1 Introduction
The steady-states are well suited to study the performance measures (including the queue length, waiting time and loss probability). However, the steady-state results do not reveal the real functioning of the system under consideration because the transient and start up effects are not taken care of [
1]. In most of applications, it is more important and useful to know how the system will operate up to some time
. We study the transient behavior of an
queueing system with heterogeneous servers, multiple vacations and impatient customers.
Queueing models with impatient customers seem to be appropriate in some service systems. Customer impatience has been dealt with in the queueing literature mainly in the context of customers abandoning the queue either due to a long wait already experienced, or a long wait anticipated upon arrival (see [
2-
5]).
There is a considerable literature on queues with customer's impatience. Palm's pioneering work (see [
6,
7]) seems to be the first to analyse queueing systems with impatient customers by considering the M/M/c queue and assuming that each customer stays in the queue as long as his waiting time does not exceed an exponentially distributed impatience time. Altman and Yechiali (see [
8,
9]) analyzed models with customers impatience when the server(s) is (are) on vacation and unavailable for service. Recently, Ammar carried out an analysis for a single server queue with impatient customers and multiple vacations where customers impatience was due to an absentee of servers upon arrival in [
10].
Queueing models with sever vacations have drawn significant attention in recent years. Levy and Yechiali first discussed the idea of sever vacations in [
11]. Then Doshi (see [
12,
13]) did several excellent surveys on these vacation models. Tian and Zhang continued devoting to this subject in [
14]. There are different types of vacation queueing systems and we refer [
15] for details. We focus on the multiple vacations in this paper (i.e., servers leave for a vacation when the system is empty and commence another vacation when severs return from a vacation and find the queue empty).
In this paper, we model the service system of interest as a two-heterogeneous servers queue, where customers leave the system and never return if the timer expires before the servers return from the vacation. In our model, the arrival process is modelled by a Poisson process. Service time is exponentially distributed where the two servers namely faster server and slower server provide heterogeneous service with different service rate. We investigate the transient analysis of such systems.
The contribution of transient analysis summarized by Ammar is mainly twofold. First, the transient analysis provides us with a deep understanding of the behaviour of a system when the parameters involved are perturbed and it can contribute to the costs and benefits of operating system. Further, such transient analysis enables us to obtain the steady state probabilities and optimal solutions which lead to the control of the system. But beyond that, we are among the few scholars who discuss the transient behaviour of queues with heterogeneous servers, multiple vacations and impatient customers.
The remainder of the paper is organized as follows. The queueing model of interest is introduced in Section 2. Section 3 gives the transient analysis of the system state and derives the time dependent probabilities along with the mean and variance of the system size. Performance measures are provided in Section 4. In Section 5, we give two special cases. Section 6 concludes the paper.
2 Model Description
We study a two-heterogeneous servers queue with multiple vacations and impatient customers. Customers arrive according to a Poisson process with parameter , and service time is exponentially distributed where the two servers namely faster server and slower server provide heterogeneous service with different service rate and such that . Moreover, the customers select servers on faster server first (FSF) basis.
When the system becomes idle, the two servers leave for a vacation together with random length exponentially distributed with parameter . Besides, if no customers are found in the queue when the two servers return from the vacation, they again leave for another vacation and continue in the same manner. The arrival processes, the service times, and the vacation times are mutually independent. In addition, the service discipline is first come first served (FCFS). When two servers are all on vacations, customers become impatient. That is, each individual customer activates an independent "impatient timer'', which is exponentially distributed with parameter . If the timer expires before the servers return from the vacation, he or she leaves the system never to return.
Let be the number of customers at time , and let be the status of the servers at time t, which is defined as follows:
Then a continuous time Markov chain (CTMC) can be defined by with a state space . Besides, we define , , and as
Then, the behaviour of the resulting system is described by a set of Kolmogorov forward equations that can be written as follow:
where . We assume that initially there are no customers in the system, i.e., .
3 Transient Analysis
In this section, we derive the transient solution for the model under consideration by employing generating function, Laplace transforms and continued fractions. Define the probability generating function
where
with initial condition . By Equations (1) and (3)–(6), we obtain the following differential equation
Solving the above equation, we get
It is well known that
where , and is the modified Bessel function of the first kind.
By Equations (10) and (11), we have
3.1 Evaluation of
Comparing the coefficients of on both sides of Equation (12) for , we obtain
The above Equation (13) holds for with left hand side replaced by zero. Using Bessel property for , we have
From Equations (13) and (14), we obtain
3.2 Evaluation of
In matrix form, the Equations (1), (3) and (4) can be written as:
where
, and .
Let denote the Laplace transform of the function . By taking Laplace transform on Equation (16), we get
with . By taking Laplace transform on Equation (8), we have
where . Comparing the constant term in both sides of Equation (10) and using Bessel property, we have
Taking Laplace transform on Equation (19) and simplifying, we get
where . By Equations (17), (18) and (20), we obtain
Let . We use the standard method to find , which is given by
where , , , and
The characteristic roots of the matrix are given by
Let be the characteristic roots of Equation (23). Such roots are
We observe that are rational algebraic functions in . The cofactor of the th element of is a Polynomial of degree . Since the characteristic roots of are all real and distinct, the inverse transforms of can be obtained by partial fraction decomposition and are given below
Using Equations (23)–(25) in Equation (21), we have
where
By routine calculations, Equation (26) can be simplified as
Then by Equation (27), we get
On Laplace inversion, we get an explicit expression for as
where is the -fold convolution of with itself. We note .
3.3 Evaluation of
By Equations (17) and (22), we obtain
and
Using Equations (30) and (31) and inverting, we have
3.4 Evaluation of
By using continued fractions, the transient-state system size probabilities are obtained. Denote with the Laplace transform of . On taking Laplace transform of Equation (2), we have
The well known identity of confluent hypergeometric function from [
16] is
which can be written as
From Equations (34) and (35), we obtain
Invoking of Equation (36), for , gives
On Laplace inversion, we have
where is the inverse Laplace transform of and denotes convolution.
3.5 Evaluation of
On taking the Laplace transform of Equation (1), we get
By Equations (3) and (4), we obtain
Thus, we have
and
Using Equations (27), (37), (39)–(41) and inverting, we get
3.6 Expression of
From Equation (37), we have
The confluent hypergeometric function is defined by the power series as
Then by resolving into partial fractions, we have
Also,
where
Using the identity given in [
17],
where and for
From Equations (43), (45) and (47), we obtain
Then, on Laplace inversion, we get
where
4 Performance Measures
In this section, we consider some important performance measures of the system.
4.1 Mean
Denote with the number of customers in the system at time . The average number of customers in the system at time is given by
and
From Equations (2)–(6), after considerable mathematical manipulations, the above equation will lead to the following differential equation
Then by Equation (51), we have
where , , and are given in Equations (15), (29), (32), (33) and (38).
4.2 Variance
Denote with the number of customers in the system at time . The variance of customers in the system at time is given by
where
Then
From Equations (2)–(6), after considerable mathematical manipulations, the above equation will lead to the following differential equation
Then by Equation (55), we have
Substituting the above equation in Equation (53), we obtain
where , , , and are given in Equations (15), (29), (32), (33), (38) and (52).
5 Special Cases
5.1 Case 1
When , vacations do not exist and the system simplifies as an queue with heterogeneously exponential servers. Equation (15) reuces to
and Equation (29) reuces to
Also Equations (32), (33) and (42) reuce to
and
5.2 Case 2
When , the system simplifies as an queue with heterogeneously exponential servers and multiple vacations. Equation (15) reuces to
and Equation (29) reuces to
Also Equations (32), (33) and (42) reuce to
and
6 Conclusion
In this paper, we stuy an multiple-vacations queueing system with two-heterogeneous exponential servers and impatient customers, whose arrival times are governed by a Markovian arrival process. We derive the transient solution for the model by using generating functions, the modified Bessel function, Laplace transforms and continued functions. Further, the explicit expressions for the mean and variance of the number of customers in the system are obtained.
{{custom_sec.title}}
{{custom_sec.title}}
{{custom_sec.content}}