0,𝑘𝑝𝜃₁𝑠^∞_1,𝑖𝑗=1𝜋_0,𝑗,𝑘+𝑝𝜃₁𝑠_1,𝑖𝜋_0,1,𝑘+1+𝑝𝜃₁𝑠_1,𝑖𝑟𝜋_1,1,𝑘+𝑝𝑎₀𝜃₁𝑟𝑠_1,𝑖𝜋_1,1,𝑘+1+𝑝𝜃₁𝑠_1,𝑖𝜋_2,1,𝑘+𝑝𝑎₀𝜃₁𝑠_1,𝑖𝜋_2,1,𝑘+1+𝑝+𝑝𝛼𝜃₁𝜋_{1,𝑖+1,𝑘}+1−𝛿_0,𝑘𝑝𝛼𝜃₁𝜋_{1,𝑖+1,𝑘−1}+1−𝛿_0,𝑘𝑝𝛼𝜃₁𝜋_{3,𝑖,1,𝑘−1}+𝑝+𝑝𝛼𝜃₁𝜋_{3,𝑖,1,𝑘}𝜋,𝑖≥1,𝑘≥0,_2,𝑖,𝑘=𝑝+𝑝𝛼𝑟𝜃₂𝑠_2,𝑖𝜋_1,1,𝑘+1−𝛿_0,𝑘×𝑝𝛼𝑟𝜃₂𝑠_2,𝑖𝜋_1,1,𝑘−1+𝑝+𝑝𝛼𝜃₂𝜋_{2,𝑖+1,𝑘}+1−𝛿_0,𝑘𝑝𝛼𝜃₂𝜋_{2,𝑖+1,𝑘−1}+1−𝛿_0,𝑘𝑝𝛼𝜃₂𝜋_{4,𝑖,1,𝑘−1}+𝑝+𝑝𝛼𝜃₂𝜋_{4,𝑖,1,𝑘}𝜋,𝑖≥1,𝑘≥0,_{3,𝑖,𝑗,𝑘}=𝛿_0,𝑘𝑝𝜃₁𝑠_1,𝑖𝑠_3,𝑗𝜋_0,0+1−𝛿_0,𝑘×𝑝𝜃₁𝑠_1,𝑖𝑠^∞_3,𝑗𝑙=1𝜋_0,𝑙,𝑘+𝑝𝜃₁𝑠_1,𝑖𝑠_3,𝑗𝜋_0,1,𝑘+1+𝑝𝜃₁𝑠_1,𝑖𝑠_3,𝑗𝑟𝜋_1,1,𝑘+𝑝𝜃₁𝑎₀𝑠_1,𝑖𝑠_3,𝑗𝑟𝜋_1,1,𝑘+1+𝑝𝜃₁𝑠_1,𝑖𝑠_3,𝑗𝜋_2,1,𝑘+𝑝𝜃₁𝑎₀𝑠_1,𝑖𝑠_3,𝑗𝜋_2,1,𝑘+1+1−𝛿_0,𝑘𝑝𝛼𝜃₁𝑠_3,𝑗𝜋_{1,𝑖+1,𝑘−1}+𝑝+𝑝𝛼𝜃₁𝑠_3,𝑗𝜋_{1,𝑖+1,𝑘}1−𝛿_0,𝑘𝑝𝛼𝜃₁𝑠_3,𝑗𝜋_{3,𝑖,1,𝑘−1}+𝑝+𝑝𝛼𝜃₁𝑠_3,𝑗𝜋_{3,𝑖,1,𝑘}+𝑝+𝑝𝛼𝜋_{3,𝑖,𝑗+1,𝑘}+1−𝛿_0,𝑘𝑝𝛼𝜋_{3,𝑖,𝑗+1,𝑘−1}𝜋,𝑖≥1,𝑗≥1,𝑘≥0,_{4,𝑖,𝑗,𝑘}=𝑝+𝑝𝛼𝑟𝜃₂𝑠_2,𝑖𝑠_4,𝑗𝜋_1,1,𝑘+1−𝛿_0,𝑘𝑝𝛼𝑟𝜃₂𝑠_2,𝑖𝑠_4,𝑗𝜋_1,1,𝑘−1+1−𝛿_0,𝑘𝑝𝛼𝜃₂𝑠_4,𝑗𝜋_{2,𝑖+1,𝑘−1}+𝑝+𝑝𝛼𝜃₂𝑠_4,𝑗𝜋_{2,𝑖+1,𝑘}+1−𝛿_0,𝑘𝑝𝛼𝜃₂𝑠_4,𝑗𝜋_{4,𝑖,1,𝑘−1}+𝑝+𝑝𝛼𝜃₂𝑠_4,𝑗𝜋_{4,𝑖,1,𝑘}+𝑝+𝑝𝛼𝜋_{4,𝑖,𝑗+1,𝑘}+1−𝛿_0,𝑘𝑝𝛼𝜋_{4,𝑖,𝑗+1,𝑘−1},𝑖≥1,𝑗≥1,𝑘≥0, (3)(4)(5)(6)(7)(8) with the normalizing condition 𝜋_0,0+^∞∞_𝑖=1𝑘=1𝜋_0,𝑖,𝑘+^2∞_𝑛=1^∞_𝑖=1𝑘=0𝜋_{𝑛,𝑖,𝑘}+^4∞_𝑙=3^∞_𝑖=1^∞_𝑗=1𝑘=0𝜋_{𝑙,𝑖,𝑗,𝑘}=1, (9) where 𝛿_0,𝑘 denotes the Kronecker delta. To resolve (3)–(8), we introduce the following generating functions: 𝜙₀(𝑥,𝑧)=^∞∞_𝑖=1𝑘=1𝜋_0,𝑖,𝑘𝑥^𝑖𝑧^𝑘,𝜙_𝑛(𝑥,𝑧)=^∞∞_𝑖=1𝑘=0𝜋_{𝑛,𝑖,𝑘}𝑥^𝑖𝑧^𝑘𝜙,𝑛=1,2,_𝑙𝑥,𝑦,𝑧=^∞∞_𝑖=1^∞_𝑗=1𝑘=0𝜋_{𝑙,𝑖,𝑗,𝑘}𝑥^𝑖𝑦^𝑗𝑧^𝑘,𝑙=3,4, (10) and the auxiliary generating functions: 𝜙_0,𝑖(𝑧)=^∞_𝑘=1𝜋_0,𝑖,𝑘𝑧^𝑘𝜙,𝑖≥1,_𝑛,𝑖(𝑧)=^∞_𝑘=0𝜋_{𝑛,𝑖,𝑘}𝑧^𝑘𝜙,𝑛=1,2,𝑖≥1,_{𝑙,𝑖,𝑗}(𝑧)=^∞_𝑘=0𝜋_{𝑙,𝑖,𝑗,𝑘}𝑧^𝑘𝜙,𝑙=3,4,𝑖≥1,𝑗≥1,_{𝑚,𝑖,𝑗}𝑦,𝑧=^∞_𝑘=0𝜋_{𝑚,𝑖,𝑗,𝑘}𝑦^𝑗𝑧^𝑘𝜓,𝑚=3,4,𝑖≥1,𝑗≥1,₁(𝑥,𝑧)=^∞∞_𝑖=1𝑘=0𝜋_{3,𝑖,1,𝑘}𝑥^𝑖𝑧^𝑘,𝜓₂(𝑥,𝑧)=^∞∞_𝑖=1𝑘=0𝜋_{4,𝑖,1,𝑘}𝑥^𝑖𝑧^𝑘. (11) Multiplying (4)–(8) by 𝑧^𝑘 and summing over 𝑘, we get the following equations: 𝜙_0,𝑖(𝑧)=𝑝𝜙_0,𝑖+1(𝑧)+𝑝𝑎_𝑖𝑟𝜙_1,1(𝑧)+𝑝𝑎_𝑖𝜙_2,1(𝑧)−𝑝𝑎_𝑖𝑟𝜋_1,1,0−𝑝𝑎_𝑖𝜋_2,1,0𝜙,𝑖≥1,_1,𝑖(𝑧)=𝑝𝜃₁𝑠_1,𝑖𝜋_0,0+𝑝𝜃₁𝑠^∞_1,𝑖𝑗=1𝜙_0,𝑗(𝑧)+𝑝𝑧𝜃₁𝑠_1,𝑖𝜙_0,1(𝑧)+𝑝𝜃₁𝑠_1,𝑖𝑟𝜙_1,1(𝑧)+𝑝𝑧𝜃₁𝑎₀𝑠_1,𝑖𝑟𝜙_1,1(𝑧)−𝜋_1,1,0+𝑝𝜃₁𝑠_1,𝑖𝜙_2,1(𝑧)+𝑝𝑧𝜃₁𝑎₀𝑠_1,𝑖𝜙_2,1(𝑧)−𝜋_2,1,0+𝑝+𝑝𝛼+𝑧𝑝𝛼𝜃₁𝜙_1,𝑖+1(𝑧)+𝑝+𝑝𝛼+𝑧𝑝𝛼𝜃₁𝜙_3,𝑖,1(𝑧)𝜙,𝑖≥1,_2,𝑖(𝑧)=𝑝+𝑝𝛼+𝑧𝑝𝛼𝜃₂𝑟𝑠_2,𝑖𝜙_1,1(𝑧)+𝑝+𝑝𝛼+𝑧𝑝𝛼𝜃₂𝜙_2,𝑖+1(𝑧)+𝑝+𝑝𝛼+𝑧𝑝𝛼𝜃₂𝜙_4,𝑖,1(𝑧)𝜙,𝑖≥1,_3,𝑖,𝑗(𝑧)=𝑝𝜃₁𝑠_1,𝑖𝑠_3,𝑗𝜋_0,0+𝑝𝜃₁𝑠_1,𝑖𝑠^∞_3,𝑗𝑙=1𝜙_0,𝑙(𝑧)+𝑝𝑧𝜃₁𝑠_1,𝑖𝑠_3,𝑗𝜙_0,1(𝑧)+𝑝𝜃₁𝑠_1,𝑖𝑠_3,𝑗𝑟𝜙_1,1(𝑧)+𝑝𝑧𝜃₁𝑎₀𝑠_1,𝑖𝑠_3,𝑗𝑟𝜙_1,1(𝑧)−𝜋_1,1,0+𝑝𝜃₁𝑠_1,𝑖𝑠_3,𝑗𝜙_2,1(𝑧)+𝑝𝑧𝜃₁𝑎₀𝑠_1,𝑖𝑠_3,𝑗×𝜙_2,1(𝑧)−𝜋_2,1,0+𝑝+𝑝𝛼+𝑧𝑝𝛼𝜃₁𝑠_3,𝑗𝜙_1,𝑖+1(𝑧)+𝑝+𝑝𝛼+𝑧𝑝𝛼𝜃₁𝑠_3,𝑗𝜙_3,𝑖,1(𝑧)+𝑝+𝑝𝛼+𝑧𝑝𝛼𝜙_{3,𝑖,𝑗+1}(𝑧)𝜙,𝑖≥1,𝑗≥1,_4,𝑖,𝑗(𝑧)=𝑝+𝑝𝛼+𝑧𝑝𝛼𝜃₂𝑟𝑠_2,𝑖𝑠_4,𝑗𝜙_1,1(𝑧)+𝑝+𝑝𝛼+𝑧𝑝𝛼𝜃₂𝑠_4,𝑗𝜙_2,𝑖+1(𝑧)+𝑝+𝑝𝛼+𝑧𝑝𝛼𝜃₂𝑠_4,𝑗𝜙_4,𝑖,1(𝑧)+𝑝+𝑝𝛼+𝑧𝑝𝛼𝜙_{4,𝑖,𝑗+1}(𝑧),𝑖≥1. (12)(13)(14)(15)(16)

By substituting (3) into (12), (13), and (15), we get 𝜙_0,𝑖(𝑧)=𝑝𝜙_0,𝑖+1(𝑧)+𝑝𝑎_𝑖𝑟𝜙_1,1(𝑧)+𝑝𝑎_𝑖𝜙_2,1(𝑧)−𝑝𝑎_𝑖𝜋_0,0𝜙,𝑖≥1,_1,𝑖(𝑧)=𝑝𝑧𝜃₁𝑠_1,𝑖𝜙_0,1(𝑧)+𝑝𝜃₁𝑠_1,𝑖𝜙₀()+1,𝑧𝑝+𝑝𝑧𝑎₀𝜃₁𝑟𝑠_1,𝑖𝜙_1,1(𝑧)+𝑝+𝑝𝑧𝑎₀𝜃₁𝑠_1,𝑖𝜙_2,1(𝑧)+𝑝+𝑝𝛼+𝑧𝑝𝛼𝜃₁𝜙_1,𝑖+1(𝑧)+𝑝+𝑝𝛼+𝑧𝑝𝛼𝜃₁𝜙_3,𝑖,1(𝑧)+𝑎1−⁰𝑧𝑝𝜃₁𝑠_1,𝑖𝜋_0,0𝜙,𝑖≥1,_3,𝑖,𝑗(𝑧)=𝑝𝜃₁𝑠_1,𝑖𝑠_3,𝑗𝜙₀(1,𝑧)+𝑝𝑧𝜃₁𝑠_1,𝑖𝑠_3,𝑗𝜙_0,1(𝑧)+𝑝+𝑝𝑧𝑎₀𝜃₁𝑟𝑠_1,𝑖𝑠_3,𝑗𝜙_1,1(𝑧)+𝑝+𝑝𝑧𝑎₀𝜃₁𝑠_1,𝑖𝑠_3,𝑗𝜙_2,1(𝑧)+𝑝+𝑝𝛼+𝑧𝑝𝛼𝜃₁𝑠_3,𝑗𝜙_1,𝑖+1(𝑧)+𝑝+𝑝𝛼+𝑧𝑝𝛼𝜃₁𝑠_3,𝑗𝜙_3,𝑖,1(𝑧)+𝑝+𝑝𝛼+𝑧𝑝𝛼𝜙_{3,𝑖,𝑗+1}(𝑧)+𝑎1−⁰𝑧𝑝𝜃₁𝑠_1,𝑖𝑠_3,𝑗𝜋_0,0,𝑖≥1,𝑗≥1. (17)(18)(19) Then, multiplying (16), (19) by 𝑦^𝑗 and summing over 𝑗, we obtain 𝜙_4,𝑖𝑦,𝑧=𝑝+𝑝𝛼+𝑧𝑝𝛼𝜃₂𝑠_2,𝑖𝑆₄𝑦𝑟𝜙_1,1(𝑧)+𝑝+𝑝𝛼+𝑧𝑝𝛼𝜃₂𝑆₄𝑦𝜙_2,𝑖+1(𝑧)+𝑝+𝑝𝜙𝛼+𝑧𝑝𝛼^4,𝑖𝑦,𝑧𝑦−𝜙_4,𝑖,1(𝑧+)𝑝+𝑝𝛼+𝑧𝑝𝛼𝜃₂𝑆₄𝑦𝜙_4,𝑖,1(𝑧),𝜙𝑖≥1,𝑗≥1,_3,𝑖𝑦,𝑧=𝑝𝜃₁𝑠_1,𝑖𝑆₃𝑦𝜙₀(1,𝑧)+𝑝𝑧𝜃₁𝑠_1,𝑖𝑆₃𝑦𝜙_0,1(𝑧)+𝑎1−⁰𝑧𝑝𝜃₁𝑠_1,𝑖𝑆₃𝑦𝜋_0,0+𝑝+𝑝𝑧𝑎₀𝜃₁𝑠_1,𝑖𝑆₃𝑦𝑟𝜙_1,1(𝑧)+𝑝+𝑝𝑧𝑎₀𝜃₁𝑠_1,𝑖𝑆₃𝑦𝜙_2,1(𝑧)+𝑝+𝑝𝛼+𝑧𝑝𝛼𝜃₁𝑆₃𝑦𝜙_1,𝑖+1(𝑧)+𝑝+𝑝𝛼+𝑧𝑝𝛼𝜃₁𝑆₃𝑦𝜙_3,𝑖,1(𝑧)+𝑝+𝑝𝜙𝛼+𝑧𝑝𝛼^3,𝑖𝑦,𝑧𝑦−𝜙_3,𝑖,1(𝑧),𝑖≥1,𝑗≥1. (20)(21) Next, multiplying (17), (18), (14), (20), and (21) by 𝑥^𝑖 and summing over 𝑖, we get 𝑥−𝑝𝑥𝜙₀(𝑥,𝑧)=𝑝𝐴(𝑥)−𝑎₀𝑟𝜙_1,1(𝑧)+𝜙_2,1(𝑧)−𝑝𝐴(𝑥)−𝑎₀𝜋_0,0−𝑝𝜙_0,1(𝑧),𝑥−𝜃₁𝜔(𝑧)𝑥𝜙₁(𝑥,𝑧)=𝑝𝑧𝜃₁𝑆₁(𝑥)𝜙_0,1(𝑧)+𝑝𝜃₁𝑆₁(𝑥)𝜙₀()+1,𝑧𝑝+𝑝𝑧𝑎₀𝜃₁𝑆₁(𝑥)𝑟−𝜃𝜔(𝑧)×𝜙_1,1(𝑧)+𝜃₁𝜔(𝑧)𝜓₁()+𝑥,𝑧𝑝+𝑝𝑧𝑎₀𝜃₁𝑆₁(𝑥)𝜙_2,1(𝑧)+𝑎1−⁰𝑧𝑝𝜃₁𝑆₁(𝑥)𝜋_0,0,𝑥−𝜃₂𝜔(𝑧)𝑥𝜙₂(𝑥,𝑧)=𝑟𝜔(𝑧)𝜃₂𝑆₂(𝑥)𝜙_1,1(𝑧)−𝜃₂𝜔(𝑧)𝜙_2,1(𝑧)+𝜃₂𝜔(𝑧)𝜓₂(),(𝑧)𝑥,𝑧𝑦−𝜔𝑦𝜙₃𝑥,𝑦,𝑧=𝑝𝜃₁𝑆₁(𝑥)𝑆₃𝑦𝜙₀()+1,𝑧𝑝𝑧𝜃₁𝑆₁(𝑥)𝑆₃𝑦𝜙_0,1(𝑧)+𝑝+𝑝𝑧𝑎₀𝜃₁𝑆₁(𝑥)𝑆₃𝑦𝑟𝜙_1,1(𝑧)+𝑝+𝑝𝑧𝑎₀𝜃₁𝑆₁(𝑥)𝑆₃𝑦𝜙_2,1(𝑧)+𝜔(𝑧)𝜃₁𝑆₃𝑦−1𝜓₁()(𝑧)𝑥,𝑧+𝜔𝜃₁𝑆₃𝑦𝜙₁(𝑥,𝑧)𝑥−𝜙_1,1(𝑧)+𝑎1−⁰𝑧𝑝𝜃₁𝑆₁(𝑥)𝑆₃𝑦𝜋_0,0,𝑦−𝜔(𝑧)𝑦𝜙₄(𝑧)𝑥,𝑦,𝑧=𝑟𝜔𝜃₂𝑆₂(𝑥)𝑆₄𝑦𝜙_1,1(𝑧)+𝜔(𝑧)𝜃₂𝑆₄𝑦−1𝜓₂()(𝑧)𝑥,𝑧+𝜔𝜃₂𝑆₄𝑦×𝜙₂(𝑥,𝑧)𝑥−𝜙_2,1(𝑧), (22)(23)(24)(25)(26) where 𝜔(𝑧)=𝑝+𝑝𝛼+𝑧𝑝𝛼.

Choosing 𝑥=1 in (22), we can get 𝜙₀(1,𝑧) and putting 𝜙₀(1,𝑧) into (23), (25), we obtain 𝑥−𝜃₁𝜔(𝑧)𝑥𝜙₁(𝑥,𝑧)=𝑝(1−𝑧)𝜃₁𝑆₁(𝑥)𝑧𝜙_0,1(𝑧)−𝑝𝑎₀(1−𝑧)𝜃₁𝑆₁(𝑥)𝑧𝜋_0,0+𝑧+𝑝𝑎₀(1−𝑧)𝑧𝑟𝑆₁(𝑥)−𝜔(𝑧)×𝜃₁𝜙_1,1(𝑧)+𝑧+𝑝𝑎₀(1−𝑧)𝑧×𝑆₁(𝑥)𝜃₁𝜙_2,1(𝑧)+𝜔(𝑧)𝜃₁𝜓₁(),(𝑧)𝑥,𝑧𝑦−𝜔𝑦𝜙₃=()𝑥,𝑦,𝑧1−𝑧𝑝𝜃₁𝑆₁(𝑥)𝑆₃𝑦𝑧𝜙_0,1(𝑧)−𝑝𝑎₀(1−𝑧)𝜃₁𝑆₁(𝑥)𝑆₃𝑦𝑧𝜋_0,0+𝑧+𝑝𝑎₀(1−𝑧)𝑧𝑟𝑆₁(𝑥)(𝑧×−𝜔)𝜃₁𝑆₃𝑦𝜙_1,1(𝑧)+𝑧+𝑝𝑎₀(1−𝑧)𝑧×𝑟𝑆₁(𝑥)𝜃₁𝑆₁(𝑥)𝑆₃𝑦𝜙_2,1(𝑧)+𝜔(𝑧)𝜃₁𝑆₃𝑦𝑥𝜙₁()(𝑧)×𝑥,𝑧−𝜔1−𝜃₁𝑆₃𝑦𝜓₁(𝑥,𝑧). (27)(28) Letting 𝑦=𝜔(𝑧) in (26), (28), we obtain 𝜓₂()=𝑆𝑥,𝑧⁴(𝜔(𝑧))𝜃₂𝜔(𝑧)1−𝜃₂𝑆₄(𝜔(𝑧×(𝑧)𝑆))𝑟𝜔₂(𝑥)𝜙_1,1(𝑧)−𝜔(𝑧)𝜙_2,1(𝑧)+𝜔(𝑧)𝑥𝜙₂(,𝜓𝑥,𝑧)₁()=𝑆𝑥,𝑧³(𝜔(𝑧))𝜃₁𝜔(𝑧)1−𝜃₁𝑆₃(𝜔(𝑧×()))1−𝑧𝑝𝑆₁(𝑥)𝑧𝜙_0,1(𝑧)−𝑝𝑎₀(1−𝑧)𝑆₁(𝑥)𝑧𝜋_0,0+𝑧+𝑝𝑎₀(1−𝑧)𝑧𝑟𝑆₁(𝑥)−𝜔(𝑧)𝜙_1,1(𝑧)+𝑧+𝑝𝑎₀(1−𝑧)𝑧×𝑆₁(𝑥)𝜙_2,1(𝑧)+𝜔(𝑧)𝑥𝜙₁(𝑥,𝑧). (29)(30) Substituting (29), (30) into (24), (27), we obtain 𝑥−𝜑₂[𝑤(𝑧)]𝑥1−𝜃₂𝑆₄(𝑤(𝑧))𝜙₂(𝑥,𝑧)=𝜃₂𝑟𝜔(𝑧)𝑆₂(𝑥)𝜙_1,1(𝑧)−𝜔(𝑧)𝜙_2,1(𝑧),𝑥−𝜑₁[𝑤(𝑧)]𝑥1−𝜃₁𝑆₃(𝑤(𝑧))𝜙₁(𝑥,𝑧)=𝜃₁1−𝑧𝑧𝑝𝑆₁(𝑥)𝜙_0,1(𝑧)−1−𝑧𝑧𝑝𝑎₀𝑆₁(𝑥)𝜋_0,0+𝑧+𝑝𝑎₀(1−𝑧)𝑧𝑆₁(𝑥)𝑟−𝜔(𝑧)𝜙_1,1(𝑧)+𝑧+𝑝𝑎₀(1−𝑧)𝑧𝑆₁(𝑥)𝜙_2,1(𝑧), (31)(32) where functions 𝜑₁(𝑧)=𝜃₁𝑧1−𝜃₁𝑆₃(𝑧),𝜑₂(𝑧)=𝜃₂𝑧1−𝜃₂𝑆₄(𝑧). (33) Letting 𝑥=𝑝 in (22), 𝑥=𝜑₂(𝜔(𝑧)) in (31), and 𝑥=𝜑₁(𝜔(𝑧)) in (32), we get 𝑝𝐴𝑝−𝑎₀𝜋_0,0=𝑝𝐴𝑝−𝑎₀×𝑟𝜙_1,1(𝑧)+𝜙_2,1(𝑧−)𝑝𝜙_0,1(𝑧),𝜙_2,1(𝑧)=𝑟𝑆₂𝜑₂(𝜔(𝑧))𝜙_1,1(𝑧),𝑝𝑎₀(1−𝑧)𝑆₁𝜑₁(𝜔(𝑧))𝑧𝜋_0,0=𝑝(1−𝑧)𝑆₁𝜑₁(𝜔(𝑧))𝑧𝜙_0,1(𝑧)+𝑧+𝑝𝑎₀(1−𝑧)𝑟𝑆₁𝜑₁(𝜔(𝑧))𝑧−𝜔(𝑧)𝜙_1,1(𝑧)+𝑧+𝑝𝑎₀(1−𝑧)𝑆₁𝜑₁(𝜔(𝑧))𝑧𝜙_2,1(𝑧). (34) Solving (34) we obtain the auxiliary generating functions 𝜙_0,1(𝑧), 𝜙_1,1(𝑧), and 𝜙_2,1(𝑧) as follows: 𝜙_0,1(𝑧)=𝑝𝑧𝐴𝑝−𝑎₀×𝜔(𝑧)−𝑆₁𝜑₁(𝜔(𝑧))𝑟+𝑟𝑆₂𝜑₂(𝜔(𝑧×(Υ(𝑧))))⁻¹×𝜋_0,0𝑝,𝜙_1,1(𝑧)=𝑝𝐴𝑝(1−𝑧)𝑆₁×𝜑₁(𝜔(𝑧))𝑟+𝑟𝑆₂𝜑₂(𝜔(𝑧×(Υ(𝑧))))⁻¹×𝜋_0,0,𝜙_2,1(𝑧)=𝑟𝑆₂𝜑₂(𝜔(𝑧×))𝑝𝐴𝑝(1−𝑧)𝑆₁×𝜑₁(𝜔(𝑧))𝑟+𝑟𝑆₂𝜑₂(𝜔(𝑧×(Υ(𝑧))))⁻¹×𝜋_0,0, (35)(36)(37) where Υ(𝑧)=𝑧+𝑝𝐴𝑝(1−𝑧)×𝑆₁𝜑₁(𝜔(𝑧))𝑟+𝑟𝑆₂𝜑₂(𝜔(𝑧))−𝑧𝜔(𝑧). (38) Next, we give two lemmas which will be used later on and their proof can be readily obtained. Thus, they are omitted.

Lemma 1. The inequality 𝑝𝐴𝑝(1−𝑧)𝑆₁𝜑₁(𝜔(𝑧))𝑟+𝑟𝑆₂𝜑₂(𝜔(𝑧𝜔(𝑧)))−𝑧−𝑆₁𝜑₁(𝜔(𝑧))𝑟+𝑟𝑆₂𝜑₂(𝜔(𝑧))>0 (39) holds, for 0≤𝑧<1, if 𝜌₁+𝑟𝜌₂<𝑝+𝑝𝐴𝑝𝛼, (40) where 𝜌₁=𝑝𝛽_1,1(1+(𝜃₁/𝜃₁)𝛽_3,1),𝜌₂=𝑝𝛽_2,1(1+(𝜃₂/𝜃₂)𝛽_4,1).

Lemma 2. The following limits are positive if 𝜌₁+𝑟𝜌₂<𝑝+𝑝𝐴(𝑝)/𝛼 as follows: lim_𝑧→11−𝑧Υ(𝑧)=1𝑝𝐴𝑝+𝛼𝑝−𝛼𝜌₁+𝑟𝜌₂,lim_𝑧→1𝜔(𝑧)−𝑆₁𝜑₁(𝜔(𝑧))𝑟+𝑟𝑆₂𝜑₂(𝜔(𝑧))Υ(𝑧)=𝛼𝜌₁+𝑟𝜌₂−𝑝𝑝𝐴𝑝+𝛼𝑝−𝛼𝜌₁+𝑟𝜌₂. (41)

From above Lemmas, we can easily get that 𝜙_0,1(𝑧) and 𝜙_1,1(𝑧) are defined for 𝑧∈[0,1) and can be extended by continuity, for 𝑧=1, if 𝜌₁+𝑟𝜌₂<𝑝+𝑝𝐴(𝑝)/𝛼. Then, we can obtain the generating functions of the stationary distribution of the system which are given by the following theorem.

Theorem 3. If 𝜌₁+𝑟𝜌₂<𝑝+𝑝𝐴(𝑝)/𝛼, the stationary distribution of the Markov chain {𝑌_𝑚,𝑚=1,2,…} has the following generating functions:𝜙₀(𝑥,𝑧)=𝑝𝑥𝑧𝜔(𝑧)−𝑆₁𝜑₁(𝜔(𝑧))𝑟+𝑟𝑆₂𝜑₂(𝜔(𝑧×(Υ(𝑧))))⁻¹×𝐴(𝑥)−𝐴𝑝𝑥−𝑝𝜋_0,0,𝜙₁()=𝑆𝑥,𝑧¹(𝑥)−𝑆₁𝜑₁(𝜔(𝑧))𝑥−𝜑₁(𝜔(𝑧×(𝑧)𝐴))𝑝𝜔𝑝(1−𝑧)𝑥𝜃₁Υ(𝑧)1−𝜃₁𝑆₃(𝜔(𝑧𝜋))^0,0,𝜙₂()=𝑆𝑥,𝑧²(𝑥)−𝑆₂𝜑₂(𝜔(𝑧))𝑥−𝜑₂(𝜔(𝑧×(𝑧)𝐴))𝑟𝑝𝜔𝑝(1−𝑧)𝑥𝜃₂𝑆₁𝜑₁(𝜔(𝑧))Υ(𝑧)1−𝜃₂𝑆₄(𝜔(𝑧𝜋))^0,0,𝜙₃=𝑆𝑥,𝑦,𝑧¹(𝑥)−𝑆₁𝜑₁(𝜔(𝑧))𝑥−𝜑₁(𝜔(𝑧×𝑆))₃𝑦−𝑆₃(𝜔(𝑧))(𝑧)×(𝑧)𝐴𝑦−𝜔𝑝𝜔𝑝(1−𝑧)𝑥𝑦𝜃₁Υ(𝑧)1−𝜃₁𝑆₃(𝜔(𝑧𝜋))^0,0,𝜙₄=𝑆𝑥,𝑦,𝑧²(𝑥)−𝑆₂𝜑₂(𝜔(𝑧))𝑥−𝜑₂(𝜔(𝑧×𝑆))₄𝑦−𝑆₄(𝜔(𝑧))(𝑧)×(𝑧)𝐴𝑦−𝜔𝑟𝑝𝜔𝑝(1−𝑧)𝑥𝑦𝜃₂𝑆₁𝜑₁(𝜔(𝑧))Υ(𝑧)1−𝜃₂𝑆₄(𝜔(𝑧))×𝜋_0,0, (42)(43)(44)(45)(46) where 𝜋_0,0=𝑝𝐴𝑝+𝛼𝑝−𝛼𝜌₁+𝑟𝜌₂𝐴𝑝𝑝+𝛼𝑝+𝛼𝜌₁+𝑟𝜌₂. (47)

Proof. Substituting (35)–(37) into (22), (31) and (32), yields 𝜙₀(𝑥,𝑧), 𝜙₂(𝑥,𝑧), and 𝜙₁(𝑥,𝑧) in Theorem 3. Further, substituting (36), (37), (43), and (31) into (29) and (30), respectively, we get 𝜓₂()=𝑆𝑥,𝑧⁴(𝜔(𝑧))𝜔(𝑧)𝑆₂(𝑥)−𝑆₂𝜑₂(𝜔(𝑧))𝑥−𝜑₂(𝜔(𝑧×(𝑧)𝐴))𝑟𝑝𝜔𝑝(1−𝑧)𝑥𝜃₂𝑆₁𝜑₁(𝜔(𝑧))Υ(𝑧)1−𝜃₂𝑆₄(𝜔(𝑧𝜋))^0,0,𝜓₁()=𝑆𝑥,𝑧³(𝜔(𝑧))𝜔(𝑧)𝑆₁(𝑥)−𝑆₁𝜑₁(𝜔(𝑧))𝑥−𝜑₁(𝜔(𝑧×(𝑧)𝐴))𝑝𝜔𝑝(1−𝑧)𝑥𝜃₁Υ(𝑧)1−𝜃₁𝑆₃(𝜔(𝑧𝜋))^0,0. (48)(49)
Inserting (35)–(37), (48)-(49) into (26) and (28), respectively, we can get 𝜙₄(𝑥,𝑦,𝑧) and 𝜙₃(𝑥,𝑦,𝑧) in Theorem 3. From normalizing condition (9), which can be written as 𝜋_0,0+𝜙₀(1,1)+²_𝑖=1𝜙_𝑖(1,1)+⁴_𝑗=3𝜙_𝑗(1,1,1)=1, (50) we can find the unknown constant 𝜋_0,0 in Theorem 3.

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.

Corollary 4. (1) The marginal generating function of the number of customers in the orbit when the server is idle is given by 𝜋_0,0+𝜙₀(1,𝑧)=𝐴𝑝𝑆₁𝜑₁(𝜔(𝑧))𝑟+𝑟𝑆₂𝜑₂(𝜔(𝑧))(1−𝑧)𝑝Υ(𝑧)−𝑧𝜔(𝑧)−𝑆₁𝜑₁(𝜔(𝑧))𝑟+𝑟𝑆₂𝜑₂(𝜔(𝑧×(Υ(𝑧))))⁻¹×𝜋_0,0. (51)
(2) The marginal generating function of the number of customers in the orbit when the server is busy with FPS is given by 𝜙₁(1,𝑧)=1−𝑆₁𝜑₁(𝜔(𝑧))1−𝜑₁(𝜔(𝑧×()))1−𝑧𝑝𝐴𝑝𝜔(𝑧)𝜃₁Υ(𝑧)1−𝜃₁𝑆₃(𝜔(𝑧𝜋))^0,0. (52)
(3) The marginal generating function of the number of customers in the orbit when the server is busy with SPS is given by 𝜙₂(1,𝑧)=1−𝑆₂𝜑₂(𝜔(𝑧))1−𝜑₂(𝜔(𝑧×(𝑧)𝐴))𝑟𝑝𝜔𝑝(1−𝑧)𝜃₂𝑆₁𝜑₁(𝜔(𝑧))Υ(𝑧)1−𝜃₂𝑆₄(𝜔(𝑧𝜋))^0,0. (53)
(4) The marginal generating function of the number of customers in the orbit when the server is under repair during FPS is given by 𝜙₃(1,1,𝑧)=1−𝑆₁𝜑₁(𝜔(𝑧))1−𝜑₁(𝜔(𝑧×))1−𝑆₃(𝜔(𝑧))(𝑧)𝐴𝛼Υ𝑝𝜔(𝑧)𝜃₁1−𝜃₁𝑆₃(𝜔(𝑧𝜋))^0,0. (54)
(5) The marginal generating function of the number of customers in the orbit when the server is under repair during SPS is given by 𝜙₄(1,1,𝑧)=1−𝑆₂𝜑₂(𝜔(𝑧))1−𝜑₂(𝜔(𝑧))1−𝑆₄(𝜔(𝑧))(𝑧)×𝐴𝛼Υ𝑝𝜔(𝑧)𝜃₂𝑟𝑆₁𝜑₁(𝜔(𝑧))1−𝜃₂𝑆₄(𝜔(𝑧𝜋))^0,0. (55)
(6) The probability generating function of the orbit size (i.e., of the variable N) is given by Ψ(𝑧)=𝜋_0,0+𝜙₀(1,𝑧)+²_𝑖=1𝜙_𝑖(1,𝑧)+⁴_𝑗=3𝜙_𝑗()=𝐴1,1,𝑧𝑝𝜔(𝑧)(1−𝛼𝑧)−𝛼𝑆₁×𝜑₁(𝜔(𝑧))𝑟+𝑟𝑆₂𝜑₂(𝜔(𝑧×((𝑧))𝛼Υ))⁻¹×𝜋_0,0. (56)
(7) The probability generating function of the number of customers in the system (i.e., of the variable L) is given by Φ(𝑧)=𝜋_0,0+𝜙₀(1,𝑧)+𝑧²_𝑖=1𝜙_𝑖(1,𝑧)+𝑧⁴_𝑗=3𝜙_𝑗()=𝐴1,1,𝑧𝑝𝛼𝑧𝜔(𝑧)+𝑆₁𝜑₁(𝜔(𝑧×))𝑟+𝑟𝑆₂𝜑₂(𝜔(𝑧))[𝜔(𝑧)(1−𝑧)−𝛼×((𝑧]𝛼Υ))⁻¹×𝜋_0,0. (57)

Corollary 5. (1) The system is idle with probability 𝜋_0,0=𝑝𝐴𝑝+𝛼𝑝−𝛼𝜌₁+𝑟𝜌₂𝐴𝑝𝑝+𝛼𝑝+𝛼𝜌₁+𝑟𝜌₂. (58)
(2) The system is occupied with probability 𝜙₀(1,1)+²_𝑖=1𝜙_𝑖(1,1)+⁴_𝑗=3𝜙_𝑗()=𝐴1,1,1𝑝𝛼𝑝+𝛼𝜌₁+𝑟𝜌₂−𝛼𝑝−𝜌₁−𝑟𝜌₂𝐴𝑝𝑝+𝛼𝑝+𝛼𝜌₁+𝑟𝜌₂. (59)
(3) The probability that the server is idle is 𝜋_0,0+𝜙₀(1,1)=𝑝+𝛼𝑝−𝛼𝜌₁+𝑟𝜌₂𝑝+𝛼𝑝+𝛼𝜌₁+𝑟𝜌₂. (60)
(4) The probability that the server is busy with FPS is 𝜙₁(1,1)=𝑝𝛽_1,1𝑝+𝛼𝑝+𝛼𝜌₁+𝑟𝜌₂. (61)
(5) The probability that the server is busy with SPS is 𝜙₂(1,1)=𝑟𝑝𝛽_2,1𝑝+𝛼𝑝+𝛼𝜌₁+𝑟𝜌₂. (62)
(6) The probability that the server is under repair during FPS is 𝜙₃(1,1,1)=𝑝𝛽_1,1𝛽_3,1𝜃₁/𝜃₁𝑝+𝛼𝑝+𝛼𝜌₁+𝑟𝜌₂. (63)
(7) The probability that the server is under repair during SPS is 𝜙₄(1,1,1)=𝑝𝛽_2,1𝛽_4,1𝜃₂/𝜃₂𝑝+𝛼𝑝+𝛼𝜌₁+𝑟𝜌₂. (64)
(8) The mean number of customers in the orbit is 𝐸(𝑁)=2𝛼𝑝−𝜌₁−𝑟𝜌₂𝑝𝐴𝑝1+𝛼𝜌₁+𝑟𝜌₂−𝑝+𝛼𝜌₁+𝑟𝜌₂×2𝑝+𝛼𝑝+𝛼𝜌₁+𝑟𝜌₂×𝑝𝐴𝑝+𝛼𝑝−𝜌₁−𝑟𝜌₂⁻¹+𝛼𝑝²𝜏𝛼+𝛼𝑝𝐴𝑝×2𝑝+𝛼𝑝+𝛼𝜌₁+𝑟𝜌₂×𝑝𝐴𝑝+𝛼𝑝−𝜌₁−𝑟𝜌₂⁻¹, (65) where 𝜏=𝛽_1,21+𝜃₁𝜃₁𝛽_3,1²+𝛽_1,1𝜃₁𝜃₁𝛽_3,2+2𝜃₁𝜃₁𝛽_3,11+𝜃₁𝜃₁𝛽_3,1+𝑟𝛽_2,21+𝜃₂𝜃₂𝛽_4,1²+𝑟𝛽_2,1𝜃₂𝜃₂𝛽_4,2+2𝜃₂𝜃₂𝛽_4,11+𝜃₁𝜃₁𝛽_4,1+2𝑟𝛽_1,1𝛽_2,11+𝜃₁𝜃₁𝛽_3,11+𝜃₂𝜃₂𝛽_4,1. (66)
(9) The arrival rate to the orbit is given by 𝑃_{𝑂𝑟𝑏𝑖𝑡}=𝛼𝑝²_𝑖=1𝜙_𝑖(1,1)+⁴_𝑗=3𝜙_𝑗()=𝜌1,1,1𝛼𝑝₁+𝑟𝜌₂𝑝+𝛼𝑝+𝛼𝜌₁+𝑟𝜌₂. (67) (10) The arrival rate to the system is given by 𝑃_{𝑆𝑦𝑠𝑡𝑒𝑚}=𝑝𝜋_0,0+𝜙₀(1,1)+𝛼𝑝²_𝑖=1𝜙_𝑖(1,1)+²_𝑗=1𝜙_𝑗()=𝑝1,1,1𝑝+𝛼𝑝𝑝+𝛼𝑝+𝛼𝜌₁+𝑟𝜌₂. (68)
(11) The loss probability of a customer is 𝑃_{𝐿𝑜𝑠𝑠}=𝑝𝛼²_𝑖=1𝜙_𝑖(1,1)+⁴_𝑗=3𝜙_𝑗()=𝑝1,1,1𝛼𝜌₁+𝑟𝜌₂𝑝+𝛼𝑝+𝛼𝜌₁+𝑟𝜌₂. (69)

Remark 6 (special cases). In the remark, we consider some special cases of our model.(i)When 𝜃₁=1, 𝑟=0, the present model reduced to a discrete-time Geo/G/1 retrial queue with balking customer. In this case, the generating function in Theorem 3 reduces to 𝜙₀()=𝜔(𝑧)𝑥,𝑧𝑝𝑥𝑧−𝑆₁(𝜔(𝑧))𝑝𝐴𝑝(1−𝑧)𝑆₁(𝜔(𝑧))−𝑧𝜔(𝑧)−𝑆₁(𝜔(𝑧×𝐴(𝑥)))−𝐴𝑝𝑥−𝑝𝜋_0,0,𝜙₁()=𝑆𝑥,𝑧₁(𝑥)−𝑆₁𝜑₁(𝜔(𝑧))(𝑧)×(𝑧)𝐴𝑥−𝜔𝑝𝜔𝑝(1−𝑧)𝑥𝑝𝐴𝑝(1−𝑧)𝑆₁(𝜔(𝑧))−𝑧𝜔(𝑧)−𝑆₁(𝜔(𝑧𝜋))^0,0, (70)  where 𝜋_0,0=𝑝𝐴𝑝+𝛼𝑝−𝛼𝜌𝐴𝑝𝑝+𝛼𝑝+𝛼𝜌,𝜌=𝑝𝛽_1,1, (71)  which coincides with the generating functions of Theorem 3.1 in Aboul-Hassan et al. [11].(ii)When 𝜃₁=1, 𝑟=0, and 𝛼=1, the present model reduced to a discrete-time Geo/G/1 retrial queue with general retrial times. In this case, the generating function in Theorem 3 reduces to 𝜙₀()=𝛾(𝑧)𝑥,𝑧𝑝𝑥𝑧−𝑆₁𝛾(𝑧)𝑝𝐴𝑝(1−𝑧)𝑆₁𝛾(𝑧)−𝑧𝛾(𝑧)−𝑆₁𝛾(𝑧)×𝐴(𝑥)−𝐴𝑝𝑥−𝑝𝜋_0,0,𝜙₁()=𝑆𝑥,𝑧₁(𝑥)−𝑆₁𝛾(𝑧)(𝑧)×(𝑧)𝐴𝑥−𝛾𝑝𝛾𝑝(1−𝑧)𝑥𝑝𝐴𝑝(1−𝑧)𝑆₁𝛾(𝑧)−𝑧𝛾(𝑧)−𝑆₁𝛾(𝑧)𝜋_0,0, (72)(73) where 𝜋_0,0=𝑝𝐴𝑝+𝑝−𝜌𝐴𝑝,𝜌=𝑝𝛽_1,1, (74) which coincides with the generating functions of Theorem 1 in Atencia and Moreno [7].

4. Stochastic Decomposition

The property of stochastic decomposition was studied firstly in queueing system with vacations; see Fuhrmann and Cooper [25]. Artalejo and Falin [26] generalized the concept of stochastic decomposition to retrial queueing system. In this section, we give a stochastic decomposition result of the system size distribution in our model. As a consequence of the stochastic decomposition result, we provide upper and lower estimates for the distance between the steady-state distribution of our discrete-time queueing system and the corresponding system without retrials.

Theorem 7. The total number of customers in the system (𝐿) can be represented as the sum of two independent random variables, one of which is the number of customers in the unreliable Geo/G/1/ ∞ queue with customer balking and second optional service (𝐿₀) and the other is the number of repeated customers given that the server is idle or down (𝑀₀). That is, 𝐿=𝐿₀+𝑀₀.

Proof. It is easy to show that Φ(𝑧) can be represented by Φ(𝑧)=Φ₁(𝑧)Φ₂(𝑧), where Φ₁(𝑧)=𝛼𝑧𝜔(𝑧)+𝑆₁𝜑₁(𝜔(𝑧))𝑟+𝑟𝑆₂𝜑₂(𝜔(𝑧×[𝜔(𝑧)−)))(1−𝑧𝛼]×(𝛼Ω(𝑧))⁻¹×𝑝+𝑝𝛼−𝛼𝜌₁+𝑟𝜌₂𝑝+𝑝𝛼+𝛼𝜌₁+𝑟𝜌₂ (75) is the generating function of Geo/G/1 queue with balking customer second optional service and unreliable server in special case (i) and Φ₂(𝑧) given by Φ₂(𝑧)=^∞_𝑘=0𝑃𝑀₀=𝑘𝑧^𝑘=𝑝+𝑝𝑧𝑆₁𝜑₁(𝜔(𝑧))𝑟+𝑟𝑆₂𝜑₂(𝜔(𝑧(𝑧)×(Υ(𝑧))−𝑧𝜔))⁻¹×𝑝𝐴𝑝+𝛼𝑝−𝛼𝜌₁+𝑟𝜌₂𝑝+𝛼𝑝−𝛼𝜌₁+𝑟𝜌₂=𝜋_0,0+𝜙₀(1,𝑧)𝜋_0,0+𝜙₀(),1,1 (76) is the probability generating function of the number of customers in the orbit given that the server is idle. This completes the proof.

The property of stochastic decomposition can be applied to give a measure of the proximity between the steady-state distributions of the unreliable Geo/G/1 queue with balking and our queueing model. The result is summarized in the following theorem.

Theorem 8. The following inequalities hold: 2𝛼𝑝−𝜌₁+𝑟𝜌₂𝐴𝑝−1𝐴𝑝𝑝+𝛼𝑝+𝛼𝜌₁+𝑟𝜌₂≤^∞_𝑗=0𝑃𝐿=𝑗−𝑃𝐿₀≤𝜌=𝑗2𝛼𝑝−₁+𝑟𝜌₂𝐴𝑝−1𝐴𝑝𝑝+𝛼𝑝−𝛼𝜌₁+𝑟𝜌₂. (77)

The proof which is similar to Aboul-Hassan et al. [11] is omitted.

5. Numerical Examples

In this section, we present some numerical examples to study the effect of the system parameters on the mean orbit size. We assume that 𝑝 is the probability that a customer arrives to the system and 𝑞 is probability that a customer receives the second phase service. Moreover, the retrial times are geometric distributions with parameters 𝑟 and thier generating function is 𝐴(𝑥)=(1−𝑟)/(1−𝑟𝑥). It is also assumed that the service times during the FPS and SPS are geometric distributions with means 𝛽_1,1, and 𝛽_2,1, respectively, and the repair times for the FPS and SPS are geometric distributions with means 𝛽_3,1 and 𝛽_4,1, respectively.

In Figures 1 and 2, the mean number of customers in the orbit 𝐸(𝑁) is plotted against 𝑞. We present three curves which correspond to 𝛼=0.5,0.7,0.9 and 𝜃₁=0.5,0.6,0.7, respectively, in Figures 1 and 2. We observe that 𝐸(𝑁) increases with the value of 𝑞. It is easily explained due to the fact that a customer choosing to receive the second-optional service increases the mean sojourn time. It is also shown that 𝐸(𝑁) is increasing as a function of 𝛼 and decreases with the value 𝜃₁ which agree with the intuitive expectations.

Figure 1 

𝐸(𝑁) versus 𝑞 for different 𝛼.

Figure 2 

𝐸(𝑁) versus 𝑞 for different 𝑅.

In Figures 3 and 4, the mean number of customers in the orbit 𝐸(𝑁) is plotted against 𝑟 which is the probability that the retrial fails. We choose 𝑝=0.1, 𝛼=0.9, 𝛽_1,1=𝛽_2,1=2, and 𝜃₁=𝜃₂=0.5 in general. Figures 3 and 4 show that 𝐸(𝑁) exhibits a stable increasing when 𝑟 approximates a threshold value 𝛾 (when 𝑟≤𝛾 the system is stable) and then a sharp increasing with the increasing of 𝑟. It is also shown in Figures 3 and 4 that 𝐸(𝑁) is increasing with the mean repair time 𝛽_3,1 and 𝛽_4,1, respectively, which agree with the intuitive expectations.

Figure 3 

𝐸(𝑁) versus 𝑟 for different 𝜃1.

Figure 4 

𝐸(𝑁) versus 𝑟 for different 𝛽3,1.

From the curves in Figures 1 and 2, we can also obtain the comparative results between our model and the corresponding discrete-time queueing model without second optional service and server breakdowns. Figure 2 also shows that the mean number of customers in the orbit 𝐸(𝑁) increases significantly compared with the model of Aboul-Hassan et al. [11]. That is, our simulation results show that the breakdown of the server has a heavy impact on the performance measures of the discrete-time retrial queueing model with balking customer. Moreover, it is also shown in our simulation results that the impact of parameter 𝑞 on 𝐸(𝑁) is more noticeable as the probability that a customer may not balk increases. In addition, Figures 3 and 4 show how the mean repair time affects the mean customer of customers in the orbit. In general, compared with the previous model of Aboul-Hassan et al. [11] and Atencia and Moreno [24], our simulation results show the impact of various practical factors on the mean number of customers in the orbit 𝐸(𝑁) simultaneously.