Study on the Departure Process of Discrete-Time 𝐺𝑒𝑜/𝐺/1 Queue with Randomized Vacations

Abstract

This paper presents an analysis of the departure process of a discrete-time 𝐺𝑒𝑜/𝐺/1 queue with randomized vacations. By using probability decomposition techniques and renewal process, the expression of expected number of departures during time interval (0⁺,𝑛⁺] is derived. The relation among departure process, server state process, and service renewal process is obtained. The relation displays the decomposition characteristic of the departure process. Furthermore, the approximate expansion of the expected number of departures is gained. Since the departure process also often corresponds to an arrival process for a downstream queue in queueing network, it is hoped that the results obtained in this paper may provide useful information for queueing network.

1. Introduction

Since discrete-time queues with vacation schedule are more suitable for its applicability in the performance analysis of telecommunication systems, it has gained extended attention. For an excellent survey of earlier works on vacation models as well as its applications, interested readers can refer to [1–5]. Keilson and Servi (1986) [6] introduced another type of important vacation—Bernoulli vacations in which, after each service completion, the server takes a vacation with probability 𝑝 or continues its busy period (if there are customers in the system) with probability 1−𝑝. Following the work of Keilson and Servi, much further research on queueing system with Bernoulli vacations was done in [7–10]. On this base, Ke and Chu (2006) [11] develop an interesting vacation discipline from actual situation, where the server takes at most 𝐽 vacations repeatedly until at least one customer is found waiting in the queue upon the server returns from a vacation. In the subsequent research of [12–14], Ke and Huang extend the vacation discipline to the randomized vacation policy with at most 𝐽 vacations where the server takes another vacation with probability 𝑝 or remains dormant within the system with probability 1−𝑝 if no customer is found waiting in the queue while the server returns from a vacation; otherwise, the server starts to serve the customers immediately if there are some customers waiting for service at the end of the vacation. This pattern does not terminate until the server has taken 𝐽 successive vacations. Wang (2010) [15] and Wang et al. (2011) [16] introduce the randomized vacation policy to the discrete-time 𝐺𝑒𝑜/𝐺/1 queue. Such a modified vacation discipline has potentially applications in practical systems [13], for example, in some stochastic production and inventory control systems such as production to orders.

This is a continuation of work by Wang et al.(2011) [16] and Luo et al. (2013) [17] where they study the queue size distribution for 𝐺𝑒𝑜/𝐺/1 with randomized vacation and at most 𝐽 vacations by using different methods, respectively. Instead of studying the queue size distribution studied in [16, 17], this paper considers another theme—the departure process in that discrete-time model. The investigation of departure process in a queueing system is primarily motivated by the need to analyze queueing network models, in which the departure process of an upstream queue is the arrival process of the downstream queue. Burke (1956) [18] has proven that the departure process of a 𝑀/𝑀/𝑠 queue is a Bernoulli process. However, it is also well known that if one goes beyond the exponential assumption, unfortunately, the departure process does not become renewal and no exact results of departure process are known for FCFS models. Motivated by the applications of queueing networks in manufacturing and communications, most of researchers adopt approximate method; for example, Whitt (1983) [19] proposes a two-moment theory by approximating the process involved by renewal processes; Harrison and Dai (1989) [20] approximate the queueing network model by a Brownian network which they then solve exactly; Zhang et al. (2005) [21] derive the departure process approximations via an exact aggregate solution technique (called ETAQA); Ferng and Chang (2000) [22] obtain the factorial moments and lag 𝑛 covariance of interdeparture times by proposing a matrix-analytic approach; and so forth.

As far as we know, most of the previous works characterize the departure process by paying close attention to interdeparture times. In this paper, being different from the previous works, we aim at the transient departure number during an arbitrary time interval (0⁺,𝑛⁺]. Our objective is to present a distinctive analysis of departure process for the discrete-time queue, obtain the transient expression (𝑧-transformation) for expected number of departures during the interval (0⁺,𝑛⁺], and subsequently discover the decomposition characteristic of the departure process.

The remainder of the paper is organized as follows. Section 2 presents the description of the model. Section 3 derives the transient expression for the server busy probability at arbitrary epoch 𝑛⁺ by using 𝑧-transformation. Section 4 gives the transient expression (𝑧-transformation) for the expected number of departures during the arbitrary interval (0⁺,𝑛⁺], denoted by 𝑀_𝑖(𝑛⁺). Section 5 gains the approximate expansion of 𝑀_𝑖(𝑛⁺). Finally, conclusions are given in Section 7.

2. Model Description

Consider a discrete-time 𝐺𝑒𝑜/𝐺/1 queue with randomized vacations. It is assumed that a potential customer arrives in system during time interval (𝑛⁻,𝑛) and a potential departure takes place during time interval (𝑛,𝑛⁺) (LAS-DA) and the input of customers is a Bernoulli process with parameter 𝜆(0<𝜆<1). Denoting the interarrival time by 𝜏, then 𝑃{𝜏=𝑗}=𝜆(1−𝜆)^𝑗−1, 𝑗≥1. Customers are served according to FCFS discipline and the service times for an accepted customer, denoted by 𝜒, are independent and identically distributed (i.i.d.) random variables with common probability mass function (p.m.f.) 𝑔_𝑗=𝑃{𝜒=𝑗}, 𝑗≥1, probability generating function (P.G.F.) 𝐺(𝑧)=∑^∞_𝑗=1𝑔_𝑗𝑧^𝑗, and mean service time 𝐸[𝜒]=𝛼. After all the customers in the queue are served exhaustively, the server operates a randomized vacation policy with at most 𝐽 vacations. As soon as the system becomes empty, the server immediately takes a vacation, denoted by 𝑉, with p.m.f. 𝑣_𝑗=𝑃{𝑉=𝑗}, 𝑗≥1 and P.G.F. 𝑣(𝑧). If there is no customer in the system when the server returns from the vacation, the server takes another vacation with probability 𝜃 or remains dormant within the system with probability 𝜃=1−𝜃. Otherwise, the server starts to serve the customers immediately if there are some customers waiting for service at the end of vacation. This pattern repeats until the server has taken 𝐽 successive vacations. If the system remains empty at the end of the 𝐽th vacation, the server keeps idle in the system until a next customer arrives, who evokes immediately service for the arrival. Furthermore, various stochastic processes involved in the system are assumed to be mutually independent. To make it clear, the various time epochs at which events occur are shown in a self-explanatory figure (see Figure 1).

Figure 1 

Various time epochs in a late arrival system with delayed access (LAS-DA).

3. Server Busy Probability at an Arbitrary Epoch 𝑛⁺

Denoting by 𝑏 the length of server busy period evoked by only one customer with P.G.F. 𝐵(𝑧), then it follows the lemma which is also obtained by Bruneel and Kim (1993) [1].

Lemma 1. In 𝐺𝑒𝑜/𝐺/1 queue, for |𝑧|<1, 𝐵(𝑧) is the root of the following equation: 𝐵(𝑧)[((𝑧,𝐸[𝑏]=𝛼=𝐺𝑧−𝑧𝜆1−𝐵))]1−𝜌,𝜌<1, (1) where 𝜌=𝜆𝛼.
Let 𝑏^<𝑖> be the length of server busy period evoked by 𝑖 customers; thus 𝑏^<𝑖> can be expressed as 𝑏^<𝑖>=∑^𝑖_𝑣=1𝑏_𝑣, where 𝑏₁,𝑏₂,…,𝑏_𝑖 are independent of each other and have the identical distribution as 𝑏. So the P.G.F. of 𝑏^<𝑖> is given by 𝐵^𝑖(𝑧).
Denote by 𝑁(𝑛⁺) the queue length at arbitrary epoch 𝑛⁺. Let 𝜉(𝑛⁺)=1 be the server busy state; that is, the server is busy at epoch 𝑛⁺. Define 𝐴_𝑖(𝑛⁺)=𝑃{𝜉(𝑛⁺)=1∣𝑁(0⁺)=𝑖} with 𝑧-transform 𝑎_𝑖(𝑢)=∑^∞_𝑛=0𝐴_𝑖(𝑛⁺)𝑢^𝑛, |𝑢|<1; thus the following expression of 𝑎_𝑖(𝑢) holds.

Theorem 2. In 𝐺𝑒𝑜/𝐺/1 queue with randomized vacations, for |𝑢|<1 and 𝑖≥1, one has 𝑎₀(𝑢)=1−𝜆𝑢1−𝜃𝑣𝜆𝑢^𝐽×𝑣(𝑢)−𝑣𝜆𝑢𝐵(𝑢)+((𝑢(𝑢)×(𝜆𝑢+𝜆𝑢1−𝐵))⋅𝑦1−𝑢)1−𝜆𝑢⋅𝑥(𝑢)−𝜆𝑢𝐵(𝑢)⋅𝑦(𝑢)⁻¹,𝑎_𝑖(𝑢)=1−𝐵^𝑖(𝑢)1−𝑢+𝐵^𝑖(𝑢)⋅𝑎₀(𝑢), (2) and conditioning on 𝜌<1, the steady state probability is given by lim_𝑛→∞𝐴₀𝑛⁺=lim_𝑛→∞𝐴_𝑖𝑛⁺=𝜌, (3) where 𝑥(𝑢)=1−𝜃𝑣−𝜆𝑢1−𝜃𝑣𝜆𝑢^𝐽𝑣𝜆𝑢𝐵(𝑢)+𝜆𝑢−𝑣𝑦(𝑢)=()𝑣𝜆𝑢1−𝜃𝜆𝑢+𝜃𝑣𝜆𝑢^𝐽1−𝑣𝜆𝑢. (4)

Proof. In consideration that the input process is Bernoulli process, the ending points of a server busy period and a vacation are all renewal points and the system is going between server busy state and unoccupied state (vacation state or potential server idle state). To derive the expressions of 𝑎_𝑖(𝑢), the following notations are introduced: 𝑆_𝑚=^𝑚_𝑘=1𝜏_𝑘,𝑆₀𝑉=0,_<𝑖>=^𝑖_𝑘=1𝑉_𝑘,𝑉^<0>=0, (5) where 𝜏₁,𝜏₂,…,𝜏_𝑚 denote the interarrival times of the Bernoulli process with rate 𝜆. 𝑉₁,𝑉₂,…,𝑉_𝑖 represent the vacations which are i.i.d. random variables with the same distribution as 𝑉. By using renewal process theory and techniques of probability decomposition, it gets 𝐴₀𝑛⁺=𝑃𝜉𝑛⁺=1∣𝑁0⁺=0=𝑃𝜏≤𝑛⁺;𝜉𝑛⁺==1_𝐽𝑖=1𝑃𝜏≤𝑛⁺;𝜉𝑛⁺=1;𝑉^<𝑖−1><𝜏≤𝑉^<𝑖>+𝑃𝜉𝑛⁺=1;𝜏≤𝑛⁺;𝑉^<𝐽>=<𝜏_𝐽𝑖=1𝜃^𝑖−1𝑃𝜉𝑛⁺=1;𝑉^<𝑖>≤𝑛⁺;𝑉^<𝑖−1><𝜏≤𝑉^<𝑖>+^𝐽_𝑖=2𝜃^𝑖−2(1−𝜃)𝑃𝜉𝑛⁺=1;𝜏≤𝑛⁺;𝑉^<𝑖−1><𝜏+𝜃^𝑛𝐽−1_𝑖=1𝑃{𝜏=𝑖}𝑃𝑉^<𝐽><𝑖⋅𝐴₁(𝑛−𝑖)⁺. (6)
In the first item of (6), the “𝑉^<𝑖>≤𝑛⁺;𝑉^<𝑖−1><𝜏≤𝑉^<𝑖>” means that the epoch 𝑛 locates behind the 𝑖th vacation and the first customer arrives in system during the 𝑖th vacation. So there would be some other potential customers arriving during the time interval [𝜏,𝑉^<𝑖>]. As a result, the first item of (6) is equal to ^𝐽_𝑖=1𝜃^∞𝑖−1_𝑚=1𝑃𝜉𝑛⁺=1;𝑉^<𝑖>≤𝑛⁺;𝑉^<𝑖−1><𝜏;𝜏+𝑆_𝑚−1≤𝑉^<𝑖><𝜏+𝑆_𝑚=^𝐽_𝑖=1𝜃^∞𝑖−1_𝑚=1𝑃𝜉𝑛⁺=1;𝑉^<𝑖>≤𝑛⁺;𝑉^<𝑖−1><𝜏;𝜏−𝑉^<𝑖−1>+𝑆_𝑚−1<≤𝑉𝜏−𝑉^<𝑖−1>+𝑆_𝑚=^𝐽_𝑖=1𝜃^{∞𝑖−1𝑛}_{𝑚=1𝑘=𝑖−1}𝑃𝑉^<𝑖−1>×=𝑘_{𝑛−𝑘𝑙=1}𝑃{}{}𝑙𝑚𝑉=𝑙⋅𝑃𝜏>𝑘𝜆^𝑚𝜆^𝑙−𝑚⋅𝐴_𝑚(𝑛−𝑘−𝑙)⁺. (7) One should note that the item of “(𝜏−𝑉^<𝑖−1>)” in the above equation represents the remaining interarrival time at the ending epoch of the (𝑖−1)th vacation. Based on the “lack of memory property” of the Bernoulli process, the “(𝜏−𝑉^<𝑖−1>)” has the same distribution as 𝜏 under the condition of 𝑉^<𝑖−1>=𝑘(𝑘=𝑖−1,𝑖,…).
The second item of (6) is given by ^𝐽_𝑖=2𝜃^𝑖−2(1−𝜃)^𝑛_𝑘=1𝑃{𝜏=𝑘}𝑃𝑉^<𝑖−1><𝑘⋅𝐴₁(𝑛−𝑘)⁺. (8) Multiplying (6) by 𝑢^𝑛 and summing over 𝑛 after substituting (7) and (8) into (6) gain 𝑎₀(𝑢)=1−𝜃𝑣𝜆𝑢^𝐽1−𝜃𝑣⋅𝜆𝑢_{∞𝑚𝑚=1𝑙=1}𝑃{𝑉=𝑚}𝑢^𝑚𝑚𝑙𝜆^𝑙𝜆^𝑚−𝑙𝑎_𝑙(𝑢)+𝑎₁(𝑢)⋅𝜆𝑢(1−𝜃)𝑣𝜆𝑢1−⋅𝜆𝑢1−𝜃𝑣𝜆𝑢^𝐽−11−𝜃𝑣𝜆𝑢+𝑎₁(𝑢)⋅𝜆𝑢⋅𝜃^𝐽−1𝑣𝜆𝑢^𝐽1−.𝜆𝑢 (9) For 1≤𝑖, it gets 𝐴_𝑖𝑛⁺=𝑃𝜉𝑛⁺=1;𝑏^<𝑖>>𝑛⁺+𝑃𝜉𝑛⁺=1;𝑏^<𝑖>≤𝑛⁺=1−^𝑛_𝑘=𝑖𝑃𝑏^<𝑖>+=𝑘_{𝑛𝑘=𝑖}𝑃𝑏^<𝑖>=𝑘𝐴₀(𝑛−𝑘)⁺. (10) Multiplying (10) by 𝑢^𝑛 and summing over 𝑛 yield 𝑎_𝑖(𝑢)=1−𝐵^𝑖(𝑢)1−𝑢+𝐵^𝑖(𝑢)⋅𝑎₀(𝑢). (11) Solving the simultaneous equations (9) and (11) leads to the expression of 𝑎_𝑖(𝑢) provided in Theorem 2.
Applying the Final Value Theorem (see [23]), it has lim_𝑛→∞𝐴_𝑖(𝑛⁺)=lim_𝑢→1−(1−𝑢)⋅𝑎_𝑖(𝑢). By using Lemma 1, the stable result of 𝐴_𝑖(𝑛⁺) given in Theorem 2 is obtained.

Remark 3. By assuming 𝑃{𝑉=0}=1, the model considered here becomes a special case where the vacation disappears. We can easily derive the corresponding expression of 𝑎_𝑖(𝑢) in classical 𝐺𝑒𝑜/𝐺/1 queueing model without vacation policy. Consider the following: 𝑎₀(𝑢)=𝜆𝑢(1−𝐵(𝑢))(1−𝑢)1−(𝑢,𝑎𝜆𝑢−𝜆𝑢𝐵)_𝑖(𝑢)=1−𝐵^𝑖(𝑢)1−𝑢+𝐵^𝑖(𝑢)⋅𝑎₀(𝑢),lim_𝑛→∞𝐴₀𝑛⁺=lim_𝑛→∞𝐴_𝑖𝑛⁺=𝜌. (12) One may note here that the steady state of 𝐴₀(𝑛⁺) has nothing to do with the vacation policy.

4. Expected Number of Departures during the Arbitrary Time Interval (0⁺,𝑛⁺]

Consider a renewal process driven by a list of service times {𝜒_𝑖,𝑖≥1} defined in Section 1. Let 𝑈_𝑛=^𝑛_𝑖=1𝜒_𝑖,𝑈₀𝐷𝑛=0,₊=sup𝑛∶𝑈_𝑛≤𝑛⁺. (13) Thus, 𝐷(𝑛⁺) is a renewal process and denotes the number of departures during time interval (0⁺,𝑛⁺] contained in the service process {𝜒_𝑖,𝑖≥1}. Let 𝑀(𝑛⁺)=𝐸[𝐷(𝑛⁺)] be the renewal function of 𝐷(𝑛⁺) with 𝑧-transform 𝑚(𝑢)=∑^∞_𝑛=0𝑀(𝑛⁺)𝑢^𝑛.

Lemma 4. For |𝑢|<1, it has 𝑚(𝑢)=𝐺(𝑢)(1−𝑢)[1−𝐺(𝑢)],lim_𝑛→∞𝑀𝑛⁺𝑛=1𝛼, (14) where 𝐺(𝑢) is the P.G.F. of distribution of service time {𝜒_𝑖,𝑖≥1} and 𝛼=𝐸[𝜒_𝑖].

Proof. One has 𝑀𝑛⁺=𝐸𝐷𝑛⁺=^∞_𝑘=0𝑘𝑃𝐷𝑛⁺==𝑘_∞𝑘=0𝑘𝑃𝜒₁+⋯+𝜒_𝑘≤𝑛⁺<𝜒₁+⋯+𝜒_𝑘+1=^∞𝑛_{𝑘=0𝑚=𝑘}𝑘𝑃𝜒₁+⋯+𝜒_𝑘=𝑚𝑃𝜒_𝑘+1>(𝑛−𝑚)⁺. (15) Taking 𝑧-transform on both sides of the above equation yields the expression of 𝑚(𝑢). Applying the essential renewal theorem (see [24]), it gets lim_𝑛→∞𝑀𝑛⁺𝑛=1𝐸𝜒_𝑖=1𝛼. (16)
On this basis, we begin to consider the expected number of departures during an arbitrary time interval (0⁺,𝑛⁺] in the queue system. For this aim, some additional notations are developed as follows: 𝐷𝑛⁺=Thenumberofdeparturesduringgeneraltimeinterval0⁺,𝑛⁺,𝑀_𝑖𝑛⁺=𝐸𝐷𝑛⁺∣𝑁0⁺=𝑖,𝑛,𝑖≥0. (17) So, 𝑀_𝑖(𝑛⁺) represents the expected departures during (0⁺,𝑛⁺] with the initial state of 𝑁(0⁺)=𝑖. One has the following: 𝑇_𝑖𝑛⁺=𝐸Thenumberofdeparturesduring0⁺,𝑏^<𝑖>;𝑏^<𝑖>≤𝑛⁺,𝑊_𝑖𝑛⁺=𝐸Thenumberofdeparturesduring0⁺,𝑛⁺;𝑏^<𝑖>>𝑛⁺𝑡,𝑖=1,2,…,_𝑖(𝑢)=^∞_𝑛=0𝑇_𝑖𝑛⁺𝑢^𝑛,𝑤_𝑖(𝑢)=^∞_𝑛=0𝑊_𝑖𝑛⁺𝑢^𝑛, (18) where 𝑏^<𝑖> denotes the length of a server busy period evoked by 𝑖 customers waiting for service.

Theorem 5. Let 𝑚_𝑖(𝑢)=∑^∞_𝑛=0𝑀_𝑖(𝑛⁺)𝑢^𝑛, for |𝑢|<1, 𝜌<1, and 𝑖≥0; it has 𝑚_𝑖(𝑢)=𝑚(𝑢)⋅(1−𝑢)𝑎_𝑖(𝑢),lim_𝑛→∞𝑀_𝑖𝑛⁺𝑛=1𝜆,𝜌<0𝛼,𝜌≥1, (19) where 𝑎_𝑖(𝑢) and 𝑚(𝑢) are determined by Theorem 2 and Lemma 4, respectively.

Proof. Since the arrival process is Bernoulli process, the beginning and ending epochs of a server busy period or a vacation are both renewal points. Thus, it can take probability decomposition techniques on 𝑀(𝑛⁺) by using 𝑏^<𝑖>. Consider 𝑀𝑛⁺=𝐸𝐷𝑛⁺=𝐸𝐷𝑛⁺;𝑏^<𝑖>>𝑛⁺+𝐸𝐷𝑛⁺;𝑏^<𝑖>≤𝑛⁺=𝑊_𝑖𝑛⁺+𝐸𝐷𝑏^<𝑖>;𝑏^<𝑖>≤𝑛⁺+^𝑛_𝑘=𝑖𝑃𝑏^<𝑖>=𝑘𝐸𝐷(𝑛−𝑘)⁺=𝑊_𝑖𝑛⁺+𝑇_𝑖𝑛⁺+^𝑛_𝑘=𝑖𝑃𝑏^<𝑖>=𝑘𝑀(𝑛−𝑘)⁺. (20) Taking 𝑧-transform on both sides of the above equation and applying Lemma 4 give 𝑡_𝑖(𝑢)+𝑤_𝑖(𝑢)=1−𝐵^𝑖(𝑢)𝐺(𝑢)((𝑢.1−𝑢)[1−𝐺)] (21) For 𝑀₀(𝑛⁺), by using the same method used for the solution of 𝑎_𝑖(𝑢)(𝑖≥0), it gets 𝑀₀𝑛⁺=𝐸𝐷𝑛⁺∣𝑁0⁺=0=𝐸𝐷𝑛⁺;𝜏≤𝑛⁺=^𝐽_𝑖=1𝐸𝐷𝑛⁺;𝜏≤𝑛⁺;𝑉^<𝑖−1><𝜏≤𝑉^<𝑖>+𝐸𝐷𝑛⁺;𝜏≤𝑛⁺;𝑉^<𝐽>=<𝜏_𝐽𝑖=1𝜃^𝑖−1𝐸𝐷𝑛⁺;𝑉^<𝑖>≤𝑛⁺;𝑉^<𝑖−1><𝜏≤𝑉^<𝑖>+^𝐽_𝑖=2𝜃^𝑖−2(1−𝜃)𝐸𝐷𝑛⁺;𝜏≤𝑛⁺;𝑉^<𝑖−1><𝜏+𝜃^𝑛𝐽−1_𝑖=1𝑃{𝜏=𝑖}𝑃𝑉^<𝐽><𝑖⋅𝑀₁(𝑛−𝑖)⁺=^𝐽_𝑖=1𝜃^{∞𝑖−1𝑛}_{𝑚=1𝑘=𝑖−1}𝑃𝑉^<𝑖−1>×=𝑘_{𝑛−𝑘𝑙=1}𝑃{}{𝑙𝑚𝑉=𝑙⋅𝑃𝜏>𝑘}𝜆^𝑚𝜆^𝑙−𝑚⋅𝑀_𝑚(𝑛−𝑘−𝑙)⁺+^𝐽_𝑖=2𝜃^𝑖−2()×1−𝜃_𝑛𝑘=1𝑃{𝜏=𝑘}𝑃𝑉^<𝑖−1><𝑘⋅𝑀₁(𝑛−𝑘)⁺+𝜃^𝐽−1×^𝑛_𝑖=1𝑃{𝜏=𝑖}𝑃𝑉^<𝐽><𝑖⋅𝑀₁(𝑛−𝑖)⁺. (22) Multiplying (22) by 𝑢^𝑛 and summing over 𝑛 result in 𝑚₀(𝑢)=1−𝜃𝑣𝜆𝑢^𝐽1−𝜃𝑣⋅𝜆𝑢_{∞𝑚𝑚=1𝑙=1}𝑃{𝑉=𝑚}𝑢^𝑚𝑚𝑙𝜆^𝑙𝜆^𝑚−𝑙𝑚_𝑙(𝑢)+𝑚₁(𝑢)⋅𝜆𝑢(1−𝜃)𝑣𝜆𝑢1−⋅𝜆𝑢1−𝜃𝑣𝜆𝑢^𝐽−11−𝜃𝑣𝜆𝑢+𝑚₁(𝑢)⋅𝜆𝑢⋅𝜃^𝐽−1𝑣𝜆𝑢^𝐽1−.𝜆𝑢 (23) For 𝑀_𝑖(𝑛⁺)(𝑖≥1), it has 𝑀_𝑖𝑛⁺=𝐸𝐷𝑛⁺∣𝑁0⁺=𝑖=𝐸𝐷𝑛⁺;𝑛⁺<𝑏^<𝑖>+𝐸𝐷𝑛⁺;𝑏^<𝑖>≤𝑛⁺=𝑊_𝑖𝑛⁺+𝐸Thenumberofdeparturesduring0⁺,𝑏^<𝑖>;𝑏^<𝑖>≤𝑛⁺+𝐸Thenumberofdeparturesduring𝑏^<𝑖>,𝑛⁺;𝑏^<𝑖>≤𝑛⁺=𝑊_𝑖𝑛⁺+𝑇_𝑖𝑛⁺+^𝑛_𝑘=𝑖𝑃𝑏^<𝑖>=𝑘𝑀₀(𝑛−𝑘)⁺. (24) Taking 𝑧-transform on both sides of (24) and applying (21) yield 𝑚_𝑖(𝑢)=1−𝐵^𝑖(𝑢)𝐺(𝑢)(1−𝑢)[1−𝐺(𝑢)]+𝐵^𝑖(𝑢)𝑚₀(𝑢). (25) Solving the simultaneous equations (23) and (25), it gets the expression of 𝑚_𝑖(𝑢)(𝑖≥0). From the expressions of 𝑚_𝑖(𝑢)(𝑖≥0) given here, 𝑎_𝑖(𝑢)(𝑖≥0) given by Theorem 2, and 𝑚(𝑢) given by Lemma 4, the relation among 𝑚_𝑖(𝑢),𝑚(𝑢), and 𝑎_𝑖(𝑢) is established as follows: 𝑚_𝑖(𝑢)=𝑚(𝑢)⋅(1−𝑢)𝑎_𝑖(𝑢). (26) Applying Theorem 2 and Lemma 4, when condition 𝜌<1 holds, it has the following steady result: lim_𝑛→∞𝑀_𝑖𝑛⁺𝑛=lim_𝑢→1−(1−𝑢)²𝑚_𝑖(𝑢)=lim_𝑢→1−(1−𝑢)²𝑚(𝑢)⋅lim_𝑢→1−(1−𝑢)𝑎_𝑖(𝑢)=lim_𝑛→∞𝑀𝑛⁺𝑛⋅lim_𝑛→∞𝐴𝑛⁺=1𝛼1⋅𝜌=𝜆,𝜌<0𝛼⋅1,𝜌≥1. (27) Thus, it finishes the proof of Theorem 5.

Remark 6. Conditioning on 𝑃{𝑉=0}=1, we can obtain the expected number of departures during an arbitrary time interval (0⁺,𝑛⁺] in classical 𝐺𝑒𝑜/𝐺/1 queue without vacation. Therefore 𝑚₀(𝑢)=𝑚(𝑢)⋅(1−𝑢)𝑎₀(𝑢)=𝐺(𝑢)((𝑢⋅((𝑢1−𝑢)[1−𝐺)]𝜆𝑢1−𝐵))1−(𝑢),𝑚𝜆𝑢−𝜆𝑢𝐵_𝑖(𝑢)=𝑚(𝑢)⋅(1−𝑢)𝑎_𝑖(𝑢)=𝐺(𝑢)((𝑢⋅1−𝑢)[1−𝐺)]1−𝐵^𝑖(𝑢)1−𝑢+𝐵^𝑖(𝑢)⋅𝑎₀(𝑢),𝑖≥1.lim_𝑛→∞𝑀_𝑖𝑛⁺𝑛=1𝛼1⋅𝜌=𝜆,𝜌<0,𝛼⋅1,𝜌≥1. (28)

5. Approximate Expansion of the Expected Number of Departures during (0⁺,𝑛⁺]

In spite of the closed-form formula for 𝑚_𝑖(𝑢) given by Theorem 5, it seems impossible to get the expected departure number during time interval (0⁺,𝑛⁺] (𝑀_𝑖(𝑛⁺)) by operating inverse 𝑧-transform of 𝑚_𝑖(𝑢). For the convenient calculation of 𝑀_𝑖(𝑛⁺), it is pretty necessary to conduct the approximate expansion of 𝑀_𝑖(𝑛⁺). Firstly, we give Lemma 7 as follows.

Lemma 7. For 𝐸[𝜒²_𝑖]<+∞ and 𝑛→+∞, the following approximate expansion of 𝑀(𝑛⁺) holds 𝑀𝑛⁺−𝑛𝛼≈𝐸𝜒²_𝑖+𝛼2𝛼²−1, (29) where {𝜒_𝑖,𝑖≥1} are service times and 𝛼=𝐸[𝜒_𝑖].

Proof. Let 𝑆_{𝐷(𝑛+)+1}=∑^{𝐷(𝑛+)+1}_𝑖=1𝜒_𝑖; thus 𝑆_{𝐷(𝑛+)+1} denotes the epoch immediately after the (𝐷(𝑛⁺)+1)th service completion. Denote by 𝑋^(𝑛+) the remain service time of a customer being served at epoch 𝑛⁺ with the corresponding steady state 𝑋₊=lim_𝑛→∞𝑋^(𝑛+) and the steady state distribution 𝑥_𝑖=𝑃{𝑋₊=𝑖}, 𝑖≥1, which has P.G.F. 𝑋^∗(𝑢)=∑^∞_𝑖=1𝑥_𝑖𝑢^𝑖. From [25], it has 𝑋^∗(𝑢)=𝑢[1−𝐺(𝑢)]𝛼(),𝐸𝑋1−𝑢₊=𝑑𝑋^∗(𝑢)𝑑𝑢_𝑢=1=𝐸𝜒²_𝑖+𝛼.2𝛼 (30) Since 𝑆_{𝐷(𝑛+)+1}=𝑛+𝑋^(𝑛+), 𝐸[𝑆_{𝐷(𝑛+)+1}]=𝑛+𝐸[𝑋^(𝑛+)],
that is, 𝐸^{𝐷(𝑛+)+1}_𝑖=1𝜒_𝑘=𝑛+𝐸𝑋^(𝑛+),𝐸𝜒_𝑖𝐸𝐷𝑛⁺+1=𝑛+𝐸𝑋^(𝑛+),𝛼𝑀𝑛⁺+1=𝑛+𝐸𝑋^(𝑛+),𝑀𝑛⁺−𝑛𝛼=𝐸𝑋^(𝑛+)𝛼−1,lim_𝑛→∞𝑀𝑛⁺−𝑛𝛼=𝐸𝑋₊𝛼𝐸𝜒−1=^2𝑖+𝛼2𝛼²−1. (31) Therefore, conditioning on 𝐸[𝜒²_𝑖]<+∞ and 𝑛→+∞, Lemma 7 holds. Now the approximate expansion of 𝑀_𝑖(𝑛⁺) can be derived by the following theorem.

Theorem 8. For 𝐸[𝜒²_𝑖]<+∞ and 𝜌<1, when 𝑛→+∞, one has 𝑀_𝑖𝑛⁺≈𝐸𝜒𝜆𝑛+𝜌^2𝑖+𝛼2𝛼²𝑛−1,𝜌<0,𝛼+𝐸𝜒²_𝑖+𝛼2𝛼²−1,𝜌≥1, (32) where 𝛼=𝐸[𝜒_𝑖].

Proof. Let 𝑄(𝑛⁺)∗𝑅(𝑛⁺)=∑^𝑛_𝑘=0𝑄(𝑘⁺)∗𝑅((𝑛−𝑘)⁺) be the fold product of 𝑄(𝑛⁺) and 𝑅(𝑛⁺). Let 𝑞(𝑢)=𝑍[𝑄(𝑛⁺)]=∑^∞_𝑛=0𝑄(𝑛⁺)𝑢^𝑛 denote the 𝑧-transform of 𝑄(𝑛⁺); then 𝑍[𝑄(𝑛⁺)∗𝑅(𝑛⁺)]=𝑍[𝑄(𝑛⁺)]⋅𝑍[𝑅(𝑛⁺)]=𝑞(𝑢)⋅𝑟(𝑢). Denote the converse 𝑧-transform of image function 𝑞(𝑢) by 𝑍⁻¹[𝑞(𝑢)]=𝑄(𝑛⁺). Applying Theorem 5 along with the limitation theory and the properties of 𝑧-transform (see [23]), it gets lim_𝑛→+∞𝑀_𝑖𝑛⁺−𝐴_𝑖𝑛⁺∗𝑍⁻¹)𝑍𝑛(1−𝑢𝛼=lim_𝑢→1−(1−𝑢)𝑍𝑀_𝑖𝑛⁺−𝐴_𝑖𝑛⁺∗𝑍⁻¹)𝑍𝑛(1−𝑢𝛼=lim_𝑢→1−(1−𝑢)𝑚_𝑖(𝑢)−()𝑍𝑛1−𝑢𝛼⋅𝑍𝐴_𝑖𝑛⁺=lim_𝑢→1−(1−𝑢)(1−𝑢)𝑎_𝑖(𝑢)𝑚(𝑢)−()𝑍𝑛1−𝑢𝛼𝑎_𝑖(𝑢)=lim_𝑢→1−(1−𝑢)𝑎_𝑖(𝑢)𝑚(𝑢)−()𝑍𝑛)(1−𝑢1−𝑢𝛼=lim_𝑢→1−(1−𝑢)𝑎_𝑖(𝑢)lim_𝑢→1−)𝑚(𝑢)−()𝑍𝑛(1−𝑢1−𝑢𝛼=lim_𝑛→∞𝐴_𝑖𝑛⁺lim_𝑛→∞𝑀𝑛⁺−𝑛𝛼𝐸𝜒=𝜌^2𝑖+𝛼2𝛼²−1. (33) Note that the final conclusion in (33) is based on Theorem 2 and (31).
Since lim_𝑛→+∞𝐴_𝑖𝑛⁺∗𝑍⁻¹[(1−𝑢)𝑍(𝑛/𝛼)]𝑛=lim_𝑢→1−(1−𝑢)²𝑍𝐴_𝑖𝑛⁺∗𝑍⁻¹)𝑍𝑛(1−𝑢𝛼=lim_𝑛→+∞(1−𝑢)𝑎_𝑖(𝑢)(1−𝑢)²𝑍𝑛𝛼=lim_𝑛→+∞𝐴_𝑖𝑛⁺⋅lim_𝑛→+∞𝑛/𝛼𝑛1=𝜌⋅𝛼, (34) when 𝑛→+∞ holds, it has 𝐴_𝑖𝑛⁺∗𝑍⁻¹)𝑍𝑛(1−𝑢𝛼𝑛≈𝜌⋅𝛼. (35) Substituting (35) into (33) yields 𝑀_𝑖𝑛⁺≈𝐴_𝑖𝑛⁺∗𝑍⁻¹)𝑍𝑛(1−𝑢𝛼𝐸𝜒+𝜌^2𝑖+𝛼2𝛼²≈𝐸𝜒−1𝜆𝑛+𝜌^2𝑖+𝛼2𝛼²𝑛−1,𝜌<1,𝛼+𝐸𝜒²_𝑖+𝛼2𝛼²−1,𝜌≥1. (36)

6. A Numerical Experiment

Based on Theorem 8 given above, the expected departure number can be calculated easily. It is one of the main distinctive highlights in the work. For showing this highlight, a numerical experiment concerning the calculation for expected departure number (𝑀_𝑖(𝑛⁺)) is carried out here to illustrate the effect of arrival rate (𝜆) and time interval ((0⁺,𝑛⁺]) on the expected departure number (see Figure 2).

Figure 2 

Effect of arrival rate and time interval on expected departure number.

In this numerical operation, some necessary parameters are taken as follows: the service time is arbitrarily distributed and the service rate is 1/𝛼=0.46; the top value of time interval (0⁺,𝑛⁺] is valued from 𝑛=10 to 𝑛=99; the arrival process is a Bernoulli process and the arrival rate is determined in the range of 𝜆∈[0.1,0.99] including the special case of 𝜌≥1.

From Figure 2, we can see the expected departure number is increasing as the length of time interval increases. It is also increasing when the customer improves the arrival rate until it edges up to the service rate 1/𝛼. The expected departure number stops increasing and maintains being unchangeable when the arrival rate is greater than the service rate; that is, 𝜌≥1. This is because the server will stay in a busy state permanently conditioning on 𝜌≥1 and in this case the departure rate is just equal to the service rate.

7. Conclusions

By using probability decomposition techniques and renewal process, this paper presents an original analysis for the departure process in discrete-time queue with random vacation policy. It gains the following conclusions.(1)The conclusion lim_𝑛→∞(𝑀_𝑖(𝑛⁺)/𝑛)=𝜆 given by Theorem 5 shows that, conditioning on 𝜌<1, the expected number of departures per unit time (called departure rate) is identical with the input rate and has nothing to do with the service rate or server’s vacation policy.(2)From the relation given in Theorem 5, that is, 𝑚_𝑖(𝑢)=𝑚(𝑢)[(1−𝑢)𝑎_𝑖(𝑢)], one sees that the expected departure number is decomposed into two parts: one is the server busy probability (𝐴_𝑖(𝑛⁺)), and another is the expected departure number during busy period. That is to say, the expected number of departures during arbitrary time interval (0⁺,𝑛⁺] is affected by both the service rate and the length of busy period. Furthermore, from Theorems 2 and 5 and Lemma 4, it implies that this decomposition still holds under steady condition; that is, lim_𝑛→∞𝑀_𝑖𝑛⁺𝑛=lim_𝑛→∞𝑀𝑛⁺𝑛⋅lim_𝑛→∞𝐴_𝑖𝑛⁺=1𝛼⋅𝜌. (37) Our further investigation indicates that this decomposition still holds in most of varieties of 𝐺𝑒𝑜/𝐺/1 with exhaustive service.(3)The approximate expansion of 𝑀_𝑖(𝑛⁺) given by Theorem 8 provides a valid method to calculate the expected departure number during arbitrary time interval (0⁺,𝑛⁺].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors sincerely appreciate the anonymous reviewers for their time and helpful comments to improve this paper’s quality. This research is supported by the National Natural Science Foundation of China (no. 71171138), the National Natural Science Foundation of China (no. 71301111), and the Fundamental Research Funds for the Central Universities (no. JBK140923).