N queue with negative customers and multiple working vacations

Journal of the Korean Statistical Society 42 (2013) 515–528

Contents lists available at ScienceDirect

Journal of the Korean Statistical Society journal homepage: www.elsevier.com/locate/jkss

Discrete-time GI X /Geo/1/N queue with negative customers and multiple working vacations Shan Gao a,b,∗ , Jinting Wang a , Deran Zhang b a

Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China

b

Department of Mathematics, Fuyang Normal College, Fuyang, Anhui 236037, China

article

info

Article history: Received 9 August 2012 Accepted 11 March 2013 Available online 3 April 2013 AMS 2000 subject classifications: primary 60K25 secondary 90B22

abstract Using the supplementary variable and the embedded Markov chain method, we consider a discrete-time batch arrival finite capacity queue with negative customers and working vacations, where the RCH killing policy and partial batch rejection policy are adopted. We obtain steady-state system length distributions at pre-arrival, arbitrary and outside observer’s observation epochs. Furthermore, we consider the influence of system parameters on several performance measures to demonstrate the correctness of the theoretical analysis. © 2013 The Korean Statistical Society. Published by Elsevier B.V. All rights reserved.

Keywords: Discrete-time finite-buffer queue Negative customers Working vacation Supplementary variable technique Embedded Markov chain

1. Introduction Recently, queueing systems with negative customers have been studied extensively, as they have been greatly motivated by some practical applications such as computer, neural networks, manufacturing systems and communication networks (see Artalejo, 2000). Negative customers can be thought of as viruses or commands to delete some transactions in a computer network or a database, inhibitor and synchronization signals in neural and high speed communication network; see Chao, Miyazawa, and Pinedo (1999). Queues with negative arrivals, called G-queues, were first introduced by Gelenbe (1989). When a negative customer arrives at the queue, it immediately removes one or more positive customers if present. Negative arrivals have no effect if the system is empty. A negative customer can remove positive customers in the queue (or being served) according to different specified killing disciplines: (i) RCH (removal of the customer from the head of the queue); (ii) RCE (removal of the customer from the end of the queue). Since 1989, there have been many studies on queues with negative customers; readers may refer to Boucherie and Boxma (1996), Chakka and Harrison (2001), Harrison, Patel, and Pitel (2000), Harrison and Pitel (1993, 1995, 1996), Li and Zhao (2004) and Wu, Liu, and Peng (2009). Although many continuous-time queueing models with negative arrivals have been studied extensively in the past years, their discrete-time counterparts received very little attention in the literature. It is well known that discrete-time queueing models are more suitable for application to digital communication systems, including mobile and broad integrated services digital networks (B-ISDN) based on asynchronous transfer mode (ATM) technology, because these protocols are operated

∗

Corresponding author at: Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China. Tel.: +86 15555992176. E-mail addresses: [email protected], [email protected] (S. Gao).

1226-3192/$ – see front matter © 2013 The Korean Statistical Society. Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jkss.2013.03.002

516

S. Gao et al. / Journal of the Korean Statistical Society 42 (2013) 515–528

Fig. 1. Various time epochs in an early arrival system.

by a unit of slot which is an equal time interval (see Bruneel & Kim, 1993). The work about negative customers in discretetime can be found in Atencia and Moreno (2004a,b, 2005), where the authors considered the single-server discrete-time queue with negative arrivals and various killing disciplines caused by the negative customers, Wang and Zhang (2009), where they presented a discrete-time single-server retrial queue with geometrical arrivals of both positive and negative customers. Other more work on negative customers can be found in Do (2011), where Do (2011) presented a bibliography on G-networks from 1989 to 2010. Recently, queueing systems with working vacations have been widely used in the performance analysis of communication systems, in which the server renders service to the customers with a lower service rate during vacation period. Working vacation policy has practical application background in optimal design of the system. When the number of customers in the system is relatively few, we set a lower speed operating period in order to economize operating cost and energy consumption. The researches about working vacation queueing models with infinite buffer size can be found in Baba (2005), Gao and Liu (2013), Kim, Choi, and Chae (2003), Li, Liu, and Tian (2010), Li and Tian (2007), Li, Tian, and Liu (2007), Li, Tian, and Ma (2008), Li, Tian, Zhang, and Luh (2009), Liu, Xu, and Tian (2007), Servi and Finn (2002), Wu and Takagi (2006), Yi, Kim, Choi, and Chae (2007) and the references therein. Though the working vacation queueing models with infinite buffer size have been studied extensively in the past years, many a time there is also need for finite buffer size. Queues with finite buffer space are more realistic in real life situations than queues with infinite buffer space as it is used to store arrived customers if server is busy. To the best of our knowledge, the works about general input working vacation queueing model with finite buffer size can be found in Banik, Gupta, and Pathak (2007), Goswami and Mund (2010), Goswami and Vijaya Laxmi (2010), Yu, Tang, and Fu (2009) and Yu, Tang, Fu, and Pan (2011), where Banik et al. (2007) discussed the GI /M /1/N queue with multiple working vacations, Yu et al. (2009) presented the GI [x] /M b /1/L queue with multiple working vacations and partial batch rejection, Goswami and Vijaya Laxmi (2010) analyzed the GI [x] /M /1/N queue with single working vacations and partial batch rejection, a finite buffer size discrete-time multiple working vacation queue was considered by Goswami and Mund (2010) and Yu et al. (2011) have introduced changeover time into the working vacation. Nevertheless, discrete-time finite buffer G-queues taking the working vacation policy into account have not been studied up to now. Thus, the contribution of this work is to analyze a discrete-time finite buffer queue with multiple working vacations and two types of customers, positive and negative, in accordance with the RCH. The rest of this paper is organized as follows. In Section 2, we give the discrete time queueing model. In Section 3, we analyze the model and obtain steady state distributions at arbitrary, pre-arrival and outside observer’s observation epochs. Various performance measures and numerical examples are presented in Section 4. 2. Model formulations Thereinafter, we denote x¯ = 1 − x for any real number x ∈ (0, 1). The GI X /Geo/1/N queue with negative customers and multiple working vacations we considered here is an early arrival system that is, a potential arrival can only take place in (n, n+ ) and a potential departure can only take place in (n− , n). We assume that the beginning and ending of vacations occurs at the instant n. Arriving customers are queued according to the first-come, first-served (FCFS) discipline. The server can serve only one customer at a time. Various stochastic processes involved in the system are independent of each other. The various time epochs at which events occur are depicted in Fig. 1. The detailed description of the model is given as follows: (1) Positive customers arrive in batches of random size X with probability mass function (p.m.f.) P (X = j) = χj , j = 1, 2, . . . , and mean E [X ] = µX . The inter-arrival times A of two successive batches of positive customers are independently identically ∞ i distributed (i.i.d.) random variables with common p.m.f. P (A = i) = ai , i ≥ 1, corresponding p.g.f.  A(z ) = i=1 ai z and −1 ′ ′    mean inter-arrival time λ = A (1), where A (1) is the first derivative of A(z ) with respect to z at z = 1. (2) Inter-arrival times B of negative customers are independent and geometrically distributed random variable with the following geometric distribution: P (B = j) = ηη¯ j−1 ,

j ≥ 1, 0 ≤ η < 1.

S. Gao et al. / Journal of the Korean Statistical Society 42 (2013) 515–528

517

Under the RCH discipline, the arrival of a negative customer does not receive service and removes the positive customer who is being served. However, when the system is empty upon the arrival, the negative customer will vanish and has no impact on the server. (3) The service time Sb in a regular busy period is geometrically distributed with p.m.f. P (Sb = i) = µµ ¯ i−1 , i ≥ 1, 0 < µ < 1. (4) The system has finite buffer space of size N, i.e., no more than N customers can present in the system. (5) The working vacation is an operating period in a lower rate, the service time Sv in a working vacation period has a geometric distribution with p.m.f. P (Sv = i) = ν ν¯ i−1 , i ≥ 1, 0 ≤ ν ≤ µ. (ν = 0 corresponds to a classical vacation queue.) (6) The server begins a working vacation at the epoch when the system becomes empty, the distribution of vacation time V is geometrically distributed with rate θ (0 < θ < 1), i.e., P (V = i) = θ θ¯ i−1 , i ≥ 1. The server serves the customers at the rate of µ and when the system is empty after a positive departure (either by service completion or by a negative arrival), the server enters into a working vacation period. Customers who arrive during a working vacation period will be served at a lower service rate ν . If there is no customer when a vacation ends, the server begins another vacation, otherwise, the server switches to the normal working level and a regular busy period starts. At the end of a vacation if there is a customer whose lower service is not yet completed, then the server switches service rate from ν to µ and a regular busy period begins. (7) Partial batch rejection policy is adopted. Since the buffer space is finite, if a batch upon arrival does not find enough space in the buffer then a part of customers fills the vacant spaces and the rest is rejected. This is known as partial batch rejection. Since the partial batch rejection policy utilizes the buffer space in an optimal manner so that the loss probability of customers is rather low, we consider only this policy in this paper. The state of the system at time n is described by the following random variables: N (n): number of customers in the system including the one in service, U (n): remaining inter-arrival time for the group going to enter into the system, J (n): the state of the server, where server may be on working vacation (J (n) = 0) or in busy period (J (n) = 1). Further define the joint probabilities as P0,0 (u, n) = P (N (n) = 0, J (n) = 0, U (n) = u), Pk,i (u, n) = P (N (n) = k, J (n) = i, U (n) = u),

u ≥ 0, 1 ≤ k ≤ N , i = 0, 1, u ≥ 0.

In steady-state, the above probabilities are denoted, respectively, as P0,0 (u) and Pk,i (u), 1 ≤ k ≤ N , i = 0, 1, u ≥ 0. 3. Analysis of the model In this section, we carry out the analytic analysis of discrete-time GI X /Geo/1/N queue with negative customers, multiple working vacations and partial batch rejection. We obtain the system length distribution at arbitrary and pre-arrival epoch. The queue is analyzed using both embedded Markov chain technique and the supplementary variable. The former one is used to obtain the state probabilities at pre-arrival epochs and the latter one is used to develop a relation between pre-arrival and arbitrary epoch probabilities. 3.1. Steady-state distribution at positive customer pre-arrival epoch Let positive customer batches arrive at time epochs T1 , T2 , . . . and the inter-arrival times Tn+1 − Tn , (n = 0, 1, . . . ; T0 = 0) be mutually independent and identically distributed random variables with common p.m.f. {au , u ≥ 1}. Let the state of the system at pre-arrival epoch of the n-th batch be defined as {(Nn− , Jn− ), n ≥ 0}, where Nn− denotes the number of customers in the queue, and Jn− indicates whether the server is busy (Jn− = 1) or on working vacation (Jn− = 0) at pre-arrival epoch of the n-th batch, that is, suppose that the n-th batch arrives at the system in (k, k+ ), then Nn− = N (k), Jn− = J (k). Then {(Nn− , Jn− ), n ≥ 0} is an embedded two-dimensional Markov chain with the state space {(i, j), j = 0, 1; j ≤ i ≤ N }. In limiting case let us define − − Pi− ,j = lim P (Nn = i, Jn = j), n→∞

j = 0, 1; j ≤ i ≤ N ,

where Pi− ,j represents the probability that there are i customers in the queue just prior to an arrival epoch of a customer batch when the server is in the state j. Before developing the transition probabilities of the two-dimensional Markov chain {(Nn− , Jn− ), n ≥ 0}, we introduce some useful notations for the future analysis results. From the assumptions of the queue model, negative customers arrive according to Bernoulli process with parameter η and eliminate the customer being served, so the numbers of the positive customers who depart the system in a unit time slot respectively during working vacation period and busy period, denoted, respectively, by D1 and D2 , satisfy the following relations: P (D1 = 0) = η¯ ν¯ ,

P (D1 = 1) = ην ¯ + ην¯ ,

P (D2 = 0) = η¯ µ, ¯

P (D2 = 1) = ηµ ¯ + ηµ, ¯

P (D1 = 2) = ην, P (D2 = 2) = ηµ.

518

S. Gao et al. / Journal of the Korean Statistical Society 42 (2013) 515–528

The generating functions  D1 (z ) and  D2 (z ) of D1 and D2 are given respectively by

 D1 (z ) = η¯ ν¯ + z (ην ¯ + ην¯ ) + z 2 ην,

 D2 (z ) = η¯ µ ¯ + z (ηµ ¯ + ην¯ ) + z 2 ηµ.

Let αk , k ≥ 0 represent the probability that there are k departures of positive customers from the system during an (n) inter-arrival time in the busy period; and Bk be the probability that there are k departures of positive customers from the system in n unit time slot in the busy period, then (n)

(n−1)

(n)

(n−1)

(n)

(n−1)

B0 = η¯ µ ¯ B0

,

B1 = η¯ µ ¯ B1

n ≥ 1,

+ (ηµ ¯ + ηµ) ¯ B(0n−1) ,

= η¯ µ ¯ Bk + (ηµ ¯ + ∞  (n) αk = an Bk , Bk

ηµ) ¯ B(kn−−11)

n ≥ 1,

+ ηµB(kn−−21) ,

n ≥ 1, k ≥ 2,

n=1

(0)

(0)

(n)

where B0 = 1, Bk = 0 (k ≥ 1), Bk = 0 (k > 2n). Next, let βk , k ≥ 0 represent the probability that V > A and there are k departures of positive customers from the system (n) during an inter-arrival time in the working vacation period; and Vk be the probability that there are k departures of positive customers from the system in n unit time slot in the working vacation period, then (n)

V0

(n)

V1

= η¯ ν¯ V0(n−1) , (n−1)

= η¯ ν¯ V1

n ≥ 1,

+ (ην ¯ + ην¯ )V0(n−1) ,

(n−1)

(n)

+ (ην ¯ + = η¯ ν¯ Vk ∞  (n) βk = an θ¯ n Vk , Vk

ην¯ )Vk(−n−1 1)

n ≥ 1,

+ ην Vk(−n−2 1) ,

n ≥ 1, k ≥ 2,

n =1

(0)

(n)

(0)

where V0 = 1, Vk = 0 (k ≥ 1), Vk = 0 (k > 2n). At last, let γk , k ≥ 0 denote the probability that V ≤ A and there are k departures of positive customers from the system during an inter-arrival time, then

γk =

∞  n=1

an

n 

θ θ¯ m−1

m=1

k 

(n−m)

Vr(m) Bk−r

.

r =0

Now we develop the transition probabilities of the two-dimensional Markov chain {(Nn− , Jn− ), n ≥ 0}. Put P(i,k)(j,m) = P {Nn−+1 = j, Jn−+1 = m|Nn− = i, Jn− = k}

 A ,   i ,j Bi,j , :=  Ci,j , Di,j ,

k k k k

= m = 0, i, j = 0, 1, 2, . . . , N , = 0, m = 1, i = 0, 1, 2, . . . , N , j = 1, 2, . . . , N , = 1, m = 0, i = 1, 2, . . . , N , j = 0, 1, 2, . . . , N , = 1, m = 1, i, j = 1, 2, . . . , N .

Observing the state of the system at two consecutive embedded points, we have the one step transition probability matrix (TPM) P of dimension (2N + 1) × (2N + 1) having four block matrices of the form:

 P =

A(N +1)×(N +1) CN ×(N +1)



B(N +1)×N . DN ×N

Blocks A = (Ai,j ) and B = (Bi,j ) denote the transition from working vacation state to working vacation state and regular busy state, respectively and are given by the following expression:

  

N −i 

∞ 

χk , 0 ≤ i ≤ N − 1, 1 ≤ j ≤ N , k=N −i+1  k=max{1,j−i} B N − 1 ,j , i = N, 1 ≤ j ≤ N.  N −i ∞     χk βi+k−j + βN −j χk , 0 ≤ i ≤ N − 1, 1 ≤ j ≤ N ,    k=max{1,j−i} k=N −i+1 N Ai,j =    1 − (Ai,k + Bi,k ), 0 ≤ i ≤ N − 1, j = 0,    k=1  AN −1,j , i = N, 0 ≤ j ≤ N. Bi,j =

χk γi+k−j + γN −j

S. Gao et al. / Journal of the Korean Statistical Society 42 (2013) 515–528

519

The second row blocks C = (Ci,j ) and D = (Di,j ) of the TPM represent the transition from regular busy state to working vacation state and regular busy state, respectively.

Ci,j =

Di,j =

 N −i i+ k −1 N −1 ∞       1 − χk αm − αm χk , k=1

m=0

   C N − 1 ,j , 0,  N −i   

m=0

k=N −i+1

χk αi+k−j + αN −j

 k=max{1,j−i} DN −1,j ,

∞ 

χk ,

k=N −i+1

1 ≤ i ≤ N − 1, j = 0, i = N , j = 0, otherwise. 1 ≤ i ≤ N − 1, 1 ≤ j ≤ N i = N, 1 ≤ j ≤ N.

Let P− = (P0−,0 , P1−,0 , P2−,0 , . . . , PN−,0 , P1−,1 , P2−,1 , . . . , PN−,1 ) be the row vector of the pre-arrival epoch probabilities which can be obtained by solving P− P = P− and P− e = 1, where e is a (2N + 1) × 1 column vector of 1’s. The system of equations can be solved by using the available software Matlab. 3.2. Steady-state distribution at arbitrary epoch To obtain the system length distribution at arbitrary epoch and performance measures of the system, we develop the difference equations using the remaining inter-arrival time as supplementary variable. Observing the state of the system at two consecutive time epochs n and (n + 1), and using probability argument, we have the following difference equations in steady-state, for u ≥ 1: P0,0 (u − 1) = P0,0 (u) + au (χ1 (η + ην) ¯ + χ2 ην)P0,0 (0) + (η + ην) ¯ P1,0 (u) + au χ1 ην P1,0 (0) + ην P2,0 (u) + (η + ηµ) ¯ P1,1 (u) + au χ1 ηµP1,1 (0) + ηµP2,1 (u),









(3.1)



n −1 n   η¯ ν¯ au Pk,0 (0)χn−k + Pn,0 (u) + (ην¯ + ην) ¯ au Pk,0 (0)χn+1−k + Pn+1,0 (u) k=0  k=0  n+1  + ην au Pk,0 (0)χn+2−k + Pn+2,0 (u) , 1 ≤ i ≤ N − 3,

Pn,0 (u − 1) = θ¯

(3.2)

k=0

   N −3  ¯ PN −2,0 (u − 1) = θ η¯ ν¯ au Pk,0 (0)χN −2−k + PN −2,0 (u) + (ην¯ + ην) ¯ k=0     N −2 N ∞    × au Pk,0 (0)χN −1−k + PN −1,0 (u) + ην au χm + PN ,0 (u) , Pk,0 (0) k=0





PN −1,0 (u − 1) = θ¯ η¯ ν¯ au

N −2 

(3.3)

m=N −k

k=0

 Pk,0 (0)χN −1−k + PN −1,0 (u)

k=0

 + (ην¯ + ην) ¯ au

N 

Pk,0 (0)

k =0

 PN ,0 (u − 1) = θ¯ η¯ ν¯ au

N 

Pk,0 (0)

k =0

 Pn,1 (u − 1) = θ



n−1 

∞ 

∞ 

 χm + PN ,0 (u)

,

(3.4)

m=N −k

 χm + PN ,0 (u) ,

(3.5)

m=N −k





n 



η¯ ν¯ au Pk,0 (0)χn−k + Pn,0 (u) + (ην¯ + ην) ¯ au Pk,0 (0)χn+1−k + Pn+1,0 (u) k=0  k=0    n+1 n−1   + ην au Pk,0 (0)χn+2−k + Pn+2,0 (u) + η¯ µ ¯ au Pk,1 (0)χn−k + Pn,1 (u) k=0 k=1   n  + (ηµ ¯ + ηµ) ¯ au Pk,1 (0)χn+1−k + Pn+1,1 (u) k=1   n+1  + ηµ au Pk,1 (0)χn+2−k + Pn+2,1 (u) , 1 ≤ i ≤ N − 3, k=1

(3.6)

520

S. Gao et al. / Journal of the Korean Statistical Society 42 (2013) 515–528







N −3 

PN −2,1 (u − 1) = θ

η¯ ν¯ au Pk,0 (0)χN −2−k + PN −2,0 (u) + (ην¯ + ην) ¯ k=0     N −2 N ∞    × au Pk,0 (0)χN −1−k + PN −1,0 (u) + ην au Pk,0 (0) χm + PN ,0 (u) m=N −k k=0  k=0  N −3  + η¯ µ ¯ au Pk,1 (0)χN −2−k + PN −2,1 (u) + (ηµ ¯ + ηµ) ¯ k=1     N −2 N ∞    × au Pk,1 (0)χN −1−k + PN −1,1 (u) + ηµ au Pk,1 (0) χm + PN ,1 (u) , k=1





k=1



N −2 

PN −1,1 (u − 1) = θ

η¯ ν¯ au Pk,0 (0)χN −1−k + PN −1,0 (u) + (ην¯ + ην) ¯ k=0     N ∞ N −2    × au Pk,0 (0) χm + PN ,0 (u) + η¯ µ ¯ au Pk,1 (0)χN −1−k + PN −1,1 (u) m=N −k k=0 k=1   N ∞   + (ηµ ¯ + ηµ) ¯ au χm + PN ,1 (u) , Pk,1 (0)

PN ,1 (u − 1) = θ η¯ ν¯ au

N 

Pk,0 (0)

∞ 





χm + PN ,0 (u) + η¯ µ ¯ au

m=N −k

k=0

(3.8)

m=N −k

k=1



(3.7)

m=N −k

N  k=1

Pk,1 (0)

∞ 

 χm + PN ,1 (u) .

(3.9)

m=N −k

(Note: χ0 = 0, a = 0, if a > b in (3.4)–(3.6) and henceforth.) We introduce the following z-transforms

b

 Pi,j (z ) =

∞ 

Pi,j (u)z u ,

j = 0, 1; j ≤ i ≤ N .

u =0

So that

 Pi,j (1) = Pi,j , where Pi,0 is the joint probability that there are i customers in the system while the server is on working vacation. Similarly, Pi,1 is the joint probability that there are i customers in the system while the server is in service period. Multiplying (3.1)–(3.9) by z u and summing over u from 1 to ∞, we obtain

(z − 1) P0,0 (z ) = A(z )(χ1 (η + ην) ¯ + χ2 ην)P0,0 (0) + (η + ην)( ¯  P1,0 (z ) − P1,0 (0)) + A(z )χ1 ην P1,0 (0) + ην( P2,0 (z ) − P2,0 (0)) + (η + ηµ)( ¯  P1,1 (z ) − P1,1 (0)) + A(z )χ1 ηµP1,1 (0) + ηµ( P2,1 (z ) − P2,1 (0)) − P0,0 (0),    n −1  (z − θ¯ η¯ ν¯ ) Pn,0 (z ) = θ¯ η¯ ν¯ A(z ) Pk,0 (0)χn−k − Pn,0 (0) k=0 

  + (ην¯ + ην) ¯ A(z ) Pk,0 (0)χn+1−k + Pn+1,0 (z ) − Pn+1,0 (0) k=0   n+1  + ην A(z ) Pk,0 (0)χn+2−k +  Pn+2,0 (u) − Pn+2,0 (0) , 1 ≤ n ≤ N − 3,

(3.10)

n 

(3.11)

k=0





(z − θ¯ η¯ ν¯ ) PN −2,0 (z ) = θ¯ η¯ ν¯ A(z )

N −3 

 Pk,0 (0)χN −2−k − PN −2,0 (0)

k=0



  + (ην¯ + ην) ¯ A(z ) Pk,0 (0)χN −1−k + PN −1,0 (z ) − PN −1,0 (0) k=0   N ∞   + ην A(z ) Pk,0 (0) χm +  PN ,0 (z ) − PN ,0 (0) , k=0

N −2 

m=N −k

(3.12)

S. Gao et al. / Journal of the Korean Statistical Society 42 (2013) 515–528





N −2 

(z − θ¯ η¯ ν¯ ) PN −1,0 (z ) = θ¯ η¯ ν¯ A(z )

521

 Pk,0 (0)χN −1−k − PN −1,0 (0)

k=0

 + (ην¯ + ην) ¯ A(z )

N 

Pk,0 (0)



∞ 

χm +  PN ,0 (z ) − PN ,0 (0)

,

  N ∞    ¯ ¯ (z − θ η¯ ν¯ )PN ,0 (z ) = θ η¯ ν¯ A(z ) Pk,0 (0) χm − PN ,0 (0) 

(z − η¯ µ) ¯  Pn,1 (z ) = θ η¯ ν¯ A(z )

(3.14)

m=N −k

k=0



(3.13)

m=N −k

k=0

n −1 

 Pk,0 (0)χn−k +  Pn,0 (u) − Pn,0 (0)

k=0



  + (ην¯ + ην) ¯ A(z ) Pk,0 (0)χn+1−k + Pn+1,0 (z ) − Pn+1,0 (0) k=0   n +1  + ην A(z ) Pk,0 (0)χn+2−k +  Pn+2,0 (z ) − Pn+2,0 (0) k=0   n −1  Pk,1 (0)χn−k − Pn,1 (0) + η¯ µ ¯ A(z ) k=1   n   + (ηµ ¯ + ηµ) ¯ A(z ) Pk,1 (0)χn+1−k + Pn+1,1 (z ) − Pn+1,1 (0) n 

k=1

 + ηµ A(z )



n +1 

Pk,1 (0)χn+2−k +  Pn+2,1 (z ) − Pn+2,1 (0) ,

1 ≤ i ≤ N − 3,

(3.15)

k=1

   N −3    (z − η¯ µ) ¯ PN −2,1 (z ) = θ η¯ ν¯ A(z ) Pk,0 (0)χN −2−k + PN −2,0 (z ) − PN −2,0 (0) k=0   N −2  + (ην¯ + ην) ¯ A(z ) Pk,0 (0)χN −1−k +  PN −1,0 (z ) − PN −1,0 (0) k=0



N 

∞ 



∞ 

  χm + PN ,1 (z ) − PN ,1 (0) ,

χm +  PN ,0 (z ) − PN ,0 (0) + ην A(z ) Pk,0 (0) m=N −k k=0   N −3  + η¯ µ ¯ A(z ) Pk,1 (0)χN −2−k − PN −2,1 (0) k=1   N −2  Pk,1 (0)χN −1−k +  PN −1,1 (z ) − PN −1,1 (0) + (ηµ ¯ + ηµ) ¯ A(z ) k=1

 + ηµ A(z )

N 

Pk,1 (0)

k=1





(z − η¯ µ) ¯  PN −1,1 (z ) = θ η¯ ν¯ A(z )

N −2 

(3.16)

m=N −k

 Pk,0 (0)χN −1−k +  PN −1,0 (z ) − PN −1,0 (0)

k=0



N 

∞ 

  χm + PN ,0 (z ) − PN ,0 (0)

+ (ην¯ + ην) ¯ A(z ) Pk,0 (0) m=N −k k=0   N −2  + η¯ µ ¯ A(z ) Pk,1 (0)χN −1−k − PN −1,1 (0) k=1   N ∞   + (ηµ ¯ + ηµ) ¯ A(z ) Pk,1 (0) χm +  PN ,1 (z ) − PN ,1 (0) , k=1

  N ∞     (z − η¯ µ) ¯ PN ,1 (z ) = θ η¯ ν¯ A(z ) Pk,0 (0) χm + PN ,0 (z ) − PN ,0 (0) m=N −k  k=0  N ∞   + η¯ µ ¯ A(z ) Pk,1 (0) χm − PN ,1 (0) . k=1

m=N −k

(3.17)

m=N −k

(3.18)

522

S. Gao et al. / Journal of the Korean Statistical Society 42 (2013) 515–528

Summing all Eqs. (3.10)–(3.18), we can obtain one important result in the form of a lemma. Lemma 1. N 

Pi,0 (0) +

i=0

N 

Pi,1 (0) = λ.

(3.19)

i=1

The left hand side denotes mean number of entrance batches into the system per unit time and is obviously equal to the mean arrival rate λ. Proof. Adding Eqs. (3.10)–(3.18) yields

  N N   A(z ) − 1    Pi,0 (z ) + Pi,1 (z ) = Pi,0 (0) + Pi,1 (0) . z−1 i =0 i=1 i =0 i=1

N 

N 

(3.20)

Taking the limit as z → 1 in (3.20) and using the normalization condition N 

 Pi,0 (1) +

i=0

N 

 Pi,1 (1) = 1,

i=1

one can obtain the desired result.



It can be seen that getting Pi,j (j = 0, 1, j ≤ i ≤ N ) from Eqs. (3.10)–(3.18) is difficult. But by observing arbitrary and arrival epochs in Fig. 1, applying Bayes’ theorem and using (3.19), one can easily get Pi− ,j =

Pi,j (0) N 

Pi,0 (0) +

i=0

N 

=

Pi,j (0)

Pi,1 (0)

λ

,

j = 0, 1; j ≤ i ≤ N .

(3.21)

i=1

The above result will be used, together with (3.10)–(3.18), to develop the relation between pre-arrival- and arbitrary-epoch probabilities. 3.3. Relations between steady-state distributions at arbitrary and pre-arrival epochs Setting z = 1 in (3.11)–(3.18) and using (3.21), after simplification, we can get relations between pre-arrival and arbitrary epoch probabilities P N ,0 =

P N − 1 ,0

P N − 2 ,0

∞ N −1  λθ¯ η¯ ν¯  − χk , Pm ,0 1 − θ¯ η¯ ν¯ m=0 k=N −m      N −2 N −1 ∞    θ¯ − − − = χm + PN ,0 , λη¯ ν¯ Pk,0 χN −1−k − PN −1,0 + (ην¯ + ην) ¯ λ Pk,0 1 − θ¯ η¯ ν¯ m=N −k k=0 k=0      N −3 N −2   θ¯ − − − − = λη¯ ν¯ Pk,0 χN −2−k − PN −2,0 + (ην¯ + ην) ¯ λ Pk,0 χN −1−k + PN −1,0 − λPN −1,0 1 − θ¯ η¯ ν¯ k=0 k=0   N −1 ∞   − + ην λ Pk,0 χm + PN ,0 , k=0

Pn,0

θ¯ = 1 − θ¯ η¯ ν¯  + ην λ

 λη¯ ν¯

m=N −k

 n−1  k=0

n+1 

 Pk−,0 χn−k

−

Pn−,0



n 

+ (ην¯ + ην) ¯ λ k=0 

Pk−,0 χn+2−k + Pn+2,0 − λPn−+2,0

,

 Pk−,0 χn+1−k

+ Pn+1,0 −

λPn−+1,0

1 ≤ n ≤ N − 3,

k=0

  N −1 N −1 ∞ ∞    λη¯ µ ¯  θ η¯ ν¯ − λ Pk,0 Pk−,1 P N ,1 = χm + PN ,0 + χm , 1 − η¯ µ ¯ 1 − η¯ µ ¯ k=1 m=N −k m=N −k k=0      N −2 N −1 ∞    θ − − − P N − 1 ,1 = η¯ ν¯ λ Pk,0 χN −1−k + PN −1,0 − λPN −1,0 + (ην¯ + ην) ¯ λ Pk,0 χ m + P N ,0 1 − η¯ µ ¯ m=N −k k=0 k=0      N −2 N −1 ∞    1 − − − + λη¯ µ ¯ Pk,1 χN −1−k − PN −1,1 + (ηµ ¯ + ηµ) ¯ λ Pk,1 χm + PN ,1 , 1 − η¯ µ ¯ m=N −k k=1 k=1

S. Gao et al. / Journal of the Korean Statistical Society 42 (2013) 515–528







523



N −3 N −1 ∞    θ η¯ ν¯ λ Pk−,0 χN −2−k + PN −2,0 − λPN−−2,0 + ην λ Pk−,0 χm + PN ,0 1 − η¯ µ ¯ m=N −k k=0  k=0  N −2  + (ην¯ + ην) ¯ λ Pk−,0 χN −1−k + PN −1,0 − λPN−−1,0 k=0      N −3 N −2   1 − − − − λη¯ µ ¯ Pk,1 χN −2−k − PN −2,1 + (ηµ ¯ + ηµ) ¯ λ Pk,1 χN −1−k + PN −1,1 − λPN −1,1 + 1 − η¯ µ ¯ k=1 k=1   N −1 ∞   + ηµ λ Pk−,1 χm + PN ,1 ,

P N − 2 ,1 =

m=N −k

k=1



Pn,1





   n  θ − − − − = η¯ ν¯ λ Pk,0 χn−k + Pn,0 − λPn,0 + (ην¯ + ην) ¯ λ Pk,0 χn+1−k + Pn+1,0 − λPn+1,0 1 − η¯ µ ¯ k=0 k=0   n+1  − − + ην λ Pk,0 χn+2−k + Pn+2,0 − λPn+2,0 k=0      n −1 n   1 − − − − + λη¯ µ ¯ Pk,1 χn−k − Pn,1 + (ηµ ¯ + ηµ) ¯ λ Pk,1 χn+1−k + Pn+1,1 − λPn+1,1 1 − η¯ µ ¯ k=1 k=1   n+1  Pk−,1 χn+2−k + Pn+2,1 − λPn−+2,1 , 1 ≤ i ≤ N − 3. + ηµ λ n −1 

k=1

N

N

One should note that P0,0 can be obtained using the normalizing condition i=0 Pi,0 + i=1 Pi,1 = 1. Meanwhile, we see that through computing the pre-arrival epoch probabilities which are given in Section 3.1, the arbitrary time epoch probabilities can be obtained from above equations. 3.4. Distribution of system size at outside observer’s observation epoch The distribution of system size at outside observer’s observation epoch is needed to evaluate average sojourn time in the system using Little’s rule. In an early arrival system, the outside observer’s observation point falls in a time interval after a potential arrival and before a potential departure. Let Pio,0 (0 ≤ i ≤ N ) and Pio,1 (1 ≤ i ≤ N ) denote the probabilities that outside observer sees i customers in the system and the server on working vacation and in the busy period, respectively. By observing arbitrary and outside observer’s observation epochs presented in Fig. 1, we have P0,0 = P0o,0 + µP1o,1 + ν P1o,0 ,

(3.22)

Pi,0 = θ¯ ν¯ Pio,0 + θ¯ ν Pio+1,0 ,

(3.23)

PN ,0 = θ¯ ν¯ Pi,1 = µ ¯

PNo ,0

Pio,1

P N ,1 = µ ¯

,

+µ

PNo ,1

1 ≤ i ≤ N − 1,

(3.24) Pio+1,1

+ θ ν¯

+ θ ν¯

PNo ,0

Pio,0

+ θν

Pio+1,0

,

1 ≤ i ≤ N − 1,

.

(3.25) (3.26)

From (3.22)–(3.26), we can obtain 1 P N ,0 , PNo ,0 =

θ¯ ν¯

Pio,0

=

θ¯ ν¯

(Pi,0 − θ¯ ν Pio+1,0 ),

1 ≤ i ≤ N − 1,

1

(PN ,1 − θ ν¯ PNo ,0 ), µ ¯  1  = Pi,1 − µPio+1,1 − θ ν¯ Pio,0 − θ ν Pio+1,0 , µ ¯

PNo ,1 = Pio,1

1

1 ≤ i ≤ N − 1,

P0o,0 = P0,0 − µP1o,1 − ν P1o,0 .

(3.27)

Remark 1. It should be noted here that by using normalization condition we can obtain P0o,0 = 1 −

N  (Pio,0 + Pio,1 ), i =1

and the numerical value of P0o,0 evaluated through (3.28) matches exactly with the one evaluated through (3.27).

(3.28)

524

S. Gao et al. / Journal of the Korean Statistical Society 42 (2013) 515–528 Table 1 Distributions of number of customers in the system at various epochs when inter-arrival time is geometric. n

Pn−,0

Pn−,1

Pn,0

Pn,1

Pno,0

Pno,1

0 1 2 3 4 5 6 7 8 Sum

0.0137 0.0028 0.0019 0.0010 0.0017 0.0008 0.0008 0.0020 0.0048 0.0296

0.0136 0.0240 0.0400 0.0663 0.1082 0.1764 0.2812 0.2606 0.9704

0.0137 0.0028 0.0019 0.0010 0.0017 0.0008 0.0008 0.0020 0.0048 0.0296

0.0136 0.0240 0.0400 0.0663 0.1082 0.1764 0.2812 0.2606 0.9704

0.0106 0.0036 0.0024 0.0012 0.0022 0.0010 0.0010 0.0025 0.0061 0.0305

0.0101 0.0194 0.0332 0.0551 0.0909 0.1480 0.2421 0.3706 0.9695

Ls = 6.1778, Los = 6.4690, WSA = 16.5820, PBF = 0.2654, PBA = 0.6067, PBL = 0.5785.

4. Performance measures As steady-state probabilities at various epochs are known, various performance measures can easily be computed. 4.1. Blocking probabilities An important performance measure of a finite buffer batch arrival single-server queueing system is the blocking probability of the first-, an arbitrary- and the last- positive customer of an arriving batch. Denoting the three kinds of blocking probabilities as PBF , PBA and PBL respectively, then we can determine the three kinds of probabilities from Section 3.1 as follows: PBF = PN−,0 + PN−,1 , PBA =

N 

∞ 

Pi− ,0

j =N −i

i =0

PBL =

N 

rj+1 +

∞ 

Pi− ,0

i=0

N 

Pi− ,1

j=N −i+1

N 

rj+1 ,

j =N −i

i=1

χj +

∞ 

Pi− ,1

i=1

∞ 

χj ,

j=N −i+1

where rj = P (X ≥ j)/µX is the probability that an arbitrary customer C occupies position j in its batch, j = 1, 2, . . . . 4.2. System length At an arbitrary epoch, the average system length when the server is on working vacation L0 , in the busy period L1 and the system length Ls are given by L0 =

N 

iPi,0 ,

L1 =

i=1

N 

iPi,1 ,

Ls = L0 + L1 .

i=1

At outside observer’s observation epoch, the average number of customers in the system when the server is on working vacation Lo0 , in busy period Lo1 and the average number of customers in the system Los are respectively given by Lo0 =

N  i=1

iPio,0 ,

Lo1 =

N 

iPio,1 ,

Los = Lo0 + Lo1 .

i=1

Let WSA denote the average sojourn time in the system of an arbitrary positive customer in a batch which is accepted upon arrival. Then by Little’s rule WSA = Los /λ′ , where λ′ = λµX (1 − PBA ). 4.3. Numerical examples of operating characteristics To show the applicability of the formulae obtained in the previous section, in this subsection, we present some numerical examples in the form of tables and graphs. We have used Matlab software for computation purpose. In Tables 1 and 2, with the same system parameters N = 8, η = 0.1, µ = 0.3, θ = 0.02, ν = 0.01, and batch size distribution χ1 = 0.4, χ2 = 0.2, χ4 = 0.2, χ7 = 0.1, χ8 = 0.1, we give the system length distributions at pre-arrival, arbitrary and outside observer’s observation epochs for two different inter-arrival times of batches of positive customers.

S. Gao et al. / Journal of the Korean Statistical Society 42 (2013) 515–528

525

Table 2 Distributions of number of customers in the system at various epochs when inter-arrival time is PHd (τ , T ). n

Pn−,0

Pn−,1

Pn,0

Pn,1

Pno,0

Pno,1

0 1 2 3 4 5 6 7 8 Sum

0.0974 0.0161 0.0101 0.0048 0.0090 0.0035 0.0037 0.0089 0.0172 0.1706

0.0467 0.0590 0.0720 0.0896 0.1078 0.1328 0.1726 0.1490 0.8294

0.1081 0.0147 0.0092 0.0045 0.0082 0.0032 0.0036 0.0084 0.0154 0.1753

0.0501 0.0623 0.0757 0.0935 0.1118 0.1352 0.1644 0.1316 0.8247

0.0951 0.0185 0.0115 0.0056 0.0103 0.0041 0.0044 0.0105 0.0195 0.1794

0.0423 0.0558 0.0697 0.0859 0.1046 0.1260 0.1537 0.1825 0.8206

Ls = 4.6496, Los = 4.8969, WSA = 14.4835, PBF = 0.1661, PBA = 0.4380, PBL = 0.4293.

Fig. 2. Effect of N on Los .

Also, various performance measures such as the blocking probabilities, three kinds of average system length and average sojourn time in the system are given at the bottom of the tables. In Tables 1 and 2, the inter-arrival times of batches of positive customers are respectively geometric distribution with λ = 0.32 and discrete phase-type distribution PHd (τ , T ), where 0.2 0.5 0.3

 τ = [0.3 0.1 0.6],

T =

0.4 0.4 0.1

0.2 0.0 . 0.3



It may be remarked that since all the results reported here were rounded to four decimal places, the sum of the probabilities may not add to one in some cases. It may be observed from Tables 1 and 2 that in case of geometric arrivals the pre-arrival and arbitrary epoch probabilities are same due to Bernoulli arrivals. In Fig. 2, we show the effect of capacity size (N) on the average system length (Los ) at outside observer’s observation epoch for various inter-batch time distributions (Geo(0.25), arbitrary distribution with a2 = 0.625, a5 = 0.2, a10 = 0.175 and deterministic distribution with a4 = 1) with same mean. The parameters are taken as λ = 0.25, µ = 0.3, ν = 0.01, η = 0.1, θ = 0.2, χ1 = 0.2, χ2 = 0.3, χ3 = 0.3 and χ4 = 0.2. It is observed that Los in case of deterministic distribution is higher as compared to geometric and arbitrary distributions, and it linearly increases with N. Further, for arbitrary and geometric distribution the difference is not significant. In Fig. 3, we compare the effects of buffer size N on the blocking probability of an arbitrary positive customer of a batch for various inter-batch time distributions (Geo(0.125), arbitrary distribution with a5 = 0.4, a8 = 0.5, a20 = 0.1 and deterministic distribution with a8 = 1) with same mean. µ = 0.3, ν = 0.01, η = 0.1, θ = 0.2, χ1 = 0.2, χ2 = 0.3, χ3 = 0.3 and χ4 = 0.2. As our expectation, due to the fact that the model becomes an infinite buffer queue, the blocking probabilities monotonically decrease as buffer size increases and finally reach the minimum value zero. It can be seen that deterministic distribution yields the lowest blocking probability and geometric distribution leads to the highest. In Fig. 4, WSA is plotted against the parameter N for different values of η = 0, 0.1, 0.15, 0.3. The inter-batch time distribution is assumed to be geometric with λ = 0.33, µ = 0.3, ν = 0.01, θ = 0.2, χ1 = 0.2, χ2 = 0.3, χ3 = 0.3 and χ4 = 0.2. From Fig. 4, we can observe that WSA increases evidently with the buffer size N, and Fig. 4 also shows that WSA is larger in case of small value of arrival rate η of negative customer which agrees with the intuitive expectations.

526

S. Gao et al. / Journal of the Korean Statistical Society 42 (2013) 515–528

Fig. 3. Effect of N on PBA .

Fig. 4. The effect of N on WSA for different values of η.

Fig. 5. Effect of θ on WSA for different values of ν in GeoX /Geo/1/10-G queue.

With the same parameters µ = 0.3, η = 0.2, χ1 = 0.2, χ2 = 0.3, χ3 = 0.3 and χ4 = 0.2, Figs. 5 and 6 compare the effect of the parameter θ on the average sojourn time in the system WSA for various values of ν = 0, 0.15, 0.2, 0.3 respectively for GeoX /Geo/1/10-G queue with λ = 0.125 and PHX /Geo/1/16-G queue whose inter-arrival time distribution is PHd (τ , T ) with 0.2 0.5 0.3

 τ = [0.3 0.3 0.4],

T =

0.4 0.4 0.1

0.2 0.0 . 0.3



S. Gao et al. / Journal of the Korean Statistical Society 42 (2013) 515–528

527

Fig. 6. Effect of θ on WSA for different values of ν in PHX /Geo/1/16-G queue.

Obviously, the average sojourn time WSA is decreasing as function of θ , that is, the bigger the probability θ , the shorter the vacation time, and then the chance that the customer is served by the normal service rate is increased which leads to the decrease of WSA . As expected, WSA decrease with increasing values of ν , which also agrees with the intuitive. We should note that when θ approaches to 1, WSA will achieve a fixed value, i.e., the average sojourn time without vacation and the fixed value is corresponding to the case that ν = 0.3(=µ), because if ν = µ, the server’s service rate during vacation is equal to the normal service, then the model is changed to GI X /Geo/1/N queue with negative customers without vacation, in this case, the value changes of θ has no effect on WSA . Remark 2 (Some Special Cases). (1) If η = 0, that is, there is no negative customer arrival, our model is reduced to the GI X /Geo/1/N queue with multiple working vacation. More specially, when η = 0, χ1 = 1, the model is changed to single arrival GI /Geo/1/N queue with multiple working vacation, which was discussed by that of Goswami and Mund (2010) and in this case, the expressions (n) (n) of Bk and Vk are, respectively, as follows: (n)

=

Bk

(n)

Vk

=

n kn 

µk µ ¯ n−k ,

0 ≤ k ≤ n, n ≥ 1,

ν k ν¯ n−k ,

0 ≤ k ≤ n, n ≥ 1,

k

(0)

(n)

= 0, k > n, and V0(0) = 1, Vk(n) = 0, k > n, the expressions of αk , βk , γk are, respectively, as follows ∞ ∞ n   (n) an µk µ ¯ n−k , k ≥ 0, αk = an Bk =

where B0 = 1, Bk

βk =

n=1

n=max{1,k}

∞ 

∞ 

(n)

an θ¯ n Vk

∞ 

an

n =1

=

an θ¯ n

n

n=max{1,k}

n =1

γk =

=

k

n 

θ θ¯ m−1

m=1

∞ 

an

n=max{1,k}

k 

k

ν k ν¯ n−k ,

k ≥ 0,

(n−m)

Vr(m) Bk−r

r =0 n 

min{m,k}



θ θ¯ m−1

m=1

r =max{0,k−n+m}

m r

ν ν¯ r

m−r



n−m k−r



µk−r µ ¯ n−m−k+r ,

k ≥ 0.

With the above notations, we can prove that the TPM P and other results in this paper are coincident with those given by Goswami and Mund (2010). But itshould  be noted that the second fj on p. 373 in Goswami and Mund (2010) should be replaced by gj =

∞

n=max{1,j}

an θ¯ n

n j

ηk η¯ n−j , k ≥ 0, where η is corresponding to ν in our paper.

(2) If ν = 0, that is the server does not provide service during vacation period, the model is changed to GI X /Geo/1/N queue with negative customers and multiple vacations. (3) If ν = µ, that is, the server’s service rate during vacation is equal to the normal service, then the model is changed to GI X /Geo/1/N queue with negative customers without vacation which is also corresponding to the case θ = 0. 5. Conclusion In this paper, we consider a discrete-time GI X /Geo/1/N queue with negative customer and multiple working vacations that have potential applications in modeling computer and telecommunication systems, computer networks, etc. We have

528

S. Gao et al. / Journal of the Korean Statistical Society 42 (2013) 515–528

obtained the queue-length distributions at pre-arrival, arbitrary and outside observer’s observation epochs for partial-batch rejection policy. The system is analyzed under the assumptions of early arrival system. Various performance measures such as the blocking probability of the first-, an arbitrary- and the last-positive customer in a batch, average system length at outside observer’s observation epoch and average sojourn time in the system have been carried out. The techniques used in this paper can be applied to analyze other similar models such as GI X /Geo/1/N − G queues with single working vacation or other working vacation policies, which are left for future investigations. Acknowledgments The authors thank the editor and the referees for their valuable comments and acknowledge that this research was supported by the National Natural Science Foundation of China (Nos 11171019, 11171179, and 11201123), Program for New Century Excellent Talents in University (No. NCET-11-0568), the Fundamental Research Funds for the Central Universities (No. 2011JBZ012), the Tianyuan Fund for Mathematics (Nos 11226200, and 11226251) and Natural Science Foundation of Education Bureau of Anhui Province (No. KJ2013Z254). References Artalejo, J. R. (2000). G-networks: a versatile approach for work removal in queueing networks. European Journal of Operational Research, 126, 233–249. Atencia, I., & Moreno, P. (2004a). The discrete-time Geo/Geo/1 queue with negative customers and disasters. Computers & Operations Research, 31, 1537–1548. Atencia, I., & Moreno, P. (2004b). A discrete-time Geo/G/1 retrial queue with general retrial time. Queueing Systems, 48, 5–21. Atencia, I., & Moreno, P. (2005). A single-server G-queue in discrete-time with geometrical arrival and service process. Performance Evaluation, 59, 85–97. Baba, Y. (2005). Analysis of a GI /M /1 queue with multiple working vacations. Operations Research Letters, 33, 201–209. Banik, A. D., Gupta, U. C., & Pathak, S. S. (2007). On the GI /M /1/N queue with multiple working vacations—analytic analysis and computation. Applied Mathematical Modelling, 31, 1701–1710. Boucherie, R. J., & Boxma, O. J. (1996). The workload in the M /G/1 queue with work removal. Probability in the Engineering and Informational Sciences, 10(2), 261–277. Bruneel, H., & Kim, B. (1993). Discrete-time models for communication systems including ATM. Boston: Kluwer Academic Publishers. Chakka, R., & Harrison, P. G. (2001). A Markov modulated multi-server queue with negative customers—the MMCPP /GE /c /LG-queue. Acta Informatica, 37, 881–919. Chao, X., Miyazawa, M., & Pinedo, M. (1999). Queueing networks: customers, signals and product form solutions. Chichester: Wiley. Do, T. V. (2011). Bibliography on G-networks, negative customers and applications. Mathematical and Computer Modelling, 53, 205–212. Gao, S., & Liu, Z. M. (2013). An M /G/1 queue with single working vacation and vacation interruption under Bernoulli schedule. Applied Mathematical Modelling, 37, 1564–1579. Gelenbe, E. (1989). Random neural networks with negative and positive signals and product form solution. Neural Computation, 1, 502–510. Goswami, V., & Mund, G. B. (2010). Analysis of a discrete-time GI /GEO/1/N queue with multiple working vacations. Journal of Systems Science and Systems Engineering, 19(3), 367–384. Goswami, V., & Vijaya Laxmi, P. (2010). Analysis of renewal input bulk arrival queue with single working vacation and partial batch rejection. Journal of Industrial and Management Optimization, 6(4), 911–927. Harrison, P. G., Patel, N. M., & Pitel, E. (2000). Reliability modelling using G-queues. European Journal of Operational Research, 126, 273–287. Harrison, P. G., & Pitel, E. (1993). Sojourn times in single-server queue with negative customers. Journal of Applied Probability, 30, 943–963. Harrison, P. G., & Pitel, E. (1995). Response time distributions in tandem G-networks. Journal of Applied Probability, 32, 224–247. Harrison, P. G., & Pitel, E. (1996). The M /G/1 queue with negative customers. Journal of Applied Probability, 28, 540–566. Kim, J. D., Choi, D. W., & Chae, K. C. (2003). Analysis of queue-length distribution of the M /G/1 queue with working vacations. In International conference on statistics and related fields, Hawaii. Li, J., Liu, W., & Tian, N. (2010). Steady-state analysis of a discrete-time batch arrival queue with working vacations. Performance Evaluation, 67, 897–912. Li, J., & Tian, N. (2007). The discrete-time GI /Geo/1 queue with working vacations and vacation interruption. Applied Mathematics and Computation, 185, 1–10. Li, J., Tian, N., & Liu, W. (2007). Discrete-time GI /Geo/1 queue with working vacations. Queueing Systems, 56, 53–63. Li, J., Tian, N., & Ma, Z. (2008). Performance analysis of GI /M /1 queue with working vacations and vacation interruption. Applied Mathematical Modelling, 32, 2715–2730. Li, J., Tian, N., Zhang, Z. G., & Luh, H. (2009). Analysis of the M /G/1 queue with exponentially working vacations—a matrix analytic approach. Queueing Systems, 61, 139–166. Li, Q. L., & Zhao, Y. Q. (2004). A MAP /G/1 queue with negative customers. Queueing Systems, 47, 5–43. Liu, W., Xu, X., & Tian, N. (2007). Stochastic decompositions in the M /M /1 queue with working vacations. Operations Research Letters, 35, 595–600. Servi, L. D., & Finn, S. G. (2002). M /M /1 queue with working vacations (M /M /1/WV ). Performance Evaluation, 50, 41–52. Wang, J. T., & Zhang, P. (2009). A discrete-time retrial queue with negative customers and unreliable server. Computers & Industrial Engineering, 56, 1216–1222. Wu, J. B., Liu, Z. M., & Peng, Y. (2009). On the BMAP /G/1G-queues with second optional service and multiple vacations. Applied Mathematical Modelling, 33(12), 4314–4325. Wu, D., & Takagi, H. (2006). M /G/1 queue with multiple working vacations. Performance Evaluation, 63, 654–681. Yi, X. W., Kim, J. D., Choi, D. W., & Chae, K. C. (2007). The Geo/G/1 queue with disasters and multiple working vacations. Stochastic Models, 23, 537–549. Yu, M. M., Tang, Y. H., & Fu, Y. H. (2009). Steady state analysis and computation of the GI [x] /M b /1/L queue with multiple working vacations and partial batch rejection. Computers & Industrial Engineering, 56, 1243–1253. Yu, M. M., Tang, Y. H., Fu, Y. H., & Pan, L. M. (2011). GI /Geom/1/N /MWV queue with changeover time and searching for the optimum service rate in working vacation period. Journal of Computational and Applied Mathematics, 235, 2170–2184.