## 1 Introduction

In recent times, queueing models with non-renewal arrival and service processes have been used to model networks of complex computer and communication systems. Traditional queueing analysis using Poisson processes is not powerful enough to capture the correlated nature of arrival (service) processes. The performance analysis of correlated type of arrival processes may be done through some analytically tractable arrival process viz., Markovian arrival process ( see Lucantoni et al. The has the property of both time varying arrival rates and correlation between inter-arrival times. To consider batch arrivals of variable capacity, Lucantoni [20] introduced batch Markovian arrival process ( . The processes and are convenient representations of a versatile Markovian point process, see Neuts [23] and Ramaswami [27]. Like the , Markovian service process is a versatile service process which can capture the correlation among the successive service times. Several other service processes, e.g., Poisson process, Markov modulated Poisson process () and phase-type () renewal process can be considered as special cases of . For details of , the readers are referred to Bocharov [3] and Albores and Tajonar [1]. The analysis of finite-buffer queue has been performed by Bocharov et al. [4]. The same queueing system with multiple servers such as has been analyzed by Albores and Tajonar [1]. Gupta and Banik [17] analyzed

queue with finite- as well as infinite-buffer capacity using a combination of embedded Markov chain and supplementary variable method.

During the last two decades, queueing systems with vacations have been studied extensively. For more details on this topic, the readers are referred to a recent book by Tian and Zhang [30] and references therein. An extensive amount of literature is available on infinite- and finite-buffer - and -type queueing models with multiple vacations, see first few chapters of [30], Karaesmen and Gupta [18] and Tian et al. [29]. However, limited studies have been done on queue with single vacation, see Chapter 4 of [30]. In the past few years, there is a growing trend to analyze queueing models with renewal or non-renewal arrival and service processes with server vacation, see, e.g., Lucantoni et al. [22] and Shin and Pearce [26]. The analysis of phase-type server vacation for the case of queue has been carried out by Chen et al. [7]. Baba [2] analyzes queue where the server is allowed to take working vacations as well as vacation interruptions. Samanta [28] discussed a discrete-time queue with single geometric vacation time. Recenly, Chaudhry et al. [9, 12] discussed queues with single and batch arrivals using the roots method, respectively.

In this paper, we carry out the analytic analysis of the queue with exponential single vacation through the calculation of roots of the denominator of the underlying vector generating function of the steady-state probabilities at pre-arrival epoch. In this connection, the readers are referred to Chaudhry et al. [9, 10, 12], Tijms [31] and Chaudhry et al. [11] who have used the roots method. The roots can be easily found using one of the several commercially available packages such as Maple and Mathematica. The algorithm for finding such roots is available in some papers, e.g., see Chaudhry et al. [11]. The purpose of studying this queueing model using roots is that we obtain computationally simple and analytically closed form solution to the infinite-buffer queue with the vacation time following exponential distribution. It may be remarked here that the matrix-geometric method (MGM) uses iterative procedure to get steady-state probabilities at the pre-arrival epochs. Further, it is well known that for the case of the MGM it is required to solve the non-linear matrix equation with the dimension of each matrix in this equation being the number of service-phases involved in a queue. In the case of the roots method, we do not have to investigate the structure of the transition probability matrices (TPM) at the embedded pre-arrival epochs. It may be mentioned here that the basic idea of correlated service was first introduced by Chaudhry [13]. Further, it may be remarked here that the analysis of the infinite-buffer queues with renewal input and exponential service time under exponential server vacation(s) has been carried out by Tian and Zhang [30], see Chapter 4. The queueing model that we are going to consider has non-renewal service () and exponential single vacation time. In addition, we discuss several other quantitative measures such as system-length distribution at a post-departure epoch and expected busy and idle periods. Later, we have established heavy- and light-traffic approximations as well as an approximation for the tail probabilities at pre-arrival epoch based on one root of the characteristic equation. Finally, some numerical results have been presented which may help researchers/practitioners to tally their results with those of ours.

## 2 Description of the model

Let us consider a single-server infinite-buffer queueing system with the server’s single vacation. The inter-arrival time of customers, the service time of a customer and the vacation time of the server are represented by the generic random variables (r.v.’s)

, and , respectively. Let denote the distribution function (D. F.) of the random variable with andthe corresponding probability density function (p.d.f.) and Laplace-Stieltjes transform (LST), respectively. The inter-arrival time

is assumed to have a general distribution with p.d.f. , D. F. and LST .Arrivals. The inter-arrival times are assumed to be independent and identically-distributed (i.i.d.) random variables and they are independent of the service process as well as vacation time. The inter-arrival time distribution

is an important special case of general distribution as the distribution possesses nice vector and matrix form representation. Several probability distributions such as Earlang, hyper-exponential, generalized Earlang, Coxian etc. can be treated as special cases of

-distribution. It may be noted here that -distribution is a special case of general distribution. If the inter-arrival times follow -type distribution with irreducible representation , where & are a vector and a matrix of dimension and , respectively, the p.d.f. and D.F. of inter-arrival times are given by(1) | |||||

(2) |

where is a non-negative vector and satisfies and is an vector with all its elements equal to 1. Throughout the paper we write a subscript as the dimension of the column vector and sometimes we write by dropping its subscript. The mean inter-arrival time during a normal busy period is given by

(3) |

Services. The customers are served singly according to the continuous-time Markovian service process () with matrix representation . The is a generalization of the Poisson process where the services are governed by an underlying -state Markov chain. For more details on , the readers are referred to recent papers by Chaudhry et al. [9, 12]. Let denote the number of customers served in units of time and the state of the underlying Markov chain at time with its state space . Then is a two-dimensional Markov process with state space . Average service rate of customers (the so called fundamental service rate) of the stationary is given by , where with denoting the steady-state probability of servicing a customer in phase . The stationary probability row-vector can be calculated from with , where . The customers are served singly according to a with steady-state mean service time .

Now, let us define as the matrix whose th element is the conditional probability defined as

Let be the matrices whose elements are . Then using Chaudhry et al. [9, 12], it may be derived that

(4) |

where and

Vacations. The server is allowed to take a single vacation whenever the system becomes empty. On return from a vacation if the server finds the system nonempty he will serve the customers present in the queue, otherwise the server waits for a customer to arrive and the system continues in this manner. For an exponential single vacation time represented by the r.v. , the LST, p.d.f. and D.F. are given as follows:

(5) |

where is assumed as the mean vacation time. The Vacation times are independent of the arrival as well as of the service processes. The traffic intensity is given by which is also independent of the vacation process.

## 3 The vector generating function of the number of customers served during an inter-arrival and other related probability matrices

Let denote the matrix of order whose th element represents the conditional probability that during an inter-arrival period customers are served and the service process passes to phase , provided at the initial instant of the previous arrival epoch there were at least customers in the system and the service process was in phase . Then

(6) |

If is the matrix-generating function of , where are the elements of , then, using (6) and (4), we get

(7) | |||||

The evaluation of the matrices can be carried out along the lines proposed by Lucantoni [20]. For the sake of completeness, we have given the procedure of obtaining , see Lucantoni [20]. One may note that the computation of using Equation (7) may be cumbersome. However, the following scheme may be efficient and is given by

(8) |

where may be obtained as proposed in Chaudhry et al. [10].

We further introduce a few more notations which are required for the rest of the analysis of the queueing model under consideration. Now from renewal theory of semi-Markov process, if we let and denote the remaining and elapsed times of an inter-arrival time, respectively, then

(9) |

which we use while deriving the expression for in Equation (12). Similar to the case of inter-arrival time, if we let denote the remaining vacation time, then

(10) |

and

(11) |

As above, we introduce the matrices of order whose th element represents the limiting probability that customers are served during an elapsed inter-arrival time of the arrival process with the service process being in phase , given that there were at least customers in the system with the service process being in phase at the beginning of the inter-arrival period. Then, from Markov renewal theory as given in Chaudhry and Templeton [8, p. 74-77], we have

(12) |

The matrices can be expressed in terms of the matrices and their relationship discussed in [12] is as follows:

(13) |

where is an indicator function and takes value 1 if the the condition is satisfied, otherwise it takes value 0.

Further, let be the conditional probability that at least customers are served in and the service process is in phase at the end of the th service completion, given that there were customers in the system and the service process was in phase at time . The probabilities then satisfy the equations

with the initial condition Rearranging the terms and taking the limit as , it reduces to

for , with the initial conditions . This system may be written in matrix notation as

(14) |

with .

Let denote the probability that exceeds
an inter-arrival time , then

(15) | |||||

Similarly, if we let denote the probability that exceeds , then

(16) |

Remark 2.1: and may be derived in slightly different way. In the following we present a slightly different derivation for and may be done similarly.

(17) | |||||

One of the frequently used inter-arrival time is phase-type renewal process which also serves as a special case of several other inter-arrival time distributions and is well-known in the literature. Therefore, we state the above formulae (17) and (16) for the case of phase-type inter-arrival time by the following theorem.

###### Theorem 3.1

In the following, we further define a few notations which are required to analyze the queueing model under consideration. If we let with , i.e., inter-arrival time is greater than remaining vacation time, then

(20) | |||||

where may be called excess inter-arrival time. Further, let us denote and as the remaining and elapsed times of the excess inter-arrival time random variable , respectively, then

(21) |

where is the mean of the random variable For an important special case of phase-type inter-arrival time the distribution of excess inter-arrival time is also phase-type which is proved in the following theorem.

###### Theorem 3.2

If inter-arrival times follow a -distribution with irreducible representation , where and are of dimension , then the distribution of is also phase-type with representation , where and are of dimension and are given by

(22) | |||||

(23) |

Proof: Simple algebraic calculation will give the proof.

Further, we introduce a few more matrices which are required for the analysis of this queueing model. With vacation ending, let denote the matrix of order whose th element represents the conditional probability that customers are served during an excess inter-arrival time and the service process passes to phase , provided at the initial instant of the previous arrival epoch there were at least customers in the system and the server was on a vacation with the service phase . Then

(24) |

Now, with vacation ending, let denote the matrix of order whose th element represents the probability that at least customers are served during an excess inter-arrival period and the service process is in phase with the server going on vacation at the end of the -th service completion, provided at the initial instant of previous arrival epoch there were exactly customers in the system and the server was on a vacation with the service phase . Then similar to the results derived above, we obtain

(25) |

Further, with vacation ending, let denote the matrix of order whose th element represents the limiting probability that customers are served during an elapsed excess inter-arrival time with the service process being in phase , given that there were at least customers in the system with the server being on a vacation with the service phase at the beginning of the inter-arrival period. Then we have the following expression for

(26) |

The relationships among the matrices , and can be derived as follows.

(27) |

and

(28) |

Further, using [12], it can be shown that

(29) |

Similarly, let denote the matrix whose th element represents the limiting probability that or more customers have been served during an elapsed excess inter-arrival time and the service process is in phase with the server going on vacation at the end of th service completion, provided at the previous arrival epoch the server was on a vacation with service phase and the arrival lead the system to state or more customers. Then we can write

(30) |

Now using the procedure discussed in [12], one may derive the following relation:

(31) |

Finally, we define a few notations which are required to analyze the queueing model under consideration. Let given that . It is needless to mention that since is exponentially distributed, the distribution of will be Erlang of order two, which is a phase-type distribution with two states. Let the phase type representation of be denoted as with To calculate the distribution function of , we need to define which denotes the probability that exceeds an inter-arrival time . Then,

(32) | |||||

(33) |

Similarly, if we let denote the probability that exceeds , then

(34) |

Further, if an inter-arrival time is following distribution with the above representation, then following the derivation as presented in Theorem 3.1, we have

(35) | |||||

(36) |

Now the distribution function of may be derived as

(37) | |||||

Similarly, let us denote and as the remaining and elapsed times of the random variable , then

(38) |

where is the mean of the random variable For an important special case of phase-type inter-arrival time, the distribution of is also phase-type whose representation can be obtained following Theorem 3.2 as , where

(39) | |||||

(40) |

Further, with another vacation ending, let denote matrix whose th element represents the conditional probability that customers are served during an excess inter-arrival time and the service process passes to phase with the server becoming idle (after completing service of the -th customer), provided at the initial instant of the previous arrival epoch there were at least customers in the system and the server was on a vacation with the service phase . Then

(41) |

Similarly, with the vacation ending, let denote the matrix of order whose th element represents the probability that at least customers are served during an excess inter-arrival period and the service process is in phase with the server becoming idle (after completing service of the -th customer), provided at the initial instant of previous arrival epoch there were exactly customers in the system and the server was on a vacation with the service phase . Then, we can define

(42) |

Also, with the vacation ending, let denote matrix whose th element represents the conditional probability that customers are served during an elapsed excess inter-arrival time and the service process passes to phase with the server becoming idle (after completing service of the -th customer), provided at the initial instant of the previous arrival epoch there were at least customers in the system and the server was on a vacation with the service phase . Then,

(43) |

Similarly, with the vacation ending, let denote the matrix whose th element represents the limiting probability that at least customers have been served during an elapsed excess inter-arrival time and the service process is in phase with the server becoming idle (after completing service of the -th customer), provided at the previous arrival epoch the server was on a vacation with service phase with the arrival leading the system to state customers. Then we can write

(44) |

One may note here that the matrices are required to obtain other matrices by using the above relations (27)-(28). The relationships among the matrices and can be similarly obtained as for the matrices and . These relationships are given below.

(45) | |||||

(46) |

(47) |

(48) |

(49) |

The matrices are calculated exactly the same way as we derive the matrices , see Chaudhry et al. [csa15]. For more information on , readers are referred to Bocharov [3], Albores and Tajonar [1] and Gupta and Banik [17].

## 4 Analysis of queue with single vacation

We consider a queueing system with single vacation as described above. In the following subsections we obtain steady-state distributions for this queueing system at different epochs considering .

### 4.1 Stationary system-length distribution at pre-arrival epoch

Consider the system just before arrival epochs which are taken as embedded points. Let be the time epochs at which arrivals occur and the time instant before . The inter-arrival times are i.i.d.r.v.’s with common distribution function . The state of the system at is defined as where is the number of customers present in the system including the one currently in service. Whereas denotes phase of the service process and or 1 indicates that the server is on vacation or busy . In the limiting case, we define the following probabilities:

where represents the probability that there are customers in the system just prior to an arrival epoch of a customer when the server is on vacation with phase of the service process . Similarly, denotes the probability that there are customers in the system just prior to an arrival epoch of a customer when the server is in a busy (when ) or dormant (when ) state with phase of the service process . Let and be the row vectors of order whose -th components are and respectively.

Observing the state of the system at two consecutive embedded points, we have an embedded Markov chain whose state space is equivalent to Observing the system at two consecutive embedded Markov point, we have the following system of vector difference equations

(50) | |||||

(51) | |||||

(52) | |||||

(53) |

Multiplying (53) by , summing from to , after adding (52) and using the vector-generating function , we obtain

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