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On the level crossing of multi-dimensional delayed renewal processes

(1997), 355-361. ON TtIE LEVEL CROSSING OF MULTIDIMENSIONAL DELAYED RENEWAL PROCESSES Department of Applied Mathematics Florida Institute of Technology Melbourne, FL 32901, USA eugene@winnie.fit.edu (Received February 1997; Revised August, 1997) The paper studies the behavior of an (1 + 3)th-dimensional, delayed renewal process dependent components, the first three (called active) of which are to cross one of their respective thresholds. More specifically, the crossing takes place when at least one of the active components reaches or exceeds its assigned level. The values of the other two active components, as well as the rest of the components (passive), are to be registered. The analysis yields the joint functional of the "crossing level" other characteristics (some of which can be interpreted as the first passage time) in a closed form, refining earlier results of the author. A brief, informal discussion of various applications to stochastic models is presented. Key words: Fluctuations, first excess level, first passage time, termination index, renewal process, marked point process, N-policy queues, D-policy queues, queues vacations. AMS subject classifications: 60K05, 60K10, 60K25, 60G25, 68M20, 90A09, 90A60, 90B22 1. Introduction The behavior of sums of independent rom variables about a fixed level is a classical theme in the theory of fluctuations. In various applied probability models, the information about the level crossing first passage time of the sum of rom variables is of significant importance. In the case of nonnegative rom variables, one can derive tame analytic formulas for corresponding functionals of the "first excess level" first passage time [1]. In a basic scenario, we consider a marked delayed renewal process dependent marking, so that the marks, as the marginal process, also form a delayed renewal process. Generally speaking, it is a two-dimensional renewal process, say (rn,Mn) mutually dependent components, the first of which represents some events on the time axis. The marked process M n (which can be both discrete- or continuousPrinted in the U.S.A. @1997 by North Atlantic Science Publishing Company valued) evolves until it crosses a fixed, non-negative threshold upon one of the events -n" Consequently, this component process is called active, while the other component (see Heyman [8], Lee et al. [9-1.1], Loris-Teghem [13-14], Muh [15]). Here, when the system is exhausted, the server leaves on vacation, which includes a series of individual vacation trips or segments. The server resumes his main work when, upon completion of one of the vacation trips, the buffer accumulates to N or more units. (This policy is aimed to reduce unwanted switchovers between busy idle modes.) Another queueing application is D-policy (see Balachran Tijms [2], Dshalalow [7], Li Niu [12]). In this case, the idle or vacationing server gets back to service when the accumulated work, in terms of the cumulative service time, hits a specified level D. A combination of the two policies, suggested studied in Dshalalow [5, 6], yielded yet another practical application; the server would resume his primary work when the number of units in the buffer or the cumulative work hit their respective levels, whichever of the two comes first. This is associated a three-dimensional renewal process, say (-n,Qn,Wn), of which the first component is passive the other two (queue length workload) are active. Other applications may be found in the stock market. For example, this analysis can predict the first fall of a stock from a bull spree (or visa versa), as well as its level, by introducing an auxiliary process, whose increments are zeros or ones, dependent on whether or not the stock gained or lost in successive trades. This paper deals a further refinement of the studies initiated by the author in [3,4]. It seemed reasonable to extend the number of active passive components, as it. turns out, the outgoing functionals of three active a finitely many passive components still yield compact formulas of a closed analytic form. The stock market can make use of the results, especially in individual, diversified packages or mutual funds dominated by two or three major stocks. Here it is worthwhile to predict the first drop of any major components of the package. Inventories queueing may also find some practical applications of these results. For instance, we may consider (batches of) units entering a servicing system along their weights, volumes, lengths, or energy. The service facility would stop vacationing when the queue length, workload, or cumulative length of all units in the system crosses its respective threshold, whichever comes first. This paper deals exclusively the derivation of formulas for the functionals of the values of a general (1 + 3)th-dimensional, delayed renewal process ( all dependent components) registered at the moment one of the three active components crosses its assigned threshold. Applications, like those mentioned above, will follow. r n is passive. A common application is queueing N-policy vacations 2. Preliminaries Suppose m,’’’,rm; rn 0,1,...) is an (1 + 3)th-dimensional, delayed renewal process defined on the probability space (f,at, P). In other words, the vector increments %m lt, m_ 1 (Xrn,Yrn,Zrn Am,...,Am) rn 0,1,... (-J-I 0), are independent (although their components are not), for rn >_ 1, identically jointly distributed. out loss of generality, we assume that the first three components are discrete, each valued in {0,1,...}, the other are continuous, each valued in +. The joint distribution of the above increments is assumed to be given in terms of the following functionals: e[ XovVo Z% 0A0 is the scalar product of 0- (1) (01,...,01) A0 --(A,...,A/0) T (2) 7(tt,V,W,0) 0A is the scalar product of 0- E[ttXlvYlwZle 0A1] (01,...,01) A 1 --(AI,...,A) T (3) Given non-negative integers p,q, r, define the rom variable min{u I -inf{i: A > p}, u 2 -inf{j: Bj > q}, u a -inf{k: C k > r}} call it the termination index. The process % will evolve until one of its first three components, A, B, or C, called active components, crosses one of their respective thresholds, p, q, or r. Consequently, the remaining components of % are referred to as passive components. In summary, the process % will be terminated (passing u "phases") once one of the active coInponents hits p, q, or r, whichever comes first. The rom vector %u is said to be the termination level, or alternatively, the first excess level, although most likely only one of the three active components will exceed p, q, or r. Note that if one of the entries of T m (vlm,.. .,V/m) T, say Vm represents time, ,1 then r 1 is referred to as the first passage time of %. u We will be interested in a closed form for the functional E[uuAuvBuwCue Oru], 0- u is the scalar product of vectors 0 r u. For functions f: N o N g" N, define the transformations (4) N03 Djf(w)-(1-w)j>oWJf(j) wl < 1, (5) (6) Dh, i, sg(X,Y, z) DhDiDsg(x,Y,Z) The functions g can be restored through the inverse operator O lirnxO k! Ox k 1 1 x ("), o, (7) used once: (Dsf(x))- f(k ), or (8) (9) repeatedly, for instance: Denote Orv](x,y,z). Theorem 1. The (10) functional ff satisfies the following expression: 1- -(,,,o) 7o(U,V,w,O) 70(ux, vy,wz,O) 1 7(ux, vy,wz,O)" (11) Proof. Obviously the events H1 {//1 <//2 <//3), H4 {2 < "3 < "1}, H7 {’ "2 < "3}, H2 {’1 < P3 < 2)’ H5 {"3 < "2 < H8 {Pl P3 < Ho {’ < H13 {1 "3}, 2 3} H {, < form a measurable partition of Consequently, I H is the indicator function of set H H3 H6 H9 "3}, HI2 {//2 < //1 < //3}’ {//3 <//1 <//2}, {//2 =//3 < //1}’ {//3 <//1 =//2}, 1 i=1 IH AB Set O"IH)(x,y,z). 0,1,..., 0,1,..., 0,1, A_ B_ C_ From Dshalalow [4], J D(Z{, )(,) A_ D(Z{, })(y) yZ_ 1 yZ, Ds(I {3 n})(z) zCn 1 zCn, n By Fubini’s theorem, 13 i=I i’ (12) (13) (14) (15) (16) (17) 0- Rjk n Il j R ljR2jR3jkR4kRSknR6n 3--1 (tX) Aj_ l(vy)B j_ I(wz)Cj_ le y Yj + (18) (19) R3j k lzZ j + Z z k ...y YkZ y Yk z k lzZk- 1 (20) (21) (22) (23) I4k 15k n being independent, we arrive at Zk+l...z Z n-1 I6n 73(z Zn + 707 23(Y z). (24) we denoted /= 7(,,v,wz,O), 70 0(,,z,e), (,v,z,O), o(,,z,O), 7a(z) 7(1,1,z,0), 723(Y,Z) 7(1,y,z,0), (25) () () which are the corresponding functionals of the marginal processes. Now, 2 can be obtained by interchanging the roles of v w y z: (y) 23(: (28) 2 21-23(Y ,z) z). 0 + 0 1 J’ [ 72(y) 7(1,y,1,0). (29) Analogously, (30) (31) (32) (33) ")’l(X) The next three 4i’s are as follows: 47 Dh, i,s-[kuAkvBkwCke-OrkI {"1 k>O n>k that yields (,v,z,O), o(,v,z,O), (,v,,o), o(,v,,o), 7(X,1,1,0), ")’12(x,y) 7(x,y,l,0), /13(X,Z) 7(x,l,z,0). (34) () () (37) Similarly, ")’0 -[- ,),12 ")’(tt,V,WZ,O), ,,2 ,O(t,V,WZ,O)" (38) ")’103 3’ ")’03 + (7 + .)/13 .),1 ,3).)/0 1 ,g3 7g (" (39) (40) -t- (7 -t- 3/23 ,,/2 73),),0 1 3" (41) The rest of 3 are subject to quite analogous calculations: 72(Y 73(z) + )1 -723(Y;)--723(Y’ z) 7 1-- 71(x) 73(z) + 1 713(x,z) 713(x,z){ 702 70 + 701 (7 1--71(x)1 - -712(x,z)72+712(x’z)" 730-70 --’)’0T_(7 (z) (42) 70? ’7 (48) (44) (45) 72_7} 7’ + (7123 -t- 71 -t- 72 -t- 73 712 713 723 7)70{ 1 (46) (47) ,,/123 ")’(t,V,w,O) 7 23 70(u,v,w,O). The summation of 31 through 313 yields formula (11). Now, the functional E[{uuAuvBuwCue Oru] E[{%Avw C e- 0f,] can be restored through xz p q 7o(U,V,w,O) -[1 7(u, v,w,O)]2x2y z r{ 7o(Ux, vy,wz,O) 7(ux, vy,wz,O) (48) Analogously, we can also arrive at cases all three active components of % are continuous or mixed. In turn, all or some of the passive components of % can be discrete. For instance, let the component C be continuous. Then substitute w in e- 0 for E[ uuAuvBu w us -Oru], 7 70 use the transformation D,f (s) in place of D h. The function I, OO oe sV f (r]) dr] (49) operator transform. -Lapl-1((.)), can be restored Lap1-1 by using the inverse Laplace-like sts for the inverse of the Laplace Given the third active component is continuous, we have the following result. Theorem 2. The functional 3(,u,v,e O,O;x,y,s) Dh, iDo[ u Uv ue -c Ue -o](,y, s) (50) (51) satisfies lhe following expression: 7o(,v,,o) 7o(, ,+ 1 7(u, v,,O) ,o) 1 7(, + ,o)" The proof of this theorem is almost identical to that of Theorem 1. The formula restoring the functional E[uuAuvBuwCue Oru] is also very similar to (48)" (52) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

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