FLUID LIMIT FOR CUMULATIVE IDLE TIME IN MULTIPHASE QUEUES. Akademijos 4, LT-08663, Vilnius, LITHUANIA 1,2 Vilnius University

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1 Iteratioal Joural of Pure ad Applied Mathematics Volume 95 No , ISSN: (prited versio); ISSN: (o-lie versio) url: doi: PAijpam.eu FLUID LIMIT FOR CUMULATIVE IDLE TIME IN MULTIPHASE QUEUES Saulius Mikevičius 1, Vladimiras Dolgopolovas 2 1,2 VU Istitute of Mathematics ad Iformatics Akademijos 4, LT-08663, Vilius, LITHUANIA 1,2 Vilius Uiversity Naugarduko 24, LT-03225, Vilius, LITHUANIA Abstract: The object of this research i the queueig theory is the Fuctioal- Strog-Law-of- Large-Numbers (FSLLN) uder the coditios of heavy traffic i Multiphase Queueig Systems (MQS). A FSLLN is kow as fluid limit or fluid approximatio. I this paper, the FSLLN is proved for values of importat probabilistic characteristics of the MQS ivestigated as well as the summary queue legth of customers ad the queue legth of customers. AMS Subject Classificatio: 60K25, 60G70, 60F17 Key Words: queueig systems, multiphase queueig systems, heavy traffic, a law of the iterated logarithm, summary queue legth of customers, queue legth of customers 1. Itroductio Iterest i the field of multiphase queueig systems was stimulated by the theoretical values of the results as well as by their possible applicatios i iformatio ad computig systems, commuicatio etworks, ad automated Received: December 29, 2013 Correspodece author c 2014 Academic Publicatios, Ltd. url:

2 124 S. Mikevičius, V. Dolgopolovas techological processes (see, for example, [13]). The methods of ivestigatio of sigle phase queueig systems are cosidered i [2], [3], etc. The asymptotic aalysis of models of queueig systems i heavy traffic is of special iterest (see, for example, [7], [8], [4], [5], etc.). The papers [9], [12] ad others desribed the begiig of the ivestigatio of diffusio approximatio to queueig etworks. Itermediate models - multiphase queueig systems - are cosidered rarer due to serious techical difficulties (see, for example, book [6]). We preset some defiitios i the theory of metric spaces (see, for example, [1]). Let C be a metric space cosistig of real cotiuous fuctios i [0, 1] with a uiform metric ρ(x,y) = sup x(t) y(t), x,y C. 0 t 1 Let D be a space of all real-valued right-cotiuous fuctios i [0,1] havig left limits ad edowed with the Skorokhod topology iduced by the metric d (uder which D is complete ad separable). Also, ote that d(x,y) ρ(x,y) for x,y D. I this paper, we will costatly use a aalog of the theorem o covergig together (see, for example, [1]): Theorem 1.1. ( ) Let ε > 0 ad X, Y, X D. If P lim d(x,x) > ε = 0 ( ) ( ) ad P lim d(x,y ) > ε = 0, the P lim d(y,x) > ε = 0. (1) We ivestigate here a k-phase multiphase queueig system (i.e., whe a customer has bee served at the j-th phase of the multiphase queue, he goes to the j+1-th phase of the multiphase queue, ad after the customer has bee served at the k-th phase of the multiphase queue, he leaves the multiphase queue). Let us deote by t the time of arrival of the -th customer, by S (j) the service time of the -th customer at the j-th phase of the multiphase queue, z = t +1 t ; by τ j, the departure of the -th customer after service at the j-th phase of the multiphase queue, j = 1,2,...,k. Let iterarrival times (z ) at the multiphase queue ad service times (S (j) ) at every phase of the multiphase queue for j = 1,2,...,k be mutually idepedet idetically distributed radom variables. Next, deote by BI j, the idle time of the -th customer at the j-th phase of the multiphase queue; I j, = BI j,l stads for a cummulative idle time of l=1

3 FLUID LIMIT FOR CUMULATIVE IDLE TIME the -th customer at the j-th phase of the multiphase queue, j = 1,2,...,k. Suppose that the idle time of a customer at each phase of the multiphase queue is ulimited, the service priciple of customers is first come, first served (FCFS). All radom variables are defied o the commo probability space (Ω,F,P). We form such a modified multiphase queue i which BI j, = 0, j = 1,2,...,k, < k. Limit distributios for the modified multiphase queue ad the usual multiphase queue workig i heavy traffic coditios are coicidetal (see, for example, [3]). Thus, later o we will ivestigate oly the modified multiphase queue ad admit that k. Whe j = 1,2,...,k, let us defie Deote S j, = 1 l=1 δ j, = { S (j) (j 1) z, if k 0, if < k. δ j,l, S 0, 0, Ŝ j, = S j 1, S j,, x j, = τ j, t, x 0, 0, ˆx j,+1 = x j, δ j,+1, ˆx 0, 0, α j = M(z S (j) ), ŷ j, = ˆx j, S j,, j = 1,2,...,k. Assume S j,0 = 0, j = 1,2,...,k. Also assume the followig coditio to be fulfilled: α k > α k 1 > α 1 > 0. (2) 2. Mai Result At first we preset oe of the mai result of paper - theorem o the FSLLN for the cumulative idle time of a customer i MQS. Theorem 2.1. If coditios (2) are fulfilled, the ( I1, ; I 2, ;...; I ) k, (α 1 ;α 2 ;...;α k ). Proof. At first we usig that for each fixed ε > 0 (see [11]) ( P lim I j, ŷ j, ) > ε = 0, j = 1,2,...,k, k. (3)

4 126 S. Mikevičius, V. Dolgopolovas First we prove that ŷ j, Ŝ i, = ŷj, ( S i, ) 0, j = 1,2,...,k, k. Usig relatios of [11] we obtai that ŷ j, = ˆx j, S j, = max 0 l (ˆx j 1,l S j,l ) = max 0 l (ˆx j 1,l S j 1,l +S j 1,l S j,l ) = max 0 l (ŷ j 1,l +S j,l ), j = 1,2,...,k, k. Thus, ŷ j, = max 0 l (ŷ j 1,l +S j,l ), j = 1,2,...,k, ŷ 0, 0, k. (5) Also, we see that (see (6)) ŷ j, Ŝ i, = max 0 l (ŷ j 1,l +Ŝj,l) j 1 Ŝ i, ŷ j 1, +Ŝj, Ŝ i, = ŷ j 1, Ŝ i,... ŷ 1, Ŝ1, = max Ŝ 1, Ŝ1, 0, (6) 0 l j = 1,2,...,k, k. But ŷ j, max + max Ŝ j,l = ŷ j 1, + max Ŝ j,l... 0 l ŷj 1,l 0 l 0 j l j = 1,2,...,k, k. Usig (7) ad (8) we get that max 0 l (4) Ŝ i,l, (7) 0 ŷ j, Ŝ i, {max 0 l Ŝ i,l Ŝi,}, (8) j = 1,2,...,k, k.

5 FLUID LIMIT FOR CUMULATIVE IDLE TIME Applyig (9) we achieve for each fixed ε > 0 ŷ j, Ŝ i, ŷ j, P > ε = P Ŝ i, > ε k {max 0 l P = P Ŝ i,l Ŝi,} Ŝ i,l Ŝi, 0 l P 0 l ( Ŝi,l) {max > ε P 0 l > ε = > ε P P Ŝ i,l Ŝi,} 0 l ( Ŝi, l) 0 l ( Ŝi,l) > ε, > ε > ε, (9) j = 1,2,...,k, k. Thus, for each fixed ε > 0 ŷ j, Ŝ i, P > ε P 0 l ( Ŝi,l) > ε, (10) j = 1,2,...,k, k. Note (see, for example, [11]) that for each fixed ε > 0 P max ( Ŝj,l) 0 l lim > ε = 0, j = 1,2,...,k, (11) if coditios (2) are fulfilled. Usig relatio Ŝ i, = S j,, j = 1,2,...,k, ad (11)-(12) we obtai that for each fixed ε > 0, ( P lim ŷ j, ( S j, ) ) > ε = 0, j = 1,2,...,k,, k. (12)

6 128 S. Mikevičius, V. Dolgopolovas So, we get for each fixed ε > 0 (see (13)) ( ) ( I j, α j I j, ŷ j, P lim > ε P lim > ε ) 3 ( ŷ j, ( S j, ) +P lim > ε ) ( ( S j, ) α j +P lim 3 = 0, > ε 3 ) (13) j = 1,2,...,k, k. Thus, if coditios (2) are fulfilled, the ( ) I j, α j P lim > ε = 0, j = 1,2,...,k, k. (14) The proof is complete. Ackowledgmets Research supported i part by the Natioal Complex Programme Theoretical ad Egieerig aspects of e-service techology creatio ad applicatio i high-performig calculatio platforms. Refereces [1] Billigsley P. (1968). Covergece of probability measures. Wiley, New York. [DOI: / ] [2] Borovkov A. (1972). Stochastic processes i queueig theory. Nauka, Moscow (i Russia).[DOI: / ] [3] Borovkov A. (1980). Asymptotic methods i theory of queues. Nauka, Moscow (i Russia). [4] Iglehart D.L., Whitt W. (1970a). Multiple chael queues i heavy traffic. I. Advaces i Applied Probability, 2, [DOI: [5] Iglehart D.L., Whitt W. (1970b). Multiple chael queues i heavy traffic. II. Sequeces, etworks ad batches. Advaces i Applied Probability, 2, [DOI:

7 FLUID LIMIT FOR CUMULATIVE IDLE TIME [6] Karpelevich F.I., Kreii A.I. (1994). Heavy traffic limits for multiphase queues. America Mathematical Society, Providece. [7] Kigma J. (1961). O queues i heavy traffic. J. R. Statist. Soc., 24, [8] Kigma J. (1962). The sigle server queue i heavy traffic. Proc. Camb. Phil. Soc., 57, [DOI: ] [9] Kobyashi H.(1974). Applicatio of the diffusio approximatio to queueig etworks. Joural of ACM, 21, [10] Mikevičius S. (1986). Weak covergece i multiphase queues. Lietuvos Matematikos Rikiys, 26, (i Russia). [11] Mikevičius S. (2005). O the full idle time i multiphase queues. Lietuvos Matematikos Rikiys, 45, (i Russia). [12] Reima M.I. (1984). Ope queueig etworks i heavy traffic. Mathematics of Operatios Research, 9, [DOI: [13] Saati T., Kers K. (1971). Aalytic plaig. Orgaizatio of systems. Mir, Moscow (i Russia).

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ANALYSIS OF THE LAW OF THE ITERATED LOGARITHM FOR THE IDLE TIME OF A CUSTOMER IN MULTIPHASE QUEUES

International Journal of Pure and Applied Mathematics Volume 66 No. 2 2011, 183-190 ANALYSIS OF THE LAW OF THE ITERATED LOGARITHM FOR THE IDLE TIME OF A CUSTOMER IN MULTIPHASE QUEUES Saulius Minkevičius