Abstract
We consider a two-heterogeneous-server queueing system with Bernoulli vacation in which customers arrive according to a Markovian arrival process (MAP). Servers returning from vacation immediately take another vacation if no customer is waiting. Using matrix-geometric method, the steady-state probability of the number of customers in the system is investigated. Some important performance measures are obtained. The waiting time distribution and the mean waiting time are also discussed. Finally, some numerical illustrations are provided.
1. Introduction
Queueing systems that allow servers to take vacation have a wide range of applications in many engineering systems such as flexible manufacturing environments, production, computers, communication networks, and telecommunication systems. Servers' vacations are useful for the systems in which the servers want to utilize their idle times for different purposes. For instance, servers' vacations may be due to lack of work, servers' failure, or some other tasks being assigned to the servers which occur in applications like computer maintenance and testing, preventive maintenance job in a production system, priority queue, and so forth (see [1]).
In general, queueing systems with vacations can be classified as systems with a single server or multiple servers involving single vacation or multiple vacations. The servers may take vacations at a random time, after serving utmost customers (-limited) or after all the customers in the queue are served (exhaustive). Also, depending on the applications, when the server finishes a vacation and there is no customer to be served, the server may take another vacation (multiple vacation model) or it may wait ready to serve until a new customer arrives (single vacation model). The queueing systems with single and multiple vacations have been first investigated by Levy and Yechiali [2].
Another interesting and important vacation model is Bernoulli vacation scheduling service. The Bernoulli vacation scheduling service discipline has been proposed by Keilson and Servi [3]. In this service discipline, when the server visits a queue, at least one customer, if any, is served. After the completion of its service, the server switches to the next vacation if there are no customers. If customers remain, however, in the queue, the next customer is served with probability , and the server repeats this procedure, or the server switches to the next vacation with probability The important merit of the Bernoulli vacation scheduling service discipline is the existence of a control parameter By adjusting the value of we can control the congestion of the system. We note that when , the Bernoulli vacation scheduling service discipline is equivalent to the exhaustive service discipline, and corresponds to the 1-limited service discipline. Servi [4] has proposed an approximate procedure to calculate the average waiting time of an asymmetric polling system under the Bernoulli scheduling service discipline. Further, Tedijanto [5] has investigated in detail the stochastic behavior of a polling system under the Bernoulli vacation scheduling service discipline and has obtained the average waiting time of a symmetric system. For comprehensive and excellent surveys/monographs on queueing systems with server vacations, see [1, 6–8], and the references therein.
In the literature, there are only a limited number of studies on multiple server vacation models. The multiserver queue with exponentially distributed vacations is first studied by Levy and Yechiali [9]. Using partial generating function technique, the system size has been obtained. Vinod [10] and Kao and Narayanan [11] have discussed the queue with multiple vacation of the servers using matrix geometric approach. Further, Chao and Zhao [12] have investigated the multiserver vacation models of both synchronous (servers taking the same vacation together) and asynchronous types and provided some algorithms for computing the stationary probability distributions and expected performance measures. The queue with phase-type synchronous vacations has been analyzed by Tian and Li [13]. An queue with multiple vacations and 1-limited service has been discussed by Tyagi et al. [14]. Recently, Krishna Kumar and Pavai Madheswari [15] have analyzed an queue with Bernoulli vacation scheduling service.
The study on multiserver queueing system generally assumes the servers to be homogeneous in which the individual service rates are the same for all the servers in the system. This assumption may be valid only when the service process is mechanically or electronically controlled. In a queueing system with human servers, the above assumptions can hardly be realized. It is common to observe servers rendering service to identical jobs at different service rates, that is, the service time distributions may be different for different servers. As noted earlier, the study of vacation queueing system incorporates the secondary jobs in the modelling. The analysis of queueing systems with heterogeneous servers and related vacation models helps to study the impact of secondary jobs on system performance.
Neuts and Takahashi [16] have pointed out that for the queueing systems with more than two heterogeneous servers, analytical results are intractable and one may have to use algorithmic approach to study even the asymptotic behavior of the performance measures like stationary distribution of system size and tail probability of waiting time of a customer in the system.
Based on the above observations, in this paper, we consider a vacation queueing system in which customers arrive according to a Markovian arrival process (MAP) involving only two heterogeneous servers availing Bernoulli vacations. The service times follow phase-type distributions (PH distributions), and vacation times of servers follow exponential distributions with heterogeneous vacation rates.
The organization of the paper is as follows. In Section 2, after recalling the definition of the MAP used as input process, we describe the model under investigation. We also study the steady-state probability of system size and some important and interesting performance measures of the system in this section. Next, the waiting time distribution and its related characterizations are discussed in Section 3. Section 4 presents some illustrative numerical examples of the system performance measures. Conclusions are given in Section 5.
2. Source Characterization and Model Description
In this section, we provide information about the Markovian arrival process (MAP) which has been assumed for the customers arrival process. We also describe the model under study.
2.1. Markovian Arrival Process
In order to allow bursty type traffic in our system, we have chosen the input process as the MAP. The MAP is particularly a tractable point process which is in general nonrenewal and which lends itself very well to modelling bursty arrival processes commonly arising in communications (see [17]). Further, the MAP has made it much easier to develop numerically tractable queueing models which take correlation into account (see [18, 19]). The MAP is a rich class of processes which includes the phase-type renewal process and the Markov-modulated Poisson process as special cases.
An MAP can be considered as a Markov process on the state space with an infinitesimal generator having the structurewhere and are matrices, has negative diagonal elements and nonnegative off-diagonal elements, is nonnegative, and , where is an -dimensional column vector of ones. An arrival process can be associated with this Markov process as follows. An arrival occurs whenever there is a state transition into a state corresponding to a block, and there is no arrival otherwise. Here, represents the number of arrivals in , and represents an axillary state or phase variable. Let be the stationary probability vector of the generator . That is, is the unique vector satisfying and The fundamental arrival rate of this process is given by The constant is the expected number of arrivals per unit time in the stationary version of the MAP.
2.2. Model Description and Analysis
Consider a two-server vacation queueing system in which the service time distributions of the servers are not identical. Customers arrive singly, according to MAP, to an infinite waiting space and form a single waiting line. Customers are served under the first-come-first-served (FCFS) discipline. There are two heterogeneous servers. The service times of server-1 and server-2 are assumed to be phase-type distributions (PH-distributions) PH(1) and PH(2) with representations of order with and of order with , respectively. We follow matrix formalism of PH-distributions as represented by Neuts [20]. The servers take Bernoulli vacation scheduling service as described by Keilson and Servi [3], that is, after each service completion, the server- takes a vacation with probability and with probability , it starts serving the next customer, if any, in the system. If the system is empty, the servers always take vacations. At the end of a vacation period, service commences if a customer is present in the queue. Otherwise, the server takes another vacation immediately and continues in the same manner until it finds at least one customer waiting upon returning from a vacation (multiple vacations). This process holds good for both servers 1 and 2.
The length of vacations' duration of the two servers is assumed to be independent and identically distributed exponential random variables with parameters, for , and is independent of the length of the service times and the arrival process. The Bernoulli vacation scheduling service queueing model with heterogeneous servers under consideration can be formulated as a continuous time Markov chain (CTMC). By appropriately keeping track of the various states, such as the number of customers in the system, the phase of the arrival process, the phases of the service processes of server-1 and server-2, and the status of the server-1 and server-2, the state space of the Markov chain describing the model is given as follows.
The set of states represents that there are customers in the system, both servers are on vacation and the phase of the arrival process is the set of states indicates that there are customers in the system, server-1 is busy in the system serving a customer in phase while server-2 is on vacation, and represents the phase of the arrival process; the set of states indicates that there are customers in the system, server-2 is busy in the system serving a customer in phase while server-1 is on vacation, and the arrival process is in phase the set of states represents that there are customers in the system, both servers 1 and 2 are busy in the system serving the customers in phases and , respectively, and the arrival process is in phase
We define levels as the set of states:where the elements of the sets are arranged in lexicographical order.
In the sequel, we use the following notations.
: a column vector of order with all its elements equal to 1.
: a column vector of order with all its elements equal to 1.
: a column vector of order with the 1st elements equal to 1, and other elements are zeros.
: a column vector of order with the to elements equal to 1, and other elements are zeros.
: a column vector of order with to elements equal to 1, and other elements are zeros.
: a column vector of order with the to elements equal to 1, and other elements are zeros.
: a column vector of order with the to elements equal to 1, and other elements are zeros.
: a column vector of order with to elements equal to 1, and other elements are zeros.
: a column vector of order with to elements equal to 1, and other elements are zeros.
Using elementary arguments, the infinite-state Markov chain for the model under study has a transition rate matrix which has a block-tridiagonal structure given byThe entries of are given by the following block matrices. The boundary matrices are defined byThe square matrices , and are of order and are given bywhere and are the Kronecker product and Kronecker sum, respectively (see [21]).
We now derive the condition for the system to reach steady state. To accomplish this, we define Then, the matrix can be written as where
It is clear that the order of the square matrix is , and it is an irreducible infinitesimal generator matrix [20, page 82], and so there exists stationary probability vector of satisfying and The vector is denoted by whose components areHere, is the stationary probability vector when both servers are on vacation, is the stationary probability vector when server-1 is busy and server-2 is on vacation, is the stationary probability vector when server-2 is busy and server-1 is on vacation and is the stationary probability vector when both servers are busy. Further, is the stationary probability that there are customers in the system, both servers are on vacation, and the underlying arrival process MAP is in phase is the stationary probability that there are customers in the system with the underlying arrival process MAP in phase the server-1 is busy serving a customer with the underlying service process PH(1) in phase , and server-2 is on vacation; is the stationary probability that there are customers in the system with the underlying arrival process MAP in phase , server-2 is busy serving a customer with the underlying service process PH(2) in phase , and server-1 is on vacation. Finally, is the stationary probability that there are customers in the system with the underlying arrival process MAP in phase both server-1 and server-2 are busy serving customers with the underlying service processes PH(1) in phase and PH(2) in phase respectively.
As the Markov process has the QBD structure, it is well known [20, page 83] that the standard drift conditionis the necessary and sufficient condition for the stability of a QBD process.
After some algebraic manipulation, the stability condition turns out to be
Remark 2.1. As the LHS of (2.9) is the rate of flow into the system, and RHS of (2.9) is the maximum rate of flow out of the system, (2.9) should be a necessary and sufficient condition for positive recurrence.
Under stability condition (2.9) of the system, there exists the steady-state probability vector satisfying The stationary probability vector partitioned as , is given byand by the normalizing equationwhereand for ,Here, refers to the joint probability that there are customers in the system, both servers are on vacation and corresponds to the phase of the arrival process; refers to the joint probability that there are customers in the system, server-1 is busy serving a customer, server-2 is on vacation, corresponds to the phase of the arrival process, and corresponds to the phase of the service process PH(1) provided by server-1; refers to the joint probability that there are customers in the system, server-2 is busy serving a customer, server-1 is on vacation, corresponds to the phase of the arrival process, and corresponds to the phase of the service process PH(2) provided by server-2; refers to the joint probability that there are customers in the system, both servers are busy serving customers, corresponds to the phase of the arrival process, and correspond to phases of the underlying service processes PH(1) and PH(2) provided by server-1 and server-2, respectively. is the identity matrix of order , and the matrix is the minimal nonnegative solution with spectral radius less than 1, that is, of the matrix quadratic equation [20, pages 82-83]:Since the system is stable, and the square matrices , and are of order is also a square matrix of order and is obtained from the above matrix quadratic equation and from the following relation: Equation (2.18) implies that the rate of transition from a state with customers to a state with matches the transition rate from to
The matrix is approximated by the following iterations:
The values of will converge, since and are positive. Hence, after each iteration, the elements of will increase monotonically. Iteration will be continued until is satisfied, where is the value of in the th iteration, and is the degree of accuracy required. The accuracy may be checked by (2.18) with [20].
The boundary probability vectors and the probability vectors , can be obtained form (2.10) to (2.14). These steady-state joint probability vectors are then used to find the following system performance measures.
2.3. Performance Measures
We will list some important performance measures which are used to bring out the qualitative behavior of the queueing model under study.
(1) The mean and second moments of the number of customers in the system can be obtained exactly aswhere is the system size at an arbitrary time. The variance of the system can also be found.(2) The probability that no customer in the system and both servers are on vacation is given by(3)The probability that both the servers are on vacation is given as(4) The probability that server-1 is busy serving a customer and server-2 on vacation is given by(5) The probability that server-2 is busy serving a customer and server-1 on vacation is given as(6) The probability that both servers are busy is obtained as(7) The mean number of customers present in the system when both the servers are on vacation is given as(8) The mean number of customers present in the system when both the servers are busy serving customers is given as(9) The mean number of customers present in the system when server-1 is busy and server-2 on vacation is given as(10) The mean number of customers present in the system when server-1 on vacation and server-2 being busy is obtained as
3. Waiting Time Distribution
We now analyze the waiting time of an arriving customer in the queue by first-passage time analysis. We then show how the mean waiting time can be determined.
Let , be the distribution function for the waiting time in the queue of an arriving (tagged) customer, that is, denotes the probability vector of order that an arriving tagged customer has to wait utmost time units until it is served. For the multiserver queue with Bernoulli vacation scheduling service, we have , because an arrival must either wait for a service completion or a vacation termination of servers. States corresponding to the number of customers are present in the system , and an absorbing state forms the state space of the CTMC. Thus, the state space of the CTMC is . Note that the customer arrival process is not required, since we are discussing an FCFS queue discipline, and hence, customers that arrive after the tagged customer do not have any impact on the waiting time of the tagged customer. Therefore, the absorbing state is a vector of order given by level 0 has the single component level is a vector of order given byand levels are vectors of order given bywhere and are the phases of the service time distributions and , respectively.
On entering the absorbing state , a tagged customer starts receiving service. Clearly, this happens at the arrival of a server either from vacation or after the completion of a service when the customer is at the head of the queue. The transition rate matrix for this absorbing Markov chain is given bywhereIt is observed that the customer arrival rates have not appeared in matrix
To obtain the distribution of waiting time of the tagged customer, the first step is to determine the stationary probability distribution of the system's state immediately after its arrival. This is actually the stationary distribution of the number of customers in the system as seen by this tagged customer at its arrival time. Denote by the probability distribution which can be obtained from by a standard method, that is, can be interpreted as a conditional probability distribution of the system's state conditioned on the occurrence of the tagged customer arrival. Due to the Markovian property of the arrival process, it is seen that the arrival-stationary probability distribution of the number of customers in the system is given bywhere is the fundamental arrival rate of the MAP as given in Section 2.1.
Now we define where is a row vector of order , when is of order and is a row vector of order , and its elements represent the probability that at time the CTMC with generator is in the respective state of level Clearly, gives the probability that the tagged customer is in the absorbing state at time Hence, the vector waiting time distribution is for In finding the waiting time distribution of the tagged customer at arrival time, it is assumed that the process starts with initial probability vector
The differential equation for reduces towhere the prime denotes the derivative of the function concerned with respect to .
The tagged customer at arrival time finds the system in level with probability , for the Laplace-Stieltjes transform (LST) of the first passage time to level is given by the row vector
As in [20], we get
Let be the LST of the absorbing time to the state given that the process starts from level On the basis of we can write the following relations for the matrices : Finally, it can be seen that the LST for the waiting time distribution is given by
3.1. Mean Waiting Time
The mean waiting time of an arriving customer is computed from The first two terms give the mean time to reach an absorbing state by the tagged customer if the system is in a level On its arrival, the third and fourth terms represent the time to reach the absorbing state if the system is in a level
To compute the mean waiting time of the tagged customer, we must calculate the value of each term in (3.12). Differentiating (3.8) with respect to and setting we getSimilarly, differentiating (3.9) and (3.10) with respect to and setting , we obtainThus, formulae (3.13)–(3.15) permit the recursive computation of the matrices Using (3.13) to (3.15) and absorbing the initial condition we can compute the first two terms of (3.12).
The value of where , is obtained by substituting in (3.7), and it needs to be evaluated numerically. Since , owing to the relation , we have that The value of can also be used, as mentioned in [11], to evaluate an approximate value of by finite summation. Using (3.13) and (3.14), we get the value of for the fourth term of (3.12).
To compute the third term of (3.12), differentiating (3.7) with respect to and setting , we obtainSince is a stochastic matrix, and using the relationship it follows thatTo obtain the value of from (3.17), we modify the technique used by Kao and Narayanan [11] and Neuts and Lucantoni [22]. Now, construct a stochastic matrix such that is nonsingular and generalized inverse of In cases where is irreducible, the matrix may be chosen as , where is the invariant probability vector of , that is, and This follows from the classical theorem on finite Markov chains given in the work of Kemeny and Snell [23, page 100].
Further, the following relation is satisfied owing to the property :Substituting (3.18) in (3.17) and simplifying, we obtainThus, all four terms of (3.12) have been computed, and the mean waiting time can be obtained.
Hence, we are able to compute the values of steady-state joint probabilities and mean waiting time using algorithms for this queueing system with Bernoulli vacation scheduling service.
4. Numerical Examples
In this section, we discuss some interesting numerical examples that qualitatively describe the performance of the queueing system under investigation. To gain an understanding of the performance measures of the Bernoulli vacation scheduling service queueing model, we study the effect of the system parameters on the following items:
(i) the expected number of customers in the system,(ii) the expected number of customers in the system when both servers are busy,(iii) the probability of no customers in the system,(iv) the probability that both servers are busy, and(v) the mean waiting time of an arriving customer.
For the arrival process, we consider the MAP input which is characterized by the matrices:In other words, the continuous-time Markov chain, which governs the input, has two states. For various values of ranging from 4 to 13, this MAP has the fundamental rate ranging from 2 to 6.5.
Next, we consider the phase-type (PH) service time processes for server-1 and server-2 asThe average intensities of the services of server-1 and server-2 are given by and , respectively.
In Figures 1 and 2, the values of the mean numbers of customers in the system and , the mean number of customers in the system when both servers being busy are plotted against the fundamental rate for chosen values of , and satisfying stability condition (2.9).
By considering higher vacation rates (of order ) in the vacation model under study, we obtain approximate results for corresponding nonvacation model. By fixing , and , the values of versus the fundamental rate are plotted in Figure 1 for the following cases:
(1) 1-limited service policy (2) Bernoulli vacation scheduling service (3) exhaustive service policy ,(4) with no vacation system.In all the cases, it is observed that steadily increases as increases and decreases with decreasing values of and . Figure 2 indicates that , the mean of the system size when both servers being busy for 1-limited service policy, Bernoulli vacation scheduling service, and exhaustive service with multiple vacation grow at a faster rate than nonvacation system for increasing values of the fundamental arrival rate
In Figure 3, the values of the probability that there is no customer in the system, and both server-1 and server-2 on vacation are plotted against the fundamental arrival rate for the chosen parametric values of the system satisfying the stability condition (2.9). It is seen that the probability steadily decreases as the values of increase, and it decreases for increasing values of and
In Figure 4, we plot the probability that both server-1 and server-2 are busy versus the fundamental arrival rate for the different values of and . It is seen that the probability exhibits the opposite trend to that of the probability in the sense that it increases with increasing as expected. However, decreases for increasing values of and . Moreover, for the case of nonvacation system, the value of is seen to be more than that for any of the vacation policies under discussion.
The trends of the average waiting time are depicted in Figures 5–7. The computation of the mean waiting time of customer at the arrival epoch for the exhaustive service policy is carried out in accordance with procedure given in the work of Kao and Narayanan [11] and in the case of 1-limited service discipline, the approach due to the work of Tyagi et al. [14] is followed. It is observed from Figure 5 that grows in an unbounded fashion for system with/without vacation for increasing values of . For nonvacation system and exhaustive service discipline , the growth of is not much faster whereas in the cases of Bernoulli vacation scheduling service and 1-limited service discipline there is a steep increase in for larger values of
Figure 6 illustrates the trend of expected waiting time of the customer at arrival epoch versus the fundamental arrival rate if server-1 follows Bernoulli vacation scheduling service and server-2 follows either of the following: (i) 1-limited service policy, (ii) Bernoulli vacation scheduling, (iii) exhaustive service discipline, and (iv) no vacation system. In all the cases, the expected waiting time increases for increasing values of Finally, Figure 7 exhibits a similar trend wherein the service disciplines of server-1 and server-2 are interchanged corresponding to Figure 6.
5. Conclusions and Further Research
A queueing system with two heterogeneous servers and Bernoulli vacation has been presented. Customers arrival pattern is described by the MAP, and service times have PH distributions. Based on the matrix geometric method, the stationary queue length distribution, mean system size, and other system performance measure have been computed. This system subsumes the 1-limited service discipline and the exhaustive service discipline as special cases. Moreover, the expected waiting time of the customer at arrival epoch has been analyzed in detail. Results of numerical experiments giving insight into behavior of the systems are presented. We expect that the method of analysis adopted in this paper can be used to discuss other complex queueing systems such as multiserver retrial queue with Bernoulli vacation policy.