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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 


Copyright © 2021 by author (s) and 
International Journal of Trend in 
Scientific Research and Development 


Journal. This is an 

Open Access article 
distributed under the cae 
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Attribution License (CC BY 4.0) 


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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 
i O(7,-7%) Ad, —1,)— ¥O(%,-7) 
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: 





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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 


+%,.,[ eM Ad- A())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) 


Ad- yo 
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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[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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