Research Article | Open Access
A. Krishnamoorthy, C. Sreenivasan, "An M/M/2 Queueing System with Heterogeneous Servers Including One with Working Vacation", International Journal of Stochastic Analysis, vol. 2012, Article ID 145867, 16 pages, 2012. https://doi.org/10.1155/2012/145867
An M/M/2 Queueing System with Heterogeneous Servers Including One with Working Vacation
This paper analyzes an queueing system with two heterogeneous servers, one of which is always available but the other goes on vacation in the absence of customers waiting for service. The vacationing server, however, returns to serve at a low rate as an arrival finds the other server busy. The system is analyzed in the steady state using matrix geometric method. Busy period of the system is analyzed and mean waiting time in the stationary regime computed. Conditional stochastic decomposition of stationary queue length is obtained. An illustrative example is also provided.
Queueing models with vacation have gained significance in the last three decades due to their wide range of applications, especially in the communication and the manufacturing systems. Doshi  provides an excellent survey of related works prior to 1986. Takagi  and Tian and Zhang  provide a good account of developments in this field since then. The literature on the vacation queueing models is growing rapidly.
In multiserver queueing models, we come across two classes of vacation mechanisms: station vacation and server vacation. In the first case, all servers take vacation simultaneously whenever the system becomes empty and they return to the system all together. Thus, station vacation is group vacation for all servers. For example, when a system consists of a number of machines operated by a single individual this scenario occurs. In such a situation, the whole station needs to be treated as a single unit for vacation, when the system is utilized for a secondary task. In the second case, each server takes its own vacation whenever it completes a service and finds no customers waiting for service. This phenomenon also occurs in practice. For example, in a post office or bank, when a clerk completes a service and finds no customer waiting, he or she might go to attend another task. This is what we refer to as the server vacation model. Analysis of a server vacation model is more complicated than that of a station vacation model. This is because at any time point, in the latter, we may have any number of servers between and on vacation. We need to track individual servers going on vacation and completing their vacation. Upon returning from a vacation some servers may find no customers waiting for service. These servers take another vacation. But if any server finds a waiting customer on returning from a vacation, it immediately starts service. For further details on queues with station and server vacations, we refer the reader to Chao and Zhao .
Most of the earlier work on multiserver queueing models deal with homogeneous servers; that is, the individual service rates are same for all servers in the system. But this assumption may be valid only when the service process is mechanically or electronically controlled. In a queueing system with human servers the above assumption is highly unrealistic. Often servers providing identical service, serves at different rates. This motivated the researchers to study multiserver queueing system with heterogeneous servers. Several authors have analyzed multiserver queues with vacation. Levy and Yechiali , Vinod , and Kao and Narayanan  discuss asynchronous multiple vacation models with exponentially distributed vacation times. They study the scenario, where any server goes on vacation whenever there are no customers waiting in the system at a service completion epoch. At a vacation termination instant, if there are no waiting customers, the server takes another vacation; if there is a customer waiting for service, the server resumes service. It should be remarked that Vinod  was the first to use matrix geometric solutions method to analyze multiserver vacation models. But none of the above models deal with heterogeneous servers.
Servi and Finn  introduced a working vacation model with the idea of offering services but at a lower rate, whenever the server is on vacation. Their model was generalized to the case of in [9, 10] and to model in . A survey of working vacation models with emphasis on the use of matrix analytic methods is given in Tian et al. . Working vacation models have a number of applications in practice. Two such examples are given in . Recently, Li and Tian  studied an queue with working vacations in which vacationing server offers services at a lower rate, for the first customer arriving during a vacation. Very recently, Zhang and Hou  studied a queue with working vacations and vacation interruptions using supplementary variable method. In this model, the authors assume that the vacation times are exponentially distributed and that the server gets back to normal service mode, when at a service (offered during a vacation) completion, the system has at least one customer waiting in the queue. The server is allowed to take multiple vacations. However, no work on multiserver working vacation model has come to our notice.
Neuts and Takahashi  observed that for queueing systems with more than two heterogeneous servers analytical results are intractable and only algorithmic approach could be used to study the steady state behavior of the system. Based on this observation, Krishna Kumar and Pavai Madheswari  analyzed queueing system with heterogeneous servers, where the servers go on vacation in the absence of customers waiting for service. In this paper, we discuss an queueing model with heterogeneous servers where one server remains idle but the other goes on working vacation in the absence of waiting customers.
2. Mathematical Model
We consider an queueing model with heterogeneous servers, server 1 and server 2. Server 1 is always available, whereas server 2 goes on vacation whenever there are no customers waiting for service. Let the service rates of servers 1 and 2 be and , respectively, where . Customers arrive to the system according to a Poisson process with parameter . The duration of vacation is exponentially distributed with parameter . At the end of a vacation, service commences if there is a customer waiting for service. Otherwise the server goes on another vacation. During vacation, if an arrival finds server 1 busy, server 2 returns to serve the customer but at a lower rate. To be precise, server 2 serves this customer at the rate , . As this vacation gets over, server 2 instantaneously switches over to the normal service rate , if there is at least one customer waiting for service. Upon completion of a service at low rate, the server will (a) continue the current vacation if it is not finished and no customer is waiting for service; (b) continue the slow service if the vacation has not expired and if there is at least one customer waiting for service. For clarity, we make it clear that if an arriving customer finds a free server, he enters service immediately. Else he joins the queue.
2.1. The QBD Process
The model discussed in Section 2 can be studied as a quasi-birth-and-death (QBD) process. First, we set up the necessary notations.
At time , let be the number of customers in the system and Let . Then is a continuous time Markov chain (CTMC) with states space where Note that when , the only possible value of is and when , has three possible values , , and .
The infinitesimal generator matrix of this Markov chain is given by where the block matrices appearing in are as follows:
3. Steady State Analysis
In this section, we will discuss the steady state analysis of the model under study.
3.1. Stability Condition
Theorem 3.1. The queueing system described above is stable if and only if where .
Proof. To establish the stability condition we use Pakes' lemma (see ). Let be the number of customers in the system immediately after the departure of the th customer. Then satisfies the equation
where is the number of arrivals during the service of th customer. Clearly is an irreducible aperiodic Markov chain. Pakes' lemma asserts that an aperiodic irreducible Markov chain is ergodic, if there exists an such that the mean drift
is finite for all and for all except perhaps for a finite number. In the present model, value of the mean drift is
Thus, if , the Markov chain is ergodic and hence the condition is sufficient.
To prove the necessity of the condition, assume that . We use Theorem 3.1 in Sennot et al. , which states that is nonergodic if it satisfies Kaplan's condition, , for and there is a such that , for . When Kaplan's condition is readily satisfied. Hence, the Markov chain is not ergodic.
3.2. Steady State Probability Vector
Let , partitioned as , be the steady state probability vector of . Note that is a scalar, and for . The vector satisfies the condition and , where is a column vector of 1's of appropriate dimension. Apparently when the stability condition is satisfied, the subvectors of , corresponding to the different levels, are given by the equation , where is the minimal nonnegative solution of the matrix quadratic equation (see ) Knowing the matrix , , and are obtained by solving the equations subject to the normalizing condition
Theorem 3.2. The matrix R of (3.4) is given by , where , and .
Proof. Since , , and are upper triangular, is essentially an upper triangular matrix. The value of , follows from the assertion that is the minimal nonnegative solution of (3.4). The rest of the proof is an easy consequence of the condition .
Remark 3.3. Though has a nice structure which enables us to make use of the properties like , for , due to the form of the expression for it may not be easy to carry out the computations required in the forthcoming discussions. Hence, we explore the possibility of algorithmic computation of . The computation of matrix can be carried out using a number of well-known methods such as logarithmic reduction. We list here only the main steps involved in logarithmic reduction algorithm for computation of . For full details on the logarithmic reduction algorithm we refer the reader to .
Logarithmic Reduction Algorithm for :
Step 0. , , , and .
Step 1. continue Step 1 until .
Step 2. .
3.3. Busy Period Analysis
For the system under study, busy period is the interval between arrival of a customer to the empty system and the first epoch thereafter when the system becomes empty again. Thus, it is precisely the first passage time from the state to the state . For the working vacation model, busy cycle for the system is the time interval between two successive departures, which leave the system empty. Thus, the busy cycle is the first return time to state with at least one visit to any other state. But before analyzing the busy period structure we need to introduce the notion of fundamental period. For the QBD process under consideration, it is the first passage time from level , where , to the level . The cases , , and , corresponding to the boundary states, need to be discussed separately. It should be noted that due to the structure of the QBD process, the distribution of the first passage time is invariant in .
Let denote the conditional probability that a QBD process starting in the state at time reaches the level for the first time no later than time , after exactly transitions to the left and does so by entering the state . For convenience we introduce the joint transform and the matrix The matrix is the unique solution to the equation (see ) The matrix takes care of the first passage times, except for the boundary states. If we know the matrix then matrix can be computed using the result (see ) Otherwise, we may use logarithmic reduction method to compute . For the boundary level states 2, 1, and 0, let , , and be the conditional probability discussed above for the first passage times from level 2 to level 1, level 1 to level 0, and the first return time to the level 0, respectively. Then as in (3.12) we get Note that is a matrix. Thus, the Laplace Stieltjes transform (LST) of the busy period is the first element of . For convenience, we use the notations Due to the positive recurrence of the QBD process, matrices , , , and are all stochastic. If we let then is the minimal nonnegative solution (see ) to the matrix equation From (3.14), (3.15), and (3.16), we get respectively. Equation (3.12) is equivalent to
Let Differentiation of (3.21) with respect to and followed by setting and leads to (see ) with as starting value for and , successive substitutions in the above equations yield the values of and . Applying an exactly similar reasoning to (3.14), (3.15), and (3.16), we get where Note that is a matrix and is a scalar. The first element of the matrix and are mean lengths of a busy period and a busy cycle, respectively. The second and third elements of the matrix are the first passage times to the state from and , respectively. With the notations it follows from (3.14) and (3.15) that The first component of the vector is the mean number of service completions in a busy period.
3.4. Stationary Waiting Time Distribution in the Queue
Let be the distribution function for the waiting time in the queue of an arriving (tagged) customer. Note that if there is no customer in the system, the arrival receives service immediately. If either of the two servers is not busy then also there would be no delay in getting service. Thus, the probability that the customer gets his service without waiting is . Hence, with probability , the customer has to wait before getting the service. The waiting time may be viewed as the time until absorption in a Markov chain with state space Here * is the absorbing state, which corresponds to taking the tagged customer into service and is obtained by lumping together the level states and . For , the level is given by . The states other than the absorbing state correspond to the number of customers present in the system as the tagged customer arrives. Once the tagged customer joins the queue, the subsequent arrivals will not affect his waiting time in the queue. Hence the parameter does not show up in the generator matrix of this Markov process, given by
Define vector where The components of the are the probabilities that at time , the CTMC with generator is in the respective states of level . Note that the scalar is the probability that the process is in the absorbing state at time . By the PASTA property, we get Clearly The LST of is given by (see ) The mean waiting time can be obtained from as where is a stochastic matrix. Hence, (3.34) can be simplified as Let Since is stochastic, we get This result can be used to find an approximate value of and hence that of the second term in (3.36) to any desired degree of accuracy. Thus, only the first term in (3.36) demands serious computation. For this we make use of the ideas in [15, 16, 21].
Now consider the matrix which has the property that Then we get By the classical theorem on finite Markov chains, the matrix is nonsingular (see ). In view of the last equation, the first term in (3.36) becomes With this simplification, we get
3.5. Conditional Stochastic Decomposition of Queue Length
In this section, we provide a stochastic decomposition of queue length in the stationary regime, subject to the condition that both servers are busy. Note that from (3.5)–(3.8) we get , , , , , and . Let be the queue length of the vacation model under study, subject to the condition that both servers are busy. Then we have the following.
Theorem 3.4. If , then , where and are two independent random variables. is the queue length of the M/M/2 queueing model with heterogeneous servers without vacation and can be interpreted as the additional queue length due to vacation and slow service, subject to the condition that both servers are busy.
Proof. Let denote the Probability that both servers are busy. Then so that where , the generating function of the queue length subject to the condition that both servers are busy, is given by By following a computational procedure similar to that of , we arrive at where From (3.49) it follows that is the generating function of an heterogeneous queuing model without vacations, which is precisely the case in . Equation (3.50) suggests that has a geometric distribution with parameter .
Remark 3.5. Due to the algorithmic approach used in the derivation stationary waiting time distribution, a similar decomposition result for the waiting time distribution is far from reality.
3.6. Key System Performance Measures
In this section, we list a number of key system performance measures along with their formulae in addition to the busy period structure and the mean waiting time discussed above. (1) The probability that the system is empty: . (2) The probability that the server 1 is idle: . (3) The probability that the server 2 is on vacation: . (4) The probability that the server 2 is working in vacation mode: . (5) The probability that the server 2 is working in normal mode: . (6) The mean number of customers in the system: .
4. Numerical Results
4.1. Illustrative Example
We analyze the effect of the parameters , and on the key performance measures. Table 1 analyzes the effect of , Table 2 explains the effect of , and Table 3 examines the effect of on the performance measures. To this end, we use the following abbreviations in addition to the notations used in Section 3.6.
: Mean waiting time in the queue.
: Mean length of a busy period.
: Mean length of a busy cycle.
: Mean number of service completions in a busy period.(i)Since and are fixed, the traffic intensity increases with . Due to this , and increase and and decrease as increases. Note that the busy period starts with the Markov chain in the state ; that is, with server 2 on vacation. Hence, initially increases with . For this reason , , and show an early downward trend. But as further increases declines as expected due to the high traffic intensity. Hence, and reverse the direction of change. Due to the effect of and , this reversal occurs only at a later stage for . It is worth comparing the values of the measures in cases A and B. Even though the net service rate in both cases, the effect of the vacation parameter becomes more predominant when . Due to this the measures and take smaller values and the measures , , , and take larger values in case B, compared to their values in case A.(ii)As increases, the mean duration of vacation 1/ decreases. But as the mean duration of vacation decreases, the probability of the expiry of the vacation without initiating the slow service increases. The chance of an early expiry of vacation always results in an increase in and and a decrease in . Note that decreases as increases and for any value of in the given range. So increases with . Thus, the proportion of time in which both servers work at the normal rate increases as increases. Hence, the measures , , , , and decrease as increases. The argument given the last paragraph holds good for the difference in magnitude of the measures in cases A and B.(iii)As increases, the service rate of the second server during vacation mode of service increases. As a result, server 2 clears out customers at an increased rate in slow service mode. This produces an increase in , , and and a decrease in as expected. Consequently, , and decrease as increases. The huge difference in the value of net service rate between cases A and B, during vacation mode of service, is the reason for the pattern of behavior of in these two cases. Increase in does not affect significantly in case A but it affects the measure in case B. This is because the effect of becomes significant only when is large compared to .
5. Concluding Remarks
In this paper, we studied an queueing model with heterogeneous servers. One server follows multiple vacation policy. But this server offers service at a lower rate during vacation if customers arrive. The other server remains in the system even when it is empty. The busy period of the system was analyzed in the stationary regime. Mean waiting time of a customer has been computed. Conditional stochastic decomposition of queue length has been derived. An illustrative numerical example to bring out the qualitative nature of the model has been presented.
The authors would like to thank the anonymous referees for their valuable suggestions and comments which greatly improved the presentation of the paper. C. Sreenivasan's research is supported by the University Grants Commission, Government of India, under Faculty Development Programme (Grant no. F.FIP/11th Plan/KLCA042TF02).
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Copyright © 2012 A. Krishnamoorthy and C. Sreenivasan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.