ON MARKOVIAN TRAFFIC WITH APPLICATIONS TO TES PROCESSES

Markov processes arc an important ingredient in a variety of stochastic applications. Notable instances include queueing systems and traffic processes offered to them. This paper is concerned with Markovian traffic, i.e., traffic processes whose inter-arrival times (separating the time points of discrete arrivals) form a real-valued Markov chain. As such this paper aims to cxtcnd the classical results of renewal traffic, where interarriva] times are assumed to be independent, identically distributed. Following traditional renewal theory, three functions are addressed: the probability of the number of arrivals in a given interval, the corresponding mean number, and the probability of the times of future arrivals. The paper derives integral equations for these functions in the transform domain. These arc then specialized to a subclass, TES +, of a versatile class of random sequences, called TES (Transform-Expan&SampIe), consisting of marginally uniform autoregressivc schemes with modu]o-i reduction, followed by various transformations. TES models arc designed to simultaneously capture both first-order and second-order statistics of empirical records, and consequently can produce high-fidelity models. Two theoretical solutions for TES + traffic functions are rived: an operator-based solution and a matric solution, both in the transform domain. A special case, permitting the conversion of the integral equations to differential equations, is illustrated and solved. Finally, the results are applied to obtain instructive closed-form representations for two measures of traffic burstincss: peakedness and index of dispersion, elucidating the relationship between them.


Introduction
Let {Xn}n=0 be a stationary non-negative Markovian stochastic process, interpreted as inter-arrival times in a traffic process.We shall refer to {Xn} as a traffic process, or interchange- (1.4) where U 0 is distributed uniformly on the interval [0, 1), and {Vn}= 1 is an arbitrary sequence of independent identically distributed random variables with common density function ,fv; the V n form a sequence of innovations, i.e., for each n > 1, V n is independent of {U0 + ,U1 + ...,Us +_ 1}, namely, the history of {Us + } to date.The auxiliary TES + processes of the form (1.4) are called the background processes, whereas the target TES + processes of the form (1.3) are called the foreground processes.
Throughout this paper, the following notational conventions are used.A plus superscript is consistently appended to mathematical objects associated with TES + sequences.To improve typographical clarity, we use the notation fx + instead of f x+, and so on.However, to economize on notation, the subscript will often be omitted; in that case, it is understood that the object is associated with the foreground sequence, {Xn + }-the focus of this paper.An exception is the transition density of the background process, {Xn + }, which will be denoted by gu + (v u), in conformance with previous notation (Jagerman and nelamed [8, 9]).Real functions are implicit- ly extended to vanish on the complement of their domains.The indicator function of a set A is denoted by 1A.A vertical bar in the argument list always denotes conditioning.The Laplace Transform of a function f is denoted by f(s)-f e-sf(y)dy; unless otherwise specified, all Laplace transforms are evaluated for a real argument s.
Since this paper is concerned with traffic modeling, we shall make throughout the reasonable assumption that D is strictly positive almost everywhere on [0,1), thereby guaranteeing that interarrival times modeled by TES + processes give rise to simple traffic.Furthermore, for a marginal distribution, F, we shall take D-F -1, where F-l(y)-inf{u:F(u)-y} is always single-valued, even if F is not one-one (F is always monotone increasing, but not necessarily strictly monotone).Distortions of the form D-F -1 ensure that {Xn + } has marginal distribution F, allowing us to match any empirical distribution; in practice, F is usually obtained from an empirical histogram of data measurements.Jagerman and Melamed [8] showed that the background TES + processes, (1.4), are Markovian with transition densities gu + (v u) 7v(i2r)e i2"(v-u), (1.5) and the corresponding foreground processes, (1.3), have autocorrelation functions (see Equation The rest of this paper is organized as follows.Section 2 presents some technical preliminaries.Section 3 develops functional equations in the transform domain for the traffic functions of interest, valid for fairly general Markovian traffic.Section 4 specializes the integral equations to TES + traffic and presents operator-based solutions, while Section 5 presents a matric solution, both in the transform domain.Section 6 shows how to solve for the traffic functions of a TES + process with exponential innovations by converting the integral equations to differential equations.Finally, Section 7 computes the peakedness measure for the burstiness of TES + processes.The resulting formula is shown to be related to, but more general than, another measure of traffic burstiness, called IDI (index of dispersion for intervals).

Prehminaries
This section presents some preliminary technical material concerning Laplace transforms of certain integrals and a theory for solving a class of Fredholm-type integral equations.
The following lemma extends the standard Laplace transform of convolution In the lemma, the (2ne-one) correspondence between a function, f, and its Laplace transform, f, is denoted by f-f (see Van Der Pol and Bremmer [15]); further, for a function g(t,x) of two variables, the Laplace transform, with respect to either variable, will be denoted by (s,x) and (t,s) as the case may be.
Then, for s >_ So, Proof: From the absolute convergence of the Laplace integrals, e sxf(X)dx e g(u,x)du e Xlg(u x 0 0 0 0 in which the double integral on the right is absolutely convergent.Setting t u -t-x in the double integral yields by Fubini's theorem i i The result follows immediately, since a Laplace integral always converges for points s >_ So, where s o is a convergence point.FI Note that if g(t,x)-g(t)is independent of x, then f g(t-x,x)f(x)dx becomes the familiar 0 convolution, and Lemma 1 yields the known result (s)f (s) on the right-hand side.
We next present a theory of Fredholm-type integral equations, to be used in the sequel to compute statistics of Markovian traffic, namely, the functions, qn(t), M(t) and pn(t) defined in Section 1.
Consider the Fredholm-type integral equation ., z(x) h(x) + z i s'z(Y)Ks(x' y)dy, 0 (2.1) in which h(x) is the forcing function and the kernel Ks(x y)is given by It will be assumed throughout that Ks(x,y E L2([0, oo)x [0, oo)), for all s > 0, namely, s>0, and that h(x) L2([0,oc)), that is (2.3) h2(x)dx < oc.0 Thus, (2.1) is a Fredholm-type integral equation (Zabreyko et al. [16]), and in terms of the Fred- holm operator %s, defined by %s[f(x)] / f(y)Ks(x y)dy, 0 f C (2.4) the integral equation (2.1) takes the form + Furthermore, since Ks(x y) E L[0, cx) as a function of y for every x > 0, the domain of %s may be enlarged, for each x, to include functions which are bounded on [0,) and summable over every compact interval.
We now return to the solution of the integral equation (2.5).The iterated kernels Ks, n(x,y), given by Ks(X,Y), gs'n(x'Y) / Wfl n- e-(w Ix)Ks, l(w,y)dw, o n-1 n>l provide an integral representation for the n th iterate, %y, of the operator %s, namely %sn[h(x)]-f h(y)Ks, n(X, y)dy.Formally, the integral equation (2.5) has the solution 0 s,z(X) (I-Z%s)-l[h(x)], (2.6) with the corresponding Neumann series representation (see Tricomi [14]) n--1 The resolvent operator, Rs(z of %s, satisfies by definition the equation [I z%s]-I + zRs(z), whence, it can be represented as Rs(Z) E zn-l%2" (2.9) n=l Substituting the right-hand side of (2.8) in (2.6) yields a solution for ps, z(X) in terms of the resolvent Rs(z), namely, + (2.10) The resolvent is also an integral operator with kernel Qs, z(X,y)-, K s, n(x y)z n-1, whence s,z(X) h(x) + z / h(y)Qs, z(X y)dy.
(2.11) 0 To investigate the radius of convergence of the Neumann expansion (2.7) or, equivalently, the expansion for the resolvent (2.9), we seek the lowest characteristic value A s and the corresponding eigenfunction s(x) satisfying Cs(x) A s ] Cs(y)e-sY fi(Y x)dy.
(2.12) 0 It is known that the Neumann series converges for z < lax (see, e.g., Tricomi [14]).Since the kernel (2.2) satisfies Ks(x y) >_ O, it is also known that s > 0 and that s(x) may be chosen to satisfy s(x) >_ O.It will now be shown that for s > 0, the circle of convergence of (2.7) and (2.10) includes the circle z 1.
Theorem 1: For s > O, the radius of convergence of the Neumann series (2.7) and the resolvent series (2.10) is greater than one.
Proof: Since A s is the radius of convergence, it suffices to show that A s > 1.To this end, write sup {s(x)} A s sup Cs(Y)e !} <_ A s sup /sup{s(Z)}e-SUfl(ylx)dy x->O/Jz>00 where the first equality follows from (2.12), and the succeeding inequality is a consequence of the positivity of the kernel and the positivity of fl(ylx) on a set of positive Lebesgue measure.
Dividing throughout by SUpz > 0{s(z)} > 0, we can write 1 < -Sfl(Ul)du < uv fl(u )d where the inequalities again follow from the positivity of the kernel and the positivity of fl(ylx) on a Borel set of positive Lebesgue measure.

Integral Equations for Markovian Traffic
Recall that by assumption, t-0 is an arrival point, -X 0 is the previous arrival point and the next arrival point.Denoting qn(tlx)-P(N(t)-n Xo-x}, define the generating function G(z,t x) E[zN(t) Xo x]-E q,(tlx) z'. n--O We now proceed to derive an integral equation for G(z, t lz ).Noting the sample path relation 1, and recalling that fl(ylx) is the 1-step transition density of {Xn} it follows from (3.1) that Unfortunately, (3.2) is neither of Volterra nor Fredholm type.For later applications it will be more convenient to transform it into a Fredholm-type integral equation.To do that, observe that the conditions of Lemma 1 are satisfied for fl(y]x) and G(z, t ly in the set {z: z _< 1}.Thus, taking Laplace transforms in (3.2) yields (z, s Ix) 1 fl(s Ix) / s +z e-SUG(z, sly)fl(Ylx)dy, s>0.
Finally, we obtain an integral equation for the Laplace transform of the generating function L(z,t Ix)- Applying Laplace transforms to (3.5) yields the relation for L (z, six ).It is interesting to note that 1 (3.7) and this relation extends the known formula for the transform of the renewal density in renewal theory (Cox [1]).
The three equations (3.3), (3.4) and (3.6), all have the generic form (2.1).We are now in a position to use the Fredholm-type integral equation theory, developed in Section 2, to solve the integral equations (3.3), (3.4)and (3.6).
For the integral equation (3.3) the forcing term, hG(X a-ll(S ix) s is assumed to belong to L2([0, cx)), or to be bounded on [0, oo] and summable over every finite interval.The solution of Since the circle of convergence of the Neumann series (2.7) includes z -1, the integral equation (3.4) for M(s Ix)is established, as noted easier, as well as itsexistence.For the integral equation (3.4), the forcing term, hi(x fl(Slx'----)s is also assumed to belong to L2([0, oc)), or to be bounded on [0, oo] and summable over every finite interval.The solution of (3.4) is M(s x) l-gf i(s x) + l -g E %r2[fa(s x)] f i(s x) + l-gRs(1)']i(s x)" (3.9) For the integral equation (a.6), the forcing term, hL( --f( Ix), is again assumed to belong to L([0,c)), or to be bounded on [0, o e] and summable over every finite interval.The solution of (a.6) is Z (Z, 8 IX) 71(8 IX)+ E znr[']l(8 Ix)I" (3.10) The coefficient of z n in (3.8) is the transform n(S x), whence ,(s x) %nI'l l(S x)l s / 1-f l ( s l Y ) K , s 0 n >_ 1. (3.11) Finally, the coefficient of z n in (3.10) is the transform n + 1( 8 IX), whence "n( 8 x) Y--1[1(8 IX)]-f Y)Ks, n-I(x,y)dy, 0 n>l. (3.12) 4. Specialization of TES + Traffic Processes We now proceed to specialize the discussion to a TES + arrival process {Xn+}, with innovation density fv" From (1.5), the transition density, gg + (v ]u) of the uniform background TES + process, {Us + }, has the representation in which the function gu on the right-hand side is given by gu(w) E fv( w + n) E 7V (i2ru)ei2w' (4.1) where the second equality follows from the Poisson summation formula (Lighthill [12]).rewrite the integral equations from Section 3 for their {Xn+} Next, counterparts we G X (z,s Ix), M+X (s Ix) and L+X (z,s Ix) in terms of the function gu(w).This has the important advantage of transforming the infinite integration range to the compact set [0, 1].To this end, observe that the probability element fl + (ylx)dy, with x D(u) and y D(v), is transformed to fl + (y x)dy f + (D(v) D(u))D'(v)dv gu(v-u)dv.
(4.5) 0 For each u, the function +x (s In(u))is analytic in the plane {s" Re[s] > 0} with a pole at s 0. The contribution of this pole (to be used in Section 7) is given in the next theorem.Theorem 2: For any TECo + process {Xn + } of the form (1.3), the asymptotic expansion of rx+ (s D(u)) at 0 is given, for each u, by for some constant bo + to be determined in Section 7.
To put the Fourier series above in standard form, we use complex conjugates to obtain On the other hand, since E v(-i2pi)'+x (i2r) ei2ru.

This
The theorem follows, since the postulated asymptotic expansion is consistent.
Since M(tlD(u)) is monotone increasing in t (for fixed D(u)), a real Tauberian theorem [15]   may be used to obtain the following asymptotic expansion at infinity, M(t D(u)) At, t---,.
One may also expect M(t D(u)) At + b + x (u), although this does not directly follow from the Tauberian theorem.
The three equations (4.3)-( 4.5) all have the generic form 1 + z / 0 where the kernel Ts(u v) is given by (v)Ts(u,v)dv (4.11) Ts(u v) e-sD(v) gu(V u). (4.12) Thus, the Fredholm-type integral equation (4.11) is a special case of the integral equation (2.1) and the kernel Ts(u v) in (4.12) is a special case of the kernel g s(x,y of (2.2).In conformance with the notational conventions of Section 2, the associated operator s, s > 0, on L2([0, oc) The iterated kernels T s, n(u v) take the form Ts, n(u,v Since the theory outline in Section 2 for Fredholm-type integral equations holds for K s special case, we may apply the solutions as well to the special case of TES + processes.
For TES + processes which ale approximately renewal, one would expect the corresponding Neumann expansions (2.7) for Gd, M d and L d to provide good approximations.This aspect of the integral equations may be used to construct analytical approximations for the solutions.

A Matric Form of Solutions for TES + Traffic Processes
The Fourier series representation of gu(w) in (4.1) may be used to construct a matric solution for the integral equation (4.11).Following Tricomi [14] to this end, substitute (4.1) for gu(w)in (4.11), yielding 1 v(i2ru U)dv.J 0 Next, we write 1 o(u) h(u) + z E fv(i2ru) e -i2uu (v)e sD(v) + i2UVdv 0 1 h(u) + z E fv( i2ru)e i2ruu j (v)e sD(v) i2rUVdv (5.1) 0 where Parseval's equality (Hardy and Rogosinski [6])justifies the interchange of integration and summation in the first equality, and complex conjugation justifies the second equality.Denoting for integer #, c.(s) / (v)e-sD(v)-i2rpVdv (5.2) 0 to be the Fourier coefficients of o(v)e-sD(v), we can rewrite (5.1) as (u) h(u) + z E 7V( i2ru)cu(s) ei2ruu" Equation (5.3) is a solution for (u)in terms of (...,c_ 1(8),C0(8),C1(8),...), which is an unknown vector with components cu(s ).In order to determine this unknown vector, substitute (5.3) where Parseval's formula again justifies the interchange of integration and summation.be the vector with components cu(s), and let h(s) be the vector with 1 hu(s f h(v)e-sD(v)-i2'rUVdv. Finally, let M(s) be the matrix with Mu, r'(s) o_ ~fv( i2rv) fl e sD(v) 4" i2t(r, U)Vdv" Equation (5.4) now takes the form 0 ct,(s ht,(s + z E Mu, u(s)ct,(s), Let c(s) components components which can be written in matric form as The solution of the matric equation above is (5.5)where I is the (infinite-dimensional) identity matrix.The solution (5.5) may be effectuated, in practice, by using an n n submatrix extracted from M(s) by symmetrical truncation.This is the same as symmetrically truncating the Fourier series (4.1) for gu(w).
6. Example: TES + Processes with Exponential Innovations When the transform fv(s) of the innovation density is rational then, in principle, it is possible to obtain the exact solution for the integral equation (4.11).In that case, gu(w) has the form of an exponential polynomial, so that a differential operator may be found to eliminate the integration; this will replace the integral equation by a differential equation.Difficulties still remain, however, since the differential equation will have variable coefficients.
The solutions for x + (s Is) and x + (zslu) are largely similar.However, for/x + (s In), we have a straightforward solution in terms of LX + (z, s Is) as given by (3.7).

The Pea&edness of TES + Traffic Processes
The peakedness functional provides a partial characterization of the burstiness of an ergodic traffic stream by gauging its effect when offered to an infinite server group.In practice, peakedness is typically used to approximate a solution for blocking and delay statistics in finite- server queues.
Let {Xn} be a stationary sequence of interarrival times with a general probability law, arrival rate A= 1/E[Xn] < ee, and expectation function M(t) (recall Section 1).Assume that the corresponding traffic stream is offered to an infinite server group consisting of independent servers with common service time distribution F. Let B(t) be the number of busy servers at time t, and assume that its limiting statistics exist.The peakedness functional, z x, associated with the traffic process {Xn} given by zx[F]_ limV_r,B,:.j,,, (.']   (7.1) tT EZ[B(t)] maps the space of all service time distributions to non-negative numbers.
Let (F} be a parametric family of service time distributions, indexed by the service rate #-1/fxdF,(x).It is convenient to standardize the F,(x) to unit rate by defining 0 Fl(X F,(x/p), and to replace the peakedness functional zx[F,] from (7.1) by the correspond- ing peakedness function ZX, FI (IA) zx[F.].
(7.2) Interestingly, if {G} is any other parametric family of service time distributions, then the corres- ponding peakedness, functions Zx, F1 (#) and zX, GI(# contain equivalent information on the traf- fic process {Xn} In the sense that the two are connected by a known transformation (see Jagerman [7]).In particular, for an exponential service time distribution, Fl(X 1-e-x, the corresponding peakedness function (7.2) is dented by zx, exp(#) and has the representation (see Eckberg [3]), Zx, exp(#) 1 -+ I/I(#). (7.3)Consider the auxiliary peakedness function Zx, exp(#,x 1 --+ (7.4)   from which (7.3) can be obtained by integration with respect to the interarrival time density fx(x).Substituting the integral equation (3.4) into (7.4)yields the integral equation (7.5) For the remainder of this section, we specialize the discussion to TES + processes {Xn + }.In this case, the peakedness value, Ze+xp(O), assumes a particularly simple form.However, before stating the main result, we shall need the following simple facts which will serve to simplify the proof.
(7.18) 0 Next, substitute (7.9) into the right-hand side of (7.18) and expand the resulting equation in powers of # around O, which yields Simplifying the above equation with the aid of (7.6) and (7.The value of ze+xp(O)is finally obtained by substituting (7.17)into (7.=(AX m2+ +(AX+ u=El-v(i2ru) 5(i2 )   (7.20)   where the second equality is justified by the fact that all quantities in (7.20) are real except for the terms in the infinite sums.
The first equality of (7.10) follows from (7.20), noting that oo 1 fv(i2r,) [n(i2r')[2 where the second equality is justified by the absolute convergence of the series, and the third by appeal to (1.6).Finally, the second equality of (7.10) follows from the identity Px + () rSx + (0)-1 2 r=l in view of (1.2).

E!
Theorem 3 reveals an interesting connection between the peakedness Ze+xp(O) and the index of dispersion notion of traffic burstiness or variability [2, 4, 5].Let {Xn} be a stationary sequence of interarrival times.The index of dispersion for intervals (IDI) is the sequence {In}, defined by Var(X
into the integral equation (4.11) yields 1 note depend on u, set in particular, u-0 in the equation above, ,V(u)) 1 ) +X / D(v)gu + (v u)dv 0 This yields the integral 1 + f ZLp(O D(v))gu + (v u)dv, o (7.12) let uf e Zexp(O,n(v))dv and conclude that Ze+xp(O,D(v)) has the Fourier series representation o 7) and dividing by # now gives the following condition equation for ze+xp(O,D(u