# 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 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 terms of the Creative Commons Attribution License (CC BY 4.0) (http://creativecommons.org/licenses/by/4.0) 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 @ IJTSRD | Unique Paper ID — IJTSRD43743 | Volume —5 | Issue—5 | Jul-Aug 2021 Page 72 International Journal of Trend in Scientific Research and Development @ www.ijtsrd.com eISSN: 2456-6470 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) @ IJTSRD | Unique Paper ID —- NTSRD43743 | Volume -—5 | Issue—5 | Jul-Aug 2021 Page 73 International Journal of Trend in Scientific Research and Development @ www.ijtsrd.com eISSN: 2456-6470 _ 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: @ IJTSRD | Unique Paper ID —- JTSRD43743 | Volume-—5 | Issue—5 | Jul-Aug 2021 Page 74 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 +%,.,[ 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. @ IJTSRD | Unique Paper ID —- NTSRD43743 | Volume -—5 | Issue—5 | Jul-Aug 2021 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. Page 76