# Full text of "The Queue Length of a GI M 1 Queue with Set Up Period and Bernoulli Working Vacation Interruption"

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```International Journal of Trend in Scientific Research and Development (IJ TSRD)
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

How to cite this paper: Li Tao "The
Queue Length of a GI/M/1 Queue with
Set-Up Period and Bernoulli Working

lower rate, if there are customers at a service completion instant, the Vacation , |

vacation can be interrupted and the server will come back to a set-up nao in

period with probability p(0< p<1)or continue the working vacation atenaatoaal Ioumaal

with probability 1— p, and when the set-up period ends, the server will of Trend in

switch to the normal working level. Using the matrix analytic Scientific Research aah
method, we obtain the steady-state distributions for the queue length and Development — YTSRD43743

at arrival epochs.

(ajtsrd), ISSN: 2456-

KEYWORDS: GI/M/1; set-up period; working vacation; vacation

interruption; Bernoulli

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

6470, Volume-5 | Issue-5, August 2021,
pp.72-76, URL:
www.ijtsrd.com/papers/ijtsrd43743.pdf

International Journal of Trend in
Scientific Research and Development

Journal. This is an

Open Access article
terms of the Creative Commons

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 set-
up period. with probability pO<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(O< 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 1, be the arrival epoch of nth customers with
T,=0. The inter-arrival times
independent and identically distributed with a general
distribution function, denoted by A(t) with a mean
1/A 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 w,7,

{z,,n>1} are

Band 4@, respectively.

Let L(t) be the number of customers in the system at
time t and, = 1(c, -0)be the number of the customers
before the nth arrival. Define J, =0, the nth arrival
occurs during a working vacation period; J, =1, the
nth arrival occurs during a set-up period; J, =2, the
nth arrival occurs during a normal service period.
Then, the process {(L,,J,),n>l}is an embedded
Markov chain with state space

Q= {0,0} U{(K, fj), k 21, 7 =0,1,2}.

In order to express the transition matrix of (L,,J,), let

Po. jckD Plas k,J Il L, i, J, J):

ntl

Meanwhile, we introduce the expressions below

a= [oo = Mi oda), k>0,

_ k
b, = in \, Be F 2 nay, k=0,

k
= J p CTO” ome "dA, k=0,

d, =|" YP ae MID tg * f Be ~Biy-x)

(ue= yy"

(k—-1)!
a= [7d Yip ly mx)
0 (-1)!

: uty
(k-1!

eM) dydxdA(t), k 20,

t
ee Bebo
x

ee! dydxdA(t), k>1,

f= J, al eae i) ae Gee F dxdA(t), k>0,

=f kl Pi. mx) ATA! 9 99 9 BUD yd A(t), k>1.
0

(k—I)!

Using the lexicographic sequence for the states, the
transition probability matrix of (L,,/,) can be written
as the Block-Jacobi matrix

n?

Bo Aor
B A A
P= B, A A A >
B, A, A A A
where
By =1-¢) -—dy— fos Agr = Co: fos4o)5
Co fo dy Gq Stk, A+
Ay = 0 A (B) by 3 A, = 0 0 b, ‘
0 0 Ay 0 0 ay
k
1-)'(c,+4,+e,4+f,4+8)-c-d- fy
i=l
k
B, = 1-}°b,-A'(B) , k=l.
i=0

k
1=> a

i=0

3. Steady-state distribution at arrival epochs
We first define

A(z) = Yaz! B@)= V2.0) = Dez
k=0 k=0 k=0

D(2)= 2! Ele)= Vez! F(2)= Y fit! G2) = Y ez
k=0 k=1 k=0 k=1

In this section, we derive the steady-state distribution
for (L,,J,) 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’ (ul— fz),

BIA (ue Hz) AB)
pplz)

C(z) = A’ (0+- pnz),

OB [A (8+—-pyz)—A'(A)]

B(z)=

2

D(z)=

B-“ud-z) O+n-pnz-B
OB [A (6+n-pyz)—A (u- Uz)]
B-MA-z)  @+pyz—(u-n)1-z)
E(z) = —PUeB_[A'(6+9~ pz) A'(B)I
p=p—2) O+n-pnz-B
pnzB [A (0+n- pynz)—A (u- wz)
B-pie<). @€+pye-Qi-7)=2)

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_ OA (B)- A (O+N- pyZ)]

F(z) =
: O+n- pnz-B
Oe pnz[A Des CE he PNZ))
O+n- pnz-B
Lemma 3.2. If A>0, the

equation z= A'(6+7-pnz)has a unique root in the
range O<z<1.

Lemma 3.3. If @>0,8>0 and p=A/y<1, then the

matrix equationR=)5’R‘A,has the minimal

k=0
nonnegative solution
ACA -h)
0 0 r,

where +r, = A'(8), 7, and +, are the unique roots in the

range O<z<1 of equations
z=A(0+n- pz) and z= A (u—z) , respectively, and
6 = (6+ pyr,)/(8+n- pyr —f), A=£6/[B-“A-n)),
y= BI[A+ pyr -(u-m)d-7)).

Moreover, we can easily verify that the Markov

chainPis positive recurrent if and only if
6>0,8>0and p<1. And the matrix
Boo An
BIR] =| =
ma SRB, LRA,
k=l k=1
I-cy-dy— fy ay fo d,
Cotto 4-H) o 1 % O%-%) fo o
K ih K K i
=| 1_ GAG =F) % 0 GAC =) Oo
Le) 4 ULE 4
a 0 0 ee
5 5
F Ad(r,- 1, O(n, — 7 O(n -1r d
with w= etn ee, aca iaasel 9 _% hasa
nh, nr nh si

positive left invariant vector
where K ia a random positive real number.

Let (L, J) be the stationary limit of the process (L,,J,) ,
and denote

Hy =Mys Me =(Myor Br Br), k21,

Xj

P{L=k,J = jj=lim P{L, =k, J, =J}, Kk Ye Q.

Theorem 3.4. If 6>0,8>0andp<1, the stationary
probability distribution of (L,/) is given by

Tyo =(1-1,)or*, k=0,
1, =(1-7,)06(r —r*), k>0,
Ty = (1-7, OIAd(s — 1) - OCF - 7), ke 1,

where

oe (= )0=%) .
(= 7 d= 1) + 50, = H+ AOU =H) =) POU FH =H)

Proof. With the Theorem 1.5.1 in [8],
(>A 9>;,4,,) 1S given by the positive left invariant
vector Eq. (1), and satisfies the normalizing condition

op + (Hyp F154, I - RY 'e = 1,

where e is a column vector with all elements equal to
one. Substituting R into the above relationship, we
can get

K= (-7)d-1)d-n)
(1-4 [0 -4) + 6, -—4)1+ AO), — 4) - OU - 14) - 7H)

=(l1-4,)o.

Therefore, we have

(MoT % 2) = 1-H )O(, 0% — 7), Ad — 4) — WO — 7) -

Using the Theorem 1.5.1 of Neuts [8], we can obtain
Hy = (Hy By» Bey) = (Big By»My)R, =k 21. (2)

Taking (z,,,7,,,7,,) and R*" 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

1) = P{L=0}= Mp» = (1-1)o,

4, =P(L=k} =4,, +4, +2.

= (1-1, )o[(l—- d)rf + 6rf + Ad(rs - 1) - YO - 7D), k 21.
The state probability of a server in the steady-state is
given by
bar = oF See SB,
k=0 1-7,

1

POU F MG —H)

PSs 2 Ga 5
ny we _ HAd(1—7,)(,-—%)- GVO— 1, (7, — 7)
HI 2) = 2 Aa as

Theorem 3.5. If 6>0,8>0and p<1, the stationary
queue length L can be decomposed asL=L,+L,,
where L, is the stationary queue length of a classical

GI/M/1 queue without vacation, and follows a
geometric distribution with parameter 7, . Additional

queue length L, has a distribution

P{L, =O}=o,

P{L, =k} =0(6-1- YON, - HK
+00(A-1)(n-n)n", =k21.

Proof. The probability generating function of L is as
follows:

@ IJTSRD | Unique Paper ID —- JTSRD43743 | Volume-—5 | Issue—5 | Jul-Aug 2021

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International Journal of Trend in Scientific Research and Development @ www.ijtsrd.com eISSN: 2456-6470

a _— if
LO=) Rot +> tae" + Y Mat
k=0 k=1 k=1

=(-n of +5 B _5 ng¢§_G (ne
I-nz = -nz\-nz) — -nz)d—-nz) © (-nz)-72)

_ 1-5 ,lonz, pa -W0-R22 , yg n)z Paci me)

l-nz 1l-nz (l-7z)A-1742z) l-nz l-nz

a8 gy GM, awe 5G Be psGra pG-De,
aa 1-7z 1-Kz 1-nz 1-7nz 1-Kz
eee ee 76) BoE es ahh
“ie rz HZ 1-r,z

= 1, (2)L, (2).

which completes the proof.

Thus, the mean queue length at the arrival epoch is
given by

o(6-1-—¥)(r, aE) o0d(A-1)(7, “h)
(l-r,)

5
es (l-7)

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(1): p, = lim P{L(t)=k}.

Theorem 4.1. If 6>0,8>0andp<1,
distribution of L(t) exists. And, we obtain

the limiting

Ad 6 a A)d(1 5) 0+ 7O\(1 4)
B O+7- pyr,

Py =1-oAL

p, =(l Pe eae V6 Par d= ~_ 1) fon (l-6+7)U-H) 44

ny ),k21.
O+N— py,

Proof. From the theory of SMP, the limiting
distribution of L(r) has the following expression (see

[9]):

2 a) a
n= >on Ty Neer aa sy “ A(1— A(t))dt

i=k-1

Se PT Bebe HE neo ge a
+ Yim, LF a dxA(1— A(t))dt

+n, 2 a ee AL A(D)at
Jo (i+1—-k)!

i=k-1

+n Hol = re = (IX) o-1 9 oy Be PO

i=k-1

mf, 8 <I-1 pf tee “1 9 fai Bebo

(-1)!

i+l-k
MHe= yy eo" dydxA(1— A(t))dt

(4i—k-i!

sitl-k xy" os -Nx —,-9x —B(t-x)
+>. Ho]. P [~~ ae Bee Pde A(1— A(t))dt

i=k-1

bee 2s n(nx)* “IX g x —B(t-x
+m ? ‘p[ =a #1 gBU-®) dy (| — A(t))dt

=b1+b2+b3+b44+ 554+ b6+b74+b8.

We compute each part of the equation and have
DON 84 Kg A‘(u- HY) 1
a M(1-1)

bl=(1 roAto

ig (= Mi) aay

1

M(1-7)
b2=(1—noafad-— A ABP) tt _ gh ps
ur) B
po (I= AHF) ery O(l-n) fy
B-ut+tur = ywd-r) B-utmr
b3 = (1 nog CL) t+ 208) sy,
B B
1-7, k-1
b4=(1-1,)oA =a!
O+1- pny,
a k-1 _ _
ee, SO (l-1,)oA Br 1 a _l-n,
B-ut+ur,O+n-pn,-B O+n-pnr B
_G=nyoaeyn* — l-5 _1- Aun)
B-ut+urn O+n-pnr. HU-7)
pon WA opmBR
B-L+ur O+n-pnr-B O+n-pnr, B
(=n Joapnyit 1-7, 1 Aner),
B-utpur 04+n-pnr Md-7)

L (l-1,)oA Or Ie ee
6+n-pnr,-B B O+n-pnr
Then, using these expressions, the theorem can be
obtained by some computation.

Let# denote the steady-state system size at an
arbitrary epoch, the mean of £ can be given by
(1-A)é 1-6+ 76
BU-r,) =n )O+n- pny)

ud-n)

ELE] = kp, = (-n)oAl
k=0

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
working vacations (M/M/1/WV)’’,
Performance Evaluation, vol. 50, pp. 41-52,
2002.

[2] Y. Baba, “Analysis of a GI/M/1 queue with
multiple working vacations’’, Operations
Research Letters, vol. 33, pp. 201-209, 2005.

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Page 75

[3]

[4]

[5]

@ IJTSRD | Unique Paper ID —- NTSRD43743 | Volume —5 | Issue—5 | Jul-Aug 2021

J. Li and N. Tian, “The M/M/1 queue with
working vacations and vacation interruption’’,
Journal of Systems Science and Systems
Engineering, vol. 16, pp. 121-127, 2007.

J. Li, N. Tian, and Z. Ma, “Performance
analysis of GI/M/1 queue with working
vacations and vacation interruption’’, Applied
Mathematical Modelling, vol. 32, pp. 2715-
2730, 2008.

G. Zhao, X. Du, and N. Tian, “GI/M/1 queue
with set-up period and working vacation and
vacation interruption’’, International Journal of
Information and Management Sciences, vol. 20,
pp. 351-363, 2009.

[6]

[7]

[8]

[9]

International Journal of Trend in Scientific Research and Development @ www.ijtsrd.com eISSN: 2456-6470

X. Bai, X. Wang, and N. Tian, “GI/M/1 queue
with set-up period and working vacations’’,
Journal of Information and Computational
Science, vol. 9, pp. 2313-2325, 2012.

H. Zhang and D. Shi, “The M/M/1 queue with
Bernoulli-schedule-controlled vacation and
vacation interruption’’, International Journal of

Information and Management Sciences, vol. 20,
pp. 579-587, 2009.

M. Neuts, “Matrix-Geometric Solutions in
Stochastic Models’’, Johns Hopkins University
Press, Baltimore, 1981.

D. Gross, C. Harris, “Fundamentals of
Queueing Theory’’, 3rd Edition, Wiley, New
York, 1998.

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