BIVARIATE EXPONENTIAL DISTRIBUTIONS USING LINEAR STRUCTURES

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1 Snkhyā : The Indin Journl of Sttistics 2002, Volume 64, Series A, Pt 1, pp BIVARIATE EXPONENTIAL DISTRIBUTIONS USING LINEAR STRUCTURES By SRIKANTH K. IYER Indin Institute of Technology, Knpur D. MANJUNATH Indin Institute of Technology, Bomby nd R. MANIVASAKAN Indin Institute of Technology, Bomby SUMMARY. We derive bivrite exponentil distributions using independent uxiliry rndom vribles. We develop seprte models for positive nd negtive correltions between the exponentilly distributed vrites. To obtin positive correltion, we define liner reltion between the vrites X nd Y of the form Y = X + Z where is positive constnt nd Z is independent of X. To obtin exponentil mrginls for X nd Y we show tht Z is product of Bernoulli nd n Exponentil rndom vribles. To obtin negtive correltions, we define X = P + V nd Y = bq + W where either P nd Q or V nd W or both re ntithetic rndom vribles. For the cse of positive correltions, we lso chrcterize bivrite Poisson process generted by using the bivrite exponentil s the interrrivl distribution. Our bivrite exponetil model is useful in introducing dependence between the interrrivls nd service times in queueing model nd in the filure process in multicomponent systems. The primry dvntge of our model is tht the resulting queueing nd relibility nlysis remins mthemticlly trctble becuse the Lplce Trnsform of the oint distribution is rtio of polynomils of s. Further, the vrites cn be very esily generted for computer simultion. 1. Introduction The single vrible exponentil distribution hs long been the fvorite of nlysts working with queueing systems, life testing nd relibility models. Pper received August 2000; revised Mrch AMS (1991) subect clssifiction. Primry 62E10, 62H05; secondry 62N05, 60K25, 62H20. Key words nd phrses. Bivrite exponentil distribution, oint distribution, Lplce trnsform, correltion.

2 bivrite exponentil distributions 157 This is becuse its memorylessness property lends itself to esy nlysis. To provide flvor of rel life models, mny distributions, like Gmm, hyper exponentil, nd Coxin, hve been derived from the exponentil distribution. Whenever there re two or more vribles in system model, in mny nlyses they re ssumed to be independent. Consider, for exmple, the interrrivl nd the service times in queuing system which re ssumed independent in queuing theory while it cn be shown tht this is not true in most relistic situtions prticulrly in pcket communiction networks. As nother exmple consider the filure nlysis of multicomponent system in which the lifetime of one component influences the lifetime of the other component in which cse it becomes necessry to introduce dependence between the two vribles. To continue to use nlyticlly menble distributions we will need to ssume them to be mrginlly exponentil nd introduce multivrite exponentil distributions tht cn model the dependence between the vribles. However, unlike the norml distribution, there is no unique nturl extension of the univrite exponentil distribution nd mny models hve been proposed. In this pper we consider fmily of bivrite distributions tht re mrginlly exponentil. Mthemticl nlysis remins firly strightforwrd with this fmily of distributions in spite of the dependence tht is introduced mong the vrites. This is not true of mny bivrite exponentil distributions in literture. Further, these vribles re very esy to simulte on computer. The most well known of the bivrite exponentil distributions ws derived by Mrshll nd Olkin(1967,b) by considering shock models. If X nd Y re the lifetimes of two components of system, then the distributions of X nd Y re exponentil with prmeters λ 1 +λ 12 nd λ 2 +λ 12 nd the oint distribution of X nd Y is F (x, y) = 1 exp( λ 1 x λ 2 y λ 12 mx(x, y)). This distribution is not bsolutely continuous but stisfies the lck of memory property (LMP), P X > t 1 + s 1, Y > t 2 + s 2 /X > s 1, Y > s 2 = P X > t 1, Y > t 2. (1) In fct it hs been shown tht the bove is the only bivrite distribution tht hs exponentil mrginls nd which stisfies LMP. Properties of this Mrshll Olkin model re described in detil in Brlow nd Proschn(1975). Block nd Bsu (1974) consider bivrite distribution whose mrginls re mixtures of exponentils nd hving n bsolutely continuous oint distribution. Arnold nd Struss (1991) chrcterize bivrite distributions where the conditionl distributions re in prescribed exponentil fmilies. A survey of vrious bivrite nd multivrite exponentil models cn be found in Bsu (1988).

3 158 sriknth k. iyer, d. mnunth, r. mnivskn In this pper we present bivrite exponentil distribution tht derives its primry motivtion from pcket communiction networks but cn be used in other models. Consider high speed pcket communiction link tht cn be modeled s single server First Come-First Served (FCFS) queue. Let X be the length of pcket tht hs ust oined the output queue of network node t its til nd Y be the time till the rrivl of the next pcket. In most networks, correltion cn be estblished between X nd Y described bove. The nture of the correltion would depend on the kind of ppliction generting the pckets. For exmple in bulk trnsfer pplictions like ftp, the correltion will be negtive. In some interctive pplictions like telnet, the correltions could be positive. In the next section we define three models - one for positive correltion nd two for negtive correltion in which the mrginls re exponentil, nd the Lplce-Steiltes trnsform (LST) of the oint distribution is rtio of polynomils. In section 3 we consider oint counting process (N, M) in which time between the increments of the M process is X nd tht between N is Y nd obtin the mss function for the oint counting process. 2. The Bivrite Exponentil The bivrite models tht we develop re bsed on Exponentil Autoregressive sequences of Gver nd Lewis (1980). While borrowing freely from the ides in the Gver nd Lewis (1980) pper, we would like to point out the differences s well. The mor difference is tht while Gver nd Lewis construct n utoregressive AR(1) sequence X n such tht X n nd X n+1 re correlted nd the X n hve identiclly distributed exponentil mrginls, our interest is to construct bivrite exponentil distribution. The primry purpose of the models introduced by Gver nd Lewis is to introduce correltion between the successive interrrivls or service times in queue. Our min interest is to introduce cross correltion between the service time nd the next interrrivl. Further, in the model proposed for introducing negtive cross correltion, we do not tke recourse to ( fictitious) ntithetic queue. 2.1 Positive cross correltion. Consider two rndom vribles X nd Y whose mrginls re exponentil with prmeters λ x nd λ y respectively. Let Y = X + Z (2) where > 0 is constnt nd Z is rndom vrible independent of X. The constnt cn be thought of s scling fctor. Let us first chrcterize Z. Let X (s), Y(s) nd Z(s) be the LST of the distributions of X, Y nd Z

4 bivrite exponentil distributions 159 respectively. From eqution (2), we see tht From this we obtin Y(s) = X (s)z(s) Z(s) = Y(s) X (s) = λ y λ x + s s + λ y λ x Define λ y /λ x = ρ. After some lgebr, for 0 ρ 1.0, we get Z(s) = ρ λ x + (s + λ y λ y ) λ y + s λ y = (1 ρ) + (ρ) (3) s + λ y It is esy to see tht (3) corresponds to the LST of the distribution of rndom vrible tht is product of two independent rndom vribles - Bernoulli rndom vrible with men (1 ρ) nd n exponentil rndom vrible with prmeter λ y. Thus the distribution of Z will be ρ + (1 ρ)(1 e λyz ), z 0. The LST of the oint distribution is esily obtined s follows. XY(s 1, s 2 ) = E e s 1X s 2 Y = E (e ) (s 1+s 2 )X )E(e s 2Z λ x λ y = (1 ρ) + (ρ) (4) λ x + s 1 + s 2 s 2 + λ y Let r xy denote the correltion between X nd Y. E(XY ), nd hence r xy is obtined s follows. E(XY ) = E(X(X + Z)) = E(X 2 ) + E(X)E(Z) = 2 λ 2 x ρ λ x λ y (5) r xy = ρ λ 2 x λ x λ y 1 λ yλ x 1 λ yλ x = ρ (6) To see the ppliction of this model to multicomponent relibility system, ssume for the moment tht = 1. Tht is, Y = X + Z. Consider system protected by two sfety devices D 1 nd D 2 tht re subect to shock from the sme source(s). The mrginl lifetime distributions of D 1 nd D 2 re exponentil with prmeters λ x nd λ y respectively. The shocks re first bsorbed by device D 1. As soon s the device D 1 fils, the device D 2 tkes over. So, the lifetime Y of D 2 is the lifetime X of D 1 plus rndom mount

5 160 sriknth k. iyer, d. mnunth, r. mnivskn Z. From (2) we see tht Z = 0 or Y = X with probbility ρ. Thus ρ is the probbility tht the shock is strong enough to destroy both the devices simultneoulsy. If the device D 2 survives the shock tht destroys D 1, then it lsts lifetime tht is exponentilly distributed with prmeter λ y. As second ppliction, consider pcket communiction network with pcket trnsmission times denoted by X nd hving n exponentil distribution. Let Y be the pcket interrrivl time which lso hve n exponentil distribution. In most networks, pcket trnsmissions re best modelled s burst processes in which during the burst, consecutive pckets hve strongly correlted rrivl times, typiclly constnt or constnt multiple of the pcket length, nd bursts rrive independently. If we use the bove model to model such pcket processes, the burst length, number of pckets in the burst, will be geometriclly distributed with men 1/(ρ) nd the time from the end of one burst till the beginning of the next one will be exponentilly distributed with prmeter λ y. During the burst, the time t which the next pcket rrives will be constnt multiple of the current pcket length. Aprt from modeling correltions in network trffic the models considered in this pper cn be used to study trnsmission control schemes over network. A tnsmitter sends pckets fter dely which is distributed s constnt multiple of the length of the previous pcket (trnsmission time) plus rndom mount. Pcket sizes re synonymous with the service times. Using the liner structure nd the fct tht the mrginls hve exponentil distributions, queueing nlysis cn be crried out. This type of control will help in incresing bndwith utilistion while reducing congestion. 2.2 Negtive cross correltion. To obtin negtive cross correltion between X nd Y, we tke different but similr pproch thn the one in the previous section. Consider the following eqution X = P + V Y = bq + W (7) Here nd b re non negtive constnts, P nd V re independent of ech other nd so re Q nd W. Further, from the previous section, if P nd Q re exponentil with prmeters λ p nd λ q respectively, nd V nd W re like Z of eqution (2), the mrginl distributions of X nd Y will be exponentil. To induce negtive correltion between X nd Y, we could mke either P nd Q or V nd W ntithetic rndom vribles nd choose the other two to mke the mrginls of X nd Y exponentil. We will consider both these options seprtely.

6 bivrite exponentil distributions 161 Model 1. First, consider P nd Q to be ntithetic exponentil rndom vribles. Further let V nd W be independent. Following Gver nd Lewis (1980), we define P nd Q through U which is uniformly distributed rndom vrible in (0, 1), s follows P = 1 λ p ln(u) Q = 1 λ q ln(1 U) (8) Under this definition of P nd Q, V is the product of Bernoulli vrible with men (1 λx λ p ) nd n exponentil rndom vrible with prmeter λ x. Similrly, W is the product of Bernoulli vrible with men (1 b λy λ q ) nd n exponentil rndom vrible with prmeter λ y. Also, nd b re to be chosen such tht 0 λx λ p 1 nd 0 b λy λ q 1. It is esy to see tht cov(x, Y ) = E(XY ) E(X)E(Y ) = bcov(p, Q). Therefore, the correltion coefficient between X nd Y, r xy, is given by r xy = bλ xλ y λ p λ q ( 1 π2 6 ) for 0 λx bλy λ p, λ q 1 nd, b 0 (9) ) From eqution 9 it follows tht (1 π2 6 r xy 0. To simplify prmeter selection, we could mke λ q = λ p nd b =. The LST of the oint distribution of X nd Y XY(s 1.s 2 ), is obtined s follows. XY(s 1, s 2 ) = E(e s 1X s 2 Y ) = E(e s 1V s 2 W )E(e s 1P bs 2 Q ) ( = 1 λ ) x λx λx + λ p s 1 + λ x λ p ( 1 bλ ) y λy bλy + λ q s 2 + λ y λ q 1 0 u s 1/λ p (1 u) bs 2/λ q du (10) Note tht the integrl on the RHS of eqution 10 corresponds to the bet function. To minimize the number of prmeters to choose, we could hve W = V, insted of ssuming them to be independent. In this cse the cov(x, Y ) = bcov(p, Q) + σ 2 v where σ 2 v is the vrince of V. Also, since P nd Q re exponentil mrginls, we could consider this model to define 4-vrite exponentil in which cse we could choose not to mke P nd Q ntithetic. In the ltter cse, cov(x, Y ) = bσ 2 v.

7 162 sriknth k. iyer, d. mnunth, r. mnivskn Model 2. In the second model to obtin negtive correltion between the exponentil mrginls of X nd Y we mke V nd W ntithetic. Once gin we follow Gver nd Lewis (1980), nd define V nd W s functions of rndom vrible U tht is uniform in (0, 1) s follows { 0 if U c V = 1 λ v ln 1 U 1 c if U > c { 0 if U d W = 1 λ w ln U d if U < d (11) Here 0 c, d 1. Note tht the rndom vribles V nd W s defined bove re of the sme form s the rndom vrible Z of eqution 2. In fct V = F1 1 (U) nd W = F2 1 (1 U), where F 1 nd F 2 re distribution functions of the sme form s the distribution of Z but with different prmeter vlues. Thus we see tht for X nd Y to be mrginlly exponentil with prmeters λ x nd λ y respectively, λ v = λ x, λ w = λ y, nd P nd Q re independent exponentil rndom vribles with λ p nd λ q such tht, c = λx λ p nd 1 d = bλ y λ q. Under this model, cov(x, Y ) = cov(v, W ) nd r xy, the correltion between X nd Y is given by { d r xy = c ln 1 u 1 c ln u d du (1 c)d if c < d (12) (1 c)d if d c The LST of the oint distribution of X nd Y will be given by XY(s 1, s 2 ) = E(e s 1X s 2 Y ) = E(e s 1P bs 2 Q )E(e s 1V s 2 W ) λ p λ s 1 q 1 u = d λ p+s 1 λ q+bs 2 c λ p λ q λ p+s 1 λ q+bs 2 d 1 c λy λx ( u d ) s 2 λy du if c < d, λ y+s 2 + (c d) + (1 c) λ x λ x+s 1 if d c. The mgnitude of negtive correltions from Model 2 cn exceed π 2 /6 1 nd hence cn be lrger thn tht of Model 1. Just like with Model 1, we could choose Q = P, insted of ssuming them to be independent nd minimize the number of prmeters. In this cse we hve cov(x, Y ) = cov(v, W ) + bσ 2 p. Further, ust like in Model 1, we could consider this model to describe trivrite exponentil in which cse we could lso choose not to mke V nd W ntithetic. In the ltter cse, cov(x, Y ) = bσ 2 p. (13) Model 3. In this model we cn mke both P nd Q nd V nd W ntithetic with P nd Q being independent of V nd W. In this cse,

8 bivrite exponentil distributions 163 cov(x, Y ) = bcov(p, Q) + cov(v, W ). Also, ust like in model 1, we could tret this to be model for 4-vrite exponentil. 3. Joint Counting Process As n ppliction we consider bivrite counting process (M t, N t ). Let the interrrivl times between the umps of M nd N be distributed s X nd Y respectively, which in turn re ech mrginlly exponentil nd the reltion between them is defined s in eqution 2. It is esy to see tht M t nd N t re mrginlly Poisson. Let (X i, Y i, Z i ), i = 1, 2,..., be n i.i.d. sequence of rndom vectors with the sme oint distribution s (X, Y, Z) of section 2.1. Observe tht n i=1 Z i hs the sme distribution s B G(B, λ y ) where B is binomil with prmeter (n, p), where p = 1 ρ, nd conditionl on B, G is Erlng (Gmm) with prmeters, B nd λ y. Define g (z) s the density of gmm rndom vrible with prmeters λ y nd, g (z) = λ y(λ y z) 1 e λyz, z > 0. ( 1)! In evluting f m,n = PrM t = m, N t = n, m, n = 0, 1, 2,..., the oint probbility mss function for (M t, N t ), we hve to consider three cses. Cse 1: > 1. In this cse f m,n is non zero only if m n, since Y i = X i + Z i X i > X i. So we restrict ourselves to the cse when m n. Thus, we hve m n PrM t m, N t n = Pr X i t, Y i t i=1 i=1 = (1 p) n m n Pr X i t, X i t i=1 i=1 n n + p (1 p) n t m n Pr X i t, X i t z g (z)dz 0 i=1 i=1 m n Pr X i t, X i t z i=1 i=1 = m 1 k=n Pr M ( t z ) = k Pr = Pr M t m, M ( t z ) n M t z (t ) m k + Pr M ( t z ) m (14)

9 164 sriknth k. iyer, d. mnunth, r. mnivskn In the bove we hve used the homogeneity nd independent increment properties of the Poisson process. Substitute for P rm s = k = exp( λ x s)(λ x s) k /k! in the bove expression nd simplify using PrM t = m, N t = n = PrM t m, N t n PrM t m + 1, N t n 0 PrM t m, N t n + 1 +PrM t m + 1, N t n + 1 to get, for m n, f m,n = (1 p) n λx t n ( (λ x t t m n n + )) t t z n ( λ x (λ x t t z )) m n g (z)dz n p (1 p) n e λxt n!(m n)! (15) The clcultions re similr for the cse 1. We only write the expression for f m,n in the remining two cses. Cse 2: 1, m > n. n f m,n = (λ x ( In this cse, it is esy to see tht n t ( p (1 p) n t(1 ) t t z λ x t z ) n )) m n g (z)dz e λxt n!(m n)! (16) Cse 3: 1, m n. As before, we will ust write down the corresponding equtions for m < n nd m = n respectively. If m < n then, ( ) t n m f m,n = (1 p) n (λ x t) m λ x t e λx t n m!(n m)! + n p (1 p) n If m = n then, f m,n = t(1 ) ( ) t z n m t z (λ x t) m λx e λ x t 0 m!(n m)! g (z)dz (17) (1 p) n (λ x t) n + t e λx n! + n ( n n ) n t ( p (1 p) n t(1 ) p (1 p) n t(1 ) 0 λ x ( t z )) n g (z)dz t z (λ x t) n g (z) e λx dz (18) n!

10 bivrite exponentil distributions 165 The oint counting processes for the models with negtive cross-correltion between the interrrivls of the processes M nd N do not yield compct formuls s bove. In this cse it is esier to write n expression for P rm t m, N t n thn in the positive correltion cse. This is due to the fct tht the interrrivls X i nd Y i stisfy two different liner equtions. The difficulty however stems from the link between X nd Y vi ntithetic vribles which mkes subsequent clcultions very messy. However, numericl simultion still remins strightforwrd. 4. Discussion nd Conclusion We hve presented bivrite distributions tht hve exponentil mrginl distributions nd positive or negtive correltion between the vrites. When the correltion is positive, liner reltion between the vrites cn be used. To obtin negtive correltion between the vrites, four more vribles, two of which re exponentil, re used nd the concept of ntithetic vribles hs been used. A mor dvntge of our description of X nd Y ccording to equtions (2) nd (7) is tht they lend themselves to esy simultion. Consider for exmple generting sequence (X n, Y n ) in which the X n nd Y n re mrginlly exponentil with positive cross correltion. We use eqution 2 nd generte X n s iid exponentil rndom vribles nd to obtin the corresponding Y n, we need to generte Bernoulli vrible, sy B n with men (1 ρ), nd nother exponentil rndom vrible Z n with men λ y nd obtin Y n = X n + B n Z n. Negtively correlted X nd Y cn be similrly generted using eqution 7. We wish to emphsize tht the distributions derived here re different from the bivrite exponentil distribution of Mrshll nd Olkin (1967,b) nd other bivrite exponentil models described in Bsu (1988). In prticulr, the bivrite exponentil described here does not stisfy the lck of memory property even though the mrginls re exponentils. Acknowledgement. The reserch of the first nd second uthors ws supported in prt by DST grnt No. III.5(20)/98-ET. References Arnold B.C. nd Struss D.J. (1991). Bivrite distributions with conditionls in prescribed exponentil fmilies. J. Roy. Stt. Soc. Series B, 53, Brlow, R.E. nd Proschn, F. (1975). Sttisticl Theory of Relibility nd Life Testing: Probbility Models. Hll. Rinehrt nd Winston, New York.

11 166 sriknth k. iyer, d. mnunth, r. mnivskn Bsu A.P. (1988). Multivrite exponentil distributions nd their pplictions in relibility. In: Krishnih, P.R. nd Ro, C.R., eds., Hndbook of Sttistics, Vol. 7. Elsevier Science Publishers B.V., Block, H.W. nd Bsu, A.P. (1974). A continuous bivrite exponentil extension, J. Amer. Sttist. Assoc., 69, Gver, D.P. nd Lewis, P.A.W (1980). First order utoregressive gmm sequences nd point processes, Adv. Appl. Probb., 12, Mrshll, A.W. nd Olkin, I. (1967). A multivrite exponentil distribution, J. Amer. Sttist. Assoc., 62, Mrshll, A.W. nd Olkin, I. (1967b). A generlized bivrite exponentil distribution. J. Appl. Probb., 4, D. Mnunth, R. Mnivskn Sriknth K. Iyer Deprtment of Electricl Engg. Deprtment of Mthemtics Indin Institute of Technology Indin Institute of Technology Mumbi, , Indi Knpur, UP Indi E-mil: dmnu@ee.iitb.ernet.in E-mil: skiyer@iitk.c.in