The Discrete-Time Bulk-Service Geo/Geo/1 Queue with Multiple Working Vacations

Abstract

This paper deals with a discrete-time bulk-service 𝐺𝑒𝑜/𝐺𝑒𝑜/1 queueing system with infinite buffer space and multiple working vacations. Considering an early arrival system, as soon as the server empties the system in a regular busy period, he leaves the system and takes a working vacation for a random duration at time 𝑛. The service times both in a working vacation and in a busy period and the vacation times are assumed to be geometrically distributed. By using embedded Markov chain approach and difference operator method, queue length of the whole system at random slots and the waiting time for an arriving customer are obtained. The queue length distributions of the outside observer’s observation epoch are investigated. Numerical experiment is performed to validate the analytical results.

1. Introduction

Recently there has been a rapid increase in the literature on discrete-time queueing system with working vacations. These queueing models have been studied extensively and applied to computer networks, communication systems, and manufacturing systems. In the classical queueing system with server vacations, the server stops working during vacation periods. Suppose, however, that a system can be staffed with a substitute server during the times the main server is taking vacations. The service rate of the substitute server is different from (and probably lower than) that of the main server. This is the notion of working vacations recently introduced by Servi and Finn [1]. They studied an 𝑀/𝑀/1 queue with multiple working vacations (𝑀/𝑀/1/𝑊𝑉). Their work is motivated by the analysis of a reconfigurable wavelength-division multiplexing (WDM) optical access network. In 2006, Wu and Takagi [2] generalized Servi and Finn’s 𝑀/𝑀/1/𝑊𝑉 queue to an 𝑀/𝐺/1/𝑊𝑉 queue. Baba [3] extended Wu and Takagi’s work to a renewal input 𝐺𝐼/𝑀/1 queue with working vacations and derived the steady-state system length distributions at an arrival and arbitrary epochs. The 𝐺𝑒𝑜/𝐺𝑒𝑜/1 queue with single and multiple working vacations have been discussed in Li and Tian [4] and Tian et al. [5], respectively. Chae et al. [6] studied the 𝐺𝐼/𝑀/1 queue and 𝐺𝐼/𝐺𝑒𝑜/1 queue with single working vacation (SWV). The discrete-time infinite buffer 𝐺𝐼/𝐺𝑒𝑜/1 queue with multiple working vacations and vacation interruption has been studied in Li et al. [7, 8]. The discrete-time finite buffer 𝐺𝐼/𝐺𝑒𝑜/1 queue with multiple working vacations has been discussed by Goswami and Mund [9]. All the above studies on discrete-time single server queues have been carried out under the assumption that a server serves singly at a time. However, there are many instances where the servers are carried out in batches to enhance the performance of the system. Over the last several years the discrete-time single server queues in batch service without vacations have been studied in Gupta and Goswami [10], Chaudhry and Chang [11], Alfa and He [12], and Yi et al. [13]. Lately, This type of queueing systems raise interest once more by many scholars such as Banerjee et al. [14, 15], Claeys et al. [16, 17].

The continuous-time infinite buffer single server batch service queue with multiple vacations has been analyzed by Choi and Han [18], Chang and Takine [19]. The 𝑀/𝐺/1 queue with bulk service and single vacation has been investigated by Sikdar and Gupta [20].

As to the research to queueing systems with batch service and working vacations (or vacation), Yu et al. [21] considered a finite capacity and bulk-arrival and bulk-service continuous-time queueing system with multiple working vacations and partial batch rejection. Vijaya Laxmi and Yesuf [22] studied a renewal input infinite buffer batch service queue with single exponential working vacation and accessibility to batches. Goswami and Mund [23] investigated a discrete-time batch service renewal input queue with multiple working vacations. Note that although all the above studies on discrete-time bulk-service queues have been carried out, but, from their models, we can only obtain the average length of the waiting for service; the average length of the whole system cannot be obtained at prearrival, arbitrary and outside observer’s observation epochs. In fact, the number of customers being served in batches is a random variable; it is difficult to compute the average length of the whole system.

This paper focuses on a discrete-time batch-service infinite buffer 𝐺𝑒𝑜/𝐺𝑒𝑜/1 queueing system with multiple working vacations in which arrivals occur according to a geometrical input. By using embedded Markov chain approach, we obtained the operation rules of the one-step transition probability matrix, the average length at random slots, and the average waiting time for an arriving customer. This model has potential applications in computer networks where jobs are processed in batches.

The rest of the paper is arranged as follows. In the next section, the model of the considered queueing system is described. In Section 3, the stationary distribution of queue length at arbitrary slot epochs is discussed. In Section 4, we study the waiting time distribution. In Section 5, we discuss the outside observer’s observation epoch probabilities distribution. In Section 6, some numerical results and the sensitivity analysis of this system are given. Section 7 concludes this paper.

2. System Description

We consider a discrete-time bulk-service infinite buffer space queueing system with server multiple working vacations according to the rule of an early arrival system. Assume that the time axis is slotted into intervals of equal length with the length of a slot being unity, and it is marked as 0,1,2,…,𝑛,…, a potential arrival occurs in the interval (𝑛,𝑛⁺) and potential batch departures occur in (𝑛⁻,𝑛). In the meantime, the interarrival times 𝑇 of customers are independent and geometrically distributed with probability mass function (p.m.f.) 𝑃{𝑇=𝑘}=𝑝𝑝^𝑘−1,𝑘≥1,𝑝=1−𝑝.

The customers are served in batches of variable capacity, the maximum service capacity for the server being 𝑎(𝑎≥1). Service times 𝑆_𝑏 during normal busy period and service times 𝑆_𝑣 during a working vacation are assumed independent and geometrically distributed with p.m.f. 𝑃{𝑆_𝑏=𝑘}=𝜇_𝑏𝜇_𝑏^𝑘−1,𝑘≥1,𝜇_𝑏=1−𝜇_𝑏 and p.m.f. 𝑃{𝑆_𝑣=𝑘}=𝜇_𝑣𝜇_𝑣^𝑘−1,𝑘≥1,𝜇_𝑣=1−𝜇_𝑣, respectively. A new arriving customer cannot go into the queue being served immediately in spite of the working vacation period and the normal busy period. The server leaves the system and takes a working vacation at epoch 𝑛 as soon as system becomes empty in regular period. The working vacation time follows a geometric distribution with parameter 𝜃(0<𝜃<1) and its p.m.f. is 𝑃{𝑉=𝑘}=𝜃𝜃^𝑘−1,𝑘≥1,𝜃=1−𝜃. If there are some customers being served after the server finishes a working vacation, the service interrupted at the end of a vacation is lost and it is restarted with service rate 𝜇_𝑏 at the beginning of the following service period, which means that the regular busy period starts. The various time epochs at which events occur are depicted in Figure 1.

Figure 1 

Various time epochs in EAS.

We assume that interarrival times, service times, and working vacation times are mutually independent.

In addition, the service discipline is first in first out (FIFO). Let 𝑄_𝑛 be the number of customers in the system at time 𝑛, and 𝐽_𝑛=0,thesystemisinaworkingvacationperiodattime𝑛,1,thesystemisinaregularbusyperiodattime𝑛. (1) Then {𝑄_𝑛,𝐽_𝑛} is a Markov chain with the state space Ω={(0,0)}𝑘,𝑗∶𝑘≥1,𝑗=0,1, (2) where state (𝑘,1),  𝑘≥1 indicates that the system is in a regular busy period; state (𝑘,0),  𝑘≥0 indicates that the system is in a working vacation period and there 𝑘 customers in the system.

Using the lexicographical sequence for the states, the one-step transition block matrix can be written as 01234⋮𝑎⋮𝐀𝐏=𝑎+1^𝟎𝟎𝐁𝐂𝐃𝐄𝐂𝐅𝐆𝐄𝐂𝐅𝟎𝐆𝐄𝐂𝐅𝟎𝟎𝐆𝐄⋮⋮⋮⋱⋱⋱⋱𝐂𝐅𝟎⋯𝟎𝟎𝐆𝐄𝟎𝐇𝐅𝟎𝟎𝟎𝟎𝐆𝐄⋮⋱⋱⋱⋱⋱⋱⋱⋱⋱, (3) where the first column in front of the matrix denotes the number of the customers for the first column block matrix in matrix 𝐏, and 𝐀_𝟎𝟎 is transition probability from (0,0) to (0,0); 𝐁 is transition probability matrix from (0,0) to {(1,0),(1,1)}; 𝐂 is transition probability matrix from {(𝑖,0),(𝑖,1)}(1≤𝑖≤𝑎) to (0,0); 𝐃 is transition probability matrix from {(1,0),(1,1)} to {(1,0),(1,1)}; 𝐄 is transition probability matrix from {(𝑖,0),(𝑖,1)}(𝑖≥1) to {(𝑖+1,0),(𝑖+1,1)}(𝑖≥1); 𝐅 is transition probability matrix from {(𝑖,0),(𝑖,1)}(2≤𝑖≤𝑎) to {(1,0),(1,1)} and from {(𝑖,0),(𝑖,1)}(𝑖≥𝑎+1) to {(𝑖−𝑎+1,0),(𝑖−𝑎+1,1)}(𝑖≥𝑎+1); 𝐆 is transition probability matrix from {(𝑖,0),(𝑖,1)}(𝑖≥2) to {(𝑖,0),(𝑖,1)}(𝑖≥2); 𝐇 is transition probability matrix from {(𝑖,0),(𝑖,1)}(𝑖≥𝑎+1) to {(𝑖−𝑎,0),(𝑖−𝑎,1)}(𝑖≥𝑎+1).

We have 𝐀_𝟎𝟎=𝑝+𝑝𝜃𝜇_𝑣,𝐁=𝑝𝜃𝜇_𝑣𝑝𝜃,𝐂=𝑝𝜃𝜇_𝑣𝑝𝜇_𝑏,𝐃=𝑝𝜃𝜇_𝑣+𝑝𝜃𝜇_𝑣0𝑝𝜃𝑝𝜇_𝑏+𝑝𝜇_𝑏,𝑝𝐄=𝜃𝜇_𝑣𝑝𝜃0𝑝𝜇_𝑏𝑝,𝐅=𝜃𝜇_𝑣00𝑝𝜇_𝑏,𝐆=𝑝𝜃𝜇_𝑣0𝑝𝜃𝑝𝜇_𝑏,𝐇=𝑝𝜃𝜇_𝑣00𝑝𝜇_𝑏. (4)

3. The Stationary Queue Size Distributions at Random Slots

Assume that (𝑄,𝐽) is the stationary limit of {𝑄_𝑛,𝐽_𝑛}, and its distribution is denoted as 𝜋_𝑘,𝑗=lim_𝑛→∞𝑃{𝑄_𝑛=𝑘,𝐽_𝑛=𝑗}=𝑃{𝑄=𝑘,𝐽=𝑗},  (𝑘,𝑗)∈Ω, we have the following theorem.

Theorem 1. If 𝜌=𝑝/𝑎𝜇_𝑏<1, 𝜌₀=𝑝/𝑎𝜇_𝑣<1, 𝜋_𝑘,𝑗(𝑘≥1,𝑗=0,1) are given by 𝜋_𝑘,0=𝑝𝜃𝜇_𝑣(1−𝜉)𝜉^𝑘−1𝜔𝜋_0,0,𝜋_𝑘,1=𝑐^′₀𝑟^𝑘+𝑝𝜃𝜃𝜇_𝑣𝑝+𝑝𝜉(1−𝜉)𝜉^𝑘−1𝜋_0,0,(),𝜔𝜛𝑘≥1 (5) where 𝜔=1−𝜃1−𝑝𝜇_𝑣−𝑝𝜃𝜇_𝑣𝜉^𝑎,𝜛=1−𝑝𝜇_𝑏𝜉−𝑝𝜇_𝑏+𝑝𝜇_𝑏𝜉^𝑎+𝑝𝜇_𝑏𝜉^𝑎+1,𝑐^′₀=𝑝𝜃𝜔𝜛+𝜃𝜇_𝑣×𝑝𝜛(1−𝜉)+𝑝+𝑝𝜉𝜛−𝑝𝜇_𝑏1−𝜉^𝑎×()×1−𝑟𝑟𝜔𝜛𝑝𝜇_𝑏1−𝑟^𝑎−1𝜋_0,0,𝜋_0,0=1+𝑝𝜃𝜇_𝑣𝜛+𝑝𝜃𝜃𝜇_𝑣𝑝+𝑝𝜉(𝜔𝜛)⁻¹+𝑝𝜃𝜔𝜛+𝜃𝜇_𝑣×𝑝𝜛(1−𝜉)+𝑝+𝑝𝜉𝜛−𝑝𝜇_𝑏1−𝜉^𝑎×𝜔𝜛𝑝𝜇_𝑏1−𝑟^𝑎−1−1, (6)0<𝜉<1 and 𝜉 is the root of the equation 𝑝𝜇_𝑣𝜃𝑧^𝑎+1+𝑝𝜇_𝑣𝜃𝑧^𝑎−(1−𝑝𝜇_𝑣𝜃)𝑧+𝑝𝜇_𝑣𝜃=0,0<𝑟<1 and 𝑟 is the root of the equation 𝑝𝜇_𝑏𝑧^𝑎+1+𝑝𝜇_𝑏𝑧^𝑎+𝑝𝜇_𝑏−(1−𝑝𝜇_𝑏)𝑧=0.

Proof. According to the one-step transition probability matrix, we can see which is not QBD process, the method of matrix-geometric solution is invalid. Based on the stationary equations obtained directly by stochastic balance, we have 𝜋_0,0=𝜋_0,0𝐀₀₀+^𝑎_𝑖=1𝜋_𝑖,0,𝜋_𝑖,1𝜋𝐂,_1,0,𝜋_1,1=𝜋_0,0𝐁+𝜋_1,0,𝜋_1,1𝐃+^𝑎_𝑖=2𝜋_𝑖,0,𝜋_𝑖,1𝐅+𝜋_𝑎+1,0,𝜋_𝑎+1,1𝜋𝐇,_𝑘,0,𝜋_𝑘,1=𝜋_𝑘−1,0,𝜋_𝑘−1,1𝐄+𝜋_𝑘,0,𝜋_𝑘,1𝐆+𝜋_{𝑎+𝑘−1,0},𝜋_{𝑎+𝑘−1,1}𝐅+𝜋_𝑎+𝑘,0,𝜋_𝑎+𝑘,1𝐇,𝑘≥2. (7) Then, we obtain the following equations: 𝜋_0,0=𝜋_0,0𝑝+𝑝𝜃𝜇_𝑣+𝑝𝜃𝜇_𝑣^𝑎_𝑖=1𝜋_𝑖,0+𝑝𝜇_𝑏^𝑎_𝑖=1𝜋_𝑖,1,𝜋_1,0=𝑝𝜃𝜇_𝑣𝜋_0,0+𝑝𝜃𝜇_𝑣𝜋_1,0+𝑝𝜃𝜇_𝑣^𝑎_𝑖=1𝜋_𝑖,0+𝑝𝜃𝜇_𝑣𝜋_𝑎+1,0,𝜋_𝑘,0=𝑝𝜃𝜇_𝑣𝜋_𝑘−1,0+𝑝𝜃𝜇_𝑣𝜋_𝑘,0+𝑝𝜃𝜇_𝑣𝜋_{𝑎+𝑘−1,0}+𝑝𝜃𝜇_𝑣𝜋_𝑎+𝑘,0𝜋,𝑘≥2,_1,1=𝑝𝜃𝜋_0,0+𝑝𝜃𝜋_1,0+𝑝𝜇_𝑏𝜋_1,1+𝑝𝜇_𝑏^𝑎_𝑖=1𝜋_𝑖,1+𝑝𝜇_𝑏𝜋_𝑎+1,1,𝜋_𝑘,1=𝑝𝜃𝜋_𝑘−1,0+𝑝𝜇_𝑏𝜋_𝑘−1,1+𝑝𝜃𝜋_𝑘,0+𝑝𝜇_𝑏𝜋_𝑘,1+𝑝𝜇_𝑏𝜋_{𝑎+𝑘−1,1}+𝑝𝜇_𝑏𝜋_𝑎+𝑘,1,𝑘≥2. (8)(9)(10)(11)(12) According to the characteristic of difference equations, let 𝜋_𝑘+𝑗,0=𝐸^𝑗𝜋_𝑘,0 [24], 𝑗∈𝑍;𝑘=0,1,2,…, where 𝐸 denotes the difference operator for the difference equation, substituting it into (10), (10) can be written as 𝑝𝜃𝜇_𝑣𝐸^𝑎+1+𝑝𝜃𝜇_𝑣𝐸^𝑎−1−𝑝𝜃𝜇_𝑣𝐸+𝑝𝜃𝜇_𝑣𝜋_𝑘−1,0=0. (13) The auxiliary equation is given by 𝑝𝜃𝜇_𝑣𝑧^𝑎+1+𝑝𝜃𝜇_𝑣𝑧^𝑎−1−𝑝𝜃𝜇_𝑣𝑧+𝑝𝜃𝜇_𝑣=0. (14) Let 𝑓(𝑧)=𝑝𝜃𝜇_𝑣𝑧^𝑎+1+𝑝𝜃𝜇_𝑣𝑧^𝑎+𝑝𝜃𝜇_𝑣𝑧+𝑝𝜃𝜇_𝑣 and 𝑔(𝑧)=−𝑧. Using Rouché's theorem, it can be shown that there is only one zero real root falls in the unit circle (Note: the root must be real root; otherwise, there are two roots at least fall in the unit circle, because the imaginary roots of an equation appear in pairs.) We denote this root by 𝜉(0<𝜉<1) and the other 𝑎 roots by 𝜉_𝑖, |𝜉_𝑖|≥1(𝑖=1,2,3,…,𝑎). So 𝜉 satisfies 𝑓(𝜉)+𝑔(𝜉)=0. Therefore, the solution of (10) can be written as 𝜋_𝑘,0=𝑐₀𝜉^𝑘+^𝑎_𝑖=1𝑐_𝑖𝜉^𝑘_𝑖,𝑘≥1. (15) Since 𝑐_𝑖(𝑖=1,2,3,…,𝑎)=0 (otherwise, the probability 𝜋_𝑘,0 tends to ∞ when 𝑘 tends to ∞), we get 𝜋_𝑘,0=𝑐₀𝜉^𝑘(𝑘≥1).
Let 𝑘=1, we get 𝑐₀=𝜉⁻¹𝜋_1,0, then 𝜋_𝑘,0=𝜋_1,0𝜉^𝑘−1,𝑘≥1. (16) Substituting (16) into (9), we obtain 𝜋_1,0=𝑝𝜃𝜇_𝑣(1−𝜉)𝜔𝜋_0,0, (17) where 𝜔=1−𝜃[1−𝑝𝜇_𝑣]−𝑝𝜃𝜇_𝑣𝜉^𝑎.
Hence, 𝜋_𝑘,0=𝑝𝜃𝜇_𝑣(1−𝜉)𝜉^𝑘−1𝜔𝜋_0,0,𝑘≥1. (18) For 𝑘≥2, the difference equation (12) can be written as 𝑝𝜇_𝑏𝜋_𝑘−1,1+𝑝𝜇_𝑏−1𝜋_𝑘,1+𝑝𝜇_𝑏𝜋_{𝑎+𝑘−1,1}+𝑝𝜇_𝑏𝜋_𝑎+𝑘,1+𝑝𝜃𝜃𝜇_𝑣𝑝+𝑝𝜉(1−𝜉)𝜉^𝑘−2𝜔𝜋_0,0=0. (19) Using 𝜋_𝑘+𝑗,1=𝐸^𝑗𝜋_𝑘,1, 𝑗∈𝑍; 𝑘=1,2,…, the auxiliary equation of (19) such that 𝑝𝜇_𝑏𝑧^𝑎+1+𝑝𝜇_𝑏𝑧^𝑎−1−𝑝𝜇_𝑏𝑧+𝑝𝜇_𝑏=0. (20) Let 𝐺(𝑧)=𝑝𝜇_𝑏𝑧^𝑎+1+𝑝𝜇_𝑏𝑧^𝑎+𝑝𝜇_𝑏𝑧+𝑝𝜇_𝑏, obviously 𝐺(1)=1, 𝐺^′(1)=(𝑎+1)𝑝𝜇_𝑏+𝑎𝑝𝜇_𝑏+𝑝𝜇_𝑏=𝑎𝜇_𝑏+1−𝑝. Since 𝜌=𝑝/(𝑎𝜇_𝑏)<1, that is, 𝑝<𝑎𝜇_𝑏, we can see that 𝐺^′(1)>1. According to Hunter [25], the equation 𝑧=𝐺(𝑧) has a unique real root in unit circle, which can be denoted by 𝑟; the other 𝑎 roots can be denoted by 𝑟_𝑖, |𝑟_𝑖|≥1(𝑖=1,2,…,𝑎). As mentioned above, the general solution of (20) can be written as 𝑧^∗=𝑐^′₀𝑟^𝑘,𝑘≥1. (21) Using the method of nonhomogeneous difference operator, the special solution of (19) can be written as 𝜋^∗_𝑘,1=𝑝𝜃𝜃𝜇_𝑣𝑝+𝑝𝜉(1−𝜉)𝜉^𝑘−1𝜋_0,0𝜔𝜛,𝑘≥1, (22) where 𝜛=[(1−𝑝𝜇_𝑏)𝜉−(𝑝𝜇_𝑏+𝑝𝜇_𝑏𝜉^𝑎+𝑝𝜇_𝑏𝜉^𝑎+1)].
From (21) and (22), we have 𝜋_𝑘,1=𝑐^′₀𝑟^𝑘+𝑝𝜃𝜃𝜇_𝑣𝑝+𝑝𝜉(1−𝜉)𝜉^𝑘−1𝜋_0,0𝜔𝜛,𝑘≥1. (23) Substituting (17), (18), and (23) into (11), we obtain 𝑐^′₀=𝑝𝜃𝜔𝜛+𝜃𝜇_𝑣𝑝𝜛(1−𝜉)+𝑝+𝑝𝜉𝜛−𝑝𝜇_𝑏1−𝜉^𝑎×(1−𝑟)𝑟𝜔𝜛𝑝𝜇_𝑏1−𝑟^𝑎−1𝜋_0,0. (24) Substituting (24) into (23), we can obtain 𝜋_𝑘,1(𝑘≥1).
Using ∑^∞_𝑖=0𝜋_𝑖,0+∑^∞_𝑖=1𝜋_𝑖,1=1, we can get 𝜋_0,0=1+𝑝𝜃𝜇_𝑣𝜛+𝑝𝜃𝜃𝜇_𝑣𝑝+𝑝𝜉(𝜔𝜛)⁻¹+𝑝𝜃𝜔𝜛+𝜃𝜇_𝑣×𝑝𝜛(1−𝜉)+𝑝+𝑝𝜉𝜛−𝑝𝜇_𝑏1−𝜉^𝑎×𝜔𝜛𝑝𝜇_𝑏1−𝑟^𝑎−1−1. (25)

Remark 2. If 𝜃→0 and 𝑎=1, this queueing system is equivalent to 𝐺𝑒𝑜𝑚/𝐺𝑒𝑜𝑚/1 queueing system with a server serves customers singly. Because of 𝜌₀=𝑝/𝑎𝜇_𝑣<1 and 1≤𝑎, then 𝑝/𝜇_𝑣<1. Hence, 0<𝜉=(𝑝𝜇_𝑣/𝑝𝜇_𝑣)<1. We have 𝜋_0,0=1−𝜉,𝜋_𝑘,0=(1−𝜉)𝜉^𝑘,𝑘≥1, (26) which are matched with the results given by Tian and Ma [26].

Corollary 3. Define 𝐽 as the state of system at random slots, the steady-state probability of this system at random slots can be written as 𝑃{}=𝑝𝐽=01+𝜃𝜇_𝑣𝜔𝜋_0,0,𝑃{𝐽=1}=𝑝𝜃𝜔𝜛+𝜃𝜇_𝑣×𝑝𝜛(1−𝜉)+𝑝+𝑝𝜉𝜛−𝑝𝜇_𝑏1−𝜉^𝑎×𝜔𝜛𝑝𝜇_𝑏1−𝑟^𝑎−1+𝑝𝜃𝜃𝜇_𝑣𝑝+𝑝𝜉(𝜔𝜛)⁻¹𝜋_0,0. (27)

Theorem 4. If |𝑧|≤1, the probability generating function (PGF) of steady-state queue length at random slots is given by 𝐿(𝑧)=1+𝑐^′₀𝑟𝑧+𝑝1−𝑟𝑧𝜃𝜇_𝑣(1−𝜉)𝑧𝜔()+1−𝜉𝑧𝑝𝜃𝜃𝜇_𝑣𝑝+𝑝𝜉(1−𝜉)𝑧()𝜋𝜔𝜛1−𝜉𝑧^0,0, (28) and the average queue length is 𝐸(𝐿)=𝑟𝑐^′₀(1−𝑟)²+𝜛+𝜃𝑝+𝑝𝜉𝑝𝜃𝜇_𝑣()𝜋𝜔𝜛1−𝜉^0,0. (29)

Proof. In the steady state, the queue length 𝐿 at random slots has marginal distribution as 𝑃{𝐿=0}=𝜋_0,0,𝑃{𝐿=𝑘}=𝜋_𝑘,0+𝜋_𝑘,1=𝑐^′₀𝑟^𝑘+𝑝𝜃𝜇_𝑣(1−𝜉)𝜉^𝑘−1𝜔𝜋_0,0+𝑝𝜃𝜃𝜇_𝑣𝑝+𝑝𝜉(1−𝜉)𝜉^𝑘−1𝜋_0,0,().𝜔𝜛𝑘≥1 (30) Using 𝐿(𝑧)=𝑃{𝐿=0}+∑^∞_𝑘=1𝑃{𝐿=𝑘}𝑧^𝑘, we can obtain (28) easily; furthermore, taking derivation to 𝐿(𝑧) and let 𝑧=1, we can get (29).

4. The Waiting Time Distribution

Let the random variable 𝑇_𝑞 be the total waiting time of an arriving customer in the queue, 𝑁_𝑤 represents the number of the customers in the system. Assume that an arriving customer finds 𝑖 customers in the system, the conditional distribution law that he waits for 𝑘 slots is subject to 𝑤_𝑖(𝑘)=𝑃{𝑇_𝑞=𝑘/𝑁_𝑤=𝑖}, 𝑖=0,1,2,…,𝑘=0,1,2,…, and PGF is 𝑊_𝑖(𝑧)=∑^∞_𝑘=0𝑤_𝑖(𝑘)𝑧^𝑘. In the steady state, the waiting time with finite mean 𝑤_𝑞 has PGF 𝑤_𝑞(𝑧)=∑^∞_𝑖=0𝜋_𝑖𝑙𝑊_𝑖(𝑧),𝑙=0,1.

Theorem 5. In the steady state, the PGF of waiting time for an arriving customer is given by 𝑤_𝑞(𝑧)=𝜋_0,0+𝑝𝜃𝜃𝜇_𝑣𝑝+𝑝𝜉1−𝜉^𝑎𝑞(𝑧)𝜋𝜔𝜛^0,0+𝑝𝜇_𝑣1−𝜉^𝑎𝑞𝜃𝑧𝜔𝜋_0,0+𝑝𝜃𝜃𝜇_𝑣𝑝+𝑝𝜉1−𝑞(𝑧)𝜉^𝑎−1−𝑞(𝑧)𝜉^𝑎−1𝜉^𝑎𝑞(𝑧)𝜔𝜛1−𝑞(𝑧)𝜉^𝑎𝜋_0,0+1−𝑞𝜃𝑧𝜉^𝑎−1−𝑞𝜃𝑧𝜉^𝑎−1𝑝𝜇_𝑣𝑞𝜃𝑧𝜉^𝑎𝜔1−𝑞𝜃𝑧𝜉^𝑎𝜋_0,0+𝑐^′₀𝑟−𝑟^2𝑎+𝑞(𝑧)𝑟1−𝑟^2𝑎1−𝑟^𝑎(1−𝑟)1−𝑞(𝑧)𝑟^𝑎𝑞(𝑧)+𝑝𝜃𝜇_𝑣𝑞(𝑧)1−𝜉^𝑎𝜉^𝑎−1𝜔1−𝜇_𝑣𝑧𝜃1−𝜉^𝑎𝑞(𝑧)1−𝜉^𝑎𝑞𝑧𝜃𝜋_0,0, (31) and the average waiting time as 𝑤_𝑞=1𝜇_𝑏1−𝑟^𝑎+𝑟^2𝑎−1𝑟(1−𝑟^𝑎)𝑐)(1−𝑟^′0+𝑝𝜃𝜃𝜇_𝑣𝑝+𝑝𝜉1−𝜉^𝑎+𝜉^2𝑎−1𝜔𝜛1−𝜉^𝑎𝜋_0,0+𝑝𝜃𝜇_𝑣𝜉^𝑎−1𝜔1−𝜇_𝑣𝜃−𝜇_𝑣𝜃𝜉^𝑎1−𝜉^𝑎𝜋_0,0+𝑝𝜇_𝑣𝜇_𝑣𝜃𝜔×1+𝜉^𝑎−1+𝜃𝜉^𝑎−1−𝜉^2𝑎−1−2𝜉^𝑎×1−𝜇_𝑣𝜃−𝜇_𝑣𝜃𝜉^𝑎−1−𝜉^𝑎𝜉^𝑎𝜉^𝑎×1−𝜇_𝑣𝜃1−𝜇_𝑣𝜃−𝜇_𝑣𝜃𝜉^𝑎2−1+1−𝜉^𝑎1−𝜇_𝑣𝜃²⁻¹𝜋_0,0+𝑝𝜃𝜃𝜇²_𝑣𝜉^𝑎−1𝜔1−𝜇_𝑣𝜃1−𝜇_𝑣𝜃−𝜇_𝑣𝜃𝜉^𝑎𝜋_0,0. (32)

Proof. Firstly, we define ⌊𝑥⌋ as a greatest integer function (floor), which returns the greatest integer less than or equal to a real number 𝑥. An arriving customer may observe the system in any one of the following two cases.
Case 1. When 𝑇_𝑞=0, this case has no customers in the system and the server is on vacation, that is, 𝑤₀(0)𝑇=𝑃^𝑞=0𝑁_𝑤=0=𝜋_0,0. (33)
Case 2. When 𝑇_𝑞=𝑚,(𝑚≥1), there are two cases as follows.
(1) The server is on a normal busy period and 𝑖 customers in the system, meantime an arriving customer cannot go into the queue being served immediately.
Under this condition, an arriving customer has to wait for one period of service for 1≤𝑖≤𝑎 and ⌊𝑖/𝑎⌋ periods of service for 𝑖>𝑎. Each period of service 𝑆_𝑏𝑖(𝑖=1,2,…) is independent and geometrically distributed with p.m.f. 𝑃{𝑆_𝑏𝑖=𝑘}=𝜇_𝑏𝜇_𝑏^𝑘−1, 𝑘≥1,𝜇_𝑏=1−𝜇_𝑏, which has PGF as 𝜇_𝑏𝑧/(1−𝜇_𝑏𝑧).
We have 𝑤_𝑖(𝑚)=𝑃𝑠_𝑏1=𝑚𝑁_𝑤𝑃𝑠=𝑖,𝑖≤𝑎,_𝑏1+𝑠_𝑏2+⋯+𝑠_{𝑏⌊𝑖/𝑎⌋}=𝑚𝑁_𝑤=𝑖,𝑖>𝑎. (34) Hence, 𝑊_𝑖(𝑧)=^∞_𝑚=1𝑤_𝑖(𝑚)𝑧^𝑚=𝜇_𝑏𝑧1−𝜇_𝑏𝑧𝑊,1≤𝑖≤𝑎,_𝑖(𝑧)=^∞_𝑚=1𝑤_𝑖(𝑚)𝑧^𝑚=𝜇_𝑏𝑧1−𝜇_𝑏𝑧^{⌊𝑖/𝑎⌋},𝑖>𝑎. (35) Let 𝑞(𝑧)=𝑢_𝑏𝑧/(1−𝜇_𝑏𝑧), the PGF of the waiting time is given by 𝑝𝜃𝜃𝜇_𝑣𝑝+𝑝𝜉1−𝜉^𝑎𝑞(𝑧)𝜋𝜔𝜛^0,0+𝑐^′₀𝑟−𝑟^2𝑎+𝑞(𝑧)𝑟1−𝑟^2𝑎1−𝑟^𝑎(1−𝑟)1−𝑞(𝑧)𝑟^𝑎𝑞(𝑧)+𝑝𝜃𝜃𝜇_𝑣𝑝+𝑝𝜉1−𝑞(𝑧)𝜉^𝑎−1−𝑞(𝑧)𝜉^𝑎−1𝜉^𝑎𝑞(𝑧)𝜔𝜛1−𝑞(𝑧)𝜉^𝑎𝜋_0,0. (36)
(2) An arriving customer finds the server is on vacation.
Let 𝑠_𝑣𝑗 be the 𝑗th length of the period of service with service rate 𝜇_𝑣 and let 𝑠_𝑣^(𝑗) be the sum of the length of 𝑗 periods of service with service rate 𝜇_𝑣, 𝑗=1,2,3,…, where 𝑠_𝑣⁽⁰⁾=0. Each period of service 𝑆_𝑣𝑖(𝑖=1,2,…) is mutually independent and geometrically distributed with p.m.f. 𝑃{𝑆_𝑣𝑖=𝑘}=𝜇_𝑣𝜇_𝑣^𝑘−1,  𝑘≥1, 𝜇_𝑣=1−𝜇_𝑣. There are two cases in this condition.
(A) A vacation is going on whereas all of the arrived customers have been served. Then, an arriving customer has to wait for one period of service for 𝑖≤𝑎 and ⌊𝑖/𝑎⌋ periods of service for 𝑖>𝑎. We have 𝑤_𝑖(𝑚)𝑠=𝑃^𝑣1=𝑚𝑁_𝑤𝑊=𝑖;𝑉≥𝑚,1≤𝑖≤𝑎,_𝑖(𝑧)=^∞_𝑚=1𝑤_𝑖(𝑚)𝑧^𝑚=𝜇_𝑣𝑧1−𝜃𝜇_𝑣𝑧𝑤,1≤𝑖≤𝑎,_𝑖(𝑚)=𝑃𝑠_𝑣1+𝑠_𝑣2+⋯+𝑠_{𝑣⌊𝑖/𝑎⌋}=𝑚𝑁_𝑤𝑊=𝑖;𝑉≥𝑚,𝑖>𝑎,_𝑖(𝑧)=^∞_𝑚=1𝑤_𝑖(𝑚)𝑧^𝑚=1𝜃𝜇_𝑣𝜃𝑧1−𝜇_𝑣𝜃𝑧^{⌊𝑖/𝑎⌋},𝑖>𝑎. (37)
Let 𝑞(𝑧)=𝜇_𝑣𝑧/(1−𝜇_𝑣𝑧). The PGF of the waiting time is given by 𝑝𝜇_𝑣1−𝜉^𝑎𝜔𝑞𝜃𝑧𝜋_0,0+1−𝑞𝜃𝑧𝜉^𝑎−1−𝑞𝜃𝑧𝜉^𝑎−1𝑝𝜇_𝑣𝑞𝜃𝑧𝜉^𝑎𝜔1−𝑞𝜃𝑧𝜉^𝑎𝜋_0,0. (38)
(B) If a vacation is over and 𝑗(𝑗=0,1≤𝑖≤𝑎;1≤𝑗<⌊𝑖/𝑎⌋,𝑖>𝑎) periods of service ended, the service rate is converted to 𝜇_𝑏 from 𝜇_𝑣 and the normal busy period begins. The waiting time of an arriving customer should be equal to the sum of the server’s vacation times and one period of service for 1≤𝑖≤𝑎 and ⌊𝑖/𝑎⌋−𝑗 periods of service for 𝑖>𝑎. The service rate of ⌊𝑖/𝑎⌋−𝑗 periods of service is 𝜇_𝑏. We have 𝑤_𝑖(𝑚)𝑇=𝑃^𝑞=𝑚𝑁=𝑖;𝑉<𝑠_𝑣1𝑤,1≤𝑖≤𝑎,_𝑖(𝑚)=^{⌊𝑖/𝑎⌋−1}_𝑗=0𝑃𝑇_𝑞=𝑚𝑁=𝑖;𝑠_𝑣^(𝑗)≤𝑉<𝑠_𝑣^(𝑗+1),𝑖>𝑎. (39) Hence, 𝑤_𝑖(𝑚)=^{⌊𝑖/𝑎⌋−1}_𝑗=0𝑃𝑇_𝑞=𝑚𝑁=𝑖;𝑠_𝑣^(𝑗)≤𝑉<𝑠_𝑣^(𝑗+1)=^{⌊𝑖/𝑎⌋−1}_𝑗=0𝑃𝑉+𝑠_𝑏1+𝑠_𝑏2+⋯+𝑠_{𝑏⌊𝑖/𝑎⌋−𝑗}=𝑚;𝑠_𝑣^(𝑗)≤𝑉<𝑠_𝑣^(𝑗+1)=^{⌊𝑖/𝑎⌋−1}_𝑗=0^{𝑚−⌊𝑖/𝑎⌋+𝑗}_𝑢=1𝑃{}𝑠𝑉=𝑢×𝑃_𝑏1+𝑠_𝑏2+⋯+𝑠_{𝑏⌊𝑖/𝑎⌋−𝑗}𝑠=𝑚−𝑢×𝑃_𝑣^(𝑗)≤𝑢<𝑠_𝑣^(𝑗+1),𝑖>𝑎. (40) The PGF of the waiting time can be given by 𝑝𝜃𝜇_𝑣𝑞(𝑧)1−𝜉^𝑎𝜉^𝑎−1𝜔1−𝜇_𝑣𝑧𝜃1−𝜉^𝑎𝑞(𝑧)1−𝜉^𝑎𝑞𝑧𝜃𝜋_0,0. (41) Adding (33)–(41), we can get (31), using (𝑑𝑤_𝑞(𝑧)/𝑑𝑧)|_𝑧=1, we can obtain (32).

5. Outside Observer’s Observation Epoch Distributions

For an early arrive system, since an outside observer’s observation epoch falls in the time interval after a potential arrival and before a potential batch departure, let, 𝜋_𝑛,0, 𝜋_𝑛,1 be 𝑛(𝑛≥0) customers in the system and the server is on vacation (including the servicing customers), 𝑛 customers in the system and the server is in regular busy period (including the servicing customers). Through observing the relationship between random slot 𝑡 and the outside observer’s observation epoch (∗), we have 𝜋_0,0=𝑝𝜋_0,0,𝜋_𝑛,0=𝑝𝜃𝜋_𝑛,0+𝑝𝜃𝜋_𝑛−1,0,(),𝜋𝑛≥1_1,1=𝑝𝜋_1,1+𝑝𝜃𝜋_0,0+𝑝𝜃𝜋_1,0,𝜋_𝑛,1=𝑝𝜋_𝑛,1+𝑝𝜋_𝑛−1,1+𝑝𝜃𝜋_𝑛,0+𝑝𝜃𝜋_𝑛−1,0,(𝑛≥2). (42)

6. Numerical Results and the Sensitivity Analysis of this System

In this section, we present some numerical results in self-explanatory tables and graphs for queue length distributions at random slots and all the numerical results have been obtained using the results derived in this paper.

We observe that 𝜋_𝑛,0 and 𝜋_𝑛,0 monotonically increase whereas 𝜋_𝑛,1 and 𝜋_𝑛,1 monotonically decrease as 𝑎 increases in Tables 1–4. This situation continues until 𝑎 is equal to some constant; all data will tend to be a steady state. The above description is consistent with actual situation. In the meantime, 𝐸(𝐿) and 𝐸(𝑤_𝑞) monotonically decease as 𝑎 increases. In Figures 2 and 3, fixing 𝑎=10, 𝑝=0.3, 𝜇_𝑏=0.5, and 𝜃=0.3,0.5,0.7, we have plotted the effect of various vacation service rates on the average queue length and the average waiting time, respectively. We observe that the average queue length and the average waiting time decrease as the vacation service rate increases. In Figure 4, fixing 𝑝=0.3,𝜇_𝑣=0.4,𝜇_𝑏=0.6, and 𝜃=0.7, the steady-state average queue length equals 0.2597 from 𝑎=4 on and the steady-state average waiting time equals 0.8421 from 𝑎=5 on. They do not change as the batch size increases.

Table 1 

Queue length distributions at random slots for 𝑎=1, 𝑝=0.3, 𝜇_𝑣=0.4, 𝜇_𝑏=0.6, and 𝜃=0.7.

Table 2 

Queue length distributions at random slots for 𝑎=4, 𝑝=0.3, 𝜇_𝑣=0.4, 𝜇_𝑏=0.6, and 𝜃=0.7.

Table 3 

Queue length distributions at random slots for 𝑎=5, 𝑝=0.3, 𝜇_𝑣=0.4, 𝜇_𝑏=0.6, and 𝜃=0.7.

Table 4 

Queue length distributions at random slots for 𝑎=10, 𝑝=0.3, 𝜇_𝑣=0.4, 𝜇_𝑏=0.6, and 𝜃=0.7.

Figure 2 

Effect of 𝜇𝑣 on the average queue length.

Figure 3 

Effect of 𝜇𝑣 on the average waiting time.

Figure 4 

Effect of 𝑎 on the average queue length and the average waiting time.

7. Conclusions

A 𝐺𝑒𝑜𝑚/𝐺𝑒𝑜𝑚^[𝑎]/1/𝑀𝑊𝑉 queueing system has been investigated. Assume that the server takes a working vacation after emptying the system in regular busy period. By using embedded Markov chain approach and the method of nonhomogeneous and homogeneous difference operator, the number of customers of the whole system at random slots has been discussed. This is different from general batch service queue literatures (excluding customers being served). The waiting time for an arriving customer and numerical results are obtained. In the future, further study such as 𝐺𝑒𝑜𝑚^[𝑎]/𝐺𝑒𝑜𝑚^[𝑏]/1/𝑀𝑊𝑉 queue will be the research topic using similar idea and method.

Acknowledgments

The authors wish to thank the referee for his careful reading of the paper and for his helpful suggestion. Meantime, this work is supported by the National Natural Science Foundation of China (no. 71171138), and the Talent Introduction Foundation of Sichuan University of Science & Engineering (2012RC23).