Transient Analysis of a Two-Heterogeneous Severs Queue with Impatient Behaviour and Multiple Vacations

Jia XU, Liwei LIU, Taozeng ZHU

Journal of Systems Science and Information ›› 2018, Vol. 6 ›› Issue (1) : 69-84.

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Journal of Systems Science and Information ›› 2018, Vol. 6 ›› Issue (1) : 69-84. DOI: 10.21078/JSSI-2018-069-16
 

Transient Analysis of a Two-Heterogeneous Severs Queue with Impatient Behaviour and Multiple Vacations

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Abstract

We consider an M/M/2 queueing system with two-heterogeneous servers and multiple vacations. Customers arrive according to a Poisson process. However, customers become impatient when the system is on vacation. We obtain explicit expressions for the time dependent probabilities, mean and variance of the system size at time t by employing probability generating functions, continued fractions and properties of the modified Bessel functions. Finally, two special cases are provided.

Key words

M/M/2 queueing system / multiple vacations / probability generating functions / continued fractions / modified Bessel functions

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Jia XU , Liwei LIU , Taozeng ZHU. Transient Analysis of a Two-Heterogeneous Severs Queue with Impatient Behaviour and Multiple Vacations. Journal of Systems Science and Information, 2018, 6(1): 69-84 https://doi.org/10.21078/JSSI-2018-069-16

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 t. We study the transient behavior of an M/M/2 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 λ, and service time is exponentially distributed where the two servers namely faster server and slower server provide heterogeneous service with different service rate μ1 and μ2 such that μ1>μ2. 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 V exponentially distributed with parameter γ. 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 ξ. If the timer expires before the servers return from the vacation, he or she leaves the system never to return.
Let {X(t),t0} be the number of customers at time t, and let Y(t) be the status of the servers at time t, which is defined as follows:
Y(t)={3,if two servers are busy at time t,2,if the slower server is busy at time t,1,if the faster server is busy at time t,0,if two servers are on vacation at time t.
Then a continuous time Markov chain (CTMC) can be defined by {X(t),Y(t),t0} with a state space S={(i,j):i=0,1,; j=0,1,2,3}. Besides, we define Pn,0(t), P1,1(t), P1,2(t) and Pn,3(t) as
P1,1(t)=Prob{Y(t)=1,X(t)=1},P1,2(t)=Prob{Y(t)=2,X(t)=1},Pn,0(t)=Prob{Y(t)=0,X(t)=n},n=0,1,,Pn,3(t)=Prob{Y(t)=3,X(t)=n},n=2,3,.
Then, the behaviour of the resulting system is described by a set of Kolmogorov forward equations that can be written as follow:
P0,0(t)=λP0,0(t)+μ1P1,1(t)+μ2P1,2(t)+ξP1,0(t),
(1)
Pn,0(t)=(λ+γ+nξ)Pn,0(t)+(n+1)ξPn+1,0(t)+λPn1,0(t),n1,
(2)
P1,1(t)=(λ+μ1)P1,1(t)+γP1,0(t)+μ2P2,3(t),
(3)
P1,2(t)=(λ+μ2)P1,2(t)+μ1P2,3(t),
(4)
P2,3(t)=(λ+μ)P2,3(t)+λ[P1,1(t)+P1,2(t)]+μP3,3(t)+γP2,0(t),
(5)
Pn,3(t)=(λ+μ)Pn,3(t)+λPn1,3(t)+μPn+1,3(t)+γPn,0(t),n3,
(6)
where μ=μ1+μ2. We assume that initially there are no customers in the system, i.e., P0,0(0)=1.

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
G(z,t)=R(t)+n=0Pn+3,3(t)zn+1,
(7)
where
R(t)=P0,0(t)+P1,1(t)+P1,2(t)+P2,3(t)
(8)
with initial condition G(z,0)=1. By Equations (1) and (3)–(6), we obtain the following differential equation
Gt=λP0,0(t)+ξP1,0(t)+γ[P1,0(t)+P2,0(t)]+λ(z1)P2,3(t)+γn=1Pn+2,0(t)zn+[(λ+μ)+λz+μz][G(z,t)R(t)].
(9)
Solving the above equation, we get
G(z,t)=λ(z1)0texp{(λ+μ)(ty)}exp{(λz+μz)(ty)}P2,3(y)dy+[(λ+μ)(λz+μz)]0texp{(λ+μ)(ty)}exp{(λz+μz)(ty)}R(y)dyλ0texp{(λ+μ)(ty)}exp{(λz+μz)(ty)}P0,0(y)dy+ξ0texp{(λ+μ)(ty)}exp{(λz+μz)(ty)}P1,0(y)dy+γ0tn=1Pn+2,0(y)znexp{(λ+μ)(ty)}exp{(λz+μz)(ty)}dyγ0texp{(λ+μ)(ty)}exp{(λz+μz)(ty)}[P1,0(y)+P2,0(y)]dy+exp{(λ+μ)t}exp{(λz+μz)t}.
(10)
It is well known that
exp[(λz+μz)t]=(βz)nIn(αt),
(11)
where α=2λμ, β=λμ and In()=In[α(ty)] is the modified Bessel function of the first kind.
By Equations (10) and (11), we have
G(z,t)=0texp{(λ+μ)(ty)}[λβ1(βz)nIn1()λ(βz)nIn()]P2,3(y)dy+0texp{(λ+μ)(ty)}[(λ+μ)(βz)nIn()λβ1(βz)nIn1()μβ(βz)nIn+1()]R(y)dy0texp{(λ+μ)(ty)}λ(βz)nIn()P0,0(y)dy+0texp{(λ+μ)(ty)}ξ(βz)nIn()P1,0(y)dy+0texp{(λ+μ)(ty)}γi=1Pi+2,0(y)βi(βz)nIni()dy0texp{(λ+μ)(ty)}γ(βz)nIn()[P1,0(y)+P2,0(y)]dy+exp{(λ+μ)t}(βz)nIn(αt).
(12)

3.1 Evaluation of Pn+2,3(t),n1

Comparing the coefficients of zn on both sides of Equation (12) for n=1,2,, we obtain
βnPn+2,3(t)=0texp{(λ+μ)(ty)}[λβ1In1()λIn()]P2,3(y)dy+0texp{(λ+μ)(ty)}[(λ+μ)In()λβ1In1()μβIn+1()]R(y)dy0texp{(λ+μ)(ty)}λIn()P0,0(y)dy+0texp{(λ+μ)(ty)}ξIn()P1,0(y)dy+0texp{(λ+μ)(ty)}γi=1Pi+2,0(y)βiIni()dy0texp{(λ+μ)(ty)}γIn()[P1,0(y)+P2,0(y)]dy+exp{(λ+μ)t}In(αt).
(13)
The above Equation (13) holds for n=1,2, with left hand side replaced by zero. Using Bessel property In()=In() for n=1,2,, we have
0=0texp{(λ+μ)(ty)}[λβ1In+1()λIn()]P2,3(y)dy+0texp{(λ+μ)(ty)}[(λ+μ)In()λβ1In+1()μβIn1()]R(y)dy0texp{(λ+μ)(ty)}λIn()P0,0(y)dy+0texp{(λ+μ)(ty)}ξIn()P1,0(y)dy+0texp{(λ+μ)(ty)}γi=1Pi+2,0(y)βiIn+i()dy0texp{(λ+μ)(ty)}γIn()[P1,0(y)+P2,0(y)]dy+exp{(λ+μ)t}In(αt).
(14)
From Equations (13) and (14), we obtain
Pn+2,3(t)=nβn0texp{(λ+μ)(ty)}In()(ty)P2,3(y)dy+γ0texp{(λ+μ)(ty)}i=1βniPi+2,0(y)×[Ini()In+i()]dy.
(15)

3.2 Evaluation of P2,3(t)

In matrix form, the Equations (1), (3) and (4) can be written as:
dH(t)dt=BH(t)+ξP1,0(t)e1+[γP1,0(t)+μ2P2,3(t)]e2+μ1P2,3(t)e3,
(16)
where
H(t)=(P0,0(t),P1,1(t),P1,2(t))T,B=(λμ1μ20(λ+μ1)000(λ+μ2)),
e1=(1,0,0)T, e2=(0,1,0)T and e3=(0,0,1)T.
Let f(s) denote the Laplace transform of the function f(t). By taking Laplace transform on Equation (16), we get
H(s)=(sIB)1{H(0)+ξP1,0(s)e1+[γP1,0(s)+μ2P2,3(s)]e2+μ1P2,3(s)e3}
(17)
with H(0)=(1,0,0)T. By taking Laplace transform on Equation (8), we have
R(s)=eH(s)+P2,3(s),
(18)
where e=(1,1,1). Comparing the constant term in both sides of Equation (10) and using Bessel property, we have
R(t)=0texp{(λ+μ)(ty)}[λβ1I1()λI0()]P2,3(y)dy+0texp{(λ+μ)(ty)}[(λ+μ)I0()λβ1I1()μβI1()]R(y)dy0texp{(λ+μ)(ty)}λI0()P0,0(y)dy+0texp{(λ+μ)(ty)}ξI0()P1,0(y)dy+0texp{(λ+μ)(ty)}γi=1Pi+2,0(y)βiIi()dy0texp{(λ+μ)(ty)}γI0()[P1,0(y)+P2,0(y)]dy+exp{(λ+μ)t}I0(αt).
(19)
Taking Laplace transform on Equation (19) and simplifying, we get
sR(s)=12P2,3(s)[ww2α22λ]λP0,0(s)+ξP1,0(s)γ[P1,0(s)+P2,0(s)]+γi=1Pi+2,0(s)(ww2α22λ)i+1,
(20)
where w=s+λ+μ. By Equations (17), (18) and (20), we obtain
P2,3(s)=[s12s(ww2α22λ)+se(sIB)1(μ2e2+μ1e3)]1×{1+ξP1,0(s)λP0,0(s)γP1,0(s)γP2,0(s)+γi=1Pi+2,0(s)(ww2α22λ)i    se(sIB)1[1+ξP1,0(s)e1+γP1,0(s)e2]}.
(21)
Let (sIB)1=(bij(s))3×3. We use the standard method to find (sIB)1, which is given by
1|D(s)|(b1(s)b2(s)μ1b2(s)μ2b1(s)0c(s)b2(s)λμ200c(s)b1(s)λμ1),
(22)
where b1(s)=s+λ+μ1, b2(s)=s+λ+μ2, c(s)=s+λ, and
|D(s)|=s3+(3λ+μ)s2+(3λ2+2λμ+μ1μ2)s+λ2μ+λμ1μ2+λ3.
The characteristic roots of the matrix B are given by
|D(s)|=0.
Let sk (k=1, 2, 3) be the characteristic roots of Equation (23). Such roots are
s1=λ,s2=(λ+μ1),s3=(λ+μ2).
We observe that bij(s) are rational algebraic functions in s. The cofactor of the (i,j)th element of (sIB) is a Polynomial of degree 2|ij|. Since the characteristic roots sk (k=1, 2, 3) of B are all real and distinct, the inverse transforms bij(t) of bij(s) can be obtained by partial fraction decomposition and are given below
e(sIB)1e1=j=13bj1(s),
(23)
e(sIB)1e2=j=13bj2(s),
(24)
e(sIB)1(μ2e2+μ1e3)=μ2j=13bj2(s)+μ1j=13bj3(s).
(25)
Using Equations (23)–(25) in Equation (21), we have
P2,3(s)=[s12s(ww2α22λ)+c2(s)]1×{1+ξP1,0(s)λP0,0(s)γP1,0(s)γP2,0(s)    +γi=1Pi+2,0(s)(ww2α22λ)i    c0(s)ξc0(s)P1,0(s)γc1(s)P1,0(s)},
(26)
where
c0(s)=s[b11(s)+b21(s)],c1(s)=s[b12(s)+b22(s)],c2(s)=s[μ2(b12(s)+b22(s))+μ1(b13(s)+b23(s)+b33(s))].
By routine calculations, Equation (26) can be simplified as
P2,3(s)=2w+w2α2{1+ξP1,0(s)λP0,0(s)γP1,0(s)γP2,0(s)+γi=1Pi+2,0(s)(ww2α22λ)ic0(s)ξc0(s)P1,0(s)γc1(s)P1,0(s)}×[1(μλ)12ww2α2α(1c2(s)μ)]1.
(27)
Then by Equation (27), we get
P2,3(s)={2α(ww2α2α)[1+ξP1,0(s)λP0,0(s)γP1,0(s)γP2,0(s)+γi=1Pi+2,0(s)(ww2α22λ)ic0(s)ξc0(s)P1,0(s)γc1(s)P1,0(s)]}×[m=0(μλ)m2(ww2α2α)m×k=0m(1)k(mk)(c2(s)μ)k].
(28)
On Laplace inversion, we get an explicit expression for P2,3(t) as
P2,3(t)=[m=0(μλ)m2k=0m(1)k(mk)(1μ)k]×{0tc2k(tu)exp{(λ+μ)u}[Im(αu)Im+2(αu)]du+(ξγ)0tc2k(tu)[0uP1,0(uv)exp{(λ+μ)v}(Im(αv)Im+2(αv))dv]duλ0tc2k(tu)[0uP0,0(uv)exp{(λ+μ)v}(Im(αv)Im+2(αv))dv]duγ0tc2k(tu)[0uP2,0(uv)exp{(λ+μ)v}(Im(αv)Im+2(αv))dv]du+γ0tc2k(tu)[0ui=1Pi+2,0(uv)βiexp{(λ+μ)v}(Im+i(αv)Im+i+2(αv))dv]du0tc2k(tu)[0uc0(uv)exp{(λ+μ)v}(Im(αv)Im+2(αv))dv]duξ0tc2k(tu)P1,0(tu)×[0uc0(uv)exp{(λ+μ)v}(Im(αv)Im+2(αv))dv]duγ0tc2k(tu)P1,0(tu)×[0uc1(uv)exp{(λ+μ)v}(Im(αv)Im+2(αv))dv]du},
(29)
where c2k is the k-fold convolution of c2(t) with itself. We note C0(t)=δ(t).

3.3 Evaluation of P1,1(t),P1,2(t)

By Equations (17) and (22), we obtain
P1,1(s)=[ξP1,0(s)+1]b21(s)+[μ2P2,3(s)+γP1,0(s)]b22(s)+μ1P2,3(s)b13(s)
(30)
and
P1,2(s)=[ξP1,0(s)+1]b31(s)+[μ2P2,3(s)+γP1,0(s)]b32(s)+μ1P2,3(s)b33(s).
(31)
Using Equations (30) and (31) and inverting, we have
P1,1(t)=b21(t)+0tξP1,0(u)b21(tu)du+0tμ1P2,3(u)b23(tu)du+0t[μ2P2,3(u)+γP1,0(u)]b22(tu)du,
(32)
P1,2(t)=b31(t)+0tξP1,0(u)b31(tu)du+0tμ1P2,3(u)b33(tu)du+0t[μ2P2,3(u)+γP1,0(u)]b32(tu)du.
(33)

3.4 Evaluation of Pn,0(t),n1

By using continued fractions, the transient-state system size probabilities Pn,0, n1 are obtained. Denote with f(s) the Laplace transform of f(t). On taking Laplace transform of Equation (2), we have
Pn,0(s)Pn1,0(s)=λ(s+λ+γ+nξ)(n+1)ξPn+1,0(s)Pn,0(s)=λ(s+λ+γ+nξ)(n+1)ξλs+λ+γ+(n+1)ξ(n+2)ξλs+λ+γ+(n+2)ξ.
(34)
The well known identity of confluent hypergeometric function from [16] is
1F1(a+1;c+1;z)1F1(a;c;z)=ccz+(a+1)zcz+1+(a+2)zcz+2+,
which can be written as
c1F1(a;c;z)1F1(a+1;c+1;z)(cz)=(a+1)zcz+1+(a+2)zcz+2+
(35)
From Equations (34) and (35), we obtain
Pn,0(s)Pn1,0(s)=λξ1F1(n+1;s+γξ+n+1;λξ)(s+γξ+n)1F1(n;(s+γξ+n);λξ).
(36)
Invoking of Equation (36), for n=1,2,, gives
Pn,0(s)=(λξ)n1Πi=1n(s+γξ+i)1F1(n+1;s+γξ+n+1;λξ)1F1(1;(s+γξ+1);λξ)P0,0(s)=Φn(s)P0,0(s).
(37)
On Laplace inversion, we have
Pn,0(t)=Φn(t)P0,0(t),
(38)
where Φn(t) is the inverse Laplace transform of Φn(s) and denotes convolution.

3.5 Evaluation of P0,0(t)

On taking the Laplace transform of Equation (1), we get
P0,0(s)=1s+λξP1,0(s)P0,0(s)μ1P1,1(s)P0,0(s)μ2P1,2(s)P0,0(s).
(39)
By Equations (3) and (4), we obtain
sP1,1(s)=(λ+μ1)P1,1(s)+μ2P2,3(s)+γP1,0(s),sP1,2(s)=(λ+μ2)P1,2(s)+μ1P2,3(s).
Thus, we have
P1,1(s)=1s+λ+μ1[μ2P2,3(s)+γP1,0(s)]
(40)
and
P1,2(s)=1s+λ+μ2μ1P2,3(s).
(41)
Using Equations (27), (37), (39)–(41) and inverting, we get
P0,0(t)=12πjσjσ+jest{1+[2w+w2α22c0(s)w+w2α2]×[1(μλ)12ww2α2α(1c2(s)μ)]1}×{[s+λ(ξ+μ1γs+λ+μ1)Φ1(s)]2w+w2α2×[(ξγξc0(s)γc1(s))Φ1(s)λ+i=1Φi+2(s)(ww2α22λ)i]×[1(μλ)12ww2α2α(1c2(s)μ)]1}1ds.
(42)

3.6 Expression of Φn(t)

From Equation (37), we have
Φn(s)=(λξ)n1Πi=1n(s+γξ+i)1F1(n+1;s+γξ+n+1;λξ)1F1(1;s+γξ+1;λξ).
(43)
The confluent hypergeometric function is defined by the power series as
1F1(n+1;s+γξ+n+1;λξ)Πi=1n(s+γξ+i)=ξnk=0(n+kk)(λ)kΠi=1n+k(s+γ+iξ).
(44)
Then by resolving into partial fractions, we have
1F1(n+1;s+γξ+n+1;λξ)Πi=1n(s+γξ+i)=ξnk=0(n+kk)(λξ)ki=1n+k(1)i1(i1)!(n+ki)!1s+γ+iξ.
(45)
Also,
1F1(1;s+γξ+1;λξ)=k=0(λ)kΠi=1k(s+γ+iξ)=k=o(λ)kak(s),a0(s)=1,
(46)
where
ak(s)=1Πi=1k(s+λ+iξ)=1ξk1r=1k(1)r1(r1)!(kr)!1s+γ+rξ,k=1,2,.
Using the identity given in [17],
[1F1(1;s+γξ+1);λξ]1=k=0bk(s)λk,
(47)
where b0(s)=1 and for k=1,2,
bk(s)=|a1(s)1a2(s)a1(s)1ak1(s)ak2(s)ak3(s)a1(s)1ak(s)ak1(s)ak2(s)a2(s)a1(s)|=i=1k(1)i1ai(s)bki(s).
From Equations (43), (45) and (47), we obtain
Φn(s)=λnj=0(λ)j(n+jj)an+j(s)k=1λkbk(s).
Then, on Laplace inversion, we get
Φn(t)=λnj=0(λ)j(n+jj)an+j(t)k=1λkbk(t),
where
ak(t)=1ξk1r=1k(1)r1(r1)!(kr)!exp{(γ+rξ)t},k=1,2,bk(t)=i=1k(1)i1ai(t)bki(t),k=2,3,,b1(t)=a1(t).
(48)

4 Performance Measures

In this section, we consider some important performance measures of the system.

4.1 Mean

Denote with X(t) the number of customers in the system at time t. The average number of customers in the system at time t is given by
m(t)=E[X(t)]=P1,1(t)+P1,2(t)+n=0(n+2)Pn+2,3(t)+n=1nPn,0(t)
(49)
and
m(t)=P1,1(t)+P1,2(t)+n=0(n+2)Pn+2,3(t)+n=1nPn,0(t).
(50)
From Equations (2)–(6), after considerable mathematical manipulations, the above equation will lead to the following differential equation
m(t)=(λ+μ)m(t)+μ2P1,1(t)+μ1P1,2(t)+n=1(n+2)(μγnξ)Pn,0(t)+γn=1nPn,0(t)+μP2,3(t)+μn=0(n+2)Pn+3,3(t)+ξn=1(n+1)(n+2)Pn+1,0(t)+λ{n=1(n+2)Pn+1,3(t)+2[P1,1(t)+P1,2(t)]+n=1(n+2)Pn+1,0(t)}.
(51)
Then by Equation (51), we have
m(t)=γn=10te(λ+μ)(tu)Pn,0(u)du+μn=0(n+2)0te(λ+μ)(tu)Pn+3,3(u)du+0te(λ+μ)(tu)[μ2P1,1(u)+μ1P1,2(u)+μP2,3(u)]du+n=1(n+2)(nγnξ)0te(λ+μ)(tu)Pn,0(u)du+λn=1(n+2)0te(λ+μ)(tu)[Pn+1,3(u)+Pn+1,0(u)]du+2λ0te(λ+μ)(tu)[P1,1(u)+P1,2(u)]du+ξn=1(n+1)(n+2)0te(λ+μ)(tu)Pn+1,0(u)du,
(52)
where P1,1(t), P1,2(t), Pn,3(t)(n2) and Pn,0(t)(n1) are given in Equations (15), (29), (32), (33) and (38).

4.2 Variance

Denote with X(t) the number of customers in the system at time t. The variance of customers in the system at time t is given by
Var[X(t)]=E[X2(t)][E[X(t)]]2,Var[X(t)]=r(t)[m(t)]2,
(53)
where
r(t)=E[X2(t)]=P1,1(t)+P1,2(t)+n=0(n+2)2Pn+2,3(t)+n=1n2Pn,0(t).
Then
r(t)=P1,1(t)+P1,2(t)+n=0(n+2)2Pn+2,3(t)+n=1n2Pn,0(t).
(54)
From Equations (2)–(6), after considerable mathematical manipulations, the above equation will lead to the following differential equation
r(t)=(λ+μ)r(t)+μ2P1,1(t)+μ1P1,2(t)+n=1(n+2)2(μγnξ)Pn,0(t)+γn=1n2Pn,0(t)+μP2,3(t)+μn=0(n+2)2Pn+3,3(t)+ξn=1(n+1)(n+2)2Pn+1,0(t)+λ{n=1(n+2)2Pn+1,3(t)+4[P1,1(t)+P1,2(t)]+n=1(n+2)2Pn+1,0(t)}.
(55)
Then by Equation (55), we have
r(t)=γn=10te(λ+μ)(tu)Pn,0(u)du+μn=0(n+2)20te(λ+μ)(tu)Pn+3,3(u)du+0te(λ+μ)(tu)[μ2P1,1(u)+μ1P1,2(u)+μP2,3(u)]du+n=1(n+2)2(μγnξ)0te(λ+μ)(tu)Pn,0(u)du+λn=1(n+2)20te(λ+μ)(tu)[Pn+1,3(u)+Pn+1,0(u)]du+4λ0te(λ+μ)(tu)[P1,1(u)+P1,2(u)]du+ξn=1(n+1)(n+2)20te(λ+μ)(tu)Pn+1,0(u)du.
(56)
Substituting the above equation in Equation (53), we obtain
Var[X(t)]=γn=10te(λ+μ)(tu)Pn,0(u)du+μn=0(n+2)20te(λ+μ)(tu)Pn+3,3(u)du+0te(λ+μ)(tu)[μ2P1,1(u)+μ1P1,2(u)+μP2,3(u)]du+n=1(n+2)2(μγnξ)0te(λ+μ)(tu)Pn,0(u)du+λn=1(n+2)20te(λ+μ)(tu)[Pn+1,3(u)+Pn+1,0(u)]du+4λ0te(λ+μ)(tu)[P1,1(u)+P1,2(u)]du+ξn=1(n+1)(n+2)20te(λ+μ)(tu)Pn+1,0(u)du[m(t)]2,
(57)
where P1,1(t), P1,2(t), Pn,3(t)(n2), Pn,0(t)(n1) and m(t) are given in Equations (15), (29), (32), (33), (38) and (52).

5 Special Cases

5.1 Case 1

When γ=ξ=0, vacations do not exist and the system simplifies as an M/M/2 queue with heterogeneously exponential servers. Equation (15) reduces to
Pn+2,3(t)=nβn0texp{(λ+μ)(ty)}In()(ty)P2,3(y)dy
(58)
and Equation (29) reduces to
P2,3(t)=[m=0(μλ)m2k=0m(1)k(mk)(1μ)k]×{0tc2k(tu)exp{(λ+μ)u}[Im(αu)Im+2(αu)]duλ0tc2k(tu)[0uP0,0(uv)exp{(λ+μ)v}(Im(αv)Im+2(αv))dv]du0tc2k(tu)[0uc0(uv)exp{(λ+μ)v}(Im(αv)Im+2(αv))dv]du.
(59)
Also Equations (32), (33) and (42) reduce to
P1,1(t)=b21(t)+0tμ1P2,3(u)b23(tu)du+0tμ2P2,3(u)b22(tu)du,
(60)
P1,2(t)=b31(t)+0tμ1P2,3(u)b33(tu)du+0tμ2P2,3(u)b32(tu)du,
(61)
and
P0,0(t)=12πjσjσ+jest{1+[2w+w2α22c0(s)w+w2α2]×[1(μλ)12ww2α2α(1c2(s)μ)]1}×{s+λ2w+w2α2[i=1Φi+2(s)(ww2α22λ)iλ]×[1(μλ)12ww2α2α(1c2(s)μ)]1}1ds.
(62)

5.2 Case 2

When ξ=0, the system simplifies as an M/M/2 queue with heterogeneously exponential servers and multiple vacations. Equation (15) reduces to
Pn+2,3(t)=nβn0texp{(λ+μ)(ty)}In()(ty)P2,3(y)dy+γ0texp{(λ+μ)(ty)}i=1βniPi+2,0(y)×[Ini()In+i()]dy
(63)
and Equation (29) reduces to
P2,3(t)=[m=0(μλ)m2k=0m(1)k(mk)(1μ)k]×{0tc2k(tu)exp{(λ+μ)u}[Im(αu)Im+2(αu)]duγ0tc2k(tu)[0uP1,0(uv)exp{(λ+μ)v}(Im(αv)Im+2(αv))dv]duλ0tc2k(tu)[0uP0,0(uv)exp{(λ+μ)v}(Im(αv)Im+2(αv))dv]duγ0tc2k(tu)[0uP2,0(uv)exp{(λ+μ)v}(Im(αv)Im+2(αv))dv]du+γ0tc2k(tu)[0ui=1Pi+2,0(uv)βiexp{(λ+μ)v}(Im+i(αv)Im+i+2(αv))dv]du0tc2k(tu)[0uc0(uv)exp{(λ+μ)v}(Im(αv)Im+2(αv))dv]duγ0tc2k(tu)P1,0(tu)×[0uc1(uv)exp{(λ+μ)v}(Im(αv)Im+2(αv))dv]du}.
(64)
Also Equations (32), (33) and (42) reduce to
P1,1(t)=b21(t)+0tμ1P2,3(u)b23(tu)du+0t[μ2P2,3(u)+γP1,0(u)]b22(tu)du,
(65)
P1,2(t)=b31(t)+0tμ1P2,3(u)b33(tu)du+0t[μ2P2,3(u)+γP1,0(u)]b32(tu)du,
(66)
and
P0,0(t)=12πjσjσ+jest{1+[2w+w2α22c0(s)w+w2α2]×[1(μλ)12ww2α2α(1c2(s)μ)]1}×{[s+λμ1γs+λ+μ1Φ1(s)]2w+w2α2×[i=1Φi+2(s)(ww2α22λ)i(γ+γc1(s))Φ1(s)λ]×[1(μλ)12ww2α2α(1c2(s)μ)]1}1ds.
(67)

6 Conclusion

In this paper, we study an M/M/2 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.

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Funding

the National Natural Science Foundation of China(11671204)
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