International Journal of Trend in Scientific Research and Development (IJTSRD)
Volume 5 Issue 5, July-August 2021 Available Online: www.ijtsrd.com e-ISSN: 2456 – 6470
The Queue Length of a GI/M/1 Queue with Set-Up
Period and Bernoulli Working Vacation Interruption
Li Tao
School of Mathematics and Statistics, Shandong University of Technology, Zibo, China
ABSTRACT
Consider a GI/M/1 queue with set-up period and working vacations.
During the working vacation period, customers can be served at a
lower rate, if there are customers at a service completion instant, the
vacation can be interrupted and the server will come back to a set-up
period with probability p(0 ≤ p ≤ 1) or continue the working vacation
with probability 1 − p , and when the set-up period ends, the server will
switch to the normal working level. Using the matrix analytic
method, we obtain the steady-state distributions for the queue length
at arrival epochs.
KEYWORDS: GI/M/1; set-up period; working vacation; vacation
interruption; Bernoulli
How to cite this paper: Li Tao "The
Queue Length of a GI/M/1 Queue with
Set-Up Period and Bernoulli Working
Vacation
Interruption"
Published
in
International Journal
of
Trend
in
Scientific Research
IJTSRD43743
and Development
(ijtsrd), ISSN: 24566470, Volume-5 | Issue-5, August 2021,
pp.72-76,
URL:
www.ijtsrd.com/papers/ijtsrd43743.pdf
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1. INTRODUCTION
Servi and Finn [1] first introduced the working
vacation models and studied an M/M/1 queue, the
server commits a lower service rate rather than
completely stopping the service during a vacation.
Baba [2] considered a GI/M/1 queue with working
vacations by the matrix-analytic method. For the
vacation interruption models, Li and Tian [3] first
introduced and studied an M/M/1 queue with working
vacations and vacation interruption. Then, Li et al. [4]
analyzed the GI/M/1 queue with working vacations
and vacation interruption by the matrix-analytic
method. Meanwhile, in some practical situations, it
needs some times to switch the lower rate to the
normal working level, which we call set-up times.
Zhao et al. [5] considered a GI/M/1 queue with set-up
period and working vacation and vacation
interruption. Bai et al. [6] studied a GI/M/1 queue
with set-up period and working vacations.
In this paper, based on the Bernoulli schedule rule we
analyze a GI/M/1 queue with set-up period and
working vacation and vacation interruption at the
same time. Zhang and Shi [7] first studied an M/M/1
queue with vacation and vacation interruption under
the Bernoulli rule. In our model, during the working
vacation period, if there are customers at a service
completion instant, the server can come back to a setup period with probability p(0 ≤ p ≤ 1) , not with
probability 1, or continue the working vacation with
probability 1 − p , which is different from the situation
many authors considered before, and when the set-up
period ends, the server will switch to the normal busy
period. Clearly, the models in [5,6] will be the special
cases of the model we consider.
2. Model description and embedded Markov
chain
Consider a GI/M/1 queue such that the arrival process
is a general distribution process. The server begins a
vacation each time when the queue becomes empty
and if there are customers arriving in a vacation
period, the server continues to work at a lower rate,
i.e., the working vacation period is an operation
period in lower speed. At a service completion
instant, if there are customers in the vacation period,
the vacation can be interrupted and the server is
resumed to a set-up period with probability
p(0 ≤ p ≤ 1) , or continues the vacation with probability
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p ( p = 1 − p ),
and when the set-up period ends, the
server will switch to the normal working level.
Otherwise, the server continues the vacation.
Meanwhile, if there is no customer when a vacation
ends, the server begins another vacation, otherwise,
he switches to the set-up period, and after the set-up
period, the server switches to the normal busy period.
Suppose τ n be the arrival epoch of nth customers with
τ 0 = 0 . The inter-arrival times { τ n , n ≥ 1 } are
independent and identically distributed with a general
distribution function, denoted by A(t ) with a mean
1 / λ and a Laplace Stieltjes transform (LST), denoted
by A* ( s) . The service times during a normal service
period, the service times during a working vacation
period, the set-up times and the working vacation
times are exponentially distributed with rate µ , η ,
β and θ , respectively.
∞
t
η (η x) k −1
0
0
(k − 1)!
g k = ∫ p k −1 p ∫
e −η x e −θ x e − β (t − x ) dxdA(t ), k ≥ 1.
Using the lexicographic sequence for the states, the
transition probability matrix of ( Ln , J n ) can be written
as the Block-Jacobi matrix
 B00

 B1
P =  B2

 B3
 M

A01
A1
A2
A3
M
A0
A1
A2
M
A0
A1
M
A0
M



,


O 
where
B00 = 1 − c0 − d 0 − f 0 ;
 c0

A0 =  0
0

A01 = (c0 , f 0 , d 0 );
f0
d0 
 ck


A ( β ) b0  ; Ak =  0
0
0
a0 

*
fk + gk
0
0
d k + ek 

bk  ,
ak 
k


1
−
 ∑ (ci + di + ei + f i + gi ) − c0 − d0 − f 0 
 i =1

k


*
1 − ∑ bi − A ( β )
Bk = 
 , k ≥ 1.
i =0


k


1 − ∑ ai


i =0


Let L(t ) be the number of customers in the system at
time t and Ln = L(τ n − 0) be the number of the customers
before the nth arrival. Define J n = 0 , the nth arrival
occurs during a working vacation period; J n = 1 , the
nth arrival occurs during a set-up period; J n = 2 , the
nth arrival occurs during a normal service period.
Then, the process {( Ln , J n ), n ≥ 1} is an embedded
Markov chain with state space
3. Steady-state distribution at arrival epochs
We
first
define
Ω = {0, 0} ∪ {(k , j ), k ≥ 1, j = 0,1, 2}.
A( z ) = ∑ ak z k , B ( z ) = ∑ bk z k , C ( z ) = ∑ ck z k ,
p(i , j ),( k ,l ) = P ( Ln +1 = k , J n +1 = ∣
l Ln = i, J n = j ).
Meanwhile, we introduce the expressions below
∞
0
bk = ∫
∞
0
( µ t )k − µt
e dA(t ), k ≥ 0,
k!
∫
t
0
β e− β x
∞
ck = ∫ p k
0
dk = ∫
0
(η x)l −η x −θ x t − β ( y − x )
e θ e ∫ βe
x
0
l!
∑p ∫
l
l =0
t
( µ (t − y )) k − l − µ ( t − y )
e
dydxdA(t ), k ≥ 0,
(k − l )!
×
ek = ∫
( µ (t − x)) k − µ (t − x )
e
dxdA(t ), k ≥ 0,
k!
(η t ) k −ηt −θ t
e e dA(t ), k ≥ 0,
k!
∞ k
∞ k
0
∑p
l −1
l =1
p∫
t
η (η x)
0
l −1
(l − 1)!
t
e−η x e −θ x ∫ β e− β ( y − x )
t
0
0
fk = ∫ p k ∫
(η x) −η x −θ x − β ( t − x )
e θe e
dxdA(t ), k ≥ 0,
k!
k
∞
∞
k =0
∞
k =0
∞
∞
k =0
k =1
D( z ) = ∑ d k z , E ( z ) = ∑ ek z , F ( z ) = ∑ f k z k , G ( z ) = ∑ g k z k .
k
k =0
k
k =1
In this section, we derive the steady-state distribution
for ( Ln , J n ) at arrival epochs using matrix-geometric
approach. In order to derive the steady-state
distribution, we need the following three lemmas.
Lemma 3.1.
A( z ) = A* ( µ − µ z ),
B( z ) =
β [ A* ( µ − µ z ) − A* ( β )]
,
β − µ (1 − z )
C ( z ) = A* (θ + η − pη z ),
D( z ) =
−
θβ
[ A* (θ + η − pη z ) − A* ( β )]
β − µ (1 − z )
θ + η − pη z − β
[ A* (θ + η − pη z ) − A* ( µ − µ z )]
θβ
,
β − µ (1 − z )
θ + pη z − ( µ − η )(1 − z )
x
( µ (t − y )) k − l − µ ( t − y )
×
e
dydxdA(t ), k ≥ 1,
(k − l )!
∞
∞
k =0
In order to express the transition matrix of ( Ln , J n ) , let
ak = ∫
∞
E ( z) =
−
pη z β [ A* (θ + η − pη z ) − A* ( β )]
β − µ (1 − z )
θ + η − pη z − β
pη z β [ A* (θ + η − pη z ) − A* ( µ − µ z )]
,
β − µ (1 − z )
θ + pη z − ( µ − η )(1 − z )
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F ( z) =
G( z) =
θ [ A* ( β ) − A* (θ + η − pη z )]
,
θ + η − pη z − β
where
σ=
pη z[ A* ( β ) − A* (θ + η − pη z )]
.
θ + η − pη z − β
Lemma 3.3. If θ > 0 , β > 0 and ρ = λ / µ < 1 , then the
matrix
equation R = ∑ R k Ak has
the
minimal
k =0
nonnegative solution
π 00 + (π 10 , π 11 , π 12 )( I − R )−1 e = 1,
where e is a column vector with all elements equal to
one. Substituting R into the above relationship, we
can get
(1 − r1 )(1 − r2 )(1 − r3 )
= (1 − r3 )σ .
(1 − r3 )[(1 − r2 ) + δ (r2 − r1 )] + ∆δ (1 − r1 )(r3 − r2 ) − γδ (1 − r2 )(r3 − r1 )
Therefore, we have
(π 10 , π 11 , π 12 ) = (1 − r3 )σ (r1 , δ (r2 − r1 ), ∆δ (r3 − r2 ) − γδ (r3 − r1 )) .
where r2 = A* ( β ) , r1 and r3 are the unique roots in the
range
0<z<1
of
equations
*
*
z = A (θ + η − pη z ) and z = A ( µ − µ z ) , respectively, and
δ = (θ + pη r1 ) / (θ + η − pη r1 − β ) ,
∆ = β / [ β − µ (1 − r2 )] ,
γ = β / [θ + pη r1 − ( µ − η )(1 − r1 )] .
Moreover, we can easily verify that the Markov
chain P is positive recurrent if and only if
θ > 0 , β > 0 and ρ < 1 . And the matrix
with ω = [
With
(π 00 , π 10 , π 11 , π 12 ) is
K=
 r1 δ (r2 − r1 ) ∆δ (r3 − r2 ) − γδ (r3 − r1 ) 


R =0
r2
∆(r3 − r2 )
,
0

0
r3


B00
A01



∞
B[ R ] =  ∞ k −1
k −1

 ∑ R Bk ∑ R Ak 
 k =1
k =1

1 − c0 − d 0 − f 0
c0


 c0 + f 0 − δ ( r2 − r1 ) − ω 1 − c0
 r1
r1
r1

=
a0 ∆ ( r3 − r2 ) b0
+
1−
0

r2 r3
r2

a0

0

r3

the Theorem 1.5.1 in [8],
given by the positive left invariant
vector Eq. (1), and satisfies the normalizing condition
Proof.
θ >0,
the
Lemma
3.2.
If
equation z = A* (θ + η − pη z ) has a unique root in the
range 0<z<1.
∞
(1 − r1 )(1 − r2 )
.
(1 − r3 )[(1 − r2 ) + δ ( r2 − r1 )] + ∆δ (1 − r1 )(r3 − r2 ) − γδ (1 − r2 )( r3 − r1 )
Using the Theorem 1.5.1 of Neuts [8], we can obtain
π k = (π k 0 , π k1 , π k 2 ) = (π 10 , π 11 , π 12 ) R k −1 , k ≥ 1. (2)
Taking (π 10 , π 11 , π 12 ) and R k −1 into Eq. (2), the theorem
can be derived.
Then, we discuss the distribution of the queue length
L at the arrival epochs. From Theorem 3.4, we have
π 0 = P{L = 0} = π 00 = (1 − r3 )σ ,
π k = P{L = k} = π k 0 + π k1 + π k 2
= (1 − r3 )σ [(1 − δ )r1k + δ r2k + ∆δ (r3k − r2k ) − γδ (r3k − r1k )], k ≥ 1.
f0
δ ( r2 − r1 ) f 0
−
r1
r1
0
0



ω


a0 ∆ ( r3 − r2 ) b0 
−
r2 r3
r2 

a

1− 0

r3

d0
∆δ (r3 − r2 ) γδ (r3 − r1 )
d
δ (r2 − r1 )
−
]a0 +
b0 − 0
r2 r3
r1r3
r1r2
r1
has a
positive left invariant vector
The state probability of a server in the steady-state is
given by
∞
P{J = 0} = ∑ π k 0 =
k =0
(1 − r3 )σ
,
1 − r1
∞
σδ (1 − r3 )(r2 − r1 )
k =1
(1 − r1 )(1 − r2 )
P{J = 1} = ∑ π k 1 =
,
∞
σ∆δ (1 − r1 )(r3 − r2 ) − σγδ (1 − r2 )(r3 − r1 )
k =1
(1 − r1 )(1 − r2 )
P{J = 2} = ∑ π k 2 =
.
π 0 = π 00 ; π k = (π k 0 , π k1 , π k 2 ), k ≥ 1,
Theorem 3.5. If θ > 0 , β > 0 and ρ < 1 , the stationary
queue length L can be decomposed as L = L0 + Ld ,
where L0 is the stationary queue length of a classical
GI/M/1 queue without vacation, and follows a
geometric distribution with parameter r3 . Additional
queue length Ld has a distribution
π kj = P{L = k , J = j} = lim P{Ln = k , J n = j}, (k , j ) ∈ Ω.
P{Ld = 0} = σ ,
Theorem 3.4. If θ > 0 , β > 0 and ρ < 1 , the stationary
probability distribution of ( L, J ) is given by
P{Ld = k} = σ (δ − 1 − γδ )(r3 − r1 )r1k −1
π k 0 = (1 − r3 )σ r1k , k ≥ 0,

k
k
π k 1 = (1 − r3 )σδ (r2 − r1 ), k ≥ 0,
π = (1 − r )σ [∆δ (r k − r k ) − γδ (r k − r k )], k ≥ 1,
3
3
2
3
1
 k2
Proof. The probability generating function of L is as
follows:
K (1, r1 , δ (r2 − r1 ), ∆δ (r3 − r2 ) − γδ (r3 − r1 )) (1)
where K ia a random positive real number.
Let ( L, J ) be the stationary limit of the process ( Ln , J n ) ,
and denote
n →∞
+σδ (∆ − 1)(r3 − r2 ) r2k −1 ,
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k ≥ 1.
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International Journal of Trend in Scientific Research and Development @ www.ijtsrd.com eISSN: 2456-6470
∞
∞
∞
k =0
k =1
k =1
∞
L( z ) = ∑ π k 0 z k + ∑ π k1 z k + ∑ π k 2 z k
1 − r3 1 − r3 z
( r − r )(1 − r3 z ) z
(r − r ) z
(r − r ) z
σ[
+δ 2 1
+ ∆δ 3 2 − γδ 3 1 ]
1 − r3 z 1 − r1 z
(1 − r1 z )(1 − r2 z )
1 − r2 z
1 − r1 z
1− r3
(r − r )z
(r − r )z (r − r )z
(r − r )z
(r − r )z
=
σ[1− 3 1 + δ 3 1 −δ 3 2 +∆δ 3 2 − γδ 3 1 ]
1− r3 z
1− r1z
1− r1z
1− r2 z
1− r2 z
1− r1z
1 − r3
(r − r ) z
(r − r ) z
=
[σ + σ (δ − 1 − γδ ) 3 1 + σδ (∆ − 1) 3 2 ]
1 − r3 z
1 − r1 z
1 − r2 z
i =k
Thus, the mean queue length at the arrival epoch is
given by
r3
σ (δ − 1 − γδ )(r3 − r1 ) σδ (∆ − 1)(r3 − r2 )
+
+
.
1 − r3
(1 − r1 ) 2
(1 − r2 ) 2
4. Steady-state distribution at arbitrary epochs
Now we consider the steady-state distribution for the
queue length at arbitrary epochs. And, denote the
limiting distribution of L(t ) : pk = lim
P{L(t ) = k} .
t →∞
Theorem 4.1. If θ > 0 , β > 0 and ρ < 1 , the limiting
distribution of L(t ) exists. And, we obtain
∆δ − γδ (1 − ∆)δ (1 − r3 ) (1 − δ + γδ )(1 − r3 )

+
+
],
 p0 = 1 − σλ[ µ
β
θ + η − pη r1



∆δ − γδ k −1 (1 − ∆)δ (1 − r2 ) k −1 (1 − δ + γδ )(1 − r1 ) k −1
 pk = (1 − r3 )σλ[
r3 +
r2 +
r1 ], k ≥ 1.
µ
β
θ + η − pη r1

Proof. From the theory of SMP, the limiting
distribution of L(t ) has the following expression (see
[9]):
pk =
∞
∑π ∫
∞
+ ∑ π i1 ∫
∞
∫
0
i = k −1
∞
i2 0
i = k −1
t
( µ t )i +1− k − µt
e λ (1 − A(t ))dt
(i + 1 − k )!
β e− β x
0
t
η (η x)i − k
0
0
(i − k )!
( µ (t − x))i +1− k − µ (t − x )
e
dxλ (1 − A(t ))dt
(i + 1 − k )!
We compute each part of the equation and have
b1 = (1 − r3 )σλ[
+π k −1,1 ∫ e
−βt
0
∞
∞
0
i = k −1
+
∞
∞ i + 1− k
∑π ∫ ∑
i = k −1
i0
−
0
l =0
(η x ) l −η x −θ x t − β ( y − x )
e θ e ∫ βe
0
x
l!
t
( µ (t − y )) i +1− k − l − µ ( t − y )
×
e
dydx λ (1 − A(t )) dt
(i + 1 − k − l )!
∞
+∑ π i 0 ∫
∞ i +1− k
0
i =k
×
∑
l =1
t
η (η x)l −1
0
(l − 1)!
p l −1 p ∫
t
e−η x e −θ x ∫ β e − β ( y − x )
x
( µ (t − y ))
e − µ ( t − y ) dydxλ (1 − A(t ))dt
(i + 1 − k − l )!
i +1− k − l
1 − A ( µ − µ r1 ) k −1
r1 ],
µ (1 − r1 )
b3 = (1 − r3 )σλ[
b4 = (1 − r3 )σλ
b5 = (1 − r3 )σ
−
b6 =
1 − A* ( µ − µ r2 ) k −1
1 − r2 k −1
r2 − ∆δ
r2
µ (1 − r2 )
β
δ (1 − r2 ) k −1 δ (1 − r2 ) k −1
r2 −
r1 ],
β
β
1 − r1
r1k −1 ,
θ + η − pη r1
(1 − r3 )σλ
θβ r1k −1
1 − r1
1 − r2
(
−
)
β − µ + µ r1 θ + η − pη r1 − β θ + η − pη r1
β
(1 − r3 )σλθγ r1k −1
1 − r1
1 − A* ( µ − µ r1 )
−
[
],
β − µ + µ r1 θ + η − pη r1
µ (1 − r1 )
(1 − r3 )σλ
pηβ r1k
1 − r1
1 − r2
(
−
)
β − µ + µ r1 θ + η − pη r1 − β θ + η − pη r1
β
−
b7 =
1 − A* ( µ − µ r2 ) k −1
r2
µ (1 − r2 )
(1 − A* ( µ − µ r1 )) k −1
δ (1 − r2 ) k −1
βδ
r1 +
r1 ],
β − µ + µ r1
µ (1 − r1 )
β − µ + µ r1
(1 − r3 )σλ pηγ r1k
1 − r1
1 − A* ( µ − µ r1 )
[
−
],
β − µ + µ r1 θ + η − pη r1
µ (1 − r1 )
(1 − r3 )σλθ r1k −1 1 − r2
1 − r1
(
−
),
θ + η − pη r1 − β β
θ + η − pη r1
Then, using these expressions, the theorem can be
obtained by some computation.
Let L denote the steady-state system size at an
arbitrary epoch, the mean of L can be given by
k =0
(η t )i +1− k −η t −θ t
e e λ (1 − A(t ))dt
(i + 1 − k )!
pl ∫
µ
r3k −1 − ∆δ
*
E[ L] = ∑ k pk = (1 − r3 )σλ[
λ (1 − A(t ))dt
+ ∑ π i 0 ∫ p i +1− k
∆δ − γδ
b2 = (1 − r3 )σλ[∆δ
∞
∞
e −η x e −θ x e − β (t − x ) dxλ (1 − A(t ))dt
= b1 + b 2 + b3 + b 4 + b5 + b6 + b7 + b8.
+ γδ
which completes the proof.
t
∞
∞
+∑ π i 0 ∫ pi −k p ∫
= L0 ( z ) Ld ( z ).
E[ L ] =
0
i = k −1
(r3 − r2 )z
(r3 − r1 )z
(r2 − r1)z
1
= (1− r3 )σ[
+δ
+∆δ
−γδ
]
1− r1z (1− r1z)(1− r2 z)
(1− r2 z)(1− r3 z)
(1− r1z)(1− r3 z)
=
(η x)i +1− k −η x −θ x − β (t − x )
e θe e
dxλ (1 − A(t ))dt
0 (i + 1 − k )!
∞
+ ∑ π i 0 ∫ p i +1− k ∫
∆δ − γδ
(1 − ∆ )δ
1 − δ + γδ
+
+
].
µ (1 − r3 )2 β (1 − r2 ) (1 − r1 )(θ + η − pη r1 )
Remark 4.2. If p = 1 , the system reduces to the model
described in [4], and if p = 0 , the system becomes a
GI/M/1 queue with set-up period and multiple
working vacations [6].
References
[1] L. Servi and S. Finn, “M/M/1 queue with
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