#### Abstract

We consider a finite-buffer single server queueing system with queue-length dependent vacations where arrivals occur according to a batch Markovian arrival process (BMAP). The service discipline is P-limited service, also called E-limited with limit variation (ELV) where the server serves until either the system is emptied or a randomly chosen limit of customers has been served. Depending on the number of customers present in the system, the server will monitor his vacation times. Queue-length distributions at various epochs such as before, arrival, arbitrary and after, departure have been obtained. Several other service disciplines like Bernoulli scheduling, nonexhaustive service, and E-limited service can be treated as special cases of the P-limited service. Finally, the total expected cost function per unit time is considered to determine locally optimal values of or a maximum limit of as the number of customers served during a service period at a minimum cost.

#### 1. Introduction

Queueing systems with vacations have found wide applications in the modelling and analysis of computer and communication networks and several other engineering systems in which single server is performing more than one type of jobs. Modelling such systems as single server queues with vacations allows one to analyze each queue in relative isolation since the time the server is attending to other jobs in the system may be modeled as vacation. For more details and versatile implementation of vacation models, one can refer to the comprehensive survey by Doshi [1]. Vacation models are distinguished by their scheduling disciplines, that is, the rules governing when a service stops and a vacation begins. Several service disciplines in combination with vacations are possible, for example, exhaustive, limited, gated, exhaustive limited (E-limited), gated limited (G-limited), and so forth. In fact, there is an extensive amount of literature available on infinite- and finite-buffer M/G/1 and GI/M/1 type vacation models that can be found in Takagi [2, 3], Tian and Zhang [4], and so forth.

Traditional teletraffic analysis using Poisson process is not powerful enough to capture the correlated and bursty nature of traffic arising in the present high-speed networks where packets or cells of voice, video, images and data are sent over a common transmission channel on statistical multiplexing basis. The performance analysis of statistical multiplexers whose input consists of superposition of several packetized sources has been done through some analytically tractable arrival process, for example, Markovian arrival process (MAP); see Lucantoni et al. [5]. To consider batch arrival of variable capacity, Lucantoni [6, 7] introduced batch Markovian arrival process (BMAP) which is a convenient representation of the versatile Markovian point process; see Neuts [8]. BMAP includes many familiar input processes such as Markovian arrival process (MAP), Markov modulated Poisson process (MMPP), PH-type renewal process, interrupted Poisson process (IPP), Poisson process. Later on, some researchers have analyzed queueing systems of finite- or infinite-buffer with MAP or BMAP arrival, for example, Akar and Arikan [9], Nishimura [10], Lee et al. [11], Dudin [12], Dudin et al. [13], and so forth. Asymptotic analysis of MAP/G/1 type queueing system has been performed by a few researchers, for example, Kim et al. [14], Gouweleeuw [15], and so forth. Chydzinski [16] obtained probability distribution of time to reach a buffer capacity in a BMAP/G/1 system. Further, Matendo [17], Ferrandiz [18], and so forth have discussed BMAP/G/1 queue with vacations. As queueing analysis of finite systems is more realistic in applications than infinite systems, the detailed study of finite capacity vacation model BMAP/G/1/*N* queue under exhaustive service discipline was performed by Niu et al. [19] where they have included setup time, close-down time, and single/multiple vacations.

In this paper, we analyze BMAP/G/1/*N* queue where the server serves until either the system empties or a randomly chosen limit of customers has been served, whichever occurs first. The server then goes for a vacation of random length of time depending on the number of customers remaining in the system. For example, the duration of vacation time will be short if there are a large number of customers present in the system, whereas duration of vacation time will be long if there are very few number of customers who are waiting for service. Returning from a vacation if the queue is empty or the limit is chosen, then the server takes another vacation of length depending on the number of customers present in the preceding vacation termination instant. The service discipline analyzed in this paper is known as P-limited service; see Takagi [3, page 398]. P-limited service was earlier studied by LaMaire [20, 21] for the case of M/G/1 infinite and finite buffer queue, respectively, and he uses the term E-limited with limit variation (ELV) for P-limited service. The analysis of MAP/G/1/*N* queue with limited service discipline is carried out by Blondia [22] and Gupta et al. [23]. Banik et al. [24, 25] discussed BMAP/G/1/*N* queue under limited service disciplines. Queue-length dependent vacation schedule has been studied by Shin and Pearce [26] for the case of infinite-buffer BMAP/G/1 system with queue-length dependent vacation and Bernoulli scheduling. In case of finite-buffer batch arrival queue, a batch which upon arrival does not find enough space in the buffer is either fully rejected or a part of that batch is rejected. Some queueing protocols are based on the former strategy, and they are known as the total batch acceptance/rejection policy. The latter one is known as the partial batch acceptance/rejection policy. For more details on different type of batch acceptance/rejection policies in the context of finite-buffer BMAP/G/1 queue, readers are referred to Dudin et al. [13]. In this paper, we analyze partial batch acceptance strategy and obtain queue-length distributions at various epochs using a combination of embedded Markov chain and the supplementary variable method. One may note here that results of several vacation policies, namely, Bernoulli scheduling, exhaustive service, pure limited service and E-limited, service can be obtained as special cases of P-limited service model. Regarding the decision making, study of cost model for a practical situation is useful to optimize the cost, for example, see Lee and Srinivasan [27], Lee and Park [28], and Wang and Ke [29]. For this reason, we calculate the total expected cost function per unit time to determine locally optimal values of and of . At the end we provide some numerical results using the analytical results obtained in this paper. Finally, some conclusions and future research directions are discussed.

#### 2. Description of the Model

Let us consider a BMAP/G/1/*N* queue wherein the server is allowed to serve a maximum of customers during a busy period, and then the server goes for a vacation whose duration depends on the number of customers present in the system. Here, is the capacity of queue excluding the one in service. That is, there is different type of vacations. The server may go for any one of the th type of vacation. If the queue has been emptied after a service, then the server will take ()-type of vacation. On the other hands if customers have been served during a service period and a total of customers remain in the system, then the server will take th type of vacation. After returning from a vacation, if the queue is empty or a limit *zero* is chosen, the server immediately goes on another vacation of th type if there are a number of customer who remain after the end of previous vacation completion instant. The limit is a random variable whose mass function is denoted by . The limit for the number of customers to be served during a service interval (i.e., a busy period) is determined at the preceding vacation termination instant. The sequence of limits chosen at these instants is independent and identically distributed random variables (i.i.d.r.vs.). By suitably choosing , one can obtain results for several other service disciplines, including Bernoulli scheduling, exhaustive service, pure limited service and E-limited service; for example, if we take and , then the service system is equivalent to E-limited service. If in the above assumption, the service system is equivalent to exhaustive service system (= pure limited service). Bernoulli scheduling system can be thought of as a special case of P-limited service system with , and , for .

The input process is BMAP where arrivals are governed by an underlying -state Markov chain which is characterized by matrices , , where th element of is the state transition rate from state to state in the underlying Markov chain without an arrival and th element of , , is the state transition rate from state to state in the underlying Markov chain with an arrival of batch size . The matrix has nonnegative off-diagonal and negative diagonal elements, and the matrix , , has nonnegative elements. Let denote the number of arrivals in , and let be the state of the underlying Markov chain at time . Then, is a two-dimensional Markov process of BMAP with state space . The infinitesimal generator of BMAP is given by As is the infinitesimal generator of the BMAP, we have , where is an column vector with all its elements equal to 1. If â€™s dimension is other than , we denote by regardless of its dimension. Further, since is irreducible infinitesimal generator of the underlying Markov chain , there exists a stationary probability vector such that , . Then, the average arrival rate and average batch arrival rate of the stationary BMAP are given by and , respectively, where .

Let us define as matrix whose th element is the conditional probability that customers accepted in the queueing system whose maximum capacity is excluding the one in service. Since we deal with finite-buffer queue with of batch arrival, one may consider different kinds batch acceptance/or rejection strategies; see Dudin et al. [13]. The matrices satisfy the following system of difference-differential equations for the case of partial batch acceptance strategy: with , where is an identity matrix of order for the above three cases and .

Let be the distribution function (DF) {probability density function (pdf)} [Laplace-Stieltjes transform (LST)] of the service time of a typical customer. Similarly, is the DF {pdf} [LST] of a typical vacation time of the server where denotes the number of customers present in the system at the starting point of a vacation. The mean service [vacation] time is , where is the th derivative of at . The service, vacation times are assumed to be i.i.d.r.vs. and each is independent of the arrival process. The traffic intensity is given by . Further, let be the probability that the server is busy. The state of the system at time is described by the r.vs., namely,(i)(ii) is the number of customers present in the queue excluding the one in service, (iii) is the state of the underlying Markov chain of BMAP, (iv) is the remaining service time of the customer in service, (v) is the remaining vacation time of the server, whereas this vacation started with customers in the system.We define for the joint probabilities of queue-length , state of the server , and the remaining service (vacation) time , respectively, by As we shall discuss the model in limiting case, that is, when the above probabilities will be denoted by and . Let us further define the row vectors of order as follows: where denotes the arbitrary epoch probability that there are customers in the queue and the state of the arrival process is when the server is serving the th customer whose remaining service time is in the present busy period consisting of the sum of the service times of or lesser number of customers. Similarly, denotes the arbitrary epoch probability that customers are in the queue and state of the arrival process is when the server is on vacation of th type with remaining vacation time .

#### 3. Queue-Length Distributions at Various Epochs

##### 3.1. Queue-Length Distribution at Service Completion and Vacation Termination Epochs

Consider the system at service completion/vacation termination epochs which are taken as embedded points. Let be the time epochs at which either service completion or vacation termination occurs. , denotes the time epoch just after a service completion or vacation termination occurs. The state of the system at is defined as , where , and are defined earlier. Therefore, indicates that the embedded point is th type vacation termination instant which started with customers in the system, whereas indicates that the embedded point is a service completion instant of the th customer in the present busy period consisting of services of or lesser number of customers. In limiting case, these probability distributions are where represents the probability that there are customers in the queue and the state of the arrival process is at service completion epoch of the th customer in the service period consisting of services of at the most customers. Similarly, represents the probability that there are customers in the queue and the state of the arrival process is at th type vacation termination epoch. Further, let us denote the row vectors of order :

Let , , , denote an matrix whose th element represents the conditional probability that customers have been accepted during a service (th type vacation, i.e., vacation started with customers) time of a customer (the server), and the underlying Markov chain is in phase at the end of the service (th type vacation) time given that the underlying Markov chain was in phase at the beginning of the service (th type vacation). and as the matrices of mass functions as defined above are given by Obviously, , will be equal to null matrix of order and similarly, for , .

Observing the system immediately after each embedded point, we have the transition probability matrix (TPM) â€‰ with four block matrices of the following form: where describes the probability of transitions among the service completion epochs. A typical service completion epoch will be denoted by the triplet of which we first consider the change in , that is, , and the second element of , that is, to describe the construction of the first block of the TPM. Then the elements of can be written as follows: where , , , are all matrices of order . is the null matrix of order given in the suffix. The matrices and are given by gives the probability of transition from any service completion epoch to the next vacation termination epochs. Vacation termination epochs are classified by the of which we consider the change in , that is, and in case of vacation termination epoch , that is, the number of customers present at the last service completion instant. The structure of is given by of TPM gives the probability of transition from every vacation termination epoch to the next service completion epochs. This block is of the given form following: of the TPM describes the probability of transitions among vacation termination epochs. This block matrix is of the following form:

The evaluation of the matrices , in general, for arbitrary service (vacation) time distribution requires numerical integration or infinite summation, and it can be carried out along the lines proposed by Lucantoni [6] for BMAP arrival. For the sake of completeness, in the following we present the derivation of finding the matrix . As presented in [6], applying uniformization argument to the matrices is of the given form following: where and is given by Now, substituting the values ofâ€‰ from (18) in (8), we obtain where . Similarly, the matrix can be found out for any arbitrary vacation time distribution function . However, when the service (vacation) time distributions are of phase type (PH-distribution), these matrices can be evaluated without any numerical integration; see Neuts [30, pages 67â€“70]. It may be noted here that various service (vacation) time distributions arising in practical applications can be approximated by PH-distribution. The following theorem gives a procedure for the computation of the matrices .

Theorem 1. *If follow PH-distribution with irreducible representation , where and are of dimension , then the matrices are given by
**
where
**
with , and the symbol denotes the Kronecker product of two matrices. Similarly, let follow a PH-distribution with irreducible representation , where and are of dimension , then the matrices are given by
**
where
**
with .*

*Proof. *See Neuts [30, pages 67â€“70], Banik and Gupta [25], and so forth.

The unknown probability vectors and can be obtained by solving the system of equations: and . We use GTH algorithm, see Grassmann et al. [31], for solving the system of equations.

Considering the departure of a customer as an embedded point excluding vacation termination epochs, one may obtain queue-length distributions at departure epoch. Distributions of number of customers in the queue at service completion and departure epochs are proportional. Let denote a row vector whose th element represents steady-state probability that there are customers in the queue and phase of the arrival process is at departure epoch of the th customer in the service period consisting of services of or lesser number of customers. Since is proportional to and , we get

##### 3.2. Queue-Length Distribution at Arbitrary Epoch

To determine queue-length distribution at arbitrary epoch, we will develop relations between distributions of number of customers in the queue at service completion (vacation termination) and arbitrary epochs. Supplementary variable method has been used, and for that we relate the states of the system at two consecutive time epochs and , and using probabilistic arguments, we have a set of partial differential equations for each phase , . Taking limit as and using matrices and vector notations, we obtain where are the respective departure rates of customers (vacation termination rates of the server). Let us define the Laplace transform of and as so that Multiplying (26) by and integrating with respect to. over to , we obtain Using equations (29)â€“(39), we will derive certain results in the form of lemmas and theorems.

Lemma 2. *Consider the following:
**
The left hand side is the mean number of entrances into the vacation states per unit of time, and the right hand side is the mean number of departure from the vacation states per unit of time. *

*Proof. *Setting in (29)â€“(34), using (28), and then post-multiplying all the equations by the vector and adding them, using after simplification, we obtain the result.

Theorem 3. *Consider the following:
** denotes the mean number of service completions per unit of time, and multiplying this by will give . Similarly, the other result can be interpreted.*

*Proof. *Differentiating (29)â€“(34) w.r.t. , setting in those equations and postmultiplying by , adding them using and Lemma 2, after simplification, we obtain (41). Similarly, post-multiplying (35)â€“(39) by , differentiating these equations w.r.t , and setting , after some algebraic manipulation, we obtain .

###### 3.2.1. Relation between Queue-Length Distributions at Arbitrary and Service Completion (Vacation Termination) Epochs

We first relate the service completion (vacation termination) epoch probabilities, and , with the rates and which are given by and similarly, where . Employing the above relations, we will determine arbitrary epoch probabilities in terms of service completion or vacation termination epoch probabilities. Setting in the (29)â€“(33) and (35)-(36), using (43) and (44), we obtain the following relations: It may be noted here that we do not have such expression for and . However, one can compute and by using Theorem 1 and is given by and , respectively. Though the vectors and are not obtained componentwise, and are sufficient to determine key performance measures (Section 4).

Lemma 4. * (probability that the server is busy) is given by
*

*Proof. *Let be the random variable denoting the length of busy {idle} period and be the mean length of a busy {idle} period, then we have
Applying Theorem 3, and then dividing numerator and denominator by , using (43) and (44), we can write the pervious ratio as
The above ratio yields the result.

*Remark 5. *One may note here that is frequently needed for calculation of the state probabilities, and it can be obtained by using and (43) in (41).

Let denote the row vector of order whose th component is the probability of customers in the queue at arbitrary epoch, and state of the arrival process is . is given by

##### 3.3. Queue-Length Distributions at Prearrival Epoch

Let be the vectors whose th components are given by which gives the probability that an arrival finds customers in the queue, and the arrival process is in state at the postarrival instant. The vectors is given by

#### 4. Performance Measures

In this section, we discuss the various performance measures which are often needed for investigating the behavior of a queueing system. As the state probabilities at which departure, arbitrary and prearrival epochs are known, the corresponding mean queue-lengths can be easily obtained. For example, the average number in the queue at any arbitrary epoch is , the average number in the queue when the server is busy is , and the average number in the queue when the server is on vacation is . It may be remarked here that for large values of , the computation of may pose some problems because of the storage of large number of probability vectors . The problem can be resolved by using main frame computer and applying efficient memory management technique. One useful performance measure is the blocking probability which is discussed below.

##### 4.1. Blocking Probabilities

Since prearrival epoch probabilities are known, the blocking probability of the first customer of an arriving batch is given by Let be the matrix of order whose element is the probability that the position of an arbitrary customer in an arriving batch is with phase changes from to . The probability that an arbitrary customer belongs to a batch of size is given by . Hence, the position of an arbitrary customer in a batch of size is with corresponding phase changes is equal to , , where denotes the probability that an arbitrary customer belongs to a batch of size . Therefore, Hence, the blocking probability of an arbitrary customer is given by Finally, the blocking probability of the last customer of a batch is given by One may note that mean waiting time (in queue) of an arbitrary customer can be obtained using Littleâ€™s law, and it is given by , where is the mean effective arrival rate and is given by

*Remark 6. *One may note here that can be obtained another way as described below. In the steady-state mean effective arrival rate of customers should be equal to the total mean departure rate of customers. From this argument, we obtain
Now, using (41) and (46) in (56), we have
Equations (55) and (57) can be used for comparison while performing numerical experiments.

##### 4.2. Optimal Policy

In this subsection, we develop the total expected cost function per unit time for this queueing system in which and are decision variables. We construct a similar cost structure that has been widely used in many works; see Lee and Srinivasan [27], Lee and Park [28], Wang and Ke [29], and so forth. For this we need explicit expression for expected length of busy and idle period. Let us assume that is the random variable denoting the length of a busy cycle and is the mean length of busy cycle; then . From the definition of the carried load (the fraction of time that the server is busy), it can be written as We first discuss the expected busy period which is comparatively easy to evaluate. Let denote the number of customers in the queue at time , let be the state of the server, that is, busy or idle , and let represent the th customer in service in the service period consisting of or lesser number of customers. Then, enters the set of busy states, at the termination of an idle (vacations) period. The conditional probability that enters , given that enters or is therefore where . Now, enters irrespective of customersâ€™ arrival during a service time, which may happen in expected time . Thus, we have Using (59) in (58), we obtain

###### 4.2.1. Total Expected Cost Function

We consider a cost function per unit time for this BMAP/G/1/*N* queueing system under P-limited service disciplines with queue-length dependent vacations, in which and are decision variables. Our objective is to determine the optimum value of the control parameter, say and , so as to minimize the expected value of this cost function. Let us define the following costs: â€‰â€‰ = holding cost per unit time per customer present in the system; â€‰â€‰â€‰ = cost incurred per unit time for keeping the server off; â€‰â€‰ = cost incurred per unit time for keeping the server on;â€‰â€‰ = start-up cost per unit time for turning the server on;â€‰â€‰ = shut-down cost per unit time for turning the server off; â€‰â€‰ = a fixed cost incurred per unit time for every lost customer.

Using the definitions of each cost element listed above, the total expected cost function per unit time is given by From the pervious formula, locally optimal values may be selected for a fixed value of as follows. For each selection of , the local optimum value of , , is determined by satisfying the following inequality: One may be interested to obtain locally optimal values and using following procedure. We first select an interval for buffer capacity and a set of limit distributions; therefore, we can evaluate for every possible ordered pairs , and . Finally, comparing each of them, we can determine locally optimal values and for which is a minimal one. This completes analytical analysis of the queueing model under consideration. Now, we present discussion of numerical results in Section 5.

#### 5. Numerical Results

To demonstrate the applicability of the results obtained in the previous sections, some numerical results have been presented in the form of graphs showing the nature of some performance measures against the variation of some critical model parameters. We have conducted an experiment on the BMAP/PH/1/*N* queue with queue-length dependent multiple vacations (vacation time follows PH-distribution) for the following input parameters.

The 2-state BMAP representation is taken as , , , , where , , and , and are null matrices of order 2. The PH-type representation of a vacation time is taken as , with . PH-type representation of service time is taken as , with , . We take the maximum value of the limit , and the limit mass function is taken as uniform: , for . Figure 1 shows the effect of on for the above described BMAP/PH/1/*N* queue where varies from 2 to 28. We have conducted this experiment two times to judge the effect of queue-length dependent multiple vacations over normal multiple vacations. Firstly, we assume that the rest of the vacation time distributions , which are not given above, are the same as service time distribution given above. Secondly, we consider . The reason behind the above assumption is that for the case of queue-length dependent multiple vacation when the system is empty, server stays longer time in vacation, whereas vacation duration is much smaller when any customer presents at the starting point of a vacation.

In Figure 1, for the case of queue-length dependent multiple vacation queue, as buffer space increases, the model behaves as infinite-buffer queue. It is obvious that a finite-buffer queue behaves like the corresponding infinite-buffer queue as the buffer size increases. Consequently, asymptotically approaches towards . We have used our method described in Section 3.1 to evaluate service completion and vacation termination epoch probabilities for BMAP/PH/1/*N* model. After that we have evaluated the quantity which closely matches with (upto four decimal places) when . Moreover, one may note that for the case of normal multiple vacation model as increases, also approaches towards with taking lesser value than queue-length dependent multiple vacation queue.

In Figure 2, we have plotted the blocking probability of an arbitrary customer against in the above described BMAP/PH/1/*N* for above described queue-length dependent vacation policy as stated in Figure 1. It is observed that as , increase blocking probabilities asymptotically approach towards zero for both the cases of queue-length dependent multiple vacations and normal multiple vacations. is slightly higher for the case of normal multiple vacations than queue-length dependent multiple vacations for any . One may observe from these experiments that queueing system behaves quite differently for the case of queue-length dependent multiple vacations.