On the exact transient solution of fluid queue driven by a birth death process with specific rational rates and absorption

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1 OPSEARCH Oct Dec : DOI.7/s THEORETICAL ARTICLE On the exact tansient solution of fluid queue diven by a bith death pocess with specific ational ates and absoption Shuti Kapoo Dhamaaja Selvamuthu Accepted: 4 Januay 25 / Published online: Febuay 25 Opeational Reseach Society of India 25 Abstact Bith death pocesses with ational bith and death ates have been studied by Maki 976. In this pape a fluid queue diven by a bith death pocess with infinite state space and absoption is studied whee the bith and death ates ae ational functions of linea polynomials. We obtain an explicit tansient solution fo the fluid queue model using continued faction appoach to solve the undelying system of patial diffeential equations. Fo specific value of the paamete the consideed model educes to the model discussed by Pathasaathy, Seicola and Vijayashee 25 and the esults coincide. We also analyze the behavio of the fluid queue in a long un. Keywods Fluid queue Bith death pocess Continued faction Intoduction and elated wok A fluid model is a mathematical model used to descibe the amount of fluid in a esevoi o dam, of infinite o finite capacity. The ate at which the fluid flows into the system can be analyzed as being contolled by an opeato which is a continuous time Makov chain and is called the backgound pocess. Such models have vaious applications in diffeent aeas. Fluid models have been used to model the pefomance of a netwok switch, a oute, the IEEE 82. potocol [6], Asynchonous Tansfe Model [2], and has applications in undestanding the busty natue of taffic in high speed netwoks []. Most of the classical eseach on stochastic fluid models in the aea of telecommunications is based on the wok of Anick, Mita and Sondhi [9]. This pape analyzes the tansient behavio of a fluid queue modulated by S. Kapoo D. Selvamuthu Depatment of Mathematics, Indian Institute of Technology Delhi, New Delhi 6, India dhama@maths.iitd.ac.in

2 OPSEARCH Oct Dec : an infinite state bith death pocess wheein the bith and death ates ae in ational fom. Many authos have found steady state solution fo vaious fluid queues. Seicola and Tuffin [8] consideed a fluid queue diven by a geneal Makovian queue with the hypothesis that only one state has a negative dift. By using the diffeential system, the fluid level distibution is obtained in tems of a seies, whose coefficients ae computed by means of ecuence elations. This study is futhe extended to the finite buffe case by Seicola [6]. Van Doon and Scheinhadt [4] analyzed fluid queues diven by infinite state bith death pocesses. Lenin and Pathasaathy have studied numeically the behavio of fluid buffe diven by tuncated bith death pocesses. Then specific models wee studied by some authos. Pathasaathy, Lenin and Vijayashee [3] foundthesteadystate solutionofafluidqueuedivenbyan M/M/ queue using the continued faction appoach. Pathasaathy and Lenin [7] found the closed fom steady state solution fo a fluid queue modulated by an M/M//N queue by finding eigenvalues and eigenvectos by using esults on tidiagonal matices. Steady state behavio of Makov diven fluid queues have been extensively studied in the liteatue. Howeve, steady state analysis is useful to study a system in long un, the tansient analysis will be of citical value in undestanding the dynamical behavio of the system. Seicola [5] gave the tansient solution fo the geneal model whee the fluid queue is modulated by a finite state Makovian queue using ecusive polynomials. Seicola and Guillemin [7] gave the tansient solution fo fluid queues diven by bith death pocess using continued faction and ecusive elations. Pathasaathy, Seicola and Vijayashee [3]foundthetansient solutionfobuffedistibution fo fluid queues diven by bith death pocess suggested by a chain sequence using the simila appoach of ecusive polynomials. The same authos also found the tansient solution fo fluid queues modulated by an M/M/ queue using ecusive polynomials [4]. Ren and Kobayashi [] found the tansient solution of buffe behavio in statistical multiplexe by finding the eigenvalues and eigenvectos using geneating functions. Maki [2] analyzed the distibution function of a bith death pocess with bith and death ates as ational functions of the state of the pocess, whee the numeato and denominato ae geneal polynomials. Also, in [5], bith death pocesses suggested by chain sequence have been studied. In this pape we assign ational bith and death ates to the backgound bith death pocess which fom a chain sequence and use the methodology of continued faction to find the buffe content distibution. The same methodology has been used by [3] to solve chain sequences, but this pape intoduces the concept of absoption. We show how the solution is affected when μ =, a common assumption in vaious othe papes. This enfoces state to be an absobing state and thus the change in solution. The est of the pape is oganized as: Section 2 descibes the backgound Makov model in detail, in Section 3 we give the detailed desciption of the fluid model, in Section 4 we study the time dependent behavio of the fluid model theeby giving the solution and also discusses the steady state behavio of the fluid model unde consideation and Section 5 illustates ou pespective of the solution obtained in the pevious section.

3 748 OPSEARCH Oct Dec : Backgound Makov model In [2], Maki consides bith death pocesses {Xt,t } with ational ates of the fom λ n = A pn B q n μ n = C n,n=,, 2,... D s n whee, n epesents the state of the bith death pocess and A p x, B q x, C x and D s x ae polynomials of degee p, q,, s espectively. It is also assumed that the polynomials ae such that A px B q x and C x D s x ae positive fo x =, 2, 3,..., A p B q >, and C D s. In [2] the autho discusses vaious conditions on p, q, and s, such that a unique solution exist fo the distibution of Xt at any time t. One of Maki s esults stated as a conjectue, assets that if p = q and = s a unique solution exist fo the distibution of Xt at any time t. Extending Maki s [2] wok to fluid queues, in ou pape, we conside a fluid queue model govened by a bith death pocess. The bith and death ates given by λ n s and μ ns espectively ae ational functions of linea polynomials of the fom λ n = A n B n μ n = C n,n=,, 2,... 2 D n With these above bith and death ates, we have p = q = and = s =. We shall futhe see that these ates will fom a chain sequence. 3 Model desciption In this pape we conside a fluid queue diven by a bith death pocess, {Xt,t } with state space S ={,,,...}. The bith and death ates as stated in Eq. 2 ae ational functions of linea polynomials given by λ n = n + + α n =,, 2, n + 2α μ n = n + α n =,, 2, n + 2α with λ =, whee α is a constant satisfying α. Note that since μ > and λ =, the state is an absobing state fo the above bith death pocess. Also the above values of bith and death ates satisfy, fo n =, 2,... λ n + μ n = λ n μ n = μ n μ n = 5 whee = 4 and thus fom a chain sequence.

4 OPSEARCH Oct Dec : Also, fom Eq. 4, μ μ 2...μ j = 2 j α α + j j =, 2,... 6 Let Ct denote the content of the buffe at time t, then{xt, Ct, t } constitutes a two-dimensional Makov pocess. When the backgound pocess Xt is in state, the net input ate into the buffe is <, when in state the the net input ate is < and when Xt is positive the net input ate is >. The buffe occupancy distibution is defined as F j t, x = P {Xt = j,ct x}; t,x,j S 7 i.e., the pobability of the backgound bith death pocess in state i and the content less than equal to x. 4 Tansient and steady state analysis 4. Tansient solution Using Eq. 5, Kolmogoov fowad equation fo the Makov pocess {Xt, Ct, t } ae given by F t, x t F t, x t and fo j =, 2,... F j t, x t + F t, x x + F t, x x + F j t, x x = μ F t, x 8 = F t, x + μ F t, x 9 = λ j F j t, x F j t, x +μ j+ F j+ t, x subject to the initial conditions, F,x=, F,x=, F j,x= fo j =, 2, 3,... and bounday conditions, F t, = q t, F t, = q t and F j t, = fo j =, 2, 3,... As, when the net input ate of fluid flow into the buffe is positive, the buffe cannot stay empty. Taking Laplace tansfom of Eqs. 8, 9 and w..t t and x we get s + wf s, w μ F s, w = q s s + w + F s, w μ F s, w = w + q s 2 fo j =, 2,... λ j F j s, w + s + + wf j s, w μ j+ s, w = 3

5 75 OPSEARCH Oct Dec : We fist solve the system of Eqs. 2 and 3. Define s, w = F s, w fj s, w = j μ μ 2...μ j j Fj s, w j =, 2,... 4 Substituting the above tansfomation, Eqs. 2 and 3 can be witten as, w + + s s, w f λ j μ j 2 λ μ f s, w + w + s + j s, w + w + s + Rewiting the above equations, s, w = w + q s f s, w s, w = j s, w = s, w j s, w + f s, w = 2 s, w = w + q s fj s, w + fj+ s, w =, j = 2, 3,... s,w, s,w + f 2 s,w, f s,w 2 + f j+ s,w fj s,w j = 2, 3,... 5 Fom the above equations, we can wite s, w in the continued faction [8]fom as, s, w = w + q s Define, 2 fs,w = = fs,w 2 Thus we get, 2 f s, w2 Solving we get, fs,w= 2... w + s + fs,w+ =

6 OPSEARCH Oct Dec : Fom the definition of fs,wand Eq. 5 we get, f s, w s, w = fs,w fj s, w fj s, w = fs,w j = 2, 3, and hence, s, w = w + q s fs,w 2 Now letting w+ s+ = θ and 2 = ν and since F s, w = f s, w,weget F s, w = q s + k θ θ 2 ν 2 k w 2 k+ k= fo f s, w w + s + < 2 Now fom Eq. 9 we get, fj s, w = j 2 fs,w s, w j =, 2, 3, Using Eq. 4 we get, Fj j j s, w = μ μ 2...μ j 2 fs,w F s, w j = q 2 μ μ 2...μ s + k θ θ 2 ν 2 k+j j w 2 k+ k= α + j = j q α s + w k θ θ 2 ν 2 k+j 2 k+ fo f s, w k= w + s + < 23 Also fom Eq. we get, F s, w = s + w q s + μ F s, w 24 Substituting the expession fo F s, w givenineqs.2, 24 and then inveting fist w..t the vaiable w and then w..t the vaiable s we get the following theoem.

7 752 OPSEARCH Oct Dec : Theoem Fo evey t and x [,t, we have F t, x = q t x + μ q te t x H t μ e x q te t x H t x +μ e t μ { H t x } e t x μ e x {H t x e t x } e t x { x e sx γ k γ e /x yνk ki k νx y 2 k= k!x y { y k e y/ H t x y y q t x y y + k e t x y/ t x y } } k dy dγ, 25 F t, x = e x q t x + e t e x e t x + x e /x yνk ki k νx y k!x y q t x y α + j k x F j t, x = j α 2 k= { y k e y/ H k= { y k e y/ H k 2 t x y } k y + k e t x y/ t x y x e /x yνj+k I j+k νx y k!x y y t x y y q t x y } k + k e t x y/ t x y whee H.denotes the heaviside function. 4.2 Steady state solution y dy, 26 dy j =, 2, 3, The steady state distibution fo buffe occupancy is defined as F j x = lim P [Xt = j,ct x], j =,,,..., x t In steady state, the Kolmogoov fowad equations fo the above model ae given by df x = μ F x dx

8 OPSEARCH Oct Dec : and fo j =, 2,... df x dx = λ F x + μ F x df j x j = λ j F j x F j x + μ j+ F j+ x dx satisfying the conditions, F =, F j =, j =,, 2,... Taking Laplace tansfom of the above equations we get, sf s F = μ F s sf s F = λ F s + μ F s j sfj s F j = λ j Fj s F j s + μ j+fj+ s, j =, 2,... Now using the bounday conditions, the above system educes to, s F s μ F s = 28 s + λ F s μ F s = 29 λ j Fj s + j s + Fj s μ j+fj+ s =,j =, 2,... 3 Solving the system of Eqs. 29 and 3, which is a homogeneous system of equations will have the zeo solution. Now using Eq. 28, we get the solution Taking invese tansfom we get, F s = s and F j s =, j =,, 2,... F x = andf j x =, fo j =,, 2,... and x Fom the above statement we conclude that the buffe content will be zeo with pobability in the long un when the backgound will be in the absobing state, as in state the net input ate is <. Thus the buffe will dain out completely when the backgound will be in the absobing state. 5 Ou pespective Ou study on the model descibed in this pape, leads us to a pespective that compises of the following:. Pathasaathy, Seicola and Vijayashee [3]have consideed a fluid queuemodel govened by a bith death pocess with state space {,,...} and bith and death ates given by U n+ λ =, μ =, λ n = 2 n =, 2,... 3 U n 2 μ n = U n 2 n =, 2, U n 2

9 754 OPSEARCH Oct Dec : whee U n is the Chebyshev polynomial of second kind of ode n. Letting α = ineqs.3 and 4 we get, λ n = n + 2 2n + 2 = U n+ 2U n n μ n = 2n + 2 = U n 2U n n =,, 2,... n =,, 2,... which ae same as Eqs. 3 and 32 fo = 4. Fom the above ates, we obseve that, λ = andμ =. Thus, the backgound bith death pocess of the undelying fluid model with the above ates has state space {,,...}. Thus substituting = 4 in Theoem 5. of [3] the following solution is obtained. F t, x = e x q t x + e t e x e t x + x e /x yνk ki k νx y k!x y { y k e y/ H t x y y q t x y y + k e t x y/ t x y } k dy, k x F j t, x = j + j e /x yνj+k I j+k νx y 2 k= k!x y { y k e y/ H t x y y q t x y y + k e t x y/ t x y } k dy j =, 2, 3,... which coincides with ou esults given by Eqs. 26 and 27 fo α =. 2. The steady state analysis of the model shows that in long un, the backgound pocess will be in absobing state. Thus in long un the function q t,which is the pobability of the buffe content being empty when the backgound is in state will appoach. Also the function q t, which gives the pobability of the buffe being empty when backgound is in absobing state will appoach, as the ate <. k= 2 k

10 OPSEARCH Oct Dec : The steady state analysis can be veified fom Eqs. 23, 24 and 25 as well. Using the above obsevation it can be seen as t appoaches, F t, x appoaches and F t, x, F j t, x appoach. 4. Fluid models of this type find vaious applications in the field of telecommunication fo modelling the netwok taffic and in the appoximation of discete stochastic queueing netwoks. Fo specific applications of this model, efe []. Acknowledgments One of the authos SD would like to thank the National Boad fo Highe Mathematics, India, fo financial suppot given to them duing the pepaation of the pape. Refeences. Kalin, S.: A Fist Couse in Stochastic Pocesses. Academic Pess, New Yok Maki, D.P.: On bith death pocesses with ational ates. SIAM J. Math. Anal. 7, Pathasaathy, P.R., Seicola, B., Vijayashee, K.V.: Tansient analysis of a fluid queue diven by a bith and death pocess suggested by a chain sequence. J. Appl. Math. Stoch. Anal. 252, Pathasaathy, P.R., Seicola, B., Vijayashee, K.V.: Exact tansient solution of an M/M/ diven fluid queue. Int. J. Comput. Math. 826, Seicola, B.: Tansient analysis of stochastic fluid models. Pef. Eval. 32, Seicola, B.: A finite buffe fluid queue diven by a Makovian queue. Queueing Syst. 38, Seicola, B., Guillemin, F.: Tansient analysis of a fluid buffe diven by a bith and death pocess. In: 9th Intenational Teletaffic Congess ITC 9, Beijing Seicola, B., Tuffin, B.: A fluid queue diven by a Makovian queue. Queueing Syst. 3, Anick, D., Mita, D., Sondhi, M.: Stochastic theoy of data-handling system with multiple souces. Bell Syst. Tech. J. 6, Ren, Q., Kobayashi, H.: Tansient solutions fo the buffe behavio in statistical multiplexing. Pefom. Eval. 2, Mao, G., Habibi, D.: Heteogeneous on-off souces in a buffeless fluid flow model. In: IFFF Intenational Confeence on Netwoks, pp Tanaka, T., Yashida, O., Takahashi, Y.: Tansient analysis of fluid models fo ATM statistical multiplexe. Pefom. Eval. 23, Pathasaathy, P.R., Vijayashee, K.V., Lenin, R.B.: An M/M/ diven fluid queue-continued faction appoach. Queuing Syst. Theoy Appl. 42, van Doon, E.A., Scheinhadt, W.R.W.: A fluid queue diven by an infinite state bith- death pocess. In: Poceedings of the ITC 5, Teletaffic Contibutions fo the Infomation Age. Elsevie, pp Lenin, R.B., Pathasaathy, P.R.: A bith-death pocess suggested by a chain sequence. J. Comput. Math. Appl. 42 3, Aunachalam, V., Vandana Gupta, S., Dhamaaja, A.: Fluid queue modulated by two independent bith death pocesses. J. Comput. Math. Appl. 6, Pathasaathy, P.R., Lenin, R.B.: Fluid queues diven by an M/M//N queue. Math. Pobl. Eng. 6, Loentzen, L., Waadeland, H.: Continued Factions with Applications. Elsevie, Nethelands 992

Fluid Queues Driven by Rogers-Ramanujan Birth and Death Processes With Catastrophes

Fluid Queues Driven by Rogers-Ramanujan Birth and Death Processes With Catastrophes Jounal of Numeical Mathematics and Stochastics, 5(1) : 46-62, 213 http://www.jnmas.og/jnmas5-4pdf JNM@S Euclidean Pess, LLC Online: ISSN 2151-232 Fluid Queues Diven by Roges-Ramanujan Bith and Death Pocesses