This paper is devoted to the analysis of retrial queue with an arbitrary distribution of service times, collisions, and non-persistent customers. Our aim is to investigate the number of customers in the orbit of the system. To this end, we use the asymptotic-diffusion method to build a diffusion approximation for the steady-state distribution of the number of customers in the orbit

Lizyura, Olga D.

Nazarov, Anatoly A.

English

Tomsk State University Repository

НАЦИОНАЛЬНЫЙ ИССЛЕДОВАТЕЛЬСКИЙ ТОМСКИЙ ГОСУДАРСТВЕННЫЙ УНИВЕРСИТЕТРОССИЙСКИЙ УНИВЕРСИТЕТ ДРУЖВЫ НАРОДОВИНСТИТУТ ПРОВЛЕМ УПРАВЛЕНИЯ им. В.А. ТРАПЕЗНИКОВА РАНИНФОРМАЦИОННЫЕ ТЕХНОЛОГИИ И МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ (ИТММ-2021)МАТЕРИАЛЫXX Международной конференции имени А. Ф. Тернугова 1^5 декабря 2021 г.ТОМСКИздательство Томского государственного университета2022 '̂ITMM -  2021DIFFUSION APPROXIMATION FOR RETRIAL QUEUE WITH COLLISIONS AND NON-PERSISTENT CUSTOMERSA. Nazarov, O. LizyuraTomsk State University, Tomsk, Russian FederationThis paper is devoted to the analysis of retrial queue with an ar­bitrary distribution of service times, collisions, and non-persistent customers. Our aim is to investigate the number of customers in the orbit of the system. To this end, we use the asymptotic-diffusion method to build a diffusion approximation for the steady-state dis­tribution of the number of customers in the orbit.Keywords: Retrial queue, collision, non-persistent customer, dif­fusion approximation.IntroductionRetrial queues arose as models of communication systems. The basic phenomenon of such systems is the retrial behavior of customers: if the server is busy upon arrival, the customer enters the orbit and repeats the attempt to access the server after a random amount of time.There are several modifications of retrial queues that reflect the system features such as collisions and non-persistent customers, which appear in various switching communication systems and CSMA-based networks [1]. In recent years, queueing systems with collisions are of interest due to the reborn of IEEE 802.11 wireless LANs. In paper [2], authors describe the markovian retrial queue with collisions and shows applications of persistence to modeling CSMA-CD protocols. In paper [3], the author consider similar markovian model and takes into account the impatience of customers.Nazarov and Sztrik with their research group have considered several models of finite-source retrial queues with collisions [4, 5, 6]. The phe­nomena of non-persistent customers in retrial queues was considered in [7]. Lakaour and his colleagues have considered markovian models with colli­sions, transmission errors and unreliable server [8, 9].We consider retrial queue with arbitrary distribution of service times, collisions and r-persistent customers. We build diffusion approximation for the number of customers in the orbit and construct the approximation of its probability distribution under the limit condition of growing delay in the orbit.Retrial Queue with Collisions and Non-Persistent Customers 951. M odel D escription and P roblem  DefinitionWe consider a retrial queue with an arbitrary distribution of service times defined by the distribution function B(x).  The input is stationary Poisson process with rate A. If the server is idle upon arrival, the incoming customer occupies it for service. Otherwise, the collision occurs and one of the customers joins the orbit. The other customer can also join the orbit with probability r or leave the system with probability (1 — r).At the orbit, a customer waits for some random time and tries again to occupy the server. The duration of delay follows an exponential distribution with rate a.Let k(t) denote the state of the server at instant t: 0, if the server is idle; 1, if the server is busy. Let i(t) denote the number of customers in the orbit at instant t. We also introduce process z(t), which represents the residual service time. Thus, process {k(t), i(t), z(t)} has variable number of components and exhaustively describes the system state. We denote the probability distribution of process {k(t), i(t), z(t)}  as follows:Po(i,t) =  P{k(t)  =  0, i(t) =  i}, Pi(i, z, t) =  P{k(t)  =  1, i(t) =  i, z(t) < z},and introduce the partial characteristic functionsHo(u, t)TOi=0ejUiPo(i,t), H i(u ,z,t)TOi=0ejuiPi(i,z,t) ,where j  is the imaginary unit. The Kolmogorov system of differential equa­tions for the partial characteristic functions has the following form:dH0(u, t) dt( t) , .adHo(u,t) , дHl (u ,0,t) , —AHo(u, t) +  j a ----- -̂--------+--------- -̂---------+du dz+Aeju(1 +  r(eju — 1))Hi(u, t) — ja(1  +  r(eju — 1)) dH1(u,t) du ,dH1(u,z,t) dH1(u,z,t) dH1(u, 0,t)+ j adt. dH1(u,z,t)dz dz— AH1 (u, z, t)++  AHo(u,t)B(z) — jae  j-in dH0(u,t) B(z).  (1)du duAfter that, we sum up the equations of system (2). Taking the limit by z ^  TO, we obtaindH(u, t)dt=  (ejn — 1)x96 А. Nazarov, О. LizyuradHo(u,t) s . dHi(u,t ) ]  . .M  j^ e ju ---- d ^  +  ^ (1 +  rej “ )Hi(M,t) -  j a r -----d ^  f ' (2)Solving system (2) and equation (4) in the limit by a ^  0, we derive drift and diffusion coefficients of approximating diffusion process.2. A sym ptotic-D iffusion  AnalysisIn system (2) and equation (4), we introduce the following notations:a =  e, u =  ew, т =  et,Hq(m, t) =  Fo(w,T,e), H i ( u , z , t ) =  Fi(w,z,T,e),  (3)and obtain the system of equationsd F Q ( w , T , e ) ^  ,dFo(w,T,e) dFi(w, 0 ,т,е)e------- -̂------- =  -AFo(w, т, e) +  j -------- -̂---------- -̂---------- -̂---------- +дт dw+Aejw (̂1 +  r(ejw  ̂ -  1))Fi(w,T,e) -  j(1  +  r(ejw  ̂ -  1)) dF1( w, z, т, e ) dF1( w, z, т, e ) dF1( w, 0, т, e)dzdF1(w, т, e)+ jдт dzdFi(w, z, т, e)dzdw ’ -  AFi(w, z,т, e)+dw + AFQ(w,т,e)B(z) -  j « - j -  дF ’ (дWT’ e) B(z).e a F ( w ^  =  (e , . ,  -  1)^дтf • -iwe дFo(w,т,e^  ̂ j^  je  ^ ^  +  A (1+ rej[ dw\ • dFi(w, т, e) ,rej )F l(w ,т,e) -  j r ------ ---------- Г' (4)We solve system (1) in the limit by e ^  0 and formulate the following theorem.Theorem  1. In considered retrial queue, under the limit condition a ^  0, the following equality holds:lim a ^ —j  I =  х (т ),where х (т ) is a solution of differential equationх '(т ) =  - х ( т  )ro +  [A +  (A +  х(т ))r]ri, values r0, r i have the following form:(5)1ro ri1 -  B*(A +  x) (6)2 -  B*(A +  x y  2 -  B*(A +  x ) 'Here B*(s) is the Laplace-Stieltjes transform (LST) of the distribution func­tion of the service times B(x).Retrial Queue with Collisions and Non-Persistent Customers 97From (4), we denote functiona{x) =  - x r o  +  (A +  (A +  x)r)ri. (7)For the second step of analysis, we make the following substitutions in equations (2)-(4):Ho(u,t) =  ej u H i ( u , z , t ) =  ejuThus, we obtain the equations for the partial characteristic functions of centered number of customers in the orbit. After that, we introduce the following substitutions:2 2 a =  e , u =  we, т =  te ,H02)(u,t) =  Fo2)(w,т,e), Hi2)(u, z,t) =  F(2)(w, z, T, e), (8)and obtain the system of equationsd F (2)( )e2 — 0—^ ^ -----+ jwea(x)F0(2)(w, T, e) =  -(A  +  x )f 02)(w , т, e)+дт(w,T,e) +  dF|2) (w, 0,T,e)(2),+ je +Bw Bz+(Aejw  ̂+  x)(1 +  r(ej w  -  1))F1(2)(w, т, e ) -dF-(2)(w, т, e)- j e ( 1 +  r(ejw" -  1))-Bwe2 дF■(2)(w, z , T, e) +  ,„ea (x )F < 2̂ (w,z,т,e) =  dF !2’ (w, z■т ,e )дт BzdF(2)(w ,0,т,e^ ^   ̂ ^  , ,^дF[ 2 ')(w,z,т,e)Bz-  (A +  x)F ( )(w,z,т,e) +  je -Bw +BF (2) ( )+(A +  x e - jw")Fo(2)(w, т, e)B(z) -  j e e -jw- 0  (W, т ’ e) B(z),Bw2 BF(2)(w, т, e ) +  jw ea(x )F (2)(w ,т,e)=  (ejw" -  1И  jee1J  jee -jwe дFo(2)(w, T, e) I Bw -  xe-jw ^F0(2)(w ,т ,e )+98 А. Nazarov, О. Lizyura+(А +  r(Aej w ^ +  x ) ) f [ ‘22 (w, T, e) — jer dF^ 22 (w, T, e) dw(9)Solving system (9) in the limit by e ^  0, we present Theorem 2.Theorem  2. Function lim F^22 (w,T, e) =  F( 2 )(w , t) has the follow-(2 )fing form:0F (2 ')(w,T) =  T(w,T )гк,where rk is given by (6), function T(w, t ) is the solution of equationd^ (w,T) d ̂ (w,T) ^  , (jw )2 .̂̂  ^---- dT----  =  w dw----a (x ) +-----2— ^(w,T)b(x).Function a(x) is defined by (5), b(x) is determined as follows:b(x) =  a(x) +  2[—(A +  x )(1 +  r)go +  xro +  rAri],where(10)(11)go(a(x) +  x )(1 — B *(A +  x )) +  (A +  x)a(x)B*' (A +  x ) (A +  x )(2 -  B*(A +  x ))2 'Here equation (10) is the Fourier transform of the Fokker-Planck equa­tion for the process approximating the number of customers in the orbit of considered retrial queue. If we make the inverse Fourier transform, we can see that the drift coefficient of the obtained diffusion limit is a(x) and diffusion coefficient if b(x).Theorem  3. Discrete function PD(i)  is the approximation of the probability distribution of the number of customers in the orbit and has the following form:D(ia)PD(i) (12)E  D(na)n= 0whereD(z) =  —  f  -  0^ dx( ) b(z )J  a b(x) .03. ConclusionWe have considered the retrial queue with collisions and r-persistent customers. For the number of customers in the orbit, we have derived the approximation of the probability distribution (12). In future, we plan to investigate the stability mode in such system based on the obtained approximation.Retrial Queue with Collisions and Non-Persistent Customers 99R E FEREN CES1. Choi B.D., Shin Y.W., Ahn W.C. Retrial queues with collision arising from unslotted CSMA/CD protocol / /  Queueing Systems. 1992. Vol. 11. P.335­356.2. Krishna Kumar B., Vijayalakshmi G., Krishnamoorthy A., Sadiq BashaS. A single server feedback retrial queue with collisions / /  Computers & Operations Research. 2010. Vol.37. P.1247-1255.3. Kim J. RETRIAL QUEUEING SYSTEM WITH COLLISION AND IMPA­TIENCE / /  Commun. Korean Math. Soc. 2010. Vol. 25^*4 . P.647-653.4. Nazarov A., Sztrik J., KvachA., Toth A. Some Features of a Finite-Source M /GI/1 Retrial Queuing System with Collisions of Customers / /  Communi­cations in Computer and Information Science. 2017. Vol. 700. P. 186-200.5. Nazarov A., Sztrik J., Kvach A., Toth A. Asymptotic sojourn time analysis of finite-source M /M /1 retrial queueing system with collisions and server subject to breakdowns and repairs / /  Annals of Operations Research. 2020. Vol. 288. P. 417-434.6. Toth A., Sztrik J. Simulation of Finite-Source Retrial Queuing Systems With Collisions, Non-Reliable Server and Impatient Customers in the Orbit / /  Proceedings of the 11 th International Conference on Applied Informatics, Eger: ICAI, 2020. P.408-419.7. Jailaxmi V., Arumuganathan R., Rathinasamy A. Performance analysis of an M x/G /1 feedback retrial queue with non-persistent customers and multiple vacations with N-policy / /  Int. J. Operational Research. 2017. Vol. 29^fs 2. P. 149-169.8. Lakaour L., Aissani D., Adel-Aissanou K., Barkaoui K. M /M /1 Retrial Queue with Collisions and Transmission Errors / /  Methodology and Computing in Applied Probability. 2019. Vol. 21. P. 1395-1406.9. Lakaour L., Aissani D., Adel-Aissanou K., Barkaoui K., Ziani S. An un­reliable single server retrial queue with collisions and transmission er­rors / /  Communications in Statistics - Theory and Methods. 2020. DOI: 10.1080/03610926.2020.1758943.Nazarov Anatoly — doctor of technical sciences, full professor, profes­sor of department of probability theory and mathematical statistics. E-mail: nazarov.tsu@gmail.comLizyura Olga — assistant professor of department of probability theory and mathematical statistics. E-mail: oliztsu@mail.ru

Diffusion approximation for retrial queue with collisions and non-persistent customers

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Diffusion approximation for retrial queue with collisions and non-persistent customers

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