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LIMITS OF ON/OFF HIERARCHICAL PRODUCT MODELS FOR DATA TRANSMISSION SIDNEY RESNICK AND GENNADY SAMORODNITSKY Abstract. A hierarchical product model seeks to model network trac as a product of independent on/o processes. Previous studies have assumed a Markovian structure for component processes amounting to assuming that exponential distributions govern on and o periods but this is not in good agreement with trac measurements. However, if the number of factor processes grows and input rates are stabilized by allowing the on period distribution to change suitably, a limiting on/of process can be obtained which has exponentially distributed on periods and whose o periods are equal in distribution to the busy period of an M/G/1 queue. We give a fairly complete study of the possible limits of the product process as the number of factors grow and oer various characterizations of the approximating processes. We also study the dependence structure of the approximations.

1. Introduction A hierarchical product model seeks to model network trac as a product of independent processes. The idea is that network dynamics depend on various mechanical and software processes and controls which operate at dierent protocol layers and time scales. The consideration of such models is motivated by the need for explanations of both large time scale long range dependence and self-similarity in measured network trac as well as perception of small time scale multifractality. See Kulkarni et al. (2001); Misra and Gong (1998); Mannersalo et al. (1999); Carlsson and Fiedler (2000); Riedi and Willinger (2000). Q The usual scheme is to consider a process fZ (n) (t) = ni=1 Ij(n)(t); t 0g where Ij(n)(); j = 1; : : : ; n are iid on/o processes or perhaps the iid structure is varied by allowing a progressive scaling of time. An on/o process is an alternating renewal processes with states f0; 1g. (For background on the role of on/o models in trac modeling see Heath et al. (1997, 1998); Leland et al. (1994); Willinger et al. (1996, 1995a,b); Taqqu et al. (1997).) The key idea is that transmission can proceed i all the component processes are in the on state. This gives an idealized representation of dierent layers and time scales though it does not fully represent in a detailed manner the network dynamics, protocols and controls. This model is a proposed balance between realism and statistical and mathematical tractability. The o periods of the factor processes provide a way to model spacings between packet arrivals due to hardware and software, TCP windowing and congestion control, server delays and eects of switches and routers. The model has the added attractive feature that it can represent other scenarios. Consider low priority trac which must traverse a node subject to cross trac of n higher priority on/o streams. See Figure 1. The low priority trac can traverse the node only when each of the n AMS 1991 subject classi cations. Primary 90B15; secondary 60K250. Key words and phrases. uid queue, M=G=1 queue, heavy tails, long range dependence, steady state distribution, product models, in nite divisibility, renewal theorems . Research partially supported by NSF grants DMS-0071073 and DMI-9713549 at Cornell University. 1

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S. RESNICK AND G. SAMORODNITSKY

Figure 1. Low priority trac encounters blocking from higher priority streams.

higher priority streams is in the o state (from the point of view of the low priority stream this is the on , or transmission enabling, state) and thus traversal is controlled by a product model. Similarly, if a low priority stream must traverse n nodes, each subject to blocking by a higher priority on/o cross stream, then the channel will be open to the low priority stream only when each node is free of the on state of the cross stream. See Figure 2. Several authors (eg, Carlsson and Fiedler (2000); Misra and Gong (1998); Kulkarni et al. (2001)) assume, in the interest of greater tractability, that the individual on/o processes are Markovian which amounts to assuming that both the on and o length distributions Fon and Fo are exponential. This permits fairly explicit calculation of moments and some queueing characteristics. In particular, Kulkarni et al. (2001) provide a compelling and stimulating account of this model applied to analysis of TCP trac traces at the packet level. The following problems are evident with the use of Markovian factors: Measured on periods as given in Kulkarni et al. (2001) are demonstrably not exponentially distributed and the factor on/o processes are probably not Markovian. A model with exponential distributions for both on and o periods cannot exhibit long range dependence, a property usually observed in network trac rates. In fact, exponential distributions imply correlations will decrease exponentially fast in the lag (see Kulkarni et al. (2001); Misra and Gong (1998) and Section 6 below). The hierarchical model already tends toward the black-box philosophy and the assumption of exponential distributions in the interests of both mathematical and statistical tractability emphasizes the black-box aspect. Reliance on statistical goodness of t by visual impression of simulated traces also emphasizes that there is is minimal structural modeling. Despite the last itemized problem, the statistical analysis presented by Kulkarni et al. (2001) is very attractive for its skill and level of detail and we wondered how important was the feature that the measured on periods were not exponentially distributed. Thus, we sought approximations to fZ (n) (t); t 0g as the number of factors n satis es n ! 1. It is evident that if just one factor process is in the o state, then so is the product so o periods of Z (n) () tend to grow with n. Thus, to get a sensible approximation as n grows, one must stabilize the overall rate. This can be done by either letting the input rate grow with n (instead of being constantly 1) or by letting the on periods grow with n. This paper concentrates on the latter mechanism. We nd in Section 3

HIERARCHICAL PRODUCT MODELS

3

Figure 2. Low priority trac encounters blocking at several nodes.

that under quite general conditions, a suitable approximation for the n-factor hierarchical model Z (n) () is an on/o model where the on periods are exponential and the o periods are M/G/1 queueing model busy periods. For such an approximation, it is easy to make sensible assumptions which guarantee long range dependence. Section 2 discusses more formally the mathematical setup and Section 3 provides a warmup to the general theory which discusses the on/o approximation mentioned in the previous paragraph. Subsequent sections deal with the general asymptotic theory of product models. Intuitively our setup can be viewed as follows. As the number of factors n in the product model grows, individual on periods become long. We will, however, allow occasional short on periods. Those can be viewed as breakdowns, or other imperfections, in the system. Our general theory describes, in particular, the eect of such \imperfections" on the limiting approximating model. Sections 4 and 5 give the necessary and sucient conditions for Z (n) () to converge to a limiting approximation Z (1) () in the sense of convergence of nite dimensional distributions and we also provide various interpretations for the limiting process Z (1) (). The last Section 6 gives information about the dependence structure of the limiting process Z (1) (). In particular we give various facts about the decay of the correlation function. 2. Preliminaries We now review necessary constructions and notations. A single channel stationary on/o process is constructed with the following ingredients. Let fXon ; Xn; n = 1; 2; : : : g be iid nonnegative random variables representing on periods and similarly let fYo ; Yn ; n = 1; 2; : : : g be iid non-negative random variables representing o periods. The X 's are assumed independent of the Y 's and the common distribution of on periods is Fon and the distribution of o periods is Fo . We assume both Fon and Fo have nite means on and o and we set = on + o . De ne

Sn(;X) =

n X i=1

Xi ; Sn(;Y ) =

n X i=1

Yi:

Consider the doubly in nite pure renewal sequence that begins with an on period at time 0: n n ?X ?X (X?i + Y?i ) n=:::;?2;?1; 0; (Xi + Yi ) n=1;2;::: : i=1

i=1

4


The interarrival distribution is Fon Fo and the mean interarrival time is . This pure renewal process has a stationary version (Resnick, 1992, pages 224) f(D? ? Sn(?;X) ? Sn(?;Y ) )n=1;2;:::; D? ; D+; (D+ + Sn(+;X) + Sn(+;Y ) )n=1;2;:::g where (D?; D+ ) is a random vector independent of fXn ; Yng with distribution Z 1 P [Xon + Yo > s] Z 1 1 ? Fon Fo(s) (2.1) ds = ds: P [D? > x; D+ > y] = x+y x+y Here is an explicit construction of the stationary on/o process (see Heath et al. (1998) for a (?;0); X (+;0)); (Y (?;0); Y (+;0)) one{sided version): De ne three independent random vectors B; (Xon on o o which are independent of fXon ; Yo ; Xn ; Yn; n 1g as follows: B is a Bernoulli random variable with values f0; 1g and mass function P [B = 1] = on = 1 ? P [B = 0] and (x > 0; y > 0)

(?;0) > x; X (+;0) > y] = P [Xon on

Let

P [Yo(?;0) > x; Yo(+;0) > y] =

Z 1 1 ? Fon(s) ds =: 1 ? Fon(0) (x + y); on Z x+y 1 1 ? Fo(s) (0) x+y

o

ds =: 1 ? Fo (x + y):

(+;0) + Yo ) + (1 ? B )Y (+;0) ; D(0) = BX (?;0) + (1 ? B )(Y (?;0) + Xon ); D+(0) = B (Xon ? on o o and de ne a delayed renewal sequence by fSn ; n = : : : ; ?1; 0; 1; 2; : : : g := f(?D?(0) ? Sn(?;X) ? Sn(?;Y ) )n=1;2;:::; ?D?(0) ; D+(0); (D+(0) + Sn(+;X) + Sn(+;Y ) )n=1;2;:::g and this delayed renewal sequence is stationary. (In our terminology S0 = D+(0).) We now de ne the indicator process of on periods I (t) to be 1 if t falls in an on period and I (t) = 0 if t is in an o period. More precisely, the process fI (t); t 2 Rg is de ned in terms of fSn ; n = : : : ; ?1; 0; 1; 2; : : : g as follows: X (2.2) I (t) = B 1[?Xon(?;0);Xon(+;0) ) (t) + (1 ? B )1[?Y (?;0) ?Xon;?Y (?;0) ) (t) + 1[Sn;Sn+Xn+1) (t) : o o n6=?1

Remark 2.1. A useful fact (Heath et al., 1998, Corollary 2.2) is that for any t 0, conditional

on I (t) = 1, the subsequent sequence of on/o periods is the same as seen from time 0 in the stationary process with B = 1. In particular, conditionally on I (0) = 1, looking forward into the future produces an on period with distribution Fon(0) and then a sequence of o and on periods with distributions Fo and Fon : The situation is similar with looking backwards, or looking both ways. Our hierarchical product model is formed by taking products of n on/o indicator processes. As n ! 1, we need to keep the overall on rate roughly constant to get useful limits. There are two ways to do this. As n increases, one can either lengthen the on periods or one can increase the individual line on -rates rather than keeping them xed at 1. We take the former approach in this paper and investigate the latter elsewhere. So we suppose we have n iid stationary on/o

HIERARCHICAL PRODUCT MODELS 5 (n) (n) indicator processes I1 ; : : : ; In . The on period distribution of each factor process depends on n (n). The o period distribution is supposed independent and is denoted by Fon(n) and has mean on of n and as usual is Fo which has mean o . We will always assume that Fo (0) = 0; this can always be assured by replacing Fon(n) by its appropriate geometric convolution power. The (0) , corresponding complementary cumulative distributions are denoted Fon(n;0) and Fo Z x 1 ? Fon(n) (s) Z x 1 ? Fo(s) (0) (n;0) Fon (x) := ds; Fo (x) := (2.3) (n) o ds; 0 0 on and Bi(n) is a Bernoulli random variable for the ith process independent of the on and o periods with distribution (n) P [Bi(n) = 1] = (n)on : on + o

The product model is

(2.4)

Z (n) (t) =

Yn

Ii(n)(t):

i=1 (n) (n) Since Z (t) = 1 i Ii (t) = 1 for i = 1; : : : ; n, we have on(n) n o ?n (n) = 1 + (n) P [Z (t) = 1] = (n) on + o on no=on(n) ?n = 1+

n

and hence, we stabilize the input rates and get (2.5) lim P [Z (n) (t) = 1] = e?o =on ; n!1 i (n) on (2.6) = on: lim n!1 n Condition (2.6) will be our standing assumption in the rest of the paper and serves as the mechanism for stabilizing the input rate of Z (n) () as n ! 1. In subsequent sections we will 1. Examine conditions under which a limit process Z (1) () exists such that (2.7) Z (n) () ) Z (1) (); in the sense of convergence of nite dimensional distributions. 2. Provide interpretations of the limit processes Z (1) (). Only in certain cases can the limit process Z (1) () be constructed from independent on/o cycles. 3. Examine the dependence structure of the limit process Z (1) (). To show convergence of the nite dimensional distributions of Z (n) () to a limit Z (1) (), it suces to show, since Z (n) () has range f0; 1g, that for any k and time points 0 = h0 h1 hk that P [Z (n) (hi) = 1; i = 1; : : : ; k] ! P [Z (1) (hi) = 1; i = 1; : : : ; k]:

6


The reason for restricting attention to the range point 1 is that n (2.8) P [Z (n) (hi) = 1; i = 1; : : : ; k] = P [Ii(n) (hi) = 1; i = 1; : : : ; k] : 3. An Illuminating Special Case. In the single channel on/o construction we replace Xi by nXi and Xi(0) by nXi(0) so that Fon(x) is replaced by Fon (x=n) = Fon(n) (x) and Fon(0) (x) is replaced by Fon(0) (x=n) =: Fon(n;0) (x). This means that Z1 (n) on = non = n xFon(dx) 0 in accordance with (2.6). Since (2.6) holds, we have (2.5) holding as well. We will assume in this section that Fon (0) = 0. The reader will nd it easy to see what changes in our calculations if this assumption does not hold. Alternatively, one can see what happens in that case from our general discussion in Section 4. With these assumptions that Fon(n) (x) = Fon(x=n) it is not hard to see that Z (n) () ) Z (1) (). We illustrate the proof by showing that for any h > 0 P [Z (n) (0) = Z (n) (h) = 1] ! P [Z (1) (0) = Z (1) (h) = 1] and defer the proof for an arbitrary nite collection of time points until the general discussion. Understanding the bivariate distributions of the limit process will already allow us to identify the limit process. De ne the ordinary renewal function (3.1)

U (n) =

1? X Fon(n) Fo n; n=0

and the delayed renewal function (3.2) V (n) = Fon(n;0) Fo U (n) : Conditional on Ii(n) (0) = 1, we have Ii(n) (h) = 1 if either the initial on period extends past h (which occurs with probability 1 ? Fon(n;0) (h)) or if the initial on period plus an o period terminate before h and then there is a last o period before h followed by an on period which covers h. Thus, we see that as n ! 1 !n Zh h ? u (0) (n) (n) ? = (n) on o 1 ? Fon (h=n) + (1 ? Fon ( n ))V (du) (3.3) P [Z (0) = Z (h) = 1] e 0 R h(1 ? F ( h?u ))nV (n)(du) !n (0) nF ( h=n ) ? on on n ? = 0 (3.4) : =e o on 1 ? n To get a limit, we obviously need to show that Zh (0) nFon (h=n) ? (1 ? Fon ( h ?n u ))nV (n) (du) (3.5) 0 converges as n ! 1. Now rst of all, as n ! 1, since Fon (0) = 0, Z h=n 1 ? Fon(s) n h=n (0) (3.6) nFon (h=n) = n on ds on = h=on: 0

HIERARCHICAL PRODUCT MODELS 7 (0) This is an expression of the regular variation of Fon () at 0 and is equivalent to the weak conver-

gence of the minimum of the n initial on times. Of course, it is this minimum which determines the end of the on time initiated at 0 by conditioning on Z (n) (0) = 1. For the integral term in (3.5), we have, as n ! 1, Zh (1 ? Fon( h ?n u ))nV (n) (du) nV (n)(h); (3.7) 0 so it suces to understand the limit of nV (n) (h). This quantity certainly remains bounded as n varies since

nV (n) (h) = nE

1 X j=0

1[nX (0) +Y +nS(X )+S(Y ) ] (h) nE j

j

and by stationarity, this is h=on: The Laplace transform of nV (n) is for any > 0

(3.8)

j=0

1[nX (0) +nS(X )] (h) j

Z1

d(0) do() e?x nV (n) (dx) = nFond(n)Fd 0 1?Fdon(n) d 1 ? Fon(n)Fo() n non Fo() = do() 1 ? Fd on (n)F do() Z 1 ?x Fo(x) = e ! F on dx; on 0

(n) () = nVd

which implies as n ! 1

1 X

nV (n) (h) !

Z h Fo(s) 0

on ds:

This leads to the following result. Proposition 3.1. If Fon(n) (x) = Fon (x=n), then Z (n) () ) Z (1) (); in the sense of convergence of nite dimensional distributions. The bivariate distributions of the limit process Z (1) () satisfy o

P [Z (1) (0) = Z (1) (h) = 1] = e? on [1+Fo (h)] : Proof. We only verify bivariate distributions converge here; the general case follows in Section 4.5. From (3.5), (3.6), (3.7) and (3.8) we conclude Rh P [Z (n) (0) = Z (n) (h) = 1] e?o =on e?[h=on ? 0 Fo (s)ds=on ] Rh =e?[o =on +o =on 0 (1?Fo (s))=o ds] (0) =e?o =on [1+Fo (h)] : (0)

8

S. RESNICK AND G. SAMORODNITSKY process. Since Fon(0) is regularly varying

3.1. Identifying the limit with index 1 at 0 as a consequence of having a density which is non-zero at 0, we get (3.6), a familiar condition from extreme value theory. Condition (3.6) implies that suitably normalized minima of n forward recurrence times converge weakly to an exponential distribution (Resnick (1987)). Conditional on Z (n) (0) = 1, this minimum is the time the initial on period ends. The subsequent o period extends as long as any of n lines is in the o state. It is also instructive to remember that condition (3.6) is also the condition that ensures that the n-initial on periods arrange themselves asymptotically as Poisson points and we may think the o periods as delays in a queueing system. The rst o period of Z (n) (), conditional on Z (n) (0) = 1, should then be related to the busy period of an M/G/1 queue.

Proposition 3.2. Suppose Fon(n) (x) = Fon (x=n). Then Z (1) () is the indicator process of a stationary on/o process where the on-distribution is exponential with parameter 1=on and the o-distribution is the busy period distribution of an M/G/1 queue whose input is a Poisson process with rate 1=on and whose service length distribution is Fo . Proof. Let C be the busy period length distribution of a stationary M/G/1 queue described in the statement of the Proposition. The Laplace transform of C is given in Takacs (1962); Hall (1988). For > 0 Z1 e?s C (ds) = 1 + on ? R 1 ?t?ono F (0) (t) 0

Z1

and the mean is

e

0

o =on

on o

dt

xC (dx) = on e ?1 : Let Z () be the indicator process of a stationary on/o process generated by an o -distribution C and an on period distribution E () which is exponential with parameter 1=on. Then for any t>0 R 1 xE(dx) 0 R = (1 + expfon = g ? 1) P [Z (t) = 1] = R 1 xE (dx 1 ) + 0 xC (dx) on o on 0 =e?o =on ; as desired. Furthermore, for any h > 0, since E (x) = E (0) (x), P [Z (0) = Z (h) = 1] =e?o =on P [Z (h) = 1jZ (0) = 1] 0

=e?o =on where the renewal function U is given by

U=

Zh 0

e?(h?s)=on U (ds);

1 X

(E C )n:

n=0

To evaluate this using transforms, we get for any > 0,

Z1 0

e?h

hZ h

s=0

i

e?(h?s)=on U (ds) dh =

Z1

s=0

e?s

hZ 1 h=s

i

e?(h?s)=on e?(h?s) dh U (ds):

HIERARCHICAL PRODUCT MODELS The inner integral evaluates to ( + 1=on )?1 and with^denoting transform, we get

9

= + 11= U^ () on 1 1 = + 1= ?1 on 1 ? C^ () on?1 +on 1 = + ?on1 ? C^ ()=on 1 = o (0) R + ?on1 ? [?on1 + ? 1= 01 e?t? on Fo (t) dt] = We conclude and therefore

Zh 0

Z1 0

o (0)

e?t e? on Fo (t) dt: o (0)

e?(h?s)on U (ds) = e? on Fo (h) o

P [Z (0) = Z (h) = 1] = e? on [1+Fo (h)] which matches the bivariate distributions of Z (1) (). Showing that Z and Z (1) have all multivariate distributions equal is omitted here. A more general statement follows from a representation theorem proven in Section 5 and is given in part (i) of Corollary 5.7. (0)

4. General Approximating Limits. Here we make use of the lessons learned from consideration of the special case in Section 3. We assume that the on -periods composing the n-component on/o processes whose product yields Z (n) () all have common distribution Fon(n) and that o -periods as usual have distribution Fo . We have seen that as n ! 1, the overall input rate must remain stable in order to obtain useful approximations and this is achieved by imposing condition (2.6). It is also apparent that in order to get convergence of bivariate distributions, it is required that the minimum of the n-iid forward recurrence times each having distribution Fon(n;0) should converge weakly. Conditions guaranteeing this weak convergence of minima are discussed next. We continue to use notation for the complementary cumulative distributions given in (2.3). Theorem 4.1. The following are equivalent. (1) and a number q, with 0 q 1 such that 1. There exists a proper distribution function Fon ( 1 ) for all points of continuity x of Fon (4.1) Fon(n) (x) ! qFon(1) (x): 2. There exists a measure which is Radon on [0; 1) such that in [0; 1) (4.2) nFon(n;0) !v ;

10

S. RESNICK AND G. SAMORODNITSKY v where \!" denotes vague convergence.R In this case, with p = 1 ? q and L being Lebesgue (1) (s))ds, measure, and m(1) [0; x] = m(1) (x) = 0x (1 ? Fon = p L + q m(1) : (4.3)

on

on

(n;0) . Then for 3. Let fXi(n;0) ; 1 i ng be iid random variables with common distribution Fon all x 0 which are continuity points of the limit distribution (4.4) P [^ni=1 Xi(n;0) x] ! e?[0;x] ; as n ! 1. (n;0) . Then the 4. Let fXi(n;0); 1 i ng be iid random variables with common distribution Fon sequence of random point processes whose nth point process has points fXi(n;0) ; 1 i ng converges weakly in the space of Radon point measures on [0; 1) to a limit Poisson point process, n X

(4.5)

i=1

Xi(n;0) ) PRM( ):

The limiting process is Poisson with mean measure and is the superposition of a homogeneous Poisson process with rate p=on and a non-homogeneous Poisson process with mean measure qm(1) =on. De ne U (n) ; V (n) as in (3.1), (3.2) and

(4.6)

U (1) =

1 X

(1) Fo )n; V (1) = Fo U (1) : (qFon

n=0

Then, any of the previous conditions (4.1){(4.5) imply pointwise convergence (4.7) nV (n) ! V (1) : Proof. The equivalence of (4.2),(4.4) and (4.5) is well known from extreme value theory (eg, Resnick (1987)). Focus on why (4.1) and (4.2) are equivalent. Given (4.1) we have as n ! 1

nFon(n;0) (x) =n

Z x (1 ? Fon(n) (u))

(n) on and by dominated convergence this converges to

!

0

du

Z x (1 ? qFon(1) (u)) 0

on

Z x (1 ? Fon(n)(u)) 0

on

du

du = px + q m(1) (x) on on

and hence (4.2) follows. Conversely, if (4.2) holds, then taking Laplace transforms yields for > 0,

!

d d (n) () (n) 1 ? F 1 ? F on on () : nFon () = n (n) on on \ (n;0)

(n) 1?Fd on ()


11

For any xed 0, f on ; n 1g is bounded in n. Hence (Feller, 1971, page 433), (n;0) nF\ on () ! ^(); where ^() is the Laplace transform of the measure on [0; 1). Thus (n) () = 1 ? ^() lim Fd on n!1 on

and therefore Fon(n) (x) ! qFon(1) (x) at points of continuity for some proper distribution Fon(1) and some 0 q 1. Finally, we show why (4.7) is true. Taking Laplace transforms we have

1?Fdon(n)() d

Fo () do() F ( ) F (onn) =n on d (n) = nV () =n d d (n) d() 1 ? Fon(n) ()Fd o() 1 ? Fon ()F o 1?qF\ (1) (1) p + q ? 1?F\ on () F on () F d do() ( ) o on on on = ! [ ( 1 ) (1) d() d 1 ? qFon ()Fo() 1 ? qF[ on ()F o P which is the Laplace transform of V (1) = F 1 (qFon(1) F )n. \ (n;0)

o

o

n=0

As an example, let

fn(x) = n2 1[0;1=n)(x): Suppose fUn ; n 1g are iid U (0; 1) random variables independent of the non-negative iid random variables fn; n 1g assumed to have nite mean. De ne Xi(n) = fn(Ui) + i . Then fn(Ui ) ) 0; so that X1(n) ) 1. Also Efn(Ui) = n2 P [Ui < 1=n] = n; so (n) = E (X (n) ) = Efn(U1) + E (1) n: on 1 ( 1 ) Hence q = 1, on = 1, Fon (x) = P [1 x] and is [0; x] =

Zx 0

P [1 > u]du = m(1) (x):

4.1. General bivariate limits. We now show under any of the equivalent conditions in Theorem 4.1 that bivariate distributions of Z (n) () converge to those of a limit process Z (1) (). However, unlike the case in Section 3, the limiting process will not, in general, be composed of alternating independent on/o periods. Using the reasoning that led to (3.3) and (3.4) we get

P [Z (n) (0) = Z (n) (h) = 1] e?o =on

R

!

nFon(n;0) (h) ? 0h (1 ? Fon(n) (h ? u))nV (n) (du) n : 1? n

12

S. RESNICK AND G. SAMORODNITSKY (n;0) Now from (4.2) we have nFon (h) ! [0; h]; and from (4.7) it follows that nV (n) (x) ! V (1) (x) for all x 0. Also 1 ? Fon(n) (x) ! 1 ? qFon(1) (x): Thus Zh Zh nFon(n;0) (h) ? (1 ? Fon(n) (h ? u))nV (n) (du) ! [0; h] ? (1 ? qFon(1) (h ? u))V (1) (du) 0 0 = [0; h] ? V (1) (h) + qFon(1) V (1) (h) = [0; h] ? Fo U (1) (h) + qFon(1) Fo U (1) (h) = [0; h] ? Fo U (1) (h) + (U (1) ? 0 ) (h) = U (1) (0 ? Fo)(h);

where 0 is the measure putting mass 1 at 0. We have proven the following result. Proposition 4.2. Suppose (2.6) and (4.1) or one of its equivalents hold. Then for any h > 0 o + U (1)(0 ?Fo )(h)] ; (n) (0) = Z (n) (h) = 1] = e?[ on lim P [ Z (4.8) n!1

where is given by (4.3) and U (1) is given by (4.6). 4.2. A special case. Consider the special case where q = 0 and p = 1. Therefore, = ?on1L and U (1) = 0. Thus the limit in (4.8) is o o 1 (1) e?[ on + U (0 ?Fo )(h)] =e?[ on + on L0(0 ?Fo )(h)] o 1 Rh =e?[ on + on 0 (1?Fo (s))ds] o

(4.9) =e?[ on (1+Fo (h))] ; which is the same limit found in Section 3. 4.3. A converse. It turns out that (2.6) and (4.2) are the exact conditions for bivariate convergence. Proposition 4.3. Suppose 1. limn!1 P [Z (n) (0) = 1] = l1 2 (0; 1): 2. For all h 0, limn!1 P [Z (n) (0) = 1; Z (n) (h) = 1] = l2(h) 2 (0; 1): Then (2.6) holds and there exists a Radon measure () on [0; 1) such that nFon(n;0) () !v (); so (4.2) holds as well. (n)

(0)

Proof. Condition 1 implies, as n ! 1 that ( (n)on+ )n ! l1 . This gives (2.6). on o Next, we have P [Z (n) (h) = 1jZ (n) (0) = 1] ! l2l(h) ; 1 and the left side equals !n R [nFon(n;0) (h) ? 0h (1 ? Fon(n) (h ? s))nV (n) (ds)] : 1? n


This implies

G0 (h) = [nF (n;0) (h) ? n

on

Zh 0

(1 ? Fon(n) (h ? s))nV (n) (ds)] ! ? log l2l(h) =: G01 (h):

After some manipulation involving the renewal function V (n) , we get G0n(h) = nFon(n;0) U (n) (0 ? Fo )(h): Note for x 0,

13

1

1 X 0 (n;0) (n) G (x) nF U (x) = n F (n;0) (F (n) Fo )j (x) n

on

n

1 X

j=0

on

on

Fon(n;0) (Fon(n))j (x)

j=0 P Fon(n;0) (Fon(n))j is a stationary renewal measure, it is Lebesgue measure divided and because 1 j=0

by the mean renewal time so that we get =n x(n) = (n)x 2x ; on on =n on for all large n. This bounding function is locally integrable and hence G0n ! G01 implies Zx Zx Zx 0 0 Gn(x) := Gn(h)dh ! G1(h)dh = ? log l2l(h) dh =: G1 (x): 1 0 0 0 Now take Laplace transforms. For > 0, Z1 G^ n() = e?u nFon(n;0) U (n) (0 ? Fo )(u)du 0

and since we get (4.10) where we set

(n;0) () do() nF\ 1 ? F on ; = (n) ()F d() 1 ? Fd on

o

d (n)

1 ? Fon () ; (n;0) F\ on () = (n) on

d G^ n() = 1 ? Fo ()

^n() ; (n) on d 1 ? Fo()[1 ? n ^n()]

Z1

e?u n (du): Note fG^ n (); n 1g is a bounded sequence for each since n = nFon(n;0)

and ^n () =

0

(n) 1?Fd on () ( n d 1 ? F ( ) on) =n G^ n() = o (n) d() 1 ? Fd on ()F o

14

S. RESNICK AND G. SAMORODNITSKY do() 2on 1 ? F = 22 : d on 1 ? Fo () We conclude (Feller, 1971, Theorem 2a, page 433) that since Gn ! G1 and fG^ n (); n bounded, that G^ n() ! G^ 1 (): Referring to (4.10), we conclude ^n() ! ^1() where

1g is

1 ? Fd ^1 () o () = G^ 1 (): 1 ? Fd ( )[1 ? ^ ( )] on 1 o Again applying Theorem 2a, page 433 of Feller (1971) we conclude n = nFon(n;0) ! 1 and the statement of the Proposition is proven with = 1 . 4.4. Asymptotic independence. In the next subsection, we will show that the convergence of bivariate limits in Proposition 4.2 can be extended to higher dimensional convergence. The following Section 4 will show various representations of processes Z (1) () which have the limiting multivariate distributions. We say that asymptotic independence holds if Z (n) ) Z (1) and o (1) (0) = Z (1) (h) = 1] = e?2 on (4.11) = P [Z (1) (0) = 1]P [Z (1) (h) = 1]: lim P [ Z h!1 Here are some special cases where it is relatively easy to resolve whether asymptotic independence holds or not. q = 0: In this case, using (4.9) we have o (1+F (0) (h)] ?2 o (1) (0) = Z (1) (h) = 1] = lim e?[ on o on lim P [ Z = e h!1 h!1 and hence asymptotic independence holds. Note = ?on1L so [0; 1) = 1. q = 1 and m(1) ( ) < : Set R x(1 ? F (1) (s))ds (1) (x) m on ( 1 ;0) Fon (x) = m(1) (1) = R 01 (4.12) (1) (s))ds : (1 ? F on 0

1 1

(1)

So = m on(1) Fon(1;0) and therefore (1) U (1) (0 ? Fo ) = m (1) Fon(1;0) U (1) (0 ? Fo ): on ( 1 ;0) ( 1 ) Note that Fon U is a (delayed) renewal function corresponding to mean interrenewal time m(1) (1) + o and hence by the key renewal theorem, the right side of the above display converges to R o m(1) (1) 01 (1 ? Fo(s))ds = m(1) (1) on m(1) (1) + o on m(1) (1) + o : Consequently i h (1) o (1) (0) = Z (1) (h) = 1] = expf? o + m (1) (4.13) lim P [ Z h!1 on on m(1) (1) + o g: Asymptotic independence does not hold. Note [0; 1) < 1 in this case.

HIERARCHICAL PRODUCT MODELS 15 ( 1 ) A particularization is obtained by assuming Fon = 0 in which case = 0 and the entire bivariate distribution is expf? o g. on ( 1 ) q = 1 and m ( ) = : Here = ?on1m(1) and we must evaluate the limit of ?on1m(1) U (1) (0 ? Fo )(h) as h ! 1. Because m(1) U (1) has a density we can write

1 1

(4.14)

m(1) U (1) (x) =

Zx Zs 0

y=0

(1 ? Fon(1) (s ? y))U (1) (dy

!

ds =

Zx 0

p(s)ds:

(Note this formula is valid whether or not m1 (1) is nite.) Considering an on/o process with on -distribution Fon(1) and o -distribution Fo and starting with an on -period, we see that p(s) is the probability of being in the on state at time s. Since m(1) (1) = 1, we have p(s) ! 1 as s ! 1. Note Zh ?on1m(1) U (1) (0 ? Fo)(h) = 1 (1 ? Fo(h ? s))p(s)ds on 0 Z 1 Z1 1 1 = (1 ? Fo (s))ds 1(s h)p(h ? s) (1 ? Fo(s))ds ! on 0 on 0 =o =on by the dominated convergence theorem. Consequently, asymptotic independence holds and (1) (0) = Z (1) (h) = 1] = e?2 o on : lim P [ Z h!1 With the experience built up in the previous cases, we now state a general result. Proposition 4.4. Under the assumptions of Proposition 4.2, we have asymptotic independence i [0; 1) = 1. Remark 4.5. The dichotomy between asymptotic independence holding and not holding corresponds to whether the limiting Poisson process in (4.5) contains in nitely or nitely many points. This dichotomy will be better understood once we discuss representations of the limit process in Section 4. Proof. Due to the consideration of special cases above, we need only consider the case when 0 < q < 1 and show asymptotic independence. First of all, Zh Zh q p (1 ? Fo (s))ds + (1 ? Fo (s))(1 ? Fon(1) (h ? s))ds (0 ? Fo )(h) = on 0 on 0 =A + B: Now A ! po=on and

Zh 0

(1 ? Fo (s))(1 ? Fon(1) (h ? s))ds =

Z1 0

(1 ? Fo (s)) 1(s h)(1 ? Fon(1) (h ? s)) ds ! 0

by the dominated convergence theorem. We conclude (4.15) v(h) := (0 ? Fo )(h) ! p o : on

16


Since U (x) " U (1) = 1=(1 ? q) we have (eg, see (Resnick, 1992, page 253)) U (0 ? Fo )(h) ! U (1)p o = o on

on

and therefore asymptotic independence holds. 4.5. Convergence of nite dimensional distributions. We now show why nite dimensional distributions of Z (n) () converge. For this section we need the quantity (4.16) pj(n)(h1; : : : ; hj ) = P [I1(n)(hi) = 1; i = 1; : : : ; j jI1(n) = 0] where we understand the conditioning to mean that the on/o process is initiated by an o -period with distribution Fo. Note for example, that p1(n) (h) =P [I1(n)(h) = 1jI1(n) = 0] =Fo U (n) (0 ? Fon(n) )(h) (4.17) !Fo U (1) (0 ? qFon(1) )(h) as n ! 1. As before (see (2.8)) we have for any k 1 and 0 = h0 < h1 < < hk n P [Z (n) (hi) = 1; i = 0; : : : ; k] = P [I1(n) (hi) = 1; i = 0; : : : ; k]

n

e?o =on P [I1(n) (hi) = 1; i = 1; : : : ; kjI1(n)(0) = 1] ;

where now the conditioning at time 0 to be in state 1 indicates an on/o process is started in the on -state with distribution Fon(n;0) . The event \ki=1[I1(n)(hi) = 1] can be realized if the initial (n;0) extends beyond h or if the initial on -period ends at time on -period with distribution Fon k u in some (hj ?1; hj ] and then a system starting with an o -period is in the on -state at times hi ? u; i = j; : : : ; k: Thus (n) n P [I1 (hi ) = 1; i = 1; : : : ; kjI1(n)(0) = 1] =

k Z hj n X (n;0) Fon(n;0) (du)pk(n)?j+1(hj ? u; : : : ; hk ? u) 1 ? Fon (hk ) + j=1 hj?1 ! P R [nFon(n;0) (hk ) ? kj=1 hhjj?1 nFon(n;0) (du)pk(n)?j+1(hj ? u; : : : ; hk ? u)] n

= 1?

: n We claim that for every j 1 and h1; : : : ; hj (except possibly countably many) (4.18) pj(n) (h1; : : : ; hj ) ! p(j1) (h1; : : : ; hj ); where p(j1)(h1; : : : ; hj ) is the probability that in a (possibly terminating) on/o process with (1) ; starting with an o -period of distribution F , o -distribution Fo and on -distribution qFon o the times h1 ; : : : ; hj are in the on -state. Assuming this claim to be true, we will have lim P [Z (n) (hj ) = 1; j = 0; : : : ; k] =: P [Z (1) (hj ) = 1; j = 0; : : : ; k] n!1

HIERARCHICAL PRODUCT MODELS 17 Z k hj X ) (h ? u; : : : ; h ? u)]g: (du)p(k1?j+1 = expf?[o =on + [0; hk ] ? j k j=1 hj?1

(4.19)

We may rewrite (4.19) as follows. Write q(1) = 1 ? p(1) and then

[0; hk ]? = (4.20)

=

k Z hj X

j=1 hj?1 k Z hj X j=1 hj?1 k Z hj X

) (h ? u; : : : ; h ? u) (du)p(k1?j+1 j k ) (h ? u; : : : ; h ? u)) (du)(1 ? p(k1?j+1 j k ) (h ? u; : : : ; h ? u) (du)qk(1?j+1 j k

j=1 hj?1 k Z hj 1 X

=

) (h ? u; : : : ; h ? u) (p + q(1 ? Fon(1) (u))du qk(1?j+1 j k

=

) (h ? u; : : : ; h ? u)du: (1 ? qFon(1) (u))qk(1?j+1 j k

on j=1 hj?1 k Z hj 1 X

Note that

on j=1 hj?1

^l

ql(1) (h1; : : : ; hl ) = P [ I1(1) (hi) = 0jI1(1) (0) = 0] i=1

is the conditional probability that at some time point the system is o and where the conditioning means start the on/o process with an o -period with distribution Fo and on -periods have distribution qFon(1) . Thus, Theorem 4.6. For any k = 0; 1; : : : and 0 = h0 < h1 < : : : hk , P [Z (1) (hj ) = 1; j = 0; : : : ; k] k?1 Z hj+1 io n h o 1 X (1) (u))q(1)(hj+1 ? u; : : : ; hk ? u)du ; (4.21) (1 ? qFon = exp ? + k?j on on j=0 hj (4.22)

k Z hl o X ) (h ; : : : ; h ; x) (dx)]g: = expf?[ + qk(1?l+1 0 k on l=1 hl?1

One sees from (4.21) that the limiting process Z (1) is characterized by a quadruple (on; q; Fon(1) ; Fo ). It remains to prove (4.18), which we do by induction. We have already veri ed the case j = 1 in (4.17) so make the induction hypothesis that (4.18) holds for all j 0 < j . Conditioning on where the rst o/on cycle ends, we decompose pj(n)(h1; : : : ; hj ) as

pj(n) (h1; : : : ;hj ) =

Z h1 0

pj(n) (h1 ? u; : : : ; hj ? u)Fo Fon(n) (du)

18

S. RESNICK AND G. SAMORODNITSKY Z hi+1?u + Fo (du) Fon(n) (dw)pj(n)?i(hi+1 ? u ? w; : : : ; hj ? u ? w) 0 hi ?u Zi=1h1 + (1 ? Fon(n) (hj ? u))Fo(du) 0 Z h1 (n) (n) pj (h1 ? u; : : : ; hj ? u)Fo Fon(n) (du): =:fj (h1; : : : ; hj ) + 0 j ?1 Z h1 X

From the induction hypothesis fj(n)(h1; : : : ; hj ) ! fj(1)(h1; : : : ; hj )

as n ! 1, for all except, perhaps, countably many h1; : : : ; hj , with fj(1)(h1; : : : ; hj ) de ned in the obvious way. Therefore,

pj(n)(h1; : : : ; hj ) =

!

Z h1 0

Z h1 0

fj(n)(h1 ? u; : : : ; hj ? u)U (n)(du)

fj(1) (h1 ? u; : : : ; hj ? u)U (1)(du) = p(j1) (h1; : : : ; hj )

and so we conclude that (4.18) holds for all, except perhaps, countably many h1; : : : ; hj as required. Remark 4.7. Here is an immediate and, perhaps, not surprising conclusion from (4.21). The limiting process Z (1) is product in nitely divisible. Indeed, let (on; q; Fon(1) ; Fo ) be the quadruple corresponding to Z (1) . For a k = 1; 2; : : : let Zj(1) ; j = 1; : : : ; k be iid processes corresponding to the quadruple (on=k; q; Fon(1) ; Fo ). Then

8k 9 < = o Y Z (1) (t); t 2 R =d : Zj(1) (t); t 2 R; ; j=1

n

in terms of equality of nite dimensional distributions, which is what we mean by in nite divisibility of Z (1) . 5. Representations of the limiting process If we think of o periods as representing interruptions of service, then we saw in Proposition 3.2 that in case q = 0, the limiting process Z (1) has a very simple representation: iid interruptions distributed according to Fo arrive according to a time homogeneous Poisson process with intensity 1=on, and Z (1) (t) is simply the indicator function of the event that at time t there are no interruptions present in the system. The duration of the interruptions form a busy period in the M/G/1 queue and Z (1) () is an on/o process. In this section we develop various representations of this type valid in more general cases. It is possible to generate representations of the limiting process Z (1) () on [0; 1) but because ( 1 Z ) () is stationary, it is also natural and illuminating to consider representations on R. We study each type of representation in turn in the next two subsections.

HIERARCHICAL PRODUCT MODELS ( 1 ) Z on [0; 1). Here is an outline of how

19

5.1. Representations of to develop a general rep( 1 ) resentation for fZ (t); t 0g. We regard this outline as suggestive of the result and do not justify all the steps. Once the representation Z () suggested by this outline is in place, we show it has the same nite dimensional distributions as Z (1) () given in (4.20). Let X ( 0) = fXj(n;0) ; 1 j ng be iid with distribution Fon(n;0) , Y (0) = fYj(0) ; 1 j ng are (0) and suppose that conditionally on X ( 0) ; Y (0) ; we have fW (n;+;) (); j iid with distribution Fo j 1g; fWj(n;?;) (); j 1g are independent with the following descriptions: Wj(n;+;x) () is a nonstationary on/o indicator process with on period distribution Fon(n) and o period distribution Fo starting with an initial on period of length x. Similarly, Wj(n;?;y) () is an on/o indicator process with on period distribution Fon(n) and o period distribution Fo starting with an initial o period of length y. We will also need Wj(+;) (); j 1; Wj(?;) (); j 1 which are independent, possibly terminating, on/o processes with Wj(+;x) () starting with an on period of length x and Wj(?;y) () starting with an o period of length y. The on period distribution is qFon(1) and the o period distribution is Fo . Consider the j th indicator process Ij(n) in (2.4). Observing Ij(n) is equivalent to observing n;

n;

(n;0)

Wj(n;+;Xj

)

and observing

(n) with probability (n)on on + o

with probability (n)o : on + o Think of Ij(n)(); 1 j n as points in the space D[0; 1) \ f0; 1g[0;1) of f0; 1g-valued cadlag functions. We thus get a sequence of point processes

Wj(n;?;Yj

(0)

)

8n 9 1=2. We conclude that

o lim sup m(1) (t)E (t) (1 ? )?(1?) ?()?(2 ? ) ; t!1

and since this is true for all 0 < < 1, we conclude that

o lim sup m(1) (t)E (t) ?()?(2 ? ) ; t!1

which together with (6.12) shows that

o lim m(1) (t)E (t) = ?()?(2 ? ) :

(6.15)

t!1

In combination with (6.3), the statement (6.15) proves the remaining part of the theorem. Remark 6.2. It follows immediate from Theorem 6.1 that

Z1

(6.16)

0

R(t) dt = 1 :

This is sometimes taken to be an indication of long range dependence (see e.g. Beran (1994)). Note, furthermore, that (6.16) always holds in the case q = 1 and (on1) = 1, whether or not the assumption of regular variation (6.9) is satis ed. Indeed, observe simply that

Z 1Z t 0

0

(1 ? Fo(t ? s))(1 ? p(s)) ds dt = o

Z1 0

(1 ? p(s)) ds = 1

since the amount of time spent in either state of a nonterminating alternating renewal process is in nite with probability 1. Now (6.16) follows from (6.3). Remark 6.3. Karamata's theorem mentioned above actually shows that if < 1 then (1) m(1) (t) 1 ?1F?on (t) ; and so Theorem 6.1 can be, in the case < 1, reformulated accordingly. Remark 6.4. It follows from Theorem 6.1 that (at least, in the range 1=2 < 1) the rate of decay of the covariance function R(t) is faster when the tail of the distribution Fon(1) is heavier. This should be contrasted with the case of on/o processes where heavier tails tend to cause the covariance function to decay slower; see e.g. Heath et al. (1998).


31

Remark 6.5. It is an open question to what extent (6.11) is true in the case 0 < 1=2. The

main ingredient in the proof of a key renewal theorem we used to obtain (6.12), the fact that 1 lim m(1) (t)(U (1)(t + h) ? U (1) (t)) = ?()?(2 (6.17) t!1 1 ? ) is false, in general, in the case 0 < 1=2, as counterexamples in Williamson (1968) demonstrate (in the arithmetic case). This, clearly, rules out the expected key renewal theorem at least for some directly Riemann intergrable functions. On the other hand, at least in the arithmetic case, (6.17) fails only on a \small" set, so there is hope that a key renewal theorem may still hold for a reasonably rich class of functions. At the very least one would expect both (6.17) and the corresponding key renewal theorem to hold under some smoothness assumptions on the distribution Fon(1) Fo and, in fact, Erickson (1971) does state a theorem of this kind, that assumes existence of a suciently regular density of Fon(1) Fo . Unfortunately, no proof is given, and we have not been able to locate the promised future publication in which the proof was to appear. It is unfortunate that there does not seem to have been much progress on heavy tailed key renewal theorems since Erickson (1970) (but the interesting recent paper of Doney (1997) may be a sign of important additional future developments). 6.3. The case q < 1. The decomposition (6.3) still holds in this case, but now p(s) is the probability of being in the on state at time s in a terminating on/o process with (defective) (1) and o -distribution F and starting with an on -period. The expressions on -distribution qFon o (6.5) and (6.6) are still valid, but now we have to replace (6.7) with (6.18)

h(t) = q

Z t 0

1 ? Fo (x) 1 ? Fo (t ? x) dx ; t 0 :

We study the decay rate of the covariance function R(t) under several dierent scenarios. We start with the case q = 0. Theorem 6.6. If q = 0 then R(t) = o expn?2 o o : (6.19) lim t!1 F (0) (t) on on o Proof. This is an immediate conclusion from (6.3). Unless speci ed otherwise, in the remainder of this section we assume that 0 < q < 1. Observe that (6.20)

h(t) 2q 1 ? Fo (t)

Z t=2 0

1 ? Fo (x) dx 2qo 1 ? Fo(t) ; t ! 1 :

It turns that under mild heavy tails assumptions on Fo the above holds as an asymptotic equivalence h(t) = 2q : lim (6.21) o t!1 1 ? F (t) o

By Fatou's lemma, the minimal asumption under which (6.21) holds is the assumption Fo 2 L, the class of long tailed distributions: G 2 L if G (t ? x)=G (t) ! 1 as t ! 1 for all x > 0. A

32


sucient condition for (6.21): there is an integrable on (0; 1) function H such that

1 ? Fo (x) 1 ? Fo (t ? x) (6.22) H (x) for all 0 < x < 2t : 1 ? Fo (t) Examples include Fo with a regularly varying tail, in which case one can take H (x) = CFo (x) for some C > 0, or, say, Fo (x) = expf?ax g for 0 < < 1, in which case one can take H (x) = expf?a(2?2 )x g. We do not know if the relation (6.21) holds under the assumption that Fo belongs to the class S of subexponential distributions: recall that G 2 S is G G(t) 2G(t) as t ! 1. We refer the reader to Embrechts et al. (1979) for information on subexponential distributions and some of their properties used below.

?

Theorem 6.7. Let Fo 2 S and let 1 ? Fon(1) (t) = o Fo (t) as t ! 1. Assume that (6.21)

holds (for which (6.22) is a sucient condition). Then (6.19) holds. Proof. We claim that under the conditons of the theorem g(t) = 2q (6.23) lim o t!1 1 ? Fo(t)

?

as well. Indeed, It follows from (6.21) and the assumption 1 ? Fon(1) (t) = o Fo(t) that

g(t) 2qo

Z t 0

1 ? Fo (t ? x)

Fon(1) (dx) = 2qo

Fo Fon(1) (t) ? Fon(1) (t)

?

as t ! 1. Using the assumptions Fo 2 S and 1 ? Fon(1) (t) = o Fo(t) , it is a standard property of subexponential distributions that Fo Fon(1) (t) Fo (t) as t ! 1 (see Embrechts et al. (1979)), and so (6.23) follows. The latter fact also implies that pU (1) which is a geometric convolution of the distributions Fo Fon(1) has a tail of the same order: 1 ? pU (1) (t) = 1 : (6.24) 1 ? Fo(t) p Therefore, using, once again, the properties of subexponential random variables we see that Z t (1) 2 q o ( 1 ) ( 1 ) E (t) 2qo 1 ? Fo(t ? x) U (dx) = p Fo pU (t) ? pU (t) 0 as t ! 1. On the other hand, (6.25)

2qp o Fo(t)

Fo (t) = o Fo(0) (t)

as t ! 1 if Fo 2 L S . Now the statement of the theorem follows from (6.25) and (6.3). This completes the proof.


33

Remark 6.8. It is easy to see by looking carefully at the above argument that the full force of (6.21) is not needed for its conclusion. For example, the same argument will establish (6.19) under the following conditions: there is a distribution G 2 S such that g(t) = O 1 ? G(t) ; 1 ? G(t) = o Fo(0) (t) as t ! 1 : (6.26) Note that the rst part of (6.26) already implies that ? ? max 1 ? Fon(1) (t); Fo(t) = O 1 ? G(t) as t ! 1. The situation is a bit dierent in the case of exponential tails. Assume that there is a > 0 such that (1) do(? ) = 1 F[ (6.27) on (? )F q

and (6.28)

Z1 0

te t Fon(1) (dt) < 1;

Z1 0

te t Fo (dt) < 1

(1) () = R 1 e?tF (1) (dt) is the Laplace transform of F (1) , and F d() is the (recall that F[ on

0

on

on

o

Laplace transform of Fo). Theorem 6.9. Assume (6.27) and (6.28) hold. Then (6.29) 2 d [ (1) (? ) F n o ( ? ) ? 1 F on o t R(t) = 1 exp ?2 o lim e : R t!1 on on F[ (1) (? ) 1 te t F (dt) + Fd (? ) R 1 te t F (1) (dt) on

on o o 0 0 Proof. It is immediate that the function Fo (t)e t is directly Riemann intergrable. Using Proposition 2.16 (d) in Cinlar (1975) twice, we see that so is the function h] (t) = e t h(t), with h given in (6.18), and, hence, so is the function g] (t) = e t g(t), with g given in (6.6). Therefore, Proposition

3.11.1 in Resnick (1992) applies, and for E (t) given in (6.5) we have R 1 e x g(x) dx 1 t lim e E (t) = R 1 0 x (1) : t!1 o q 0 xe Fon Fo (dx) Notice that under the asumptions of the theorem Fo(0) (t) = o e? t as t ! 1 : Since Z1 2 d (1) e x g(x) dx = q F[ on (? ) F o (? ) ? 1 o 0 and Z Z1 Z1 1 [ ( 1 ) x ( 1 ) t d xe Fon Fo (dx) = Fon (? ) te Fo (dt) + Fo (? ) te t Fon(1) (dt) ; 0

0

the statement of the theorem follows from (6.3).

0

34


Remark 6.10. It is interesting to note that in the case q < 1 the statement (6.16) indicating

that the covariance function is not integrable holds if and only if the second moment of Fo is in nite. Indeed, this is a direct consequence of (6.3) and the fact that

Z 1Z t 0

0

(1 ? Fo(t ? s))(1 ? p(s)) ds dt = o

Z1 0

(1 ? p(s)) ds < 1

since the amount of time spent in the o state of a terminating alternating renewal process with a nite mean o distribution is nite, regardless of the on distribution. Compare this with Remark 6.2. References J. Beran (1994): Statistics for Long{Memory Processes . Chapman and Hall, New York. P. Carlsson and M. Fiedler (2000): Multifractal products of stochastic processes: uid ow

analysis. In Proceedings of 15th Nordic Teletrac Seminar NTS-15, August 22-24, 2000; Lund, Sweden . pp. 173{184. E. C inlar (1975): Introduction to Stochastic Processes . Prentice{Hall, Englewood Clis, N.J. R. Doney (1997): One{sided local large deviation and renewal theorems in the case of in nite mean. Probability Theory and Related Fields 107:451{465. P. Embrechts, C. Goldie and N. Veraverbeke (1979): Subexponentiality and in nite divisibility. Zeitschrift fur Wahrscheinlichkeitstheorie und verwandte Gebiete 49:335{347. K. Erickson (1970): Strong renewal theorems with in nite mean. Transactions of the American Mathematical Society 151:263{291. K. Erickson (1971): A renewal theorem for distributions on R1 without expectation. Bulletin of the American Mathematical Society 77:406{410. W. Feller (1971): An Introduction to Probability Theory and its Applications , volume 2. Wiley, New York, 2nd edition. P. Hall (1988): Introduction to the Theory of Coverage Processes . Wiley, New York. D. Heath, S. Resnick and G. Samorodnitsky (1997): Patterns of buer over ow in a class of queues with long memory in the input stream. The Annals of Applied Probability 7:1021{1057. D. Heath, S. Resnick and G. Samorodnitsky (1998): Heavy tails and long range dependence in on/o processes and associated uid models. Mathematics of Operations Research 23:145{ 165. V. Kulkarni, J. Marron and F. Smith (2001): A cascaded on-o model for TCP connection traces. Technical Report, Department of Statistics, University of North Carolina, Chapel Hill, NC. W. Leland, M. Taqqu, W. Willinger and D. Wilson (1994): On the self-similar nature of Ethernet trac (extended version). IEEE/ACM Transactions on Networking 2:1{15. P. Mannersalo, I. Norros and R. Riedi (1999): Multifractal products of stochastic processes: a preview. Technical Document COST257TD(99)31, September, 1999; available at http://www.vtt. /tte/sta2/petteri/research.html. V. Misra and W. Gong (1998): A hierarchical model for teletrac. In Proceedings of 37th IEEE Conference of Decision and Control, December 1998, Tempa, Fl . S. Resnick (1987): Extreme Values, Regular Variation and Point Processes . Springer-Verlag, New York. S. Resnick (1992): Adventures in Stochastic Processes . Birkhauser, Boston.


35

R. H. Riedi and W. Willinger (2000): Toward an Improved Understanding of Network Trac

Dynamics. In Self-similar Network Trac and Performance Evaluation . Wiley.

L. Takacs (1962): An Introduction to Queueing Theory . Oxford University Press, New York. M. Taqqu, W. Willinger and R. Sherman (1997): Proof of a fundamental result in self-

similar trac modeling. Computer Communications Review 27:5{23. J. Williamson (1968): Random walks and Riesz kernels. Paci c Journal of Mathematics 25:393{ 415. W. Willinger, M. Taqqu and A. Erramilli (1996): A bibliographical guide to self-similar trac and performance modeling for modern high-speed networks. In Stochastic Networks: Theory and Applications , S. Z. F.P. Kelly and I. Ziedins, editors. Clarendon Press (Oxford University Press), Oxford, pp. 339{366. W. Willinger, M. Taqqu, M. Leland and D. Wilson (1995a): Self{similarity in high{speed packet trac: analysis and modelling of ethernet trac measurements. Statistical Science 10:67{85. W. Willinger, M. Taqqu, M. Leland and D. Wilson (1995b): Self{similarity through high variability: statistical analysis of ethernet LAN trac at the source level. Computer Communications Review 25:100{113. Proceedings of the ACM/SIGCOMM'95, Cambridge, MA. School of Operations Research and Industrial Engineering, and Department of Statistical Science, Cornell University, Ithaca, NY 14853

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