A Novel Analytic Technique for the Service Station Reliability in a Discrete-Time Repairable Queue

Abstract

This paper presents a decomposition technique for the service station reliability in a discrete-time repairable Geom^𝑋/G/1 queueing system, in which the server takes exhaustive service and multiple adaptive delayed vacation discipline. Using such a novel analytic technique, some important reliability indices and reliability relation equations of the service station are derived. Furthermore, the structures of the service station indices are also found. Finally, special cases and numerical examples validate the derived results and show that our analytic technique is applicable to reliability analysis of some complex discrete-time repairable bulk arrival queueing systems.

1. Introduction

Queues with service station subject to failures and repairs, so-called repairable queues, are often encountered in many practical applications such as in computer, manufacturing systems, and communication networks. Because system performance deteriorates seriously by service station failures and the limitation of repair capacity, the study of queues with service station failures and repairs is not only important for theoretic investigations, but it is also necessary for engineering applications.

Relative to continuous-time repairable queues, their discrete-time counterparts received very little attention in the literature. Moreover, most of discrete-time repairable queueing models were investigated from a queueing theory viewpoint only, and some quantities relative to queueing theory, such as queue size, waiting time, and busy period, were obtained. But the reliability quantities of the service station, such as the unavailability and failure frequency of the service station, were neglected (see, e.g., [1–11]).

On the other hand, to the best of our knowledge, the existing work about the discrete-time repairable queues was studied with the help of the discrete supplementary variable technique, the matrix-geometric solution method, and the decomposition method. The discrete supplementary variable technique has important applications in queueing theory. With this method, various repairable queueing systems were investigated in the existing literatures (see, e.g., [1–8]). The matrix-geometric solution method studied by Yu [9] and other researchers is a more powerful tool. For example, Yu et al. [10] applied this method to analyze the characteristics of the MAP/PH(PH/PH)/1 queue with repairable server where lifetime of server, service time, and repair time of server are all discrete phase-type random variables. The decomposition technique first introduced by Tang et al. [11] is a novel probabilistic method, and it is based on renewal process theory and 𝑧 transforms and is different from the above two methods. Tang et al. [11] use this method to analyze the queue length distribution and its stochastic decomposition property for a discrete-time Geo^𝑋/G/1 repairable queueing system with multiple adaptive delayed vacations. Recently, they have also obtained several reliability quantities of the service station [12].

The purpose of this paper is to introduce Tang's decomposition technique for the reliability of a discrete-time repairable Geom^𝑋/G/1 queueing system, in which the server takes exhaustive service and multiple adaptive delayed vacation discipline, and the service station is subject to failures and repairs. According to such a novel analytic technique, we discuss the main reliability indices of the service station, including the probability that the time 𝑛 is during server busy period, the unavailability and the average failure number during (0,𝑛]. What is more, we derive some important reliability relation equations, which indicate the structures of the service station indices. It should be noted that these reliability relation equations are new results and are not obtained by the existing discrete analytic techniques, such as the discrete supplementary variable technique [1–8] or matrix-geometric solution method [9, 10]. Some special cases and numerical examples validate the derived results and show that Tang's decomposition technique is applicable to reliability analysis of some complex discrete-time repairable bulk arrival queues.

The rest of the paper is organized as follows. The next section gives the model assumptions and some preliminaries. In Section 3, we use Tang's decomposition technique to study the main reliability indices of the service station, including the probability that the time 𝑛 is during server busy period, the unavailability and the average failure number during (0,𝑛]. We also derive some important reliability relation equations, which indicate the structures of the service station indices. Sections 4 and 5 present some special cases and numerical examples. Conclusions are finally drawn in Section 6.

2. Assumptions and Preliminaries

We consider the following discrete-time Geom^𝑋/G/1 queueing system with multiple adaptive delayed vacations and repairable service station.

(1) Let the time axis be slotted into intervals of equal length with the length of a slot being unity. To be more specific, let the time axis be marked by 0,1,…,𝑛,…. Here, we discuss the model for a late arrival system with delayed access (LAS-DA), and therefore a potential batch arrival takes place in (𝑛⁻,𝑛), and a potential departure occurs in (𝑛,𝑛⁺); for details, see Hunter [13]. The various time epochs involved in our model can be viewed through a self-explanatory figure (see Figure 1). The interarrival times between bulk customers, {𝜏_𝑖,𝑖≥1}, are independent identically distributed (i.i.d) random variables generated by a geometric distribution 𝑃{𝜏_𝑖=𝑗}=𝑝(1−𝑝)^𝑗−1, (𝑗≥1, 0<𝑝<1). Each group size 𝐷 is a random variable satisfying distribution 𝑃{𝐷=𝑘}=𝑒_𝑘, 𝑘≥1 with finite mean 𝑒 and probability-generating function (PGF) 𝐴(𝑧)=∑^∞_𝑘=1𝑒_𝑘𝑧^𝑘,|𝑧|<1.

Figure 1 

Various time epochs in a late arrival system with delayed access.

(2) The customers in different bulk arrivals are served in FCFS order, and the order in one bulk arrival is arbitrary. At a time, the server only serves a customer at the service station. The service times, {𝜒_𝑛,𝑛≥1}, are i.i.d random variables having distribution 𝑔_𝑗=𝑃{𝜒_𝑛=𝑗}, 𝑗≥1 with PGF 𝐺(𝑧)=∑^∞_𝑗=1𝑔_𝑗𝑧^𝑗, |𝑧|<1 and average service time 𝜇.

(3) The server will take at most 𝐻 vacations when the system becomes empty. Before each vacation, the server will spend some time for preparation. If a batch of customers arrives in the preparation (delayed) time, the server will abort the upcoming vacation and begin to serve the customers immediately. A vacation will be taken if no customer arrives in delayed time. Upon returning from a vacation, the server immediately serves if there is a waiting queue at the service station or prepares for another vacation if there is an empty queue at the service station, and the number of vacations taken is less than 𝐻. If the total 𝐻 vacations have been finished and no arrival occurs, the server will stay idle and wait for the customer arrival.

(4) Assume that 𝐻 obeys an arbitrary distribution 𝑃{𝐻=𝑗}=ℎ_𝑗, 𝑗≥1 with PGF 𝐻(𝑧)=∑^∞_𝑗=1ℎ_𝑗𝑧^𝑗, |𝑧|<1. Denote the server's the 𝑖th vacation time by 𝑉_𝑖, 𝑖≥1. Suppose that 𝑉_𝑖, 𝑖≥1 are independent and follow an identical arbitrary distribution 𝑣_𝑗=𝑃{𝑉_𝑖=𝑗}, 𝑗≥1 with PGF 𝑣(𝑧)=∑^∞_𝑗=1𝑣_𝑗𝑧^𝑗, |𝑧|<1. The delayed times, denoted by {𝑊_𝑖,𝑖≥1}, are i.i.d discrete random variables having an arbitrary distribution 𝑤_𝑗=𝑃{𝑊_𝑖=𝑗}, 𝑗≥1 with PGF 𝑤(𝑧)=∑^∞_𝑗=1𝑤_𝑗𝑧^𝑗, |𝑧|<1.

(5) The service station may possibly fail when and only when it is serving a customer. The failed service station will be repaired by the server immediately. After repair, the service station is as good as new and continues to serve the customer whose service has not been finished yet. We assume that the service time for a customer is cumulative.

(6) The lifetime 𝑋 of the service station has a geometric distribution 𝑃{𝑋=𝑗}=𝛼(1−𝛼)^𝑗−1, (𝑗≥1,0<𝛼<1), and its repair time 𝑌 obeys an arbitrary distribution 𝑦_𝑗=𝑃{𝑌=𝑗},𝑗≥1 with PGF 𝑦(𝑧)=∑^∞_𝑗=1𝑦_𝑗𝑧^𝑗, |𝑧|<1 and a mean repair time 𝛽.

(7) All random variables are mutually independent. At the initial time 𝑛=0, the server begins to serve when the number of customers presented in the system 𝑁(0)>0, or the server is idle and waits for the first arrival when 𝑁(0)=0.

Remark 1. From the assumptions (3) and (7), we know that after the first busy period, the server will begin to prepare for a vacation. Obviously, the assumption (7) is practical and reasonable. However, with Tang's decomposition technique, we will later prove that the steady-state reliability indices of the service station are independent of the initial state 𝑁(0)=𝑖,𝑖≥0, and they have nothing to do with multiple adaptive delayed vacations.

Remark 2. The multiple adaptive delayed vacation presented by this paper is introduced by Tang et al. [11], which is a generalization of multiple adaptive vacations [14]. The analysis in this work shows that Tang's method is applicable to reliability analysis of some complex discrete-time repairable queueing systems with generalized vacation.

Throughout this paper, we adopt the following notations. 𝐸(𝑋) is the mean of random variable 𝑋. 𝑃(𝑄) is the probability of event 𝑄. 𝐶^𝑘_𝑙=𝑙!/(𝑘!(𝑙−𝑘)!),0≤𝑘≤𝑙.

Definition 3. The “generalized service time of a customer” 𝜒 denotes the time interval from the time when the server begins to serve a customer until the service of this customer ends, which includes some possible repair times of the service station due to its failures in the process of serving this customer.

Lemma 4 (see [11]). The PGF and the mean of 𝜒 are given, respectively, by 𝐺(𝑧)=^∞_𝑗=0𝑃𝜒=𝑗𝑧^𝑗(𝑧)(,|𝑧|𝐸𝜒=𝑑𝐺(𝑧)=𝐺𝑧𝛼𝑦+𝑧1−𝛼)<1,𝑑𝑧_𝑧=1=𝜇1+𝛼𝛽, (1) where 𝐺(𝑧)=∑^∞_𝑗=1𝑔_𝑗𝑧^𝑗 and 𝑦(𝑧)=∑^∞_𝑗=1𝑦_𝑗𝑧^𝑗 are given by assumptions (2) and (6).

Definition 5. The “server busy period” denotes the time interval from the time when the server begins to serve a customer until the system becomes empty, which contains some possible repair times of service station due to its failures in the process of service.
Let 𝑏̃ denote the server busy period initiated with one customer and let its PGF be 𝑏̃(𝑧)=∑^∞_𝑗=0𝑃(𝑏̃=𝑗)𝑧^𝑗,|𝑧|<1. According to [15], the following lemma holds.

Lemma 6. 𝑏̃(𝑧) satisfies the equation 𝑏̃(𝑧)=𝐺[(1−𝑝)𝑧+𝑝𝐴(𝑏̃(𝑧))𝑧], and 𝐸𝑏̃=𝜇1+𝛼𝛽,1−𝑝𝑒𝜇1+𝛼𝛽𝜌<1,∞,𝜌≥1, (2) where 𝜌=𝑝𝑒𝜇(1+𝛼𝛽) is the traffic intensity of the considered queueing system.
The server busy period initiated with 𝑖 customers is denoted by 𝑏̃^⟨𝑖⟩. Then, 𝑏̃^⟨𝑖⟩ can be expressed as 𝑏̃^⟨𝑖⟩=𝑏̃₁+⋯+𝑏̃_𝑖, where 𝑏̃₁,…,𝑏̃_𝑖 are independent of each other and have the same distribution as 𝑏̃. Thus, the PGF of  𝑏̃^⟨𝑖⟩ is 𝑏̃^𝑖(𝑧).

Definition 7. The “idle period of the system” denotes the time interval from the time the system becomes empty until bulk customers arrive and enter the system.
Let 𝐼_𝑘 denote the 𝑘th idle period of the system, and then according to the system assumptions, {𝐼_𝑘,𝑘≥1} are mutually independent and follow an identical geometric distribution 𝑃(𝐼_𝑘=𝑗)=𝑝(1−𝑝)^𝑗−1,𝑘,𝑗=1,2,3,….

Lemma 8 (see [13]). If |𝑧|<1,𝑐_𝑛>0 and 𝑐(𝑧)=∑^∞_𝑛=0𝑐_𝑛𝑧^𝑛, then lim_𝑧↑1(1−𝑧)𝑐(𝑧)=𝑐<∞⟺lim_𝑛→∞1𝑛^𝑛_𝑘=1𝑐_𝑘=𝑐<∞. (3)

Lemma 9 (see [13]). If lim_𝑛→∞𝑐_𝑛=𝑐<∞, then lim_𝑛→∞(1/𝑛)∑^𝑛_𝑘=1𝑐_𝑘=𝑐.

3. Reliability Analysis of the Service Station

In this section, with the help of Tang's decomposition technique, we will discuss main reliability indices of the service station and analyze their structures.

3.1. The Probability That the Time 𝑛 Is during Server Busy Period

Theorem 10. Let 𝐴_𝑖(𝑛)=𝑃(the time 𝑛 is during server busy period ∣𝑁(0)=𝑖),𝑖≥0, and let 𝑎_𝑖(𝑧)=∑^∞_𝑛=0𝐴_𝑖(𝑛)𝑧^𝑛, |𝑧|<1, and then for 𝑖≥0, the PGF of 𝐴_𝑖(𝑛) is 𝑎₀(𝑧)=𝜏(𝑧)𝐴𝑏̃(𝑧1−𝑧1−)1−𝑤𝑝𝑧𝑣(𝑧)−𝑤𝑝𝑧𝐻(𝜃(𝑧))𝑣𝑝𝑧−𝑣(𝑧)(𝑧)+𝐴𝑏̃(𝑧)𝜏(𝑧1−𝑡)1−𝑤𝑝𝑧+𝐻(𝜃(𝑧))𝑤𝑝𝑧−𝜃(𝑧)(𝑧),𝑎1−𝑡_𝑖(𝑧)=1𝑏̃1−𝑧1−^𝑖(𝑧)1−𝑤𝑝𝑧𝑣(𝑧)−𝑤𝑝𝑧𝐻(𝜃(𝑧))𝑣𝑝𝑧−𝑣(𝑧)(𝑧)+𝑏̃1−𝑡_𝑖(𝑧)𝜏(𝑧)1−𝑤𝑝𝑧+𝐻(𝜃(𝑧))𝑤𝑝𝑧−𝜃(𝑧)1−𝑡(𝑧),𝑖≥1. (4)(5)
Further, one has lim_𝑛→∞𝐴_𝑖(𝑛)=lim_𝑧↑1(1−𝑧)𝑎_𝑖(𝑧)=𝑝𝑒𝜇1+𝛼𝛽,𝜌<1,1,𝜌≥1, (6) where 𝑝=1−𝑝,𝜏(𝑧)=𝑝𝑧1−(𝑧)𝑝𝑧,𝜃=𝑤𝑝𝑧𝑣,Λ(𝑧)=𝑝𝑧𝑏̃(𝑧),𝑡(𝑧)𝑏̃(𝑧)𝜏(𝑧𝑝𝑧+𝑝𝑧𝐴=𝐴)1−𝑤𝑝𝑧+𝐻(𝜃(𝑧))𝑤𝑝𝑧−𝜃(𝑧)+𝑤𝑝𝑧𝑣(Λ(𝑧))+𝑤𝐻(𝜃(𝑧×𝑣𝑝𝑧))𝑝𝑧−𝑣(Λ(𝑧)). (7) And 𝑤(𝑧), 𝑣(𝑧), 𝐴(𝑧), 𝐻(𝑧), 𝑏̃(𝑧), and 𝜌 are given by assumptions (1)–(6), Definition 5, and Lemma 6, respectively.

Proof. Let 𝑠_𝑘=^𝑘_𝑖=1𝑊_𝑖+𝑉_𝑖,𝛿_𝑘=^𝑘_𝑖=1𝜏_𝑖,𝑘≥1,𝑠₀=𝛿₀=0,_{∞𝑚[𝑖]=𝑖}=^∞_𝑚1=1⋯^∞_𝑚𝑖=1,𝑚[𝑖]=^𝑖_𝑗=1𝑚_𝑗, (8) and 𝜉(𝑛)=1 represents the event that the time 𝑛 is during server busy period. Denote by 𝐼_𝑖 the 𝑖th idle period of the system with a geometric distribution (see Definition 7), and 𝑏̃^⟨𝑘⟩ is the server busy period initiated with 𝑘 customers with PGF 𝑏̃^𝑘(𝑧) (see Lemma 6). Since the ending points of server busy period and server vacation are renewal points, by noting that the server stays idle and waits for the first arrival when 𝑁(0)=0, we get 𝐴₀(𝑛)=𝑃𝐼₁(𝑛)=≤𝑛,𝜉=1_∞𝑘=1𝑒_𝑘𝑃𝐼₁≤𝑛<𝐼₁+𝑏̃^⟨𝑘⟩+^𝑛_𝑘=1𝑒_𝑘^∞_𝑗=1ℎ_𝑗𝑃𝐼₁+𝑏̃^⟨𝑘⟩+𝐼₂≤𝑛,𝑠_𝑗<𝐼₂(𝑛)+,𝜉=1_𝑛𝑘=1𝑒_𝑘^∞_𝑗=1ℎ_𝑗^𝑗_𝑙=1𝑃𝐼₁+𝑏̃^⟨𝑘⟩+𝐼₂𝑠≤𝑛,_𝑙−1<𝐼₂≤𝑠_𝑙−1+𝑊_𝑙(𝑛)+,𝜉=1_𝑛𝑘=1𝑒_𝑘^∞_𝑗=1ℎ_𝑗^𝑗∞_𝑙=1𝑖=1𝑃𝑠_𝑙−1+𝑊_𝑙<𝐼₂≤𝑠_𝑙,𝐼₂+𝛿_𝑖−1<𝑠_𝑙≤𝐼₂+𝛿_𝑖,𝑛≥𝐼₁+𝑏̃^⟨𝑘⟩+𝑠_𝑙(𝑛)=,𝜉=1_∞𝑘=1𝑒_𝑘^𝑛_𝑖=1𝑃𝐼₁=𝑖1−^𝑛−𝑖_𝑗=𝑘𝑃𝑏̃^⟨𝑘⟩+=𝑗_𝑛𝑘=1𝑒_𝑘^∞_𝑗=1ℎ_𝑗^∞_𝑚=1𝑒_𝑚×^𝑛_𝑢=𝑘+1^𝑛−𝑢_𝑟=1𝑃𝐼₁+𝑏^⟨𝑘⟩=𝑢𝐴_𝑚()𝐼𝑛−𝑢−𝑟×𝑃₂=𝑟^𝑟−1_{𝑤=2𝑗−2}𝑃𝑠_𝑗+=𝑤_𝑛𝑘=1𝑒_𝑘^∞_𝑗=1ℎ_𝑗^𝑗∞_𝑙=1𝑚=1𝑒_𝑚×^𝑛_𝑢=𝑘+1^𝑛−𝑢_𝑟=1𝑃𝐼₁+𝑏^⟨𝑘⟩=𝑢𝑃𝐼₂=𝑟𝐴_𝑚()×𝑛−𝑢−𝑟_{𝑟−1𝑤=2𝑙−2}𝑃𝑠_𝑙−1=𝑤−^𝑟−1_{𝑤=2𝑙−1}𝑃𝑠_𝑙−1+𝑊_𝑙+=𝑤_𝑛𝑘=1𝑒_𝑘^∞_𝑗=1ℎ_𝑗^𝑗∞_𝑙=1^∞_{𝑖=1𝑚[𝑖]=𝑖}𝑒_𝑚1⋯𝑒_𝑚𝑖×^𝑛_𝑢=𝑘+1^𝑛−𝑢_{𝑡=2𝑙−1}^{𝑛−𝑢−𝑡}_𝑟=1𝐴_𝑚[𝑖]()𝐼𝑛−𝑢−𝑡−𝑟×𝑃₁+𝑏^⟨𝑘⟩=𝑢𝑃𝑠_𝑙−1+𝑊_𝑙=𝑡𝑃𝑉_𝑙=𝑟×𝐶^𝑖_𝑟𝑝^𝑖1−𝑝^𝑟−𝑖1−𝑝^𝑡. (9) By the same way, for 𝑖≥1, we have 𝐴_𝑖(𝑛)=1−^𝑛_𝑗=𝑖𝑃𝑏̃^⟨𝑖⟩=𝑗^∞_𝑗=1ℎ_𝑗^∞_𝑚=1𝑒_𝑚^𝑛_𝑢=𝑖^𝑛−𝑢_𝑟=1𝑃𝑏̃^⟨𝑖⟩𝐼=𝑢×𝑃₁=𝑟𝐴_𝑚(𝑛−𝑢−𝑟)^𝑟−1_𝑤=2𝑗𝑃𝑠_𝑗+=𝑤_𝑛𝑘=1𝑒_𝑘^∞_𝑗=1ℎ_𝑗^𝑗∞_𝑙=1𝑚=1𝑒_𝑚×^𝑛_𝑢=𝑖^𝑛−𝑢_𝑟=1𝑃𝑏^⟨𝑖⟩=𝑢𝐴_𝑚()𝐼𝑛−𝑢−𝑟×𝑃₁×=𝑟_{𝑟−1𝑤=2𝑙−2}𝑃𝑠_𝑙−1=𝑤−^𝑟−1_{𝑤=2𝑙−1}𝑃𝑠_𝑙−1+𝑊_𝑙+=𝑤_∞𝑗=1ℎ_𝑗^𝑗∞_𝑙=1^∞_{𝑖=1𝑚[𝑖]=𝑖}𝑒_𝑚1⋯𝑒_𝑚𝑖×^𝑛_𝑢=𝑖^𝑛−𝑢_{𝑡=2𝑙−1}^{𝑛−𝑢−𝑡}_𝑟=1𝐴_𝑚[𝑖]()𝑏𝑛−𝑢−𝑡−𝑟×𝑃^⟨𝑖⟩=𝑢𝑃𝑠_𝑙−1+𝑊_𝑙=𝑡𝑃𝑉_𝑙=𝑟×𝐶^𝑖_𝑟𝑝^𝑖1−𝑝^𝑟−𝑖1−𝑝^𝑡. (10)
Taking 𝑧-transforms of (9) and (10), respectively, we get 𝑎₀(𝑧)=𝜏(𝑧)̃𝑏(𝑧)+[𝜏(𝑧1−𝑧1−𝐴)]²𝐴𝑏̃(𝑧)𝐻(𝜃(𝑧))^∞_𝑚=1𝑒_𝑚𝑎_𝑚(𝑧)+[𝜏(𝑧)]²𝐴𝑏̃(𝑧)1−𝑤×(𝜃(𝑧𝑝𝑧1−𝐻))1−𝜃(𝑧)^∞_𝑚=1𝑒_𝑚𝑎_𝑚(𝑧)+𝜏(𝑧)𝐴𝑏̃(𝑧)𝑤(𝜃(𝑧𝑝𝑧1−𝐻))(𝑧)×1−𝜃_{∞∞𝑖=1𝑚[𝑖]=𝑖}𝑒_𝑚1⋯𝑒_𝑚𝑖𝑎_𝑚[𝑖](𝑧)^∞_𝑟=1𝑣_𝑟𝑧^𝑟𝐶^𝑖_𝑟𝑝^𝑖1−𝑝^𝑟−𝑖,𝑎_𝑖(𝑧)=1−𝑏̃^𝑖(𝑧)(𝑧)𝑏̃1−𝑧+𝜏^𝑖(𝑧)𝐻(𝜃(𝑧))^∞_𝑚=1𝑒_𝑚𝑎_𝑚(𝑧)+𝜏(𝑧)𝑏̃^𝑖(𝑧)1−𝑤(𝜃(𝑧𝑝𝑧1−𝐻))1−𝜃(𝑧)^∞_𝑚=1𝑒_𝑚𝑎_𝑚(𝑧)+𝑏̃^𝑖(𝑧)𝑤(𝜃(𝑧𝑝𝑧1−𝐻))(𝑧)×1−𝜃_{∞∞𝑖=1𝑚[𝑖]=𝑖}𝑒_𝑚1⋯𝑒_𝑚𝑖𝑎_𝑚[𝑖](𝑧)×^∞_𝑟=1𝑣_𝑟𝑧^𝑟𝐶^𝑖_𝑟𝑝^𝑖1−𝑝^𝑟−𝑖,𝑖≥1. (11)(12) It follows from (11) and (12) that 𝑎_𝑖(𝑧)=1−𝑏̃1−𝑧^𝑖(𝑧)𝜏(𝑧)−()𝑎1−𝑧1−𝑧₀(𝑧)𝜏(𝑧)𝐴𝑏̃(𝑧),𝑖≥1. (13) Substituting (13) into (11) and noting ^∞∞_{𝑖=1𝑚[𝑖]=𝑖}𝑒_𝑚1⋯𝑒_𝑚𝑖𝑏̃^𝑚[𝑖](𝑧)^∞_𝑟=1𝑣_𝑟𝑧^𝑟𝐶^𝑖_𝑟𝑝^𝑖1−𝑝^𝑟−𝑖=^∞_𝑟=1𝑣_𝑟𝑧^𝑟∞_𝑖=1𝐶^𝑖_𝑟𝑝𝐴𝑏̃(𝑧)^𝑖1−𝑝^𝑟−𝑖=𝑣𝑝𝑧+𝑝𝑧𝐴𝑏̃(𝑧)−𝑣𝑝𝑧, (14) we easily get (4). Equation (5) is obtained by (4) and (13). By Lemmas 8 and 9, we have lim_𝑛→∞𝐴_𝑖(𝑛)=lim_𝑧↑1(1−𝑧)𝑎_𝑖(𝑧). So, (6) follows by (4), (5), L' Hospital's rule, and direct calculations.

To study the unavailability and the failure number during (0,𝑛] of the service station, we first consider a classical discrete-time single-unit repairable system. Whenever the unit fails, the system breaks down. The failed unit is repaired immediately, and it is as good as new after repair. As soon as the repair of the failed unit is completed, the system starts to operate immediately. The lifetime 𝑋 of the unit is geometrically distributed with parameter 𝛼,0≤𝛼<1, and its repair time 𝑌 obeys an arbitrary discrete-time distribution with a mean repair time 𝛽. Suppose that the unit is new at the initial time 𝑛=0. 𝑋 and 𝑌 are mutually independent. For 𝑛≥0, let Φ(𝑛)𝑀(𝑛)((],=𝑃,=𝐸thefailurenumberofthesystemduring0,𝑛 (15) and 𝜑(𝑧)=∑^∞_𝑛=0Φ(𝑛)𝑧^𝑛,  𝑚(𝑧)=∑^∞_𝑛=0𝑀(𝑛)𝑧^𝑛, |𝑧|<1.

Similar to the discussions in a continuing-time single-unit system [16], we have the following lemma.

Lemma 11. For |𝑧|<1, then 𝜑(𝑧)=𝑧𝛼1−𝑦(𝑧)((𝑧),𝑚(𝑧)=1−𝑧)1−𝑧+𝑧𝛼1−𝑦𝑧𝛼((𝑧,1−𝑧)1−𝑧+𝑧𝛼1−𝑦)lim_𝑛→∞Φ(𝑛)=lim_𝑧↑1(1−𝑧)𝜑(𝑧)=𝛼𝛽,1+𝛼𝛽lim_𝑛→∞𝑀(𝑛)𝑛=lim_𝑧↑1(1−𝑧)²𝑚(𝑧)=𝛼,1+𝛼𝛽 (16) where 𝑦(𝑧)=∑^∞_𝑗=0𝑃(𝑌=𝑗)𝑧^𝑗, |𝑧|<1.

Remark 12. Let 𝑀_𝑓(𝑘)=𝑀(𝑘)−𝑀(𝑘−1), 𝑘≥1, and let 𝑚_𝑓(𝑧)=∑^∞_𝑛=0𝑀_𝑓(𝑛)𝑧^𝑛, |𝑧|<1, and then according to Lemmas 8 and 9, we have lim_𝑛→∞𝑀(𝑛)𝑛=lim_𝑧↑1(1−𝑧)𝑚_𝑓(𝑧)=lim_𝑧↑1(1−𝑧)²𝑚(𝑧). (17)

3.2. The Unavailability of the Service Station

The unavailability of the service station at time 𝑛, that is, the probability that the service station is repaired at time 𝑛.

Theorem 13. Let Φ_𝑖(𝑛)=𝑃 (the service station is repaired at time 𝑛∣𝑁(0)=𝑖), 𝑖≥0, and let 𝜑_𝑖(𝑧)=∑^∞_𝑛=0Φ_𝑖(𝑛)𝑧^𝑛, |𝑧|<1, and then the PGF of Φ_𝑖(𝑛) is 𝜑_𝑖(𝑧)=𝜑(𝑧)(1−𝑧)𝑎_𝑖(𝑧),𝑖≥0, (18) and the steady-state unavailability of the service station is given by lim_𝑛→∞Φ_𝑖(𝑛)=lim_𝑧↑1(1−𝑧)𝜑_𝑖(𝑧)=𝑝𝑒𝜇𝛼𝛽,𝜌<1𝛼𝛽,1+𝛼𝛽𝜌≥1, (19) where 𝜑(𝑧), 𝑎_𝑖(𝑧), 𝑖≥0, and 𝜌 are given by Lemma 11, Theorem 10, and Lemma 6, respectively.

Proof. (i) Let 𝜂(𝑛)=1 denote the event that the service station is repaired at time 𝑛. The service station is repaired at time 𝑛 if and only if the time 𝑛 is during one server busy period, and the service station is repaired at time 𝑛. Therefore, by total probability decomposition and renewal point technique, Φ₀(𝑛) is decomposed as Φ₀(𝑛)=^∞_𝑘=1𝑒_𝑘𝑃𝐼₁≤𝑛<𝐼₁+𝑏̃^⟨𝑘⟩(𝑛)+,𝜂=1_𝑛𝑘=1𝑒_𝑘^∞_𝑗=1ℎ_𝑗𝑃𝐼₁+𝑏̃^⟨𝑘⟩+𝐼₂≤𝑛,𝑠_𝑗<𝐼₂(𝑛)+,𝜂=1_𝑛𝑘=1𝑒_𝑘^∞_𝑗=1ℎ_𝑗^𝑗_𝑙=1𝑃𝐼₁+𝑏̃^⟨𝑘⟩+𝐼₂𝑠≤𝑛,_𝑙−1<𝐼₂≤𝑠_𝑙−1+𝑊_𝑙(𝑛)+,𝜂=1_𝑛𝑘=1𝑒_𝑘^∞_𝑗=1ℎ_𝑗^𝑗∞_𝑙=1𝑖=1𝑃𝑠_𝑙−1+𝑊_𝑙≤𝐼₂≤𝑠_𝑙,𝐼₂+𝛿_𝑖−1<𝑠_𝑙≤𝐼₂+𝛿_𝑖,𝑛≥𝐼₁+𝑏̃^⟨𝑘⟩+𝑠_𝑙(𝑛)=,𝜂=1_∞𝑘=1𝑒_𝑘^𝑛_𝑖=1𝑃𝐼₁=𝑖𝑆_𝑘()+𝑛−𝑖_𝑛𝑘=1𝑒_𝑘^∞_𝑗=1ℎ_𝑗^∞_𝑚=1𝑒_𝑚^𝑛_𝑢=𝑘+1^𝑛−𝑢_𝑟=1𝑃𝐼₁+𝑏^⟨𝑘⟩=𝑢×Φ_𝑚(𝑛−𝑢−𝑟)𝑃𝐼₂=𝑟^𝑟−1_𝑤=2𝑗𝑃𝑠_𝑗+=𝑤_𝑛𝑘=1𝑒_𝑘^∞_𝑗=1ℎ_𝑗^𝑗∞_𝑙=1𝑚=1𝑒_𝑚×^𝑛_𝑢=𝑘+1^𝑛−𝑢_𝑟=1𝑃𝐼₁+𝑏^⟨𝑘⟩=𝑢𝑃𝐼₂=𝑟Φ_𝑚()×𝑛−𝑢−𝑟_{𝑟−1𝑤=2𝑙−2}𝑃𝑠_𝑙−1=𝑤−^𝑟−1_{𝑤=2𝑙−1}𝑃𝑠_𝑙−1+𝑊_𝑙+=𝑤_𝑛𝑘=1𝑒_𝑘^∞_𝑗=1ℎ_𝑗^𝑗∞_𝑙=1^∞_{𝑖=1𝑚[𝑖]=𝑖}𝑒_𝑚1⋯𝑒_𝑚𝑖×^𝑛_𝑢=𝑘+1^𝑛−𝑢_{𝑡=2𝑙−1}^{𝑛−𝑢−𝑡}_𝑟=1Φ_𝑚[𝑖]()𝐼𝑛−𝑢−𝑡−𝑟×𝑃₁+𝑏^⟨𝑘⟩=𝑢𝑃𝑠_𝑙−1+𝑊_𝑙=𝑡𝑃𝑉_𝑙=𝑟×𝐶^𝑖_𝑟𝑝^𝑖1−𝑝^𝑟−𝑖1−𝑝^𝑡, (20) where 𝑆_𝑘(𝑛)=𝑃(0≤𝑛<𝑏̃^⟨𝑘⟩, and the service station is repaired at time n), 𝑘≥1.
Similarly, for 𝑖≥1, we have the decomposition of Φ_𝑖(𝑛) as follows: Φ_𝑖(𝑛)=𝑆_𝑖(𝑛)+^∞_𝑗=1ℎ_𝑗^∞_𝑚=1𝑒_𝑚×^𝑛_𝑢=𝑖^𝑛−𝑢_𝑟=1𝑃𝑏̃^⟨𝑖⟩=𝑢Φ_𝑚(𝑛−𝑢−𝑟)𝑃𝐼₁×=𝑟_{𝑟−1𝑤=2𝑗}𝑃𝑠_𝑗=𝑤+^∞_𝑗=1ℎ_𝑗^𝑗∞_𝑙=1𝑚=1𝑒_𝑚×^𝑛_𝑢=𝑖^𝑛−𝑢_𝑟=1𝑃𝑏^⟨𝑖⟩=𝑢Φ_𝑚()𝐼𝑛−𝑢−𝑟×𝑃₁×=𝑟_{𝑟−1𝑤=2𝑙−2}𝑃𝑠_𝑙−1=𝑤−^𝑟−1_{𝑤=2𝑙−1}𝑃𝑠_𝑙−1+𝑊_𝑙+=𝑤_∞𝑗=1ℎ_𝑗^𝑗∞_𝑙=1^∞_{𝑖=1𝑚[𝑖]=𝑖}𝑒_𝑚1⋯𝑒_𝑚𝑖×^𝑛_𝑢=𝑖^𝑛−𝑢_{𝑡=2𝑙−1}^{𝑛−𝑢−𝑡}_𝑟=1Φ_𝑚[𝑖]()𝑏𝑛−𝑢−𝑡−𝑟×𝑃^⟨𝑖⟩=𝑢𝑃𝑠_𝑙−1+𝑊_𝑙=𝑡𝑃𝑉_𝑙=𝑟×𝐶^𝑖_𝑟𝑝^𝑖1−𝑝^𝑟−𝑖1−𝑝^𝑡. (21)
(ii) For 𝑖≥1, 𝑆_𝑖(𝑛)=Φ(𝑛)−^𝑛_𝑘=𝑖𝑃𝑏̃^⟨𝑖⟩=𝑘Φ(𝑛−𝑘_∞𝑛=0),𝑆_𝑖(𝑛)𝑧^𝑛=𝜑(𝑧)1−𝑏̃^𝑖(𝑧),|𝑧|<1, (22) where Φ(𝑛) and 𝜑(𝑧) are given by Lemma 11.
In fact, Φ(𝑛) can be decomposed as Φ(𝑛)=𝑃thesystemisrepairedattime𝑛,𝑏̃^⟨𝑖⟩𝑏̃≤𝑛+𝑃thesystemisrepairedattime𝑛,^⟨𝑖⟩=>𝑛_{𝑛𝑘=𝑖}𝑃𝑏̃^⟨𝑖⟩=𝑘Φ(𝑛−𝑘)+𝑆_𝑖(𝑛). (23) So, (22) is easily obtained.
(iii) Taking 𝑧-transforms of (20) and (21), respectively, and by means of (22), we have the following results: 𝜑₀(𝑧)(𝑧)𝜑(𝑧𝑏̃(𝑧+[𝜏(𝑧=𝜏)1−𝐴))]²𝐴𝑏̃(𝑧)𝐻(𝜃(𝑧))^∞_𝑚=1𝑒_𝑚𝜑_𝑚(𝑧)+[𝜏(𝑧)]²𝐴𝑏̃(𝑧)1−𝑤(𝜃(𝑧𝑝𝑧1−𝐻))(𝑧)×1−𝜃_∞𝑚=1𝑒_𝑚𝜑_𝑚(𝑧)+𝜏(𝑧)𝐴𝑏̃(𝑧)𝑤(𝜃(𝑧𝑝𝑧1−𝐻))(𝑧)×1−𝜃_{∞∞𝑖=1𝑚[𝑖]=𝑖}𝑒_𝑚1⋯𝑒_𝑚𝑖𝜑_𝑚[𝑖](𝑧)×^∞_𝑟=1𝑣_𝑟𝑧^𝑟𝐶^𝑖_𝑟𝑝^𝑖1−𝑝^𝑟−𝑖,𝜑_𝑖(𝑧)=𝜑(𝑧)1−𝑏̃^𝑖(𝑧)+𝜏(𝑧)𝑏̃^𝑖(𝑧)𝐻(𝜃(𝑧))^∞_𝑚=1𝑒_𝑚𝜑_𝑚(𝑧)+𝜏(𝑧)𝑏̃^𝑖(𝑧)1−𝑤(𝜃(𝑧𝑝𝑧1−𝐻))1−𝜃(𝑧)^∞_𝑚=1𝑒_𝑚𝜑_𝑚(𝑧)+𝑏̃^𝑖(𝑧)𝑤(𝜃(𝑧𝑝𝑧1−𝐻))(𝑧)×1−𝜃_{∞∞𝑖=1𝑚[𝑖]=𝑖}𝑒_𝑚1⋯𝑒_𝑚𝑖𝜑_𝑚[𝑖](𝑧)×^∞_𝑟=1𝑣_𝑟𝑧^𝑟𝐶^𝑖_𝑟𝑝^𝑖1−𝑝^𝑟−𝑖,𝑖≥1. (24) Performing similar operations in the proof of Theorem 10 on (24), we can complete the proof by Theorem 10 and Lemma 11 above.

3.3. The Mean Failure Number of Service Station during (0,𝑛]

Theorem 14. Let 𝑀_𝑖(𝑛)=𝐸 (the failure number of service station during (0,𝑛]∣𝑁(0)=𝑖), 𝑖≥0, and let 𝑚_𝑖(𝑧)=∑^∞_𝑛=0𝑀_𝑖(𝑛)𝑧^𝑛, |𝑧|<1, and then the PGF of 𝑀_𝑖(𝑛) is 𝑚_𝑖(𝑧)=𝑚(𝑧)(1−𝑧)𝑎_𝑖(𝑧),𝑖≥0, (25) and the steady-state failure frequency of service station is given by lim_𝑛→∞𝑀_𝑖(𝑛)𝑛=lim_𝑧↑1(1−𝑧)²𝑚_𝑖(𝑧)=𝛼𝑝𝑒𝜇𝛼,𝜌<1,1+𝛼𝛽𝜌≥1, (26) where 𝑚(𝑧), 𝑎_𝑖(𝑧), 𝑖≥0, and 𝜌 are given by Lemma 11, Theorem 10, and Lemma 6, respectively.

Proof. (1) For 𝑖≥1, let 𝐽_𝑖(𝑛)=𝐸0≤𝑛<𝑏̃^⟨𝑖⟩(],𝐿,thefailurenumberofservicestationduring0,𝑛_𝑖(𝑛)=𝐸𝑏̃^⟨𝑖⟩𝑏̃≤𝑛,thefailurenumberofservicestationduring0,^⟨𝑖⟩. (27) Then, similar to (22) above, we have 𝐽_𝑖(𝑛)+𝐿_𝑖(𝑛)=𝑀(𝑛)−^𝑛_𝑘=𝑖𝑃𝑏̃^⟨𝑖⟩=𝑘𝑀(𝑛−𝑘),𝑖≥1,_∞𝑛=0𝐽_𝑖(𝑛)+𝐿_𝑖(𝑛)𝑧^𝑛=𝑚(𝑧)1−𝑏̃^𝑖(𝑧),|𝑧|<1, (28) where 𝑀(𝑛) and 𝑚(𝑧) are given by Lemma 11.
(2) Using total probability decomposition and renewal point technique, we have the decomposition of 𝑀₀(𝑛) as follows: 𝑀₀(𝑛)=^∞_𝑘=1𝑒_𝑘𝐸𝐼₁≤𝑛<𝐼₁+𝑏̃^⟨𝑘⟩𝐼,thefailurenumberofservicestationduring₁,𝑛+𝐸𝑛≥𝐼₁+𝑏̃^⟨𝑘⟩𝐼,thefailurenumberofservicestationduring₁,𝐼₁+𝑏̃^⟨𝑘⟩+^𝑛_𝑘=1𝑒_𝑘^∞_𝑗=1ℎ_𝑗𝐸𝐼₁+𝑏̃^⟨𝑘⟩+𝐼₂≤𝑛,𝑠_𝑗<𝐼₂,𝐼thefailurenumberofservicestationduring₁+𝑏̃^⟨𝑘⟩+𝐼₂+,𝑛_𝑛𝑘=1𝑒_𝑘^∞_𝑗=1ℎ_𝑗×^𝑗_𝑙=1𝐸𝑠_𝑙−1<𝐼₂≤𝑠_𝑙−1+𝑊_𝑙,𝐼₁+𝑏̃^⟨𝑘⟩+𝐼₂𝐼≤𝑛,thefailurenumberofservicestationduring₁+𝑏̃^⟨𝑘⟩+𝐼₂+,𝑛_𝑛𝑘=1𝑒_𝑘^∞_𝑗=1ℎ_𝑗×^𝑗∞_𝑙=1𝑖=1𝑃𝑠_𝑙−1+𝑊_𝑙≤𝐼₂≤𝑠_𝑙,𝐼₂+𝛿_𝑖−1<𝑠_𝑙≤𝐼₂+𝛿_𝑖,𝑛≥𝐼₁+𝑏̃^⟨𝑘⟩+𝑠_𝑙𝐼,thefailurenumberofservicestationduring₁+𝑏̃^⟨𝑘⟩+𝑠_𝑙=,𝑛_∞𝑘=1𝑒_𝑘^𝑛_𝑖=1𝑃𝐼₁=𝑖𝐽_𝑘(𝑛−𝑖)+𝐿_𝑘(+𝑛−𝑖)_𝑛𝑘=1𝑒_𝑘^∞_𝑗=1ℎ_𝑗^∞_𝑚=1𝑒_𝑚×^𝑛_𝑢=𝑘+1^𝑛−𝑢_𝑟=1𝑃𝐼₁+𝑏^⟨𝑘⟩=𝑢𝑀_𝑚(𝑛−𝑢−𝑟)𝑃𝐼₂×=𝑟_{𝑟−1𝑤=2𝑗}𝑃𝑠_𝑗=𝑤+^𝑛_𝑘=1𝑒_𝑘^∞_𝑗=1ℎ_𝑗^𝑗∞_𝑙=1𝑚=1𝑒_𝑚×^𝑛_𝑢=𝑘+1^𝑛−𝑢_𝑟=1𝑃𝐼₁+𝑏^⟨𝑘⟩𝐼=𝑢×𝑃₂=𝑟𝑀_𝑚()×𝑛−𝑢−𝑟_{𝑟−1𝑤=2𝑙−2}𝑃𝑠_𝑙−1=𝑤−^𝑟−1_{𝑤=2𝑙−1}𝑃𝑠_𝑙−1+𝑊_𝑙+=𝑤_𝑛𝑘=1𝑒_𝑘^∞_𝑗=1ℎ_𝑗^𝑗∞_𝑙=1^∞_{𝑖=1𝑚[𝑖]=𝑖}𝑒_𝑚1⋯𝑒_𝑚𝑖×^𝑛_𝑢=𝑘+1^𝑛−𝑢_{𝑡=2𝑙−1}^{𝑛−𝑢−𝑡}_𝑟=1𝑀_𝑚[𝑖]()𝐼𝑛−𝑢−𝑡−𝑟×𝑃₁+𝑏^⟨𝑘⟩=𝑢𝑃𝑠_𝑙−1+𝑊_𝑙=𝑡𝑃𝑉_𝑙=𝑟×𝐶^𝑖_𝑟𝑝^𝑖1−𝑝^𝑟−𝑖1−𝑝^𝑡. (29) Similarly, 𝑀_𝑖(𝑛),𝑖≥1 are decomposed as 𝑀_𝑖(𝑛)=𝐽_𝑖(𝑛)+𝐿_𝑖(𝑛)+^∞_𝑗=1ℎ_𝑗^∞_𝑚=1𝑒_𝑚×^𝑛_𝑢=𝑖^𝑛−𝑢_𝑟=1𝑃𝑏̃^⟨𝑖⟩=𝑢𝑀_𝑚(𝑛−𝑢−𝑟)𝑃𝐼₁×=𝑟_{𝑟−1𝑤=2𝑗}𝑃s_𝑗=𝑤+^∞_𝑗=1ℎ_𝑗^𝑗∞_𝑙=1𝑚=1𝑒_𝑚×^𝑛_𝑢=𝑖^𝑛−𝑢_𝑟=1𝑃𝑏^⟨𝑖⟩=𝑢𝑀_𝑚()𝐼𝑛−𝑢−𝑟×𝑃₁×=𝑟_{𝑟−1𝑤=2𝑙−2}𝑃𝑠_𝑙−1=𝑤−^𝑟−1_{𝑤=2𝑙−1}𝑃𝑠_𝑙−1+𝑊_𝑙+=𝑤_∞𝑗=1ℎ_𝑗^𝑗∞_𝑙=1^∞_{𝑖=1𝑚[𝑖]=𝑖}𝑒_𝑚1⋯𝑒_𝑚𝑖×^𝑛_𝑢=𝑖^𝑛−𝑢_{𝑡=2𝑙−1}^{𝑛−𝑢−𝑡}_𝑟=1𝑀_𝑚[𝑖]()𝑏𝑛−𝑢−𝑡−𝑟×𝑃^⟨𝑖⟩=𝑢𝑃𝑠_𝑙−1+𝑊_𝑙=𝑡𝑃𝑉_𝑙=𝑟×𝐶^𝑖_𝑟𝑝^𝑖1−𝑝^𝑟−𝑖1−𝑝^𝑡. (30) Taking 𝑧-transforms of (29) and (30), and performing similar operations in the proof of Theorem 10, we can complete the proof by Theorem 10 and Lemma 11 above.

Remark 15. Equations (18) and (25) show that studying the unavailability and the mean failure number during (0,𝑛] of the service station is equivalent to studying the corresponding indices in the classical discrete-time single-unit system and the probability that the time 𝑛 is during server busy period presented in this paper. It is important that (18) and (25) indicate the structures of the service station indices.

Remark 16. It is seen from (6), (19), (26), and Lemma 11 that the steady-state relation equations lim_𝑛→∞Φ_𝑖(𝑛)=lim_𝑛→∞Φ(𝑛)lim_𝑛→∞𝐴_𝑖(𝑛) and lim_𝑛→∞𝑀_𝑖(𝑛)/𝑛=lim_𝑛→∞(𝑀(𝑛)/𝑛)lim_𝑛→∞𝐴_𝑖(𝑛),   𝑖≥0 also hold. Further, the two steady-state results are independent of the initial state 𝑁(0)=𝑖, 𝑖≥0, and they have nothing to do with server vacations and delayed times.

Remark 17. It should be noted that (18), (25), and two steady-state relation equations derived in Remark 16 are new results, and these important reliability relation equations are not obtained by the existing discrete analytic techniques, such as the discrete supplementary variable technique [1–8] or matrix-geometric solution method [9, 10].

4. Special Cases

Case 1. If 𝐻(𝑧)=𝑧^𝐽 and 𝑃(𝑊_𝑖=0)=1,𝑖≥1, then our model becomes a discrete-time Geom^𝑋/G/1 queue with an unreliable service station and at most 𝐽 vacations. In this special case, for 𝜌=𝑝𝑒𝜇(1+𝛼𝛽)<1, the busy probabilities of the server lim_𝑛→∞𝐴_𝑖(𝑛), the unavailability lim_𝑛→∞Φ_𝑖(𝑛), and the failure frequency lim_𝑛→∞𝑀_𝑖(𝑛)/𝑛 are as follows: lim_𝑛→∞𝐴_𝑖(𝑛)=𝑝𝑒𝜇1+𝛼𝛽=𝜌,lim_𝑛→∞Φ_𝑖(𝑛)=𝑝𝑒𝜇𝛼𝛽,lim_𝑛→∞𝑀_𝑖(𝑛)𝑛=𝑝𝑒𝜇𝛼. (31) These results are very similar to those for the continuous-time counterpart, which also have nothing to do with server vacations (see [17, Sections 3 and 4, p 9-10] and the following fact).
In fact, for the continuous-time counterpart, that is, the M^𝑋/G/1 queue with an unreliable service station and at most 𝐽 vacations (setting 𝑝=0 in Ke's model [17]), under steady-state condition 𝜌_𝐻<1  (𝜌_𝐻=𝜆𝐸[𝑋]𝐸[𝑆](1+𝛼𝐸[𝑅]), the busy probability of the server 𝑃_busy, the unavailability 𝑃_𝑟, and the failure frequency 𝑀_𝑓 are given, respectively, by 𝑃_busy=𝐸[𝐻]𝐸[𝐶]=𝜆𝐸[𝑋]𝐸[𝑆](1+𝛼𝐸[𝑅])=𝜌_𝐻,𝑃_𝑟=𝐸[𝐷]𝐸[𝐶]=𝜆𝐸[𝑋]𝐸[𝑆]𝛼𝐸[𝑅],𝑀_𝑓=𝜆𝐸[𝑋]𝐸[𝑆]𝛼, (32) where 𝐸[𝐻]=1−𝛾^𝐽₀𝜆𝐸[𝑉]1−𝛾₀+𝛾^𝐽₀𝐸[𝑋]𝐸[𝑆](1+𝛼𝐸[𝑅])1−𝜌_𝐻 (33) is the expected length of completion period (here, 𝛾₀=∫^∞₀𝑒^−𝜆𝑡𝑑𝑉(𝑡), and 𝑉(𝑡) and 𝐸[𝑉] are the vacation distribution function and the expected length of each vacation), 𝐸[𝐷]=1−𝛾^𝐽₀𝜆𝐸[𝑉]1−𝛾₀+𝛾^𝐽₀𝐸[𝑋]𝐸[𝑆]𝛼𝐸[𝑅]1−𝜌_𝐻 (34) is the expected length of breakdown period of service station, 𝐸[𝐶]=1−𝛾^𝐽₀𝜆𝐸[𝑉]1−𝛾₀+𝛾^𝐽₀1𝜆1−𝜌_𝐻 (35) is the expected length of busy cycle, and 𝜆,𝐸[𝑋],𝐸[𝑆],𝛼, and 𝐸[𝑅] are corresponding to 𝑝, 𝑒, 𝑢, 𝛼, and 𝛽 in our model, respectively, (see [17, Sections 3 and 4, p 9-10] for details).

Case 2. If 𝐻(𝑧)=𝑧, 𝑃(𝐷=1)=1 and 𝑃(𝑊_𝑖=0)=1, 𝑖≥1, then our model becomes a discrete-time Geom/G/1 queue with an unreliable service station and a single vacation. In this case, for 𝜌=𝑝𝜇(1+𝛼𝛽)<1, we get lim_𝑛→∞𝐴_𝑖(𝑛)=𝑝𝜇1+𝛼𝛽=𝜌,lim_𝑛→∞Φ_𝑖(𝑛)=𝑝𝜇𝛼𝛽,lim_𝑛→∞𝑀_𝑖(𝑛)𝑛=𝑝𝜇𝛼, (36) which are very similar to those for the continuous-time counterpart [18].

5. Numerical Examples

To demonstrate our model and analysis results, the switched-virtual-connection- (SVC-) based IP-over-ATM network [19] is presented as a practical example of our model. In such a system, packets arrive in batch and can be modelled as a Bernoulli process with rate 𝑝. The distribution for batch size 𝐷 is taken as 𝑃(𝐷=𝑘)=1/𝑛, 𝑘=1,2,…,𝑛. The service time of each packet is assumed to follow a geometric distribution with mean 𝜉=1.2. Whenever the last packet is served, IP controller (server) releases the SVC (vacations). Before each release, the server checks the packets (delayed vacation). If no packets occur, a decision is made for another release to be performed next. Otherwise, if packets occur, service restarts. Moreover, the service may be interrupted (server breakdown) due to some unpredictable events. Suppose that unpredictable events occur according to a Bernoulli process with rate 𝛼 and broken server is immediately recovered. The recovery time obeys negative binomial with generating function 𝑌(𝑥)=(2𝑥/(3−𝑥))² with mean 𝛽=3. The service will continuously start when the interruption is recovered.

Several numerical results in Tables 1–3 analyze the effects of ATM network parameters on IP controller (server) performance, including the busy probability of IP controller lim_𝑡→∞𝐴_𝑖(𝑛), the unavailability lim_𝑡→∞Φ_𝑖(𝑛), and the failure frequency lim_𝑡→∞𝑀(𝑛)/𝑛. In addition, the system load 𝜌 is also considered.

Table 1 

The effects of batch packet arrival rate 𝑝 on IP controller (𝑛=1,𝛼=0.25).

Table 2 

The effects of batch size parameter 𝑛 on IP controller (𝑝=0.15,𝛼=0.25).

Table 3 

The effects of unpredictable events arrival rate 𝛼 on IP controller (𝑝=0.15,𝑛=1).

With analysis results in Section 3, the effects of batch packet arrival rate 𝑝 on IP controller indices are reported in Table 1, where we set (𝑛,𝛼)=(1,0.25). It is seen from Table 1 that as 𝑝 increases, four performance indices all increase, but when 𝑝≥0.55, the system load 𝜌 is more than one, and the other three indices do not change. This is because the queueing system becomes unstable. Table 2 shows the effects of batch size on IP controller indices, where we set (𝑝,𝛼)=(0.15,0.25). We observe that the influence of batch size on four indices is very similar to that of 𝑝. The effects of unpredictable events arrival rate 𝛼 on IP controller indices are shown in Table 3 for the set of parameters (𝑝,𝑛)=(0.15,1). We note that all IP controller indices increase monotonously as the value 𝛼 increases. Moreover, as 𝛼≥1.65, IP controller is always busy.

Finally, we conclude that batch packet arrival, batch size, and unpredictable events arrival affect IP controller performance under the stability condition; that is, 𝜌<1. As 𝜌≥1, IP controller indices will not be affected by batch packet arrival and batch size, and IP controller is always busy. This phenomenon is due to the unstability property of the ATM network.

6. Conclusions

In this paper, we use Tang's decomposition technique for reliability indices of a discrete-time repairable Geom^𝑋/G/1 queue, in which the server takes exhaustive service and multiple adaptive delayed vacation discipline, and the service station is subject to failures and repairs. According to such a novel analytic technique, we discuss main reliability indices of the service station, such as the probability that the time 𝑛 is during server busy period, the unavailability and the average failure number during (0,𝑛]. Also, we derive some important reliability relation equations, which indicate the structures of the service station indices. It should be noted that these reliability relation equations are new results and are not obtained by existing discrete analytic techniques, such as the discrete supplementary variable technique [1–8] or matrix-geometric solution method [9, 10]. Some special cases and numerical examples validate the derived results and show that Tang's analytic technique is applicable to reliability analysis of complex discrete-time repairable bulk arrival queues.

Acknowledgments

The authors would like to thank the referees and editor for their valuable comments and suggestions. This work is supported by the Youth Foundation of Chongqing University of Technology of China (No. 2010ZQ14) and the National Natural Science Foundation of China (No. 71171138).