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Discrete Dynamics in Nature and Society
Volume 2015, Article ID 160697, 15 pages
http://dx.doi.org/10.1155/2015/160697
Research Article

Study on the Queue-Length Distribution in Queue with Working Vacations

1School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China
2Research Institute of Economics and Management, Southwestern University of Finance and Economics, Chengdu 611130, China
3School of Mathematics and Software Science, Sichuan Normal University, Chengdu, Sichuan 610066, China

Received 4 April 2015; Revised 8 June 2015; Accepted 14 June 2015

Academic Editor: Gabriella Bretti

Copyright © 2015 Chuanyi Luo et al. 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.

Abstract

This paper analyzes a finite buffer size discrete-time queue with multiple working vacations and different input rate. Using supplementary variable technique and embedded Markov chain method, the queue-length distribution solution in the form of formula at arbitrary epoch is obtained. Some performance measures associated with operating cost are also discussed based on the obtained queue-length distribution. Then, several numerical experiments follow to demonstrate the effectiveness of the obtained formulae. Finally, a state-dependent operating cost function is constructed to model an express logistics service center. Regarding the service rate during working vacation as a control variable, the optimization analysis on the cost function is carried out by using parabolic method.

1. Introduction

Discrete-time queues with classical vacation policies have been explored more in depth during the last few decades due to their widespread application in telecommunication system, electronic information network, production system, and so on (see Takagi [1], Tian et al. [2], Alfa [3], and the references therein). In the queueing systems with classical vacation policies, the server is assumed completely inactive (does not provide any service) during his vacation time. Motivated by the study of reconfigurable wavelength-division multiplexing optical access network, Servi and Finn [4] developed a different vacation policy against the classical one, working vacation (WV), in which the server did not stop service for customers; instead, he remained semiactive. From then on, a large number of researchers were attracted to study queueing systems with working vacation policy. By using matrix-analytic method, Li et al. [5], Li and Tian [6], and Li [7] studied the discrete-time queues with working vacations. Yi [8] considered disasters in queue with working vacations. Li et al. [9] and Gao and Liu [10] added the bach-arrival schedule to a discrete-time queue with working vacations. On the basis of working-vacation queue, Goswami and Selvaraju [11] extended the working-vacation policy to queueing model with Markovian arrival process and general phase-type distributed service time.

These mentioned research works all concentrated on working-vacation queueing models with infinite buffer size; however, the finite buffer size counterparts received little attention. In real situations, queues with finite buffer size are more suitable than queues with infinite buffer space as it is used to store arrived customers if server is busy. Among the existing references, very few papers considered the working-vacation queue with finite buffer size; see Goswami and Samanta [12], Yu et al. [13], Yu et al. [14], Zhang and Hou [15], Gao et al. [16], and Banerjee et al. [17]. Nevertheless, the existing research topics with finite buffer size and working vacations mentioned above all concentrate on queueing system and its different varieties. We note that the finite buffer queue with working vacations has not been studied up to now. It motivates us to fill this gap.

In addition, as far as the queueing systems with working vacations are concerned, the assumption assumes in general that the customers arrive in system at a fixed rate. However, the customer’s choice of entering into system or not usually depends on the system’s status what they see at the arrival epoch. For example, in a make-to-order production system where the system information such as server’s status and queue length is fully observable to an arriving customer, the arriving customer with rate of may choose to leave the system when he finds that the server is not active, or the service rate is lower than the normal rate, or too many customers accumulate in front of the server, and so on. Thus, we assume in this paper that the customers arrive in system at different rates. Consequently, the changeable arrival rate of order brings influence on some critical performance measures associated with operating cost such as queue length, waiting times, delay probability for an order, and blocking probability (see Bhaskar and Lallement [18]).

On account of the introduction mentioned above, we study the finite buffer queue with working vacations and different arrival rates, denoted by . Using supplementary variable method and embedded Markov chain techniques, the queue-length distribution in the form of formula at arbitrary epoch is obtained. Some performance measures associated with operating cost are also discussed under the achievement of the queue-length distribution solution. To demonstrate the effectiveness of the achieved formulae, a numerical experiment is carried out with respect to a state-dependent operating cost function.

The rest of this paper is organized as follows. Section 2 describes the queueing model. Section 3 analyzes the queue-length distribution at arbitrary epoch. In Section 4, various performance measures are obtained. Section 5 focuses on the numerical performance characteristics. To demonstrate the application of the model studied in this paper, Section 6 constructs a state-dependent cost function from an express logistics service center and discusses a cost minimization problem concerning the function. Finally, some conclusions and topics for future research are mentioned in Section 7.

2. Model Description

We consider a discrete-time single server queue with vacations, in which a potential customer arrives in time interval with delayed access and departs in (we call it LAS-DA discipline). Upon the server working at a normal service rate, customers arrive in system according to a Bernoulli process with parameter of . We denote the arrival interval in this case by random variable ; that is, , where ; in another case, when the server is on vacation, customers arrive in system according to another Poison process with parameter of . The arrival intervals in this case are denoted by ; that is, .

Service and Vacation Rules. The server serves the waiting customers (if there is any) according to FCFS (first come, first served) discipline. The service begins with a normal rate when the first customer arrives in system and ends when the system becomes empty. We call this time interval the “normal busy period.” The duration of the service time for a customer, denoted by , is a random variable with arbitrary probability mass function (PMF) , , probability generating function (PGF) , and finite mean service time . When the system becomes empty, the server takes multiple working vacations [2]. The length of a working vacation, denoted by , is geometrically distributed with PMF . The length of the service time for a customer during working vacation period, denoted by , is also a random variable following another arbitrary distribution with PMF , and PGF and finite mean service time . If the server finds that the system is nonempty upon comes back from a working vacation, he returns the service rate to the normal level and restarts the service interrupted at the end of vacation from the beginning; otherwise, he takes the next working vacation. To avoid confusion, the different time epochs at which events occur are shown in Figure 1. Finally, we assume the service, arrival, and the vacations are mutually independent.

Figure 1: Various time epochs in a late arrival system with delayed access (LAS-DA).

To describe the system state, the following random variables are introduced:: queue length (including the customer in service) at epoch ;: remaining service time of the customer being served at epoch ;, server is in working vacation at epoch ; 1, server is in normal busy period at epoch .Let  ; then is a Markov process with state space . We define the joint probabilities byAs , the mentioned probabilities above are denoted by and , ; ; , respectively.

3. The Queue-Length Solutions in the Form of Formula

In this section, by combining embedded Markov chain and supplementary variable methods, the queue-length distribution at arbitrary epoch is obtained.

3.1. Steady State Queue Length at an Arbitrary Epoch

Firstly, we develop the usual Chapman-Kolmogorov (C-K) difference equations by regarding the remaining service time as the supplementary variable. Generally, using one-step transition probability, the system can get to the state of (assume at epoch ) from four types of the up-step state (assume at epoch ): one is from state of by probability ; that is, the system is in vacation and one customer arrives during time interval ; the second is from state of by probability ; that is, the system is in vacation and no customer arrives during time interval , and the customer being served at epoch completes his service at ; the third is from state of by probability ; that is, the system is in vacation and one customer arrives during time interval , and the customer being served at epoch completes his service at ; the last one is from state of by probability , that is, the system is in vacation and no customer arrives during time interval . So, under steady state (), we getLet , respectively; we get the degenerate equationsSimilarly, we also obtain the C-K equations when system is in normal busy period:Define the -transforms of and , respectively:We propose the following notations:Multiplying (2) and (4)–(7) by and summing over from 1 to , we obtain after some simplificationAdding (10) over all possible values of and using (3), we getSetting in (11), we haveSubstituting (13) into (11) and letting lead toCalculating limit on both sides of (12) as after substituting (13) into (12) yieldswhere , .

Substituting (14) and (15) into the normalization condition, , we getFrom (3), (13), (14), and (16), the terms of and can be expressed by and as follows:

Let , be the departure epoch and let denote the queue length immediately after the epoch . denotes the joint probability of and with steady state , ; .

For , , we haveCalculating limit on both sides of (18) as leads toSimilarly, for , , we haveFrom (3), (17), and (19), we getwhere .

In order to find the solution of , we would adopt matrix equations to express (20)–(23). Firstly, some vector notations are introduced as follows:Using (24), the group equations of (20)-(21) and (22)-(23) are expressed in matrix form, respectively:where

is determined by (28):Solving the simultaneous matrix equations (28) and (29) leads toLet in (10); we haveThe equations of (34)–(37) and (38)–(41) are expressed by the following matrix forms, respectively:where and are determined by (25) and (26), respectively:Substituting (25)-(26) and (32)-(33) into (42)-(43), respectively, and solving the simultaneous matrix equations (42)-(43), we havewhere , , is an identity matrix of degree , and the remaining notations are consistent with the previous ones.

One may note that if we could obtain the queue-length distribution at departure epoch, , , and , the arbitrary epoch probabilities, and , can be derived from (45) and (46). Its main steps are introduced as follows:(a)Gain the probabilities of and expressed by after substituting , , and into (26) and (33), respectively.(b)Obtain the arbitrary epoch probabilities of and expressed by from (45) and (46) as well as the arbitrary epoch probability of expressed by from the normalization condition .(c)Get the value of after substituting , , , and into (15).(d)Achieve the values of and after substituting the value of back into (45) and (46), as well as the value of through normalization condition.

So, in the following subsection, using the embedded Markov chain technique, we investigate the queue-length distribution at a departure epoch.

3.2. Steady State Queue Length at a Departure Epoch

Let denote the queue length immediately after the departure epoch and define the variable of :Thus, is a two-dimensional Markov chain with state space . Denote the one step transition probability by and define , ; , , where the random variables of , , denote the arrival intervals with the same distribution as , . Then the one step transition probability matrix (TPM), denoted by , will be obtained in the following work.

The state transition occurs if the length of vacation (or remaining vacation) is greater than a service time in working vacation; that is, , and customers arrive during the service time ; that is, ; then it getsSimilarly, for , we have

The state transition occurs if no customer arrives during a service time in normal busy period; that is, . So we have

The state transition occurs if the length of vacation (or remaining vacation) is not less than a service time in working vacation and no customer arrives during the service time; that is, ; , or the length of vacation (or remaining vacation) is less than a service time in working vacation and no customer arrives during the vacation time and the normal service time follows the vacation; that is, ; ; . Then we have

The state transition occurs under two cases. First, the vacation does not end immediately after the first arrival epoch behind the previous departure and the length of vacation (or remaining vacation) is not less than a service time in working vacation and no customer arrives during the service time, or the length of vacation (or remaining vacation) is less than a service time in working vacation and no customer arrives during the vacation time and the service time in normal busy period follows the vacation. Second, the vacation just ends immediately after the first arrival epoch behind the previous departure (thus, the first arrival will obtain a normal service) and no arrival occurs during the normal service time. Then we have

The state transition occurs if customers arrive during a service time in normal busy period; that is, . It yields

The state transition occurs if the length of vacation (or remaining vacation) is less than a service time in working vacation; that is, , and customers arrive during the vacation time and the service time in normal busy period (assuming that customers arrive during the vacation and customers arrive during the service time in normal busy period); that is, ; , or the length of vacation (or remaining vacation) equals a service time in working vacation and customers arrive during the service time; that is, ; . Then it leads toSimilarly, for , we have

To obtain the TPM , some additional notations are introduced as follows:

Using lexicographical sequence for the states, the structure of is given by

Let be a column vector of departure epoch probabilities; is a row vector of dimension of ; then we have the system linear equations , which can be directly converted into the following equations:where is an identity square matrix with dimensions and is a column vector with dimensions and all of its’ entries are equal to zero.

Subsequently, the queue-length probabilities at departure epoch, that is, , are obtained by using software of MATLAB. Moreover, we finally work out the queue-length distribution at arbitrary epoch.

3.3. Steady State Queue Length at Other Epochs

To find the queue-length distributions at a potential arrival epoch , prearrival epoch, arbitrary epoch , and outside observer’s observation epoch, we define the following additional notations. or is the steady state probability of customers waiting in system at an arbitrary epoch during a working vacation or normal busy period. or is the transient probability of customers waiting in system at a prearrival epoch during a working vacation or normal busy period: or is the steady state probability of customers waiting in system at an outside observer’s observation epoch during a working vacation or normal busy period. or is the steady state probability of customers waiting in system at an arbitrary epoch during a working vacation or normal busy period.

From Figure 1, it is clear that

The prearrival epoch probabilities are determined by the following relations:Similarly, we have

We can derive the conclusion of from (62) and (63) under the condition of ; it means the GASTA (geometric arrivals see time average) property does not hold if .

The relations between and can also be conducted by considering the arbitrary epochs and in Figure