JOURNAL OF INDUSTRIAL AND MANAGEMENT OPTIMIZATION Volume 11, Number 3, July 2015
doi:10.3934/jimo.2015.11.779 pp. 779–806
ON A DISCRETE-TIME GIX /GEO/1/N-G QUEUE WITH RANDOMIZED WORKING VACATIONS AND AT MOST J VACATIONS
Shan Gaoa,b and Jinting Wanga1 a Department
of Mathematics, Beijing Jiaotong University Beijing, 100044, China b Department of Mathematics, Fuyang Normal College Fuyang, Anhui 236037, China
Abstract. This paper considers a discrete-time GIX /Geo/1/N-G queue with randomized working vacations, where upon arrival, a negative customer removes one positive (ordinary) customer in service if any is present and disappears immediately; otherwise, it has no effect on the system if the system is empty. As soon as the system becomes empty, the server immediately takes a working vacation. If there are no customers in the system at the end of the working vacation, the server takes another working vacation with probability p or remains dormant in the system with probability 1 − p. Otherwise, the server starts to serve the customers with the normal service rate immediately if there are some customers at the end of a working vacation. This pattern does not terminate until the server has taken J successive working vacations. Steady-state system length distributions at various epochs such as, pre-arrival, arbitrary and outside observer’s observation epochs have been obtained. Based on the various system length distributions, we also give some important performance measures including blocking probabilities, mean queue length, probability mass function of waiting time and other performance measures along with some numerical examples. Then, we use the parabolic method to search the optimum value of the normal service rate under a established cost function.
1. Introduction. Pioneered by Gelenbe [13, 14], queues with negative customers called G-queues have been studied extensively due to their extensively applications, such as computer, communication networks and manufacturing systems etc. A negative customer immediately removes one or more positive customers if present and has no effect if the system is empty. If the negative customer deletes all present positive customers, it is called a disaster. In literature, the studies on G-queues can be mainly divided into two lines, one is that the negative customer only eliminates positive customers and has no other effect on the server, the other is that the negative customer not only eliminates positive customers but also causes the server breakdown. Readers may refer to Atencia and Moreno [1], Chae et al. [7], Chakka and Harrison [8], Dimitriou [9], Harrison et al. [16], Li and Zhao [18], Wang et al. [22], Wang et al. [23], Wang et al. [24], Wu et al. [26], among others. More 2010 Mathematics Subject Classification. Primary: 60K25; Secondary: 90B22, 68M20. Key words and phrases. Discrete time queues, performance evaluation, randomized vacations, working vacations. The reviewing process of the paper was handled by Wuyi Yue and Yutaka Takahashi as Guest Editors. 1 Corresponding author. Fax: +86-10-51840433.
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work on negative customers can be found in Do [10] where the author presented a bibliography on G-networks from 1989 to 2010. Recently, queues with working vacations have attracted considerable interests and have been widely used in the performance analysis of communication systems. The characteristic feature of working vacations is that the server renders service to the customers with a relatively lower service rate during vacation period rather than completely stop serving the customers, which can economize operating cost and energy consumption. The studies on queues with working vacations can go back to Servi and Finn [21], where an M/M/1 queue with multiple working vacations was studied. Later Liu et al. [19] obtained the stochastic decomposition structures of the system indices in the M/M/1 queue with working vacations. Subsequently, queues with various working policies were studied, for example, Baba [2] studied a GI/M/1 queue with multiple working vacations, in which the vacation times and service times are independent and exponential distributions. Wu and Takagi [25] analyzed multiple working vacations for M/G/1 queue. Li et al. [17] studied the GI/Geo/1 queue with working vacations and vacation interruption, Chae et al. [6] introduced the single working policy into the GI/M/1 queue and GI/Geo/1 queue. Gao and Liu [11] analyzed an M/G/1 queue with single working vacation and vacation interruption under Bernoulli schedule. Compared to many studies of the infinite buffer queue with working vacation, however, studies on finite buffer size queue with working vacations are comparatively few, see Goswami and Mund [15], Yu et al. [27], Yu et al. [28]. Recently, Zhang and Hou [29] presented a GI/M/1/N queue with working vacations, in which the server will take another working vacation if there are no customers in the system upon returning from a vacation and is permitted to take consecutively at most H working vacations (called variant working vacations). Gao et al. [12] presented a discrete-time GIX /Geo/1/N-G queue with multiple working vacations, where the authors obtained steady-state system length distributions at pre-arrival, arbitrary and outside observer’s observation epochs by using embedded Markov Chain and supplementary variable method. The purpose of this work is to extend the works [12] and [29] to the discrete-time GIX /Geo/1/N/RWVS(J)G queue, where RWVS(J) presents randomized working vacations and at most J vacations. The rest of this paper is organized as follows. In Section 2, we give the detailed queueing model and a practical application of the model. In Section 3, we analyze the model and obtain steady state distributions at arbitrary, pre-arrival and outside observer’s observation epochs. In Section 4, various performance measures such as the blocking probabilities and the distribution of the waiting time of an arbitrary tagged positive customer, etc. are obtained. Further, some numerical results in the form of tables and graphs are presented in Section 5. In Section 6, the expected cost function per unit time is established to determine the optimal normal service rate µ∗ at the minimum cost and a numerical example is given to explain the parabolic method to find out the value of µ∗ . 2. Model descriptions. 2.1. Assumptions of the model. Throughout this paper, we denote x ¯ = 1−x for any real number x ∈ (0, 1). We consider a discrete-time GI X /Geo/1/N queueing system with negative customers and randomized working vacations, where N is the capacity of the system including the one who is in service. Throughout this paper, we adopt the EAS
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system. A potential (positive or negative) arrival occurs in (n, n+ ), and we assume that the location of a potential negative arrival is immediately after a potential positive arrival. A potential departure (due to the completion of a service) can only take place in (n− , n). Positive customers arrive in batch of random size X with probability mass function (p.m.f.) P (X = j) = χj , j = 1, 2, · · · , and mean E[X] = µX . The interarrival times {An , n ≥ 1} of two successive batches are independently identically distributed (i.i.d.) random variables and have the distribution P (An = i) = ai , i ≥ 1, E[An ] = λ
−1
, A(z) =
∞ X
ai z i .
i=1
Since the buffer size N is finite, if a batch upon arrival doesn’t find enough space in the buffer, then a part of positive customers fills the vacant spaces and the rest is rejected. This is known as partial batch rejection. Positive customers are served according to a first-come-first-served (FCFS) discipline. The normal service time Sb in a regular busy period and the service time Sv in a working vacation period are geometrically distributed, respectively, with rate µ and ν, 0 ≤ ν ≤ µ < 1. Inter-arrival times {Bn , n ≥ 1} of negative customers are independent and geometrically distributed with rate η, 0 ≤ η < 1. Upon arrival, a negative customer only eliminates the positive customer who is either being served or about to be served and has no other effect on the system. However, when the system is empty upon its arrival, the negative customer will vanish and has no impact on the system. The server begins a working vacation at the epoch when a positive customer is served completely by the normal service rate µ and the queue becomes empty, the vacation time V is assumed to be geometrically distributed with rate θ, 0 < θ < 1. Suppose the beginning and ending of vacation occur at the instant n. Customers arrived during the vacation will be served at the rate of ν in the order of arriving at the system. Upon completion of a vacation, if the server finds the system is empty, he will take another working vacation with probability p (0 ≤ p ≤ 1) or remain dormant within the system with probability 1 − p. Otherwise, the server switches service rate from ν to µ and a regular busy period starts. Thus, a regular busy period is the duration in which the server works at a rate of µ continuously. This pattern does not terminate until the server has taken J successive working vacations. If the system remains empty at the end of the Jth vacation, the server keeps idle in the system until a next arrival occurs. The state of the system at time n is described by the following random variables: • N (n) = the 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. We take J(n) = 0 when the server is in normal service period or J(n) = j when the server is on the j-th working vacation period, j = 1, 2, · · · , J. Further define the joint probabilities as Pk,j (u, n) = P (N (n) = k, J(n) = j, U (n) = u), 0 ≤ k ≤ N ; j = 0, 1, 2, · · · , J; u ≥ 0. In steady-state, the above probabilities are denoted as Pk,j (u), 0 ≤ k ≤ N ; j = 0, 1, · · · , J; u ≥ 0.
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Remark 1. In comparison with Refs. [12] and [27], our model has new characteristics as follows: We adopt the randomized working vacations and at most J vacations in the paper, which is more general than those in Refs. [12] and [27]. More specifically, if we let p = 1 and J → ∞, our queueing model in this paper becomes that in [12]. If p = 1, η = 0, and P (X = 1) = 1, then our model is the discrete-time counterpart corresponding to [27]. 2.2. Practical application of the model. Such a modified working vacation discipline has potential applications in the area of computer processing, production and inventory control system. Take a computer processing system for example. In a computer processing system, the buffer size used to store messages is finite and the messages (positive customers) arrive in batch, the processor (server) is in charge of processing messages, where the system is subjective to the perturbation of the arrival of viruses (negative customers). A virus just infects one or more messages and the infected messages may not be recoverable. To avoid transmitting the infected message to other processors, the infected message should generally be deleted. If the processor is available indicating that it is not currently working on a task and then a message is processed. The messages are temporarily stored in a buffer to be served some time later according to FCFS if the processor is unavailable. If a batch of messages upon arrival doesn’t find enough space in the buffer, then a part of messages are admitted to fill the vacant spaces and the rest is blocked and loss. To enhance the computer performance, whenever all messages are processed and no new messages arrive, the processor will perform a sequence of finite essential or optional maintenance jobs, such as virus scan, where the first maintenance is essential task (referred to an essential vacation) and the subsequent maintenances are referred optional vacations. Meanwhile, during the maintenance period, the processor can deal with the messages at the slower rate to economize the cost (working vacation period). Upon completion of each maintenance, the processor checks the messages and decides whether or not to resume the normal service rate. If at this moment no message is in the system, a decision may be made for another maintenance. This modified working vacation discipline is a good approximation of such computer processing system. 3. Analysis of the Model. In this section, we carry out the analysis of discretetime GI X /Geo/1/N/RW V S(J) with negative customers and partial batch rejection. 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 system, and Jn− indicates whether the server is in normal service period (Jn− = 0) or on j-th working vacation (Jn− = j, j = 1, 2, · · · , J) 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).
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Then {(Nn− , Jn− ), n ≥ 0} is an embedded two-dimensional Markov chain with the state space {(k, j), 0 ≤ k ≤ N ; j = 0, 1, 2, · · · , J}. In limiting case, define − Pk,j = lim P (Nn− = k, Jn− = j), 0 ≤ k ≤ N ; j = 0, 1, 2, · · · , J, n→∞
− where Pk,j represents the probability that there are k customers in the system 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 later use: (n)
(n−1)
B0
=¯ ηµ ¯B0
(n) B1
(n−1) =¯ ηµ ¯B1
(n) Bk
(n−1) =¯ ηµ ¯Bk
(0)
where B0
, n ≥ 1, (n−1)
+ (¯ ηµ + ηµ ¯)B0 + (¯ ηµ +
(0)
(n−1) ηµ ¯)Bk−1
(n)
= 1; Bk = 0, k ≥ 1; Bk (n)
=¯ η ν¯V0
V1
(n)
=¯ η ν¯V1
(n) Vk
(n−1) =¯ η ν¯Vk
V0
(n−1) (n−1)
(0)
, n ≥ 1, (n−1)
+ ηµBk−2 , n ≥ 1, k ≥ 2,
= 0, k > 2n.
, n ≥ 1, (n−1)
+ (¯ η ν + η¯ ν )V0 + (¯ ην +
(0)
(n−1) η¯ ν )Vk−1
, n ≥ 1, (n−1)
+ ηνVk−2 , n ≥ 1, k ≥ 2,
(n)
where V0 = 1; Vk = 0, k ≥ 1; Vk = 0, k > 2n. (n) (n) Then we can see that Bk and Vk are, respectively, the probabilities that there are k departures of positive customers from the system during n slots in a regular busy period and a working vacation period either by service completions or by negative arrivals. Next, let αk =
∞ X
(n)
an Bk , k ≥ 0,
n=1
βk =
∞ X
(n) an θ¯n Vk , k ≥ 0,
n=1
γk =
∞ X
n X
an
n=1
θθ¯m−1
m=1
0 α1,k = pk−1
∞ X
(n−m)
Vr(m) Bk−r
, k ≥ 0,
r=0
an
n−k+1 X j=1
n=k 0 αm,k = pk−1
k X
∞ X
n − j k−1 ¯n−j−(k−1) η¯µ θ θ , 1 ≤ k ≤ J, k−1 " n−k+1 X (j−1) (j−1) an Bm−1 η¯µ + Bm−2 ηµ (j−1)
B0
n=d m 2 e+k−1
j=d m 2 e
# n − j k−1 ¯n−j−(k−1) × θ θ , k−1 m ≥ 2, 1 ≤ k ≤ J,
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SHAN GAO AND JINTING WANG
0
β1 = 0
βk =
∞ X
an θ¯n
n=1 ∞ X
n X
(j−1)
V0
(¯ η ν + η),
j=1 n X
an θ¯n
n=d k 2e
(j−1) (j−1) Vk−1 (¯ η ν + η) + Vk−2 ην , k ≥ 2,
j=d k 2e ∞ X
0 γ1,k = pk−1
an
n=k+1
n−k X
(j)
θθ¯j−1 V0
j=1
n−k+1 X
(m−j−1)
B0
m=j+1
n − m k−1 ¯n−m−(k−1) η¯µ θ θ , k−1
1 ≤ k ≤ J, 0 γm,k
∞ X
k−1
=p
an
n−k X
¯j−1
θθ
j=1
n=k+1
n−k+1 X
" m−2 X
u=j+1
r=0
(u−j−1) (u−j−1) Vr(j) Bm−r−1 η¯µ + Bm−r−2 ηµ
#
Γ1,k
n − u k−1 ¯n−u−(k−1) (j) (u−j−1) + Vm−1 B0 η¯µ θ θ , m ≥ 2, 1 ≤ k ≤ J, k−1 j ∞ n−k+1 X X X n − j k−1 ¯n−j−(k−1) (h−1) = pk an θθ¯j−1 V0 (¯ η ν + η) θ θ , k−1 j=1 n=k
h=1
1 ≤ k ≤ J − 1, ∞ X
Γm,k = pk
an
n=d m 2 e+k−1
×
n−k+1 X j=d m 2 e
θθ¯j−1
j X
(h−1)
(h−1)
Vm−1 (¯ η ν + η) + Vm−2 ην
h=d m 2 e
n − j k−1 ¯n−j−(k−1) θ θ , m ≥ 2, 1 ≤ k ≤ J − 1, k−1
where αk (k ≥ 0) is the probability that during a batch of positive customers’ interarrival time, k positive customers depart the system in the regular busy service period; βk (k ≥ 0) represents the probability that during a batch of positive customers’ inter-arrival time, k positive customers depart the system in a working vacation period and the residual vacation doesn’t ends; γk (k ≥ 0) denotes the probability that k positive customers depart the system during a batch of positive customers’ interarrival time which is not less than the residual vacation time and the server enters 0 normal service period; αm,k (m ≥ 1, 1 ≤ k ≤ J) denotes the probability that during the inter-arrival time of a batch of positive customers, m positive customers depart the system in the normal service period and after the m-th positive departure (by the normal service completion) the system become empty, then server consecutively enters working vacation period till the k-th vacation and the k-th doesn’t end during the remaining inter-arrival time; βk0 (k ≥ 1) denotes the probability that the residual vacation time is longer than the inter-arrival time and k positive customers depart the system during the vacation period which leads the system become empty; 0 γm,k (m ≥ 1, 1 ≤ k ≤ J) denotes the probability that the residual vacation time is shorter than the inter-arrival time and m positive customers depart the system– there are r(0 ≤ r ≤ m − 1) positive departures during the working vacation and m − r positive departures during the following normal service period, and the m-th positive customer is served completely by the normal service which leads to the system become empty and then server consecutively enters working vacation period till the k-th vacation and the k-th doesn’t end during the remaining inter-arrival time; Γ0m,k (m ≥ 1, 1 ≤ k ≤ J) denotes the probability that the residual vacation
ON A DISCRETE-TIME GIX /GEO/1/N-G QUEUE
785
time is shorter than the inter-arrival time and m positive customers depart the system during the residual working time and the system becomes empty after the m-th departure, and the server consecutively enters the following k vacations and the k-th vacation doesn’t end during the remaining inter-arrival time. Now we develop the transition probabilities of the two-dimensional Markov chain {(Nn− , Jn− ), n ≥ 0}. For 0 ≤ i, j ≤ J, 0 ≤ k, m ≤ N, define − − (Pi,j )m,n = P {Nn+1 = n, Jn+1 = j|Nn− = m, Jn− = i}.
By observing the state of the system at two consecutive embedded points, we have the one step transition probability matrix (TPM) P with (J + 1)2 block matrices as follows: P0,0 P0,1 P0,2 · · · P0,J P1,0 P1,1 P1,2 · · · P1,J P = P2,0 P2,1 P2,2 · · · P2,J , .. .. .. . .. .. . . . . PJ,0
PJ,1
PJ,2
···
PJ,J
where the blocks Pi,j , i, j = 0, 1, 2, · · · , J are all matrices of order (N + 1) × (N + 1) denoting the transition probabilities from state i to the state j. For j = 1, 2, · · · , J, the structure of P0,j is given by (P0,j )m,n PN −m P∞ 0 0 k=1 χk αm+k,j + αN,j k=N −m+1 χk , = 0, (P0,j )N −1,n ,
0 ≤ m ≤ N − 1, n = 0, 0 ≤ m ≤ N − 1, 1 ≤ n ≤ N, m = N, 0 ≤ n ≤ N.
and then (P0,0 )m,n PN −m k=max{1,n−m} χk αm+k−nP ∞ +αN −n k=N −m+1 χk , 0 ≤ m ≤ N − 1, 1 ≤ n ≤ N, = PN PJ 0 ≤ m ≤ N − 1, n = 0, 1 − n=1 (P0,0 )m,n − j=1 (P0,j )m,0 , (P0,0 )N −1,n , m = N, 0 ≤ n ≤ N. For i = 1, 2, · · · , J, (Pi,i )m,n PN −m k βm+k−n k=max{1,n−m} χP ∞ +β N −n k=N −m+1 χk , PN −m 0 0 = ) k=1 χk (βm+k + γm+k,i P∞ 0 0 +(β + γ ) N N,i k=N −m+1 χk , (Pi,i )N −1,n ,
0 ≤ m ≤ N − 1, 1 ≤ n ≤ N, 0 ≤ m ≤ N − 1, n = 0, m = N, 0 ≤ n ≤ N.
For 1 ≤ j < i ≤ J, (Pi,j )m,n PN −m P∞ 0 0 k=1 χk γm+k,j + γN,j k=N −m+1 χk , = 0, (Pi,j )N −1,n ,
0 ≤ m ≤ N − 1, n = 0, 0 ≤ m ≤ N − 1, 1 ≤ n ≤ N, m = N, 0 ≤ n ≤ N.
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SHAN GAO AND JINTING WANG
For 1 ≤ i < j ≤ J, (Pi,j )m,n PN −m 0 ) k=1 χk (Γm+k,j−i + γm+k,j P∞ 0 +(ΓN,j−i + γN,j ) k=N −m+1 χk , = 0, (Pi,j )N −1,n ,
0 ≤ m ≤ N − 1, n = 0, 0 ≤ m ≤ N − 1, 1 ≤ n ≤ N, m = N, 0 ≤ n ≤ N.
For i = 1, 2, · · · , J, (Pi,0 )m,n PN −m k γm+k−n k=max{1,n−m} χ P ∞ +γN −n k=N −m+1 χk , = PN PJ PN 1 − k=1 (Pi,0 )m,k − j=1 k=0 (Pi,j )m,k , (Pi,j )N −1,n ,
0 ≤ m ≤ N − 1, 1 ≤ n ≤ N, 0 ≤ m ≤ N − 1, n = 0, m = N, 0 ≤ n ≤ N.
Let us further define the row vectors of order 1 × (N + 1) as follows: P− j = − − − − − − (P0,j , P1,j , · · · , PN,j ), j = 0, 1, 2, · · · , J and let P− = (P− , P , P , · · · , P ). The 0 1 2 J − unknown probability vector P , which is the pre-arrival epoch probabilities, can be obtained by solving the system of equations: P− P = P− and the normalization condition P− e = 1, where e is a 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 t and (t+1), and using probability argument, we have the following difference equations in steady-state, for u ≥ 1: P0,0 (u − 1) = P0,0 (u) + ηP1,0 (u) + au χ1 ηP0,0 (0) J−1 X + θ p¯ P0,j (u) + P0,J (u) + θau (χ1 (η + η¯ν) + χ2 ην) j=1
J−1 J−1 X X × p¯ P0,j (0) + P0,J (0) + θ(η + η¯ν) p¯ P1,j (u) + P1,J (u) j=1
j=1
J−1 J−1 X X + θην au χ1 p¯ P1,j (0) + P1,J (0) + p¯ P2,j (u) + P2,J (u) . (1) j=1
j=1
For convenience, let us define Qk,J (u) =
J X
Pk,j (u), 0 ≤ k ≤ N, u ≥ 0.
j=1
Then for u ≥ 1, we have (
"
Pn,0 (u − 1) = θ η¯ν¯ au
n−1 X k=0
# Qk,J (0)χn−k + Qn,J (u)
ON A DISCRETE-TIME GIX /GEO/1/N-G QUEUE
"
n X
+ (η¯ ν + η¯ν) au
787
# Qk,J (0)χn+1−k + Qn+1,J (u)
k=0
" +ην au
#)
n+1 X
Qk,J (0)χn+2−k + Qn+2,J (u)
k=0
"
#
n−1 X
+ η¯µ ¯ au
Pk,0 (0)χn−k + Pn,0 (u)
k=0
"
#
n X
+ (η µ ¯ + η¯µ) au
Pk,0 (0)χn+1−k + Pn+1,0 (u)
k=0
"
#
n+1 X
+ ηµ au
Pk,0 (0)χn+2−k + Pn+2,0 (u) , 1 ≤ n ≤ N − 3,
(2)
k=0
( PN −2,0 (u − 1) = θ
N −3 X
" η¯ν¯ au
# Qk,J (0)χN −2−k + QN −2,J (u)
k=0
" +(η¯ ν + η¯ν) au
N −2 X
# QPk,J (0)χN −1−k + QN −1,J (u)
k=0
"
N X
+ην au
∞ X
Qk,J (0)
+ η¯µ ¯ au
χm + QN,J (u)
m=N −k
k=0
"
#)
N −3 X
# Pk,0 (0)χN −2−k + PN −2,0 (u)
k=0
" + (η µ ¯ + η¯µ) au
N −2 X
# Pk,0 (0)χN −1−k + PN −1,0 (u)
k=0
" + ηµ au
N X
Pk,0 (0)
PN −1,0 (u − 1) = θ
" η¯ν¯ au
# χm + PN,0 (u) ,
(3)
m=N −k
k=0
(
∞ X
N −2 X
# Qk,J (0)χN −1−k + QN −1,J (u)
k=0
" +(η¯ ν + η¯ν) au
N X
Qk,J (0)
+ η¯µ ¯ au
N −2 X
#) χm + QN,J (u)
m=N −k
k=0
"
∞ X
# Pk,0 (0)χN −1−k + PN −1,0 (u)
k=0
" + (η µ ¯ + η¯µ) au
N X
Pk,0 (0)
k=0
" PN,0 (u − 1) = θ¯ η ν¯ au
N X k=0
Qk,J (0)
∞ X m=N −k
∞ X
# χm + PN,0 (u) ,
m=N −k
# χm + QN,J (u)
(4)
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SHAN GAO AND JINTING WANG
" + η¯µ ¯ au
N X
∞ X
Pk,0 (0)
# χm + PN,0 (u) ,
(5)
m=N −k
k=0
P0,1 (u − 1) = θ¯ P0,1 (u) + au (χ1 (η + η¯ν) + χ2 ην)P0,1 (0) + (η + η¯ν)P1,1 (u) + au χ1 ηνP1,1 (0) + ηνP2,1 (u) + ηµP2,0 (u) + au χ1 η¯µ + χ2 ηµ P0,0 (0) + η¯µP1,0 (u) + au χ1 ηµP1,0 (0), (6) ¯ P0,j (u − 1) = θ P0,j (u) + au (χ1 (η + η¯ν) + χ2 ην)P0,j (0) + (η + η¯ν)P1,j (u) + au χ1 ηνP1,j (0) + ηνP2,j (u) + pθ P0,j−1 (u) + au (χ1 (η + η¯ν) + χ2 ην)P0,j−1 (0) + (η + η¯ν)P1,j−1 (u) + au χ1 ηνP1,j−1 (0) + ηνP2,j−1 (u) , 2 ≤ j ≤ J, (7) (
"
Pn,j (u − 1) = θ¯ η¯ν¯ au
n−1 X
# Pk,j (0)χn−k + Pn,j (u) + (η¯ ν + η¯ν)
k=0
" × au
n X
# Pk,j (0)χn+1−k + Pn+1,j (u) + ην
k=0
" × au
n+1 X
#) Pk,j (0)χn+2−k + Pn+2,j (u)
, 1 ≤ j ≤ J, 1 ≤ n ≤ N − 3,
k=0
(8) (
"
PN −2,j (u − 1) = θ¯ η¯ν¯ au
N −3 X
# Pk,j (0)χN −2−k + PN −2,j (u)
k=0
" +(η¯ ν + η¯ν) au
N −2 X
# Pk,j (0)χN −1−k + PN −1,j (u)
k=0
" +ην au
N X
Pk,j (0)
"
PN −1,j (u − 1) = θ¯ η¯ν¯ au
#) , 1 ≤ j ≤ J,
χm + PN,j (u)
(9)
m=N −k
k=0
(
∞ X
N −2 X
# Pk,j (0)χN −1−k + PN −1,j (u)
k=0
" +(η¯ ν + η¯ν) au
N X k=0
∞ X
Pk,j (0)
#) χm + PN,j (u)
, 1 ≤ j ≤ J,
m=N −k
(10) " # N ∞ X X ¯ PN,j (u − 1) = θ¯ η ν¯ au Pk,j (0) χm + PN,j (u) , 1 ≤ j ≤ J, k=0
m=N −k
(11)
ON A DISCRETE-TIME GIX /GEO/1/N-G QUEUE
(Note: χ0 = 0,
Pb
a
789
= 0, if a > b.)
Remark 2. To interpret the above Eq.s (1)-(11), we take Eq.(2) for an example. We consider the epochs t and t + 1. For the the system to exist in state (n, 0, u − 1) (1 ≤ n ≤ N − 3, u ≥ 1) at the epoch t + 1, that is, there are n customers in the system and the server is in normal service period, the remaining inter-arrival time is u − 1, the following conditions must hold: • At the epoch t the system is in state (k, j, 0), (0 ≤ k ≤ n − 1; j = 1, 2, · · · , J) ( i.e., k customers are in the system and the server is in the j-th working vacation, a batch of n − k customers arrives in the time interval (t, t+ )) and the inter-arrival time of the next batch is u (with probability au ) or at the epoch t the system is in state (n, j, u), (j = 1, 2, · · · , J) ( i.e., n customers are in the system and the server is in the J-th working vacation and the remaining inter-arrival time of the next batch is u), no negative customer arrives in the time interval (t, t+ ) (with probability η¯), the customer being served doesn’t leave the system in ((t + 1)− , t + 1) (with probability ν¯), and the vacation ends at the epoch t + 1 (with probability θ). • At the epoch t the system is in state (k, j, 0), (0 ≤ k ≤ n; j = 1, 2, · · · , J) ( i.e., k customers are in the system and the server is in the j-th working vacation, a batch of n + 1 − k customers arrives in the time interval (t, t+ )) and the inter-arrival time of the next batch is u (with probability au ) or at the epoch t, the system is in state (n + 1, j, u), (j = 1, 2, · · · , J) ( i.e., n + 1 customers are in the system and the server is in the J-th working vacation and the remaining inter-arrival time of the next batch is u), no negative customer arrives in the time interval (t, t+ ) (with probability η¯) and the customer being served leaves the system in ((t + 1)− , t + 1) (with probability ν) or a negative customer arrives in the time interval (t, t+ ) (with probability η) and removes the customer in service and the next customer being served doesn’t leave the system in ((t + 1)− , t + 1) (with probability ν¯), the vacation ends at the epoch t + 1 (with probability θ). • At the epoch t the system is in state (k, j, 0), (0 ≤ k ≤ n + 1; j = 1, 2, · · · , J) ( i.e., k customers are in the system and the server is in the j-th working vacation, a batch of n + 2 − k customers arrives in the time interval (t, t+ )) and the inter-arrival time of the next batch is u (with probability au ) or at the epoch t, the system is in state (n + 2, j, u), (j = 1, 2, · · · , J) ( i.e., n + 2 customers are in the system and the server is in the j-th working vacation and the remaining inter-arrival time of the next batch is u), a negative customer arrives in the time interval (t, t+ ) (with probability η) and removes the customer in service and the next customer being served leaves the system in ((t + 1)− , t + 1) (with probability ν), the vacation ends at the epoch t + 1 (with probability θ). • At the epoch t the system is in state (k, 0, 0), (0 ≤ k ≤ n−1) ( i.e., k customers are in the system and the server is in the normal service period, a batch of n − k customers arrives in the time interval (t, t+ )) and the inter-arrival time of the next batch is u (with probability au ) or at the epoch t the system is in state (n, 0, u),( i.e., n customers are in the system and the server is in the normal service period and the remaining inter-arrival time of the next batch is u), no negative customer arrives in the time interval (t, t+ ) (with probability η¯), the customer being served doesn’t leave the system in ((t+1)− , t+1) (with probability µ ¯).
790
SHAN GAO AND JINTING WANG
• At the epoch t the system is in state (k, 0, 0), (0 ≤ k ≤ n) ( i.e., k customers are in the system and the server is in the normal service period, a batch of n + 1 − k customers arrives in the time interval (t, t+ )) and the inter-arrival time of the next batch is u (with probability au ) or at the epoch t, the system is in state (n + 1, 0, u), ( i.e., n + 1 customers are in the system and the server is in the normal service period and the remaining inter-arrival time of the next batch is u), no negative customer arrives in the time interval (t, t+ ) (with probability η¯) and the customer being served leaves the system in ((t + 1)− , t + 1) (with probability µ) or a negative customer arrives in the time interval (t, t+ ) (with probability η) and removes the customer in service and the next customer being served doesn’t leave the system in ((t+1)− , t+1) (with probability µ ¯). • At the epoch t the system is in state (k, 0, 0), (0 ≤ k ≤ n + 1) ( i.e., k customers are in the system and the server is in the normal service period, a batch of n + 2 − k customers arrives in the time interval (t, t+ )) and the inter-arrival time of the next batch is u (with probability au ) or at the epoch t, the system is in state (n + 2, 0, u) ( i.e., n + 2 customers are in the system and the server is in the normal service period and the remaining inter-arrival time of the next batch is u), a negative customer arrives in the time interval (t, t+ ) (with probability η) and removes the customer in service and the next customer being served leaves the system in ((t + 1)− , t + 1) (with probability µ). To get the system length distributions at arbitrary epoch, we introduce the following z-transforms Pen,j (z) =
∞ X
Pn,j (u)z u , 0 ≤ j ≤ J; 0 ≤ n ≤ N.
u=0
So that Pen,j (1) = Pn,j , where Pn,j is the joint probability that there are n positive customers in the system while the server is in state j. Define Qk,J (0) =
J X
Pk,j (0), Qk,J =
j=1
J X
Pk,j , Q− k,J =
j=1
J X
− Pk,j , 0 ≤ k ≤ N.
j=1
Multiplying (1) to (11) by z u and summing over u from 1 to ∞, we arrive at (z − 1)Pe0,0 (z) J−1 X e = η(P1,0 (z) − P1,0 (0)) + A(z)χ1 ηP0,0 (0) + θ p¯ (Pe0,j (z) − P0,j (0)) j=1
J−1 X e +P0,J (z) − P0,J (0) + θA(z)(χ1 (η + η¯ν) + χ2 ην) p¯ P0,j (0) + P0,J (0) j=1
J−1 X e e +θ(η + η¯ν) p¯ (P1,j (z) − P1,j (0)) + P1,J (z) − P1,J (0) j=1
ON A DISCRETE-TIME GIX /GEO/1/N-G QUEUE
791
J−1 J−1 X X +θην A(z)χ1 p¯ P1,j (0) + P1,J (0) + p¯ (Pe2,j (z) − P2,j (0)) j=1
j=1
+Pe2,J (z) − P2,J (0) − P0,0 (0), ( " # n−1 X e n,J (z) − Qn,J (0) (z − η¯µ ¯)Pen,0 (z) = θ η¯ν¯ A(z) Qk,J (0)χn−k + Q
(12)
k=0
" +(η¯ ν + η¯ν) A(z)
# e Qk,J (0)χn+1−k + Qn+1,J (z) − Qn+1,J (0)
n X k=0
" +ην A(z)
#)
n+1 X
e n+2,J (z) − Qn+2,J (0) Qk,J (0)χn+2−k + Q
k=0
" +¯ ηµ ¯ A(z)
n−1 X
# Pk,0 (0)χn−k − Pn,0 (0)
k=0
" +(η µ ¯ + η¯µ) A(z)
n X
# e Pk,0 (0)χn+1−k + Pn+1,0 (z) − Pn+1,0 (0)
k=0
" +ηµ A(z)
n+1 X
# Pk,0 (0)χn+2−k + Pen+2,0 (z) − Pn+2,0 (0) ,
k=0
1 ≤ n ≤ N − 3, ( (z − η¯µ ¯)PeN −2,0 (z) = θ
" η¯ν¯ A(z)
N −3 X
(13) #
e N −2,J (z) − QN −2,J (0) Qk,J (0)χN −2−k + Q
k=0
" +(η¯ ν + η¯ν) A(z)
N −2 X
# e N −1,J (z) − QN −1,J (0) Qk,J (0)χN −1−k + Q
k=0
" +ην A(z)
N X
Qk,J (0)
+¯ ηµ ¯ A(z)
#) e N,J (z) − −QN,J (0) χm + Q
m=N −k
k=0
"
∞ X
N −3 X
# Pk,0 (0)χN −2−k − PN −2,0 (0)
k=0
" +(η µ ¯ + η¯µ) A(z)
N −2 X
# Pk,0 (0)χN −1−k + PeN −1,0 (z) − PN −1,0 (0)
k=0
" +ηµ A(z)
N X
Pk,0 (0)
# e χm + PN,0 (z) − PN,0 (0) ,
m=N −k
k=0
(z − η¯µ ¯)PeN −1,0 (z) ( " =
∞ X
θ η¯ν¯ A(z)
N −2 X
# e Qk,J (0)χN −1−k + QN −1,J (z) − QN −1,J (0)
k=0
" +(η¯ ν + η¯ν) A(z)
N X k=0
Qk,J (0)
∞ X m=N −k
#) e χm + QN,J (z) − QN,J (0)
(14)
792
SHAN GAO AND JINTING WANG
" +¯ ηµ ¯ A(z)
N −2 X
# Pk,0 (0)χN −1−k − PN −1,0 (0)
k=0
" +(η µ ¯ + η¯µ) A(z)
N X
Pk,0 (0)
=
θ¯ η ν¯ A(z)
N X
+¯ ηµ ¯ A(z)
∞ X
Qk,J (0)
(15)
# e N,J (z) − QN,J (0) χm + Q
m=N −k
k=0
"
# e χm + PN,0 (z) − PN,0 (0) ,
m=N −k
k=0
(z − η¯µ ¯)PeN,0 (z) "
∞ X
N X
∞ X
Pk,0 (0)
# χm − PN,0 (0) ,
(16)
m=N −k
k=0
¯ Pe0,1 (z) (z − θ) = θ¯ A(z)(χ1 (η + η¯ν) + χ2 ην)P0,1 (0) + (η + η¯ν)(Pe1,1 (z) − P1,1 (0)) +A(z)χ1 ηνP1,1 (0) + ην(Pe2,1 (z) − P2,1 (0)) − P0,1 (0) +ηµ(Pe2,0 (z) − P2,0 (0)) + A(z) χ1 η¯µ + χ2 ηµ P0,0 (0) +¯ η µ(Pe1,0 (z) − P1,0 (0)) + A(z)χ1 ηµP1,0 (0),
(17)
¯ Pe0,j (z) (z − θ) ¯ = θ A(z)(χ1 (η + η¯ν) + χ2 ην)P0,j (0) + (η + η¯ν)(Pe1,j (z) − P1,j (0)) e +A(z)χ1 ηνP1,j (0) + ην(P2,j (z) − P2,j (0)) − P0,j (0) +pθ Pe0,j−1 (z) − P0,j−1 (0) +A(z)(χ1 (η + η¯ν) + χ2 ην)P0,j−1 (0) + (η + η¯ν)(Pe1,j−1 (z) − P1,j−1 (0)) e +A(z)χ1 ηνP1,j−1 (0) + ην(P2,j−1 (z) − P2,j−1 (0)) , 2 ≤ j ≤ J, (18) ¯η ν¯)Pen,j (z) (z − θ¯ ( " # n−1 X ¯ = θ η¯ν¯ A(z) Pk,j (0)χn−k − Pn,j (0) + (η¯ ν + η¯ν) k=0
" × A(z)
n X
# Pk,j (0)χn+1−k + Pen+1,j (z) − Pn+1,j (0)
k=0
" +ην A(z)
n+1 X
Pk,j (0)χn+2−k
k=0
#) +Pen+2,j (z) − Pn+2,j (0)
¯η ν¯)PeN −2,j (z) (z − θ¯
, 1 ≤ j ≤ J, 1 ≤ n ≤ N − 3,
(19)
ON A DISCRETE-TIME GIX /GEO/1/N-G QUEUE
( =
"
θ¯ η¯ν¯ A(z)
N −3 X
793
# Pk,j (0)χN −2−k − PN −2,j (0)
k=0
" +(η¯ ν + η¯ν) A(z)
N −2 X
# e Pk,j (0)χN −1−k + PN −1,j (z) − PN −1,j (0)
k=0
" +ην A(z)
N X
∞ X
Pk,j (0)
#) e χm + PN,j (z) − PN,j (0) ,
(20)
m=N −k
k=0
¯η ν¯)PeN −1,j (z) (z − θ¯ ( " # N −2 X = θ¯ η¯ν¯ A(z) Pk,j (0)χN −1−k − PN −1,j (0) k=0
" +(η¯ ν + η¯ν) A(z)
N X
Pk,j (0)
∞ X
#) χm + PeN,j (z) − PN,j (0)
,
m=N −k
k=0
1 ≤ j ≤ J, ¯ e (z − θ¯ η ν¯)PN,j (z) " # N ∞ X X ¯ = θ¯ η ν¯ A(z) Pk,j (0) χm − PN,j (0) , 1 ≤ j ≤ J.
(21)
(22)
m=N −k
k=0
Summing all Eqs. (12)-(22), we obtain J
N J X X
N
A(z) − 1 X X Pn,j (0). Pen,j (z) = z − 1 j=0 n=0 j=0 n=0
Letting z → 1 and using lim
z→1
A(z) − 1 1 = , z−1 λ
and the normalization condition J X N X
Pen,j (1) = 1,
j=0 n=0
we have that J X N X
Pn,j (0) = λ.
(23)
j=0 n=0
Applying Bayes’ theorem and using (23), one can easily get Pn,j (0) Pn,j (0) = , j = 0, 1, 2, · · · , J; 0 ≤ n ≤ N. PN λ j=0 k=0 Pk,j (0)
− Pn,j = PJ
(24)
Setting z = 1 in (13)-(22) and using (24), we can get the expressions for Pn,j (0 ≤ − j ≤ J; 0 ≤ n ≤ N ) through the pre-arrival epoch probabilities Pn,j (0 ≤ j ≤ J; 0 ≤ n ≤ N ) as follows.
794
SHAN GAO AND JINTING WANG
For 1 ≤ j ≤ J, −1 ∞ X ¯η ν¯ NX λθ¯ − P χk , m,j ¯η ν¯ 1 − θ¯ m=0 k=N −m ( "N −2 # X θ¯ − − η ν¯ Pk,j χN −1−k − PN −1,j = ¯η ν¯ λ¯ 1 − θ¯ k=0 " N −1 #) ∞ X X − +(η¯ ν + η¯ν) λ Pk,j χm + PN,j ,
PN,j =
PN −1,j
m=N −k
k=0
PN −2,j
θ¯ = ¯η ν¯ 1 − θ¯
( λ¯ η ν¯
"N −3 X
# − Pk,j χN −2−k
−
PN−−2,j
k=0 N −2 X
" +(η¯ ν + η¯ν) λ
# − Pk,j χN −1−k + PN −1,j − λPN−−1,j
k=0
" +ην λ
N −1 X
Pn,j
#) χm + PN,j
,
m=N −k
k=0
θ¯ = ¯η ν¯ 1 − θ¯
∞ X
− Pk,j
( λ¯ η ν¯
"n−1 X
# − Pk,j χn−k
−
− Pn,j
k=0
"
n X
+(η¯ ν + η¯ν) λ
# − Pk,j χn+1−k
+ Pn+1,j −
− λPn+1,j
k=0
" +ην λ
#)
n+1 X
− − Pk,j χn+2−k + Pn+2,j − λPn+2,j
, 1 ≤ n ≤ N − 3,
k=0
PN,0
" N −1 X θ¯ η ν¯ = λ Q− k,J 1 − η¯µ ¯ k=0
∞ X
# χm + QN,J
m=N −k
∞ N −1 λ¯ ηµ ¯ X − X Pk,0 χm , 1 − η¯µ ¯ k=0 m=N −k ( " N −2 # X θ − − = η¯ν¯ λ Qk,J χN −1−k + QN −1,J − λQN −1,J 1 − η¯µ ¯ k=0 " N −1 #) ∞ X X − +(η¯ ν + η¯ν) λ Qk,J χm + QN,J
+
PN −1,0
m=N −k
k=0
1 + 1 − η¯µ ¯
( λ¯ ηµ ¯
"N −2 X
# − Pk,0 χN −1−k
k=0
" +(η µ ¯ + η¯µ) λ
N −1 X
− Pk,0
k=0
PN −2,0
θ = 1 − η¯µ ¯
−
PN−−1,0
(
"
η¯ν¯ λ
N −3 X k=0
∞ X
#) χm + PN,0
,
m=N −k
# Q− k,J χN −2−k
+ QN −2,J −
λQ− N −2,J
ON A DISCRETE-TIME GIX /GEO/1/N-G QUEUE
" +ην λ
N −1 X
∞ X
Q− k,J
795
# χm + QN,J
m=N −k
k=0
N −2 X
" +(η¯ ν + η¯ν) λ
#) Q− k,J χN −1−k
+ QN −1,J −
λQ− N −1,J
k=0
1 + 1 − η¯µ ¯
( λ¯ ηµ ¯
"N −3 X
# − Pk,0 χN −2−k
−
PN−−2,0
k=0
" +(η µ ¯ + η¯µ) λ
N −2 X
# − Pk,0 χN −1−k
+ PN −1,0 −
λPN−−1,0
k=0
" +ηµ λ
N −1 X
− Pk,0
Pn,0
#) χm + PN,0
,
m=N −k
k=0
(
θ = 1 − η¯µ ¯
∞ X
" η¯ν¯ λ
n−1 X
# − Q− k,J χn−k + Qn,J − λQn,J
k=0
" +(η¯ ν + η¯ν) λ
n X
# Q− k,J χn+1−k
+ Qn+1,J −
λQ− n+1,J
k=0
" +ην λ
n+1 X
#) Q− k,J χn+2−k
+ Qn+2,J −
λQ− n+2,J
k=0
1 + 1 − η¯µ ¯
( λ¯ ηµ ¯
"n−1 X
# − − Pk,0 χn−k − Pn,0
k=0
" +(η µ ¯ + η¯µ) λ
n X
# − Pk,0 χn+1−k
+ Pn+1,0 −
− λPn+1,0
k=0
" +ηµ λ
n+1 X
#) − Pk,0 χn+2−k
+ Pn+2,0 −
− λPn+2,0
, 1 ≤ n ≤ N − 3,
k=0
θ¯ − − P0,1 = λP0,1 (χ1 (η + η¯ν) + χ2 ην − 1) + (η + η¯ν)(P1,1 − λP1,1 ) θ 1 − − − + λχ1 ηνP1,1 + ην(P2,1 − λP2,1 ) + λ(χ1 η¯µ + χ2 ηµ)P0,0 θ − − − + η¯µ(P1,0 − λP1,0 ) + λχ1 ηµP1,0 + ηµ(P2,0 − λP2,0 ) , θ¯ − − P0,j = λP0,j χ1 (η + η¯ν) + χ2 ην − 1 + (η + η¯ν)(P1,j − λP1,j ) θ − − − + λχ1 ηνP1,j + ην(P2,j − λP2,j ) + p P0,j−1 + λP0,j−1 − × χ1 (η + η¯ν) + χ2 ην − 1 + (η + η¯ν)(P1,j−1 − λP1,j−1 ) − − + λχ1 ηνP1,j−1 + ην(P2,j−1 − λP2,j−1 ) , 2 ≤ j ≤ J, P0,0 = 1 −
N X n=0
Qn,J −
N X n=1
Pn,0 .
796
SHAN GAO AND JINTING WANG
3.3. Steady-state distribution 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. o (0 ≤ n ≤ N, 0 ≤ j ≤ N ) denote the probabilities that outside observer Let Pn,j PJ o sees n customers in the system and the server in state j. Put Qok,J = j=1 Pk,j ,0 ≤ k ≤ N. By observing arbitrary and outside observer’s observation epochs, we have P0,0 =
o P0,0
+ θ p¯
J−1 X
o P0,j
+
o P0,J
J−1 ! X o o + ν p¯ P1,j + P1,J ,
j=1
Pn,0 =
o µ ¯Pn,0
+
o µPn+1,0
j=1
+θ
ν¯Qon,J
+
νQon+1,J
, 1 ≤ n ≤ N − 1,
o PN,0 = µ ¯PN,0 + θ¯ ν QoN,J , ¯ o + µP o + θνP ¯ o , P0,1 = θP 0,1 1,0 1,1 o ¯ o + θνP ¯ o + θpP o P0,j = θP 0,j 1,j 0,j−1 + θνpP1,j−1 , 2 ≤ j ≤ J,
¯ν P o + θνP ¯ o Pn,j = θ¯ n,j n+1,j , 1 ≤ n ≤ N − 1, 1 ≤ j ≤ J, ¯ν P o , 1 ≤ j ≤ J. PN,j = θ¯ N,j From the above equations, we can obtain 1 o PN,j = ¯ PN,j , 1 ≤ j ≤ J, θ¯ ν 1 o ¯ o Pn,j = ¯ (Pn,j − θνP n+1,j ), 1 ≤ n ≤ N − 1, 1 ≤ j ≤ J, θ¯ ν 1 o ν QoN,J ), PN,0 = (PN,0 − θ¯ µ ¯ 1 o o Pn,0 = Pn,0 − µPn+1,0 − θ¯ ν Qon,J − θνQon+1,J , 1 ≤ n ≤ N − 1, µ ¯ 1 o o ¯ o ), P0,1 = ¯ (P0,1 − µP1,0 − θνP 1,1 θ 1 o o o ¯ o ), 2 ≤ j ≤ J, − θνpP1,j−1 − θνP P0,j = ¯ (P0,j − θpP0,j−1 1,j θ ! J−1 J−1 X X o o o o o P0,0 = P0,0 − θ p¯ P0,j + P0,J + ν p¯ P1,j + P1,J . j=1
(25)
j=1
Remark 3. It should be noted here that by using normalization condition we can obtain o P0,0 =1−
N X i=0
Qoi,J −
N X
o Pi,0 ,
(26)
i=1
o and the numerical value of P0,0 evaluated through (26) matches exactly with the one evaluated through (25). As steady-state probabilities at various epochs are known, various performance measures can be readily derived. We will study them in the following sections.
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4. Performance measures. 4.1. Blocking probabilities. An important performance measure of a finite buffer batch arrival single server queueing system is the blocking probability. Denoting the blocking probabilities of the first-, an arbitrary- and the last- positive customer of an arriving batch, respectively, as PBF , PBA and PBL , then we can determine the three kinds of probabilities from Section 3.1 as follows: PBF =
J X
− PN,j ,
j=0
PBA =
N X J X
PBL =
n=0 j=0
χ− r ,
r=N −n
n=0 j=0 N X J X
∞ X
− Pn,j
∞ X
− Pn,j
χr ,
r=N −n+1
P∞ where χ− r = i=r+1 χi /µX is the probability that there are r customers ahead of an arbitrary customer in his batch, r = 0, 1, 2, · · · . 4.2. Queue length distribution. To find the system length distribution (including the one in service) at pre-arrival, arbitrary and outside observer’s observation epoch, we define the following notations. πk− ≡ the steady-state probability that there are k positive customers in the system at a pre-arrival (of a batch of positive) epoch; πk ≡ the steady-state probability that there are k positive customers in the system at arbitrary epoch; πko ≡ the steady-state probability that there are k positive customers in the system at outside observer’s observation epoch. Then we have that πk− =
J X j=0
− Pk,j , πk =
J X
Pk,j , πko =
j=0
J X
o Pk,j , 0 ≤ k ≤ N.
j=0
At an arbitrary epoch, the average queue length (excluding the one being served) when the server is on working vacation LV , in the normal service period LB and the queue length Lq are given by LB =
N N X J X X (i − 1)Pi,0 , LV = (i − 1)Pi,j , Lq = LB + LV . i=1
i=1 j=1
At outside observer’s observation epoch, the average queue length (excluding the one being served) when the server is on working vacation LoV , in busy period LoB and the average number of customers in the system Loq are respectively given by LoB =
N X i=1
WqA
o (i − 1)Pi,0 , LoV =
N X J X o (i − 1)Pi,j , Loq = LoB + LoV . i=1 j=1
Let denote the average waiting time in the queue of an accepted arbitrary 0 positive customer in a batch. Then by Little’s rule WqA Little = Loq /λ , where 0 λ = λµX (1 − PBA ) is the effective arrival rate of the positive customers, WqA Little denotes the average waiting time WqA evaluated through Little’s rule.
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4.3. Probabilities of the server’s states. Let Pb denote the probability the the server is in normal busy period at outside observer’s observation epoch, and Pv denote the probability the the server is in working vacation period at outside observer’s observation epoch,then Pb =
N X
o Pi,0 ,
Pv =
i=0
J X N X
o Pi,j .
j=1 i=0
4.4. Waiting time analysis of an arbitrary positive customer. In this subsection, we focus on the waiting-time analysis of an arbitrary positive customer of a batch. For this purpose, we should take into account the number of customers in front of an arbitrary positive customer in his batch. Note that in partial-batch rejection policy, up to (N − n) customers of the whole batch will be accepted if n customers are already present in the system, then for an arbitrary customer who is allowed to join the system, there may be l (0 ≤ l ≤ N − 1 − n) customers ahead of him in his batch. Let TqA be the waiting time of an arbitrary positive customer that is put in the queue of an arriving batch, which is given by the interval from the instant at which A it enters in the system to the instant when it begins its service, let Wk,j = P (TqA = − k, J = j)(j = 0, 1, 2, · · · , J) is the probability that the server is in state j prior to an arriving batch and the accepted arbitrary positive customer’s waiting time in PJ A the queue is k. Put Wk,V = j=1 P (TqA = k, J − = j), then 4
A A WkA = P (TqA = k) = Wk,0 + Wk,V , k = 0, 1, 2, · · · .
PJ − Considering various possible cases and in view of the notation of Q− j=1 Pk,J , k,J = we can have that − − 1 − − − − A χ0 P0,0 + η(P0,0 χ1 + P1,0 χ0 ) , W0,0 = 1 − PBA ( 1 2 X X 1 (k−1) (k−1) − − − − A Pn,0 χ2−n B1 η¯(µ + µ ¯η) P χ B η¯(µ + µ ¯η) + Wk,0 = 1 − PBA n=0 n,0 1−n 0 n=0 NX −1 X n (k−1) (k−1) (k−1) − − ¯η) + Bn−2 (ηµ + B0 (ηµ + η¯µη + η µ ¯η) + Pl,0 χn−l Bn−1 η¯(µ + µ n=3 l=0 (k−1)
+ η¯µη + η µ ¯η) + Bn−3 ηµη
) , k ≥ 1,
and n o 1 − − − − − χ− Q + η(Q χ + Q χ ) , 0 0,J 0,J 1 1,J 0 1 − PBA ( 1 2 X X 1 − − − A W1,V = Q χ η¯(ν + ν¯η) + Q− ¯νη + η¯ ν η) n,J χ2−n (ην + η 1 − PBA n=0 n,J 1−n n=0 ) 3 X − − + Qn,J χ3−n ηνη ,
A W0,V =
n=0
ON A DISCRETE-TIME GIX /GEO/1/N-G QUEUE
A Wk,V =
θ¯k−1 1 − PBA
(
1 X
(k−1)
− Q− n,J χ1−n V0
η¯(ν + ν¯η) +
n=0
2 X
799
− Q− n,J χ2−n
n=0
−1 X n i NX (k−1) (k−1) − × V1 η¯(ν + ν¯η) + V0 (ην + η¯νη + η¯ ν η) + Q− l,J χn−l
h
n=3 l=0
) (k−1) (k−1) (k−1) × Vn−1 η¯(ν + ν¯η) + Vn−2 (ην + η¯νη + η¯ ν η) + Vn−3 ηνη ( 1 k−1 X X 1 (m) (k−m−1) − m−1 θθ¯ Q− B0 η¯(µ + µ ¯η) + n,J χ1−n V0 1 − PBA m=1 n=0 2 X (m) (k−m−1) (k−m−1) − − + Qn,J χ2−n V0 B1 η¯(µ + µ ¯η) + B0 (ηµ + η¯µη + η µ ¯η) n=0
+
(m) (k−m−1) V 1 B0 η¯(µ
+
(k−m−1) Bn−r−2 (ηµ
" n−3 NX n −1 X X (k−m−1) − − Ql,J χn−l Vr(m) Bn−r−1 η¯(µ + µ ¯η) +µ ¯η) + r=0
n=3 l=0
(k−m−1)
+ B0
+ η¯µη + η µ ¯η) +
(k−m−1) Bn−r−3 ηµη
(m) (k−m−1) + Vn−2 B1 η¯(µ + µ ¯η) #)
(m) (k−m−1) (ηµ + η¯µη + η µ ¯η) + Vn−1 B0 η¯(µ + µ ¯η)
,
k ≥ 2,
Remark 4. It may be noted here that the numerical value of the mean waiting time in the queue of an accepted arbitrary customer of an arriving batch evaluated through WqA =
∞ X
A A k(Wk,0 + Wk,V )
(27)
k=1
matches exactly with the one obtained earlier using Little’s rule, as it should be.
5. Numerical examples. First, In Tables 1 and 2, we give the computation results of the steady-state system length distribution at various epochs and other performances measures for various arrival time distributions with the same system parameters µ = 0.55, ν = 0.1, η = 0.04, p = 0.002, θ = 0.30, N = 10, J = 5, and batch size distribution χ1 = 0.3, χ2 = 0.2, χ4 = 0.2, χ6 = 0.1, χ8 = 0.1, χ10 = 0.1. The inter-arrival times of batches of positive customers are respectively geometric distribution with λ = 0.15 and arbitrary distribution a1 = 0.55, a3 = 0.25, a5 = 0.2. It may be remarked that since all the results reported in Table 1 and Table 2 are rounded to four decimal places on the Matlab software package, the sum of the probabilities may not add to one in some cases. From Table 1, we can see that the pre-arrival and arbitrary epoch probabilities are same in the case of Bernoulli arrivals, which coincides with the BASTA (i.e. Bernoulli arrivals see time averages) property and indicates the correctness of the theoretical analysis in the previous section and the reliability of the computation program. On the other hand, we can see from Tables 1 and 2 that the average waiting time in the queue evaluated by Eq. (27) exactly matches with that obtained from Little’s rule, which shows that the theoretical analysis and numerical experiment in this paper is correct.
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Table 1. Steady-state system length distribution at various epochs for queue with Geometric inter-arrival time
k
πk−
πk
πko
0 1 2 3 4 5 6 7 8 9 10
0.2645 0.0736 0.0701 0.0708 0.0766 0.0748 0.0793 0.0803 0.0839 0.0835 0.0426 PBF = 0.0426 WqA = 8.3015
0.2645 0.0736 0.0701 0.0708 0.0766 0.0748 0.0793 0.0803 0.0839 0.0835 0.0426 PBA = 0.2532 A Wq little = 8.3015
0.2278 0.0743 0.0706 0.0661 0.0781 0.0717 0.0790 0.0776 0.0845 0.0820 0.0883 PBL = 0.2732
Table 2. Steady-state system length distribution at various epochs for queue with arbitrary inter-arrival time
k
πk−
πk
πko
0 1 2 3 4 5 6 7 8 9 10
0.0009 0.0016 0.0034 0.0072 0.0158 0.0370 0.0765 0.1358 0.1780 0.3368 0.2069 PBF = 0.2069 WqA = 13.2745
0.0007 0.0013 0.0029 0.0061 0.0133 0.0299 0.0650 0.1275 0.2102 0.3491 0.1939 P BA = 0.6524 WqA little = 13.2745
0.0003 0.0008 0.0017 0.0037 0.0081 0.0175 0.0401 0.0854 0.1620 0.2496 0.4307 PBL = 0.6387
In the following series of numerical examples, we study the effects of various parameters on the main system characteristics, such as the server’s state probabilities Pb and Pv , average waiting time WqA and the blocking probability PBA . For convenience, we assume that the inter-batch time distribution is assumed to be geometrical distribution with rate λ = 0.15 and batch size distribution χ1 = 0.3, χ2 = 0.2, χ4 = 0.2, χ6 = 0.1, χ8 = 0.1, χ10 = 0.1 , and the same system parameters N = 6, η = 0.04, µ = 0.55, θ = 0.3. This model is denoted by GeoX /(Geo1 , Geo2 )/1/N/RW V S(J)-G queue. Firstly, in Figure 1, given ν = 3 we compare the effect of J on the server’s state probabilities Pb and Pv for different values of p = 0, 0.45, 1. Figure 1 shows that the probability that the server, Pb , is in normal service period decreases as J increases,
ON A DISCRETE-TIME GIX /GEO/1/N-G QUEUE
801
0.7 p=0, Pb p=0, Pv
0.65
Server’s state probabilities
p=0.45,Pb p=0.45,Pv
0.6
p=1,Pb p=1,Pv
0.55
0.5
0.45
0.4
0.35
2
4
6
8 J
10
12
14
Figure 1. The effect of J on Pb and Pv . 5.8 5.6
ν=0 ν=0.15 ν=0.35 ν=0.55
5.4
WA q
5.2 5 4.8 4.6 4.4 4.2
0
0.2
0.4
0.6
0.8
1
p
Figure 2. The effect of p on mean waiting time WqA .
but the larger Pb is as the smaller p is. However, the probability that the server is in working period, Pv is increases as J increases and increases as p increases, which agree to our expectation. On the other hand, we can find an interesting situation that when J ≥ 6, the effect of J on Pb and Pv (also on WqA , PBA , see Figures 4 and 5) almost vanishes. In Figures 2 and 3, assume that J = 5, for different values of ν = 0, 0.15, 0.35, 0.55, we show the effect of p on WqA and PBA , where p is the probability with which the server continues the next working vacation if the server finds no positive customers present in the system at the end of a vacation. Figures 2 and 3 show the following results: (1) When the value of the lower service rate equals that of the normal service rate, i.e., ν = µ = 0.55, the effect of p can be ignored. This is because when ν = µ, the model is changed to the batch arrival G-queue without vacation, and then the effect of p on the system turns to zero. (2) As our expectations, Figures 1-2 show that (i) given ν and 0 ≤ ν < µ, WqA and PBA increase as p increases. The reason is that as p increases, the chances of the server on vacation increase, which causes the system size to increase and then leads to WqA and PBA to increase. (ii) given p and 0 ≤ p ≤ 1, WqA and PBA decrease as ν increases. We should note that all curves always locate between ν = 0 and
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SHAN GAO AND JINTING WANG
0.42 ν=0 ν=0.15 ν=0.35 ν=0.55
0.41
0.4
PBA
0.39
0.38
0.37
0.36
0.35
0
0.2
0.4
0.6
0.8
1
p
Figure 3. The effect of p on blocking probability PBA .
5.9 p=0 p=0.3 p=0.5 p=1
5.8
5.7
WA q
5.6
5.5
5.4
5.3
5.2
2
4
6
8 J
10
12
14
Figure 4. The effect of J on mean waiting time WqA .
0.42
0.415 p=0 p=0.3 p=0.5 p=1
PBA
0.41
0.405
0.4
0.395
2
4
6
8 J
10
12
14
Figure 5. The effect of J on blocking probability PBA .
ν = µ = 0.55. The reason is that ν = 0 implies that the server doesn’t provide service to the positive customers, then the values of WqA and PBA are the highest. But ν = µ means that at any time the server doesn’t take vacation, so the waiting time and the blocking probability are the lowest.
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Finally, under the assumption ν = 0.3, Figures 4 and 5 show the effect of the number of vacations J for 1 ≤ J ≤ 15 on WqA and PBA for different values of p, from which we see that all curves start from identical initial point, i.e., when J = 1, the effect of p on WqA and PBA can be ignored. This is because when J = 1, the model is decreased to the batch arrival G-queue with single working vacation, which is also corresponding to the case p = 0, so as p = 0, the effect of J becomes disappearing. As our expectations, Figures 4 and 5 also show that WqA and PBA increase as J increases. The reason is that the bigger J is, the longer the vacation period becomes, accordingly which increases the mean queue length because of the lower service rate during vacation period. 6. Analysis of cost optimization. In this section we develop the expected cost function per unit time for the GI X /Geo/1/N (RWVS(J))-G queue, where µ is a decision variable. Our main objective is to determine the optimal value of µ, say µ∗ , so that the decision maker could use this optimal value to minimize the expected operating cost function per unit time. Define the following cost elements: Ch0 ≡ unit time cost of every customer in the system when the server is in a normal busy period; Ch1 ≡ unit time cost of every customer in the system when the server is on a working vacation; Cµ ≡ fixed service cost per unit time during the normal busy period; Cν ≡ fixed service cost per unit time during a working vacation period. Using the above cost elements and the corresponding performance measures, the expected cost function per unit time is as follows T C(µ) = Ch0 Lo0 + Ch1 Lo1 + Cµ µ + Cν ν, PN PN PJ o o , Lo1 = i=1 j=1 iPi,j . where Lo0 = i=1 iPi,0 It would be an arduous task to develop optimal solution µ∗ analytically because the cost function T C(µ) is highly non-linear and complex, where T C(µ∗ ) = min T C(µ). In order to find the optimal value µ∗ , we will use the parabolic ν≤µ x(m) , go to Step 5. Step 4 (Left). If f (x(m) ) is superior to f (x(q) ) (less for a minimize, greater for a
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SHAN GAO AND JINTING WANG X
Geo /Geo/1/N with randomized working vacations and at most J vacations 58 56 54
TC(µ)
52 50 48 46 44 42 40
0.1
0.2
0.3
0.4
0.5 µ
0.6
0.7
0.8
0.9
1
Figure 6. Effect of ν on expected operating cost per unit time.
maximize), then update x(q) → x(l) . Otherwise, replace x(m) → x(r) , x(q) → x(m) . Either way, advance i = i + 1, and return to Step 2. Step 5 (Right). If f (x(m) ) is superior to f (x(q) ) (less for a minimize, greater for a maximize), then update x(q) → x(r) . Otherwise, replace x(m) → x(l) , x(q) → x(m) . Either way, advance i = i + 1, and return to Step 2. In the following numerical example, we then apply the procedure of the parabolic method to search the optimal normal service rate µ∗ . Table 3. The parabolic method in searching the optimum solution of GeoX /Geo/1/N (RWVS(J))-G No. of iterations
µ(l) µ(m) µ(r) T C(µ(l) ) T C(µ(m) ) T C(µ(r) ) µ(q) T C(µ(q) ) Tolerance
0
1
2
3
4
5
0.300000 0.350000 0.400000 41.210395 41.133800 41.370565 0.337222 41.122035 0.012778
0.300000 0.337222 0.350000 41.210395 41.122035 41.133800 0.336625 41.121998 5.9690×10−4
0.300000 0.336625 0.337222 41.210395 41.121998 41.122035 0.336453 41.121996 1.7209×10−4
0.300000 0.336453 0.336625 41.210395 41.121996 41.121998 0.336443 41.121996 9.7248×10−6
0.300000 0.336443 0.336453 41.210395 41.121996 41.121996 0.336441 41.121996 2.3053×10−6
0.300000 0.336441 0.336443 41.210395 41.121996 41.121996 0.336441 41.121996 1.5226×10−7
Example 1. (GeoX /Geo/1/N (RWVS(J))-G) Assume that the inter-arrival time of a batch is geometric with rate λ = 0.05, batch size distribution is χ1 = 0.3, χ2 = 0.2, χ4 = 0.2, χ6 = 0.1, χ8 = 0.1, χ1 0 = 0.1, and ν = 0.05, θ1 = 0.3, η2 = 0.04, p = 0.002, N = 6, J = 5, Ch0 = 5, Ch1 = 3, Cµ = 45, Cν = 15. With the information of Fig.6, we assume the stopping tolerance ε = 10−6 and select the initial 3-point pattern µ(l) = 0.3, µ(m) = 0.35, µ(r) = 0.4. Then applying the parabolic method as given above, after 5 iterations, see Table 3 for details, we can obtain that the minimum expected operating cost per unit time converges to the solution µ∗ = 0.336441 with value 41.121996. 7. Conclusions. Using embedded Markov chain and supplementary variable technique, we have carried out completely an analysis of a general batch arrival finitebuffer G-queue with a randomized working vacations and at most J vacations policy. We have obtained the system length distributions at pre-arrival, arbitrary and
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outside observer’s observation epochs and some important performance measures. Actual waiting time distribution of an arbitrary tagged positive customer in the queue is also presented. The effect of various parameters on the system performance measures and cost optimization are also illustrated numerically. For future investigation, we can consider more complex model such as batch or single arrival GI/Geo/1/N -G queue with phase-type working vacation times by using the techniques discussed in this paper. Acknowledgments. The authors thank the referees for their valuable comments and acknowledge that this research was supported by the National Natural Science Foundation of China (Nos. 11171019, 71390334 and 11171179), Program for New Century Excellent Talents in University (No. NCET-11-0568), China Postdoctoral Science Foundation funded project (No. 2013M540041), Natural Science Foundation of Anhui Higher Education Institutions (No. KJ2014ZD21). The authors would like to thank Prof. Wei Wayne Li at Department of Computer Science, Texas Southern University for his constructive comments which improve the presentation of this paper. REFERENCES [1] I. Atencia and P. Moreno, The discrete-time Geo/Geo/1 queue with negative customers and disasters, Comput. Oper. Res., 31 (2004), 1537–1548. [2] Y. Baba, Analysis of a GI/M/1 queue with multiple working vacations, Oper. Res. Lett., 33 (2005), 201–209. [3] A. D. Banik, U. C. Gupta and S. S. Pathak, On the GI/M/1/N queue with multiple working vacations-analytic analysis and computation, Appl. Math. Model., 31 (2007), 1701–1710. [4] A. D. Banik, M. L. Chaudhry and U. C. Gupta, On the finite buffer queue with renewal input and batch Markovian service process: GI/BMSP/1/N, Methodol. Comput. Appl. Probab., 10 (2008), 559–575. [5] A. D. Banik, Stationary distributions and optimal control of queues with batch Markovian arrival process under multiple adaptive vacations, Comput. Ind. Eng., 65 (2013), 455–465. [6] K. C. Chae, D. E. Lim and W. S. Yang, The GI/M/1 queue and the GI/Geo/1 queue both with single working vacation, Perform. Eval., 66 (2009), 356–367. [7] K. C. Chae, H. M. Park and W. S. Yang, A GI/Geo/1 queue with negative and positive customers, Appl. Math. Model., 34 (2010), 1662–1671. [8] R. Chakka and P. G. Harrison, A Markov modulated multi-server queue with negative customers—the MM CPP/GE/c/L G-queue, Acta Inform., 37 (2001), 881–919. [9] I. Dimitriou, A mixed priority retrial queue with negative arrivals, unreliable server and multiple vacations, Appl. Math. Model., 37 (2013), 1295–1309. [10] T. V. Do, Bibliography on G-networks, negative customers and applications, Math. Comput. Model., 53 (2011), 205–212. [11] S. Gao and Z. Liu, An M/G/1 queue with single working vacation and vacation interruption under bernoulli schedule, Appl. Math. Model., 37 (2013), 1564–1579. [12] S. Gao, J. Wang and D. Zhang, Discrete-time GIX /Geo/1/N queue with negative customers and multiple working vacations, J. Korean. Stat. Soc., 42 (2013), 515–528. [13] E. Gelenbe, Random neural networks with negative and positive signals and product form solution, Neural Comput., 1 (1989), 502–510. [14] E. Gelenbe, Product-form queueing networks with negative and positive customers, J. Appl. Probab., 28 (1991), 656–663. [15] V. Goswami and G. B. Mund, Analysis of discrete-time batch service renewal input queue with multiple working vacations, Comput. Ind. Eng., 61 (2011), 629–636. [16] P. G. Harrison, N. M. Patel and E. Pitel, Reliability modelling using G-queues, Eur. J. Oper. Res., 126 (2000), 273–287. [17] J.-H. Li and N. Tian, The discrete-time GI/Geo/1 queue with working vacations and vacation interruption, Appl. Math. Comput., 185 (2007), 1–10.
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Received September 2013; 1st revision February 2014; final revision May 2014. E-mail address: sgao−
[email protected] E-mail address:
[email protected] 
