Convergence results for ant routing

Transcription

1 Convergence resuts for ant routing Joon-Hyuk Yoo, Richard J. La and Armand M. Makowski Abstract We study the convergence property of a famiy of distributed routing agorithms based on the ant coony metaphor, namey the uniform and reguar ant routing agorithms discussed by Subramanian et a. [8]. For a simpe two-node network, we show that the probabiistic routing tabes converge in distribution resp. in the a.s. sense for the uniform resp. reguar case. To the best of the authors knowedge, the resuts given here appear to be the first forma convergence resuts for ant routing agorithms. Athough they hod ony for a very imited cass of networks, their anaysis aready provides some usefu essons for extending the resuts to more compicated networks. We aso discuss impementation issues that naturay arise from the convergence anaysis. I. INTRODUCTION In the past decade, severa authors have used the ant coony metaphor to design distributed adaptive routing agorithms in both datagram networks [5] and teephone networks [7]; a good survey of these efforts is given in Chapter 2 of [1]. More recenty, these ideas have been proposed in the context of mobie ad-hoc networks MANETs, e.g., [6]. A common feature shared by these agorithms is their attempt to reproduce the coective behavior of a swarm of insects to sove a reativey compex probem in a distributed manner, typicay using ony oca information. This new paradigm has the potentia to deiver a new way of designing robust, scaabe, adaptive, and distributed agorithms for network resource management. Athough many ant coony optimization ACO-based agorithms [1] have been proposed and studied in the past, itte is known about their convergence properties. There are severa difficuties in estabishing such convergence resuts. First, it is not aways easy to identify a suitabe mathematica framework for anayzing these agorithms at some eve of generaity, thereby perhaps expaining the ack of a genera theory of ant routing agorithms. Secondy, even when such a framework is avaiabe, many state variabes which are required to capture the dynamics of the agorithms, are couped in their evoution. Often this couping severey imits the usefuness of traditiona techniques of convergence anaysis. In this paper we are concerned specificay with the convergence properties of the cass of ant routing agorithms discussed by Subramanian et a. [8]. These agorithms are randomized agorithms that impement a form of backward earning in response to short contro messages caed ants. A. Ant routing Consider the situation where hosts are provided connectivity through a network of R routers. Two routers are said to be neighbors if there exists a bidirectiona point-to-point ink between them, and et N r denote the set of routers which are neighbors of router r r 1,..., R. Router r maintains a probabiistic routing tabe with a separate vector entry d, i, p i, i N r for each host destination d. For each neighboring router i in N r, we understand p i as the probabiity with which router r uses ink r, i when forwarding a data packet destined for d. There is a cost c ri associated with the use of ink r, i; this cost is assumed symmetric i.e., c ri c ir and is known to router r. An ant is a contro message of the form d, s c where d and s are distinct hosts and c is some numerica vaue to be updated in due course. We refer to hosts d and s as the destination and source, respectivey, and regard c as an estimate of the cost-togo for reaching host d. Periodicay, host d generates an ant d, s c which is destined for some randomy seected host s d with c initiay set to zero. The ant is forwarded to the source host s over the network of interconnected routers and on the way updates the routing tabes at intermediary routers in a way to specified shorty. When ant d, s c arrives at the intermediary router r coming from router i through ink i, r, the cost estimate c for reaching d from router i is incremented by the cost of the reverse ink r, i with c c + c ri, 1 and this new vaue of c thus provides an estimate of the costto-go to reach d from router r. Next, the vector entry in the routing tabe for destination d is updated according to p i p i + 1 +, p j p j 1 + j N r \ {i} 2 where fc and fc is a decreasing function of the just incremented vaue of c. Consequenty, the probabiity of using the reverse ink over which the ant arrived at router r has been increased reative to that of other inks, whie the probabiity of using the other inks is discounted. Upon competing these various updates, router r forwards the updated ant d, s c to one of its neighboring router i in N r. The ant d, s c eventuay reaches its destination s with c now giving the end-to-end cost vaue of sending a message from s to d, and is destroyed. In fact, the fina vaue of c is simpy the cost of traversing the network from s to d aong the reverse path foowed by the ant. B. Convergence resuts The manner in which ant forwarding is carried out distinguishes the various types of ant routing agorithms studied by Subramanian et a. [8]. This reference deas mosty with a simpe two-router network, announces some convergence resuts but provides no forma proofs. To the best of the

2 authors knowedge, this is the extent of the current state of knowedge concerning the mathematica convergence of these ant agorithms. In this paper we take some steps towards remedying this state of affairs. Beow we review the convergence statements of [8] and [9], and present our resuts. Proofs are outined in the body of the paper, with additiona detais avaiabe in [10]. 1. Uniform ant agorithms are designed with muti-path routing in mind, and require that ants arriving, say at router r, be forwarded to the next neighboring router with equa probabiity L 1 r where L r is the number of routers in N r. 1 As a resut, uniform ants wi utiize each and every path between a source and a destination with a positive probabiity. It is caimed [8, Prop. 2] [9] that the routing tabes converge in an unspecified mode of convergence to constant vaues. Here, for the two-node network, we show that i convergence takes pace in distribution, and ii not to constants. This ast fact has impications for the impementation of such ant agorithms [Section V]. The key observation behind the convergence resut for uniform ants is the identification of the iterates of the uniform ant agorithm as the output of a very simpe coection of iterated random functions. A arge iterature is avaiabe on the convergence of these iterative schemes, and the survey in [4] and references therein provides a nice introduction to this topic. We expoit the very simpe structure of the underying coection of random functions to give a simpe and sef-contained proof. 2. On the other hand, reguar ant agorithms attempt to find the east cost or shortest paths. This is achieved by forwarding the ants according to the probabiistic routing tabes used to impement the routing of data packets. For the two-node network the agorithm is caimed [8, Prop. 1] [9] to converge presumaby in the a.s. mode of convergence to constant vaues, but no proof is provided in that reference. Here, in the case of synchronous instantaneous updates, we show by martingae techniques that reguar ant agorithm wi indeed ead asymptoticay to east path discovery, in the twonode network, as desired! True, whie the resuts given here are perhaps the first forma convergence resuts for ant routing, they hod ony for a very imited cass of networks, i.e., a two-node network. Yet, the underying anayses given beow aready provide some usefu essons for extending the resuts to more compicated networks. This is discussed at the end of Sections VI and VII, respectivey. The paper is organized as foows: Ant agorithms are formay described in Section II for a simpe two-ink network. Generaized uniform ant and reguar ant agorithms are introduced in Sections III and IV, respectivey. The main convergence resuts are given in Theorems 1 and 2 which are estabished in Sections VI and VII, respectivey. Some impementation issues are pointed out in Section V. 1 More generay, L r is the number of inks going out of r in the case of mutipe inks. A word on the notation: For any integer L, the th component of any eement x in IR L is denoted by x, 1,..., L, so that x x 1,..., x L. A simiar convention is used for IR L -vaued random variabes rvs. II. THE TWO-NODE MODEL The network is composed of two routers or nodes, thereafter abeed node i 0 and node i 1, connected by a set of L parae bidirectiona inks. 2 A hosts are attached to either node i 0 or node i 1. Each ink 1,, L, has a transmission cost of c > 0 in either direction. Without oss of generaity we assume the inks to be abeed in order of increasing cost with c 1 c 2 c L. 3 Pick node i i 0, 1. Destination hosts attached to node i generate ants at times {t i n, n 1, 2,...} with t i n < t i n+1 for each n 1, 2,.... Forwarding the n th ant to node 1 i requires that one of the L inks from node i to node 1 i, say 1 i n, be seected. The n th ant is then sent over ink 1 i n, and arrives at node 1 i, say at time a i n with t i n a i n. For simpicity of exposition we assume the non-overtaking condition a i n < a i n+1, n 1, 2,... 4 and for convenience we take a i 0 arbitrary such that a i 0 a i 1. Node i maintains a probabiistic forwarding tabe p i n : p i 1n,, p i Ln 5 with 0 p i n 1 1,..., L and L 1 pi n 1. The entry p i n is interpreted as the probabiity that during the interva [a 1 i n, a 1 i n+1, node i forwards a data packet from node i to node 1 i over ink. Its precise meaning wi be carified in due time. When, at time a i n+1, node 1 i receives the n + 1 st ant, say over ink i 1 n + 1, it immediatey updates its probabiistic forwarding tabe according to and k n + 1 n + 1 p1 i n + C 1 i 1 + C 1 i 6 k n, k, k 1,..., L C 1 i for constants C 1 i > 0. The seection of these constants is discussed ater. The cost update 1 is superfuous here due to the simpified structure of this two-node network. To competey specify ant agorithms we need to provide the rue by which the inks { 0 n, 1 n, n 1, 2,...} are seected. This wi be done in Sections III and IV for uniform and reguar ant routing, respectivey. 2 These inks can be aso thought of as disjoint paths.

3 Before doing so, we introduce some notation: For each i 0, 1, and for each ink 1,, L, define the mapping φ i : [0, 1] L [0, 1] L by { p +C i φ i,kp : 1+C i k p k 1+C i k, p [0, 1]L 8 for the constants C i > 0 appearing in 6 and 7. These updating rues can now be written more compacty as p i n + 1 φ i p i n if i n + 1, n 0, 1,... 9 with p i 0 denoting the forwarding probabiity vector initiay stored at node i i.e., at time a i 0. III. GENERALIZED UNIFORM ANTS The cass of ant agorithms we now introduce is somewhat more genera than the one discussed in [8]. We specify how ants are propagated by assuming that the {1,..., L}-vaued rvs { 0 n, 1 n, n 1, 2,...} are mutuay independent rvs which are taken to be independent of the initia conditions p 0 0 and p 1 0. Moreover, for each i 0, 1, the rvs { i n, n 1, 2,...} are taken to be i.i.d. rvs distributed according to some probabiity mass function pmf v i v1 i,..., vi L on {1,..., L}. Under these assumptions, it is pain from 6 and 7 that the probabiistic forwarding tabes of the nodes are updated independenty of each other, whence the evoutions of the tabes {p 0 n, n 0, 1,...} and {p 1 n, n 0, 1,...} are decouped. Their ong-term behavior is summarized in the next resut where n denotes convergence in distribution with n going to infinity. Theorem 1: Under the foregoing assumptions we have: i For each i 0, 1, the imit p i im n φ i i1 φi i2... φi in p 10 exists for each p in [0, 1] L, and is independent of p; ii Moreover, there exists a pair of independent [0, 1] L - vaued rvs p 0 and p 1 distributed ike p 0 and p 1, respectivey, such that p 0 n, p 1 n n p 0, p 1 11 regardess of the initia condition p 0 0, p 1 0. The proof of Theorem 1 is given in Section VI. We emphasize that the resut hods for a cass of agorithms which is somewhat more genera than the one introduced in [8]: Indeed, the positive constants entering 6-7 are arbitrary and need not be constrained to C i fc : i 0, 1, 1,..., L 12 for some stricty decreasing function f : IR 0,. Next, when we take the pmfs v 0 and v 1 to be the uniform pmf on {1,..., L}, we recover the case discussed in [8], hence the name uniform ant agorithm. Finay, the assumptions enforced on the reception times {a i n, n 1, 2,...} i 1, 0 can be weakened consideraby: These times coud in principe be random and Theorem 1 woud sti hod. The non-overtaking assumption 4 can aso be dropped if we now interpret 1 i n as the identity of the ink from node i to node 1 i which was traversed by the n th ant received at node 1 i at the possiby random time a i n. Then, under the aforementioned independence assumptions, Theorem 1 wi sti hod. IV. REGULAR ANTS WITH SYNCHRONOUS UPDATES With reguar ant agorithms, the ants are forwarded according to the routing tabe at the router, and the update processes at the two nodes are now couped. This renders the anaysis of reguar ant agorithms more deicate than that of uniform ant agorithms. To make progress, we need assumptions on the reative timing of the various events. Here we consider the idea situation where both nodes generate the ants in a synchronized manner and the updates take pace simutaneousy and instantaneousy. Formay, we require t 0 n a 0 n t 1 n a 1 n, n 1, 2, For each n 0, 1,..., we introduce the σ-fied F n generated by the rvs {p 0 0, p 1 0, p 0 k, p 1 k, 0 k, 1 k, k 1, 2,..., n}. The rvs 0 n + 1 and 1 n + 1 are then specified to be conditionay independent given F n with P [ i n + 1 F n ] n 14 for each 1,..., L and i 0, 1. The prescription 14 encodes the fact that ants are routed through the network according to the very routing tabes they hep update. The constants entering 6-7 wi be assumed of the form 12. The entry for ink at node 1 i is increased proportionay to fc, which is a stricty decreasing function of c, whie those of the other inks are discounted by 1+fc 1. Combining 14 with 6-7, we can write the evoution of the routing tabes in the more suggestive yet somewhat soppy manner: Fix i 0, 1. For each 1,..., L, we have p i n + 1 p i n+fc 1+fc p i n 1+fc j w.p. n w.p. j n, j 15 for a n 0, 1,.... This formuation ceary shows the couping of the entries p 0 n and p 1 n in the routing tabes. Writing pn p 0 n, p 1 n, n 0, 1,... we readiy see from 9 and 14 that the sequence of successive routing tabes {pn, n 0, 1,...} form a time-homogeneous Markov chain on the non-countabe state space [0, 1] L [0, 1] L. A natura question is whether the Markov chain converges to the desired operating point, i.e., 1, 0,, 0, 1, 0,, 0, which corresponds to the eastcost path. This is the content of the next resut.

4 Theorem 2: Assume f to be stricty decreasing and that c 1 < c for a 2,, L. For each i 0, 1, the routing tabes {p i n, n 0, 1,...} converge a.s. with im n pi n 1, 0,, 0 a.s. provided the initia conditions are seected such that p p 1 10 > Under the enforced assumptions, there is ony one ink with minima cost and it is ink 1. V. IMPLEMENTATION ISSUES We briefy discuss some of the impementation issues associated with the ant routing agorithms discussed earier. For simpicity, we do so for the case first described in [8] with L 2 where the constants in the probabiity updates 6 and 7 are seected according to 12. A. Uniform ant routing Take the pmfs v 0 and v 1 to be the uniform pmf on {1, 2}. It is easy to see that the probabiistic routing tabes evove according to p i n + 1 [ ] p i 1 n , pi 2 n 1+ 1 [ ] p i 1 n 1+ 2, pi 2 n w.p. 1 2 w.p where the constants 1, 2 are given by 12. Seecting the proper vaues of fc 1, 2 is crucia here for good performance. Indeed, if the parameters are seected such that 1 < 1, then the iterates produced by 17 exhibit an osciatory behavior with successive vaues possiby bouncing around between the non-overapping intervas 0, and , 1. This behavior eads to undesirabe osciations in the routing tabes, and is cear evidence that the convergence of Theorem 1 cannot be in the a.s. sense. In fact, convergence takes pace in distribution not a.s. to a imiting random variabe whose distribution has a nonconnected support on the interva [0, 1]. Subramanian et a. [8, Prop. 2] caim the convergence im n pi n L i, i 0, 1 for some constants L 0 and L 1, without further indication of the mode of convergence used for this convergence statement which invove rvs. As the previous discussion impies, this cannot be correct. B. Reguar ant routing By Theorem 2 the reguar ant routing agorithm converges to the desired operating point, namey the east-cost ink. However, if the agorithm is impemented on a machine with finite precision, as is aways the case with computer impementations, there is a positive probabiity that the agorithm wi converge to the path with a arger cost. Indeed, suppose that the product space [0, 1] [0, 1] is approximated by a two-dimensiona grid with a finite number of points, say k K, K with k, 0,..., K. We can then approximate p 0 1n, p 1 1n using the cosest point in the grid, with resuting output p 0 1n, p 1 1n evoving according to a timehomogeneous finite-state Markov chain described by projecting the right hand sides of 15 onto the grid. Regardess of the vaue of K, the corner point 0, 0 is now an absorbing state, and the probabiity of reaching it is stricty positive, so that the probabiity of converging to the shortest path is stricty smaer than one. VI. A PROOF OF THEOREM 1 As remarked earier, because the forwarding probabiity tabes evove independenty of each other, we need ony estabish the convergence p i n n p i for each i 0, 1. Thus fix i 0, 1 and pick n 0, 1,.... Iterating yieds the reation p i n φ i in+1 φi... in φi i2 φi i1 p i 0. The key observation is the stochastic equivaence 3 p i n st φ i i1 φi... i2 φi in φi in+1 p 0 0 which hods by virtue of the i.i.d. assumption on the sequence { i n, n 1, 2,...} and its independence from p i 0. The convergence p i n n p i wi foow if we can estabish the pointwise convergence im φ i n i1 φi... i2 φi in p 0 0 p i 20 with p i defined through the pointwise convergence 10. To estabish 10 with p i independent of the initia condition, we proceed as foows: Equip IR L with the norm defined by x : L 1 x for any vector x in IR L. This norm is equivaent to the usua Eucidean norm, but easier to use here. 3 Two IR vaued rvs X and Y are said to be equa in aw if they have the same distribution, a fact we denote by X st Y.

5 Fix 1,..., L. For arbitrary x and y in [0, 1] L, we get with φ i x φ i y L φ i,k x φ i,ky k1 x k y k 1 + C i + x + C i y + C i 1 + C i k Therefore, x y 1 + C i K i x y K i : max 1 + C i 1 < 1. 1,...,L max 1,...,L φi x φ i y K i x y. 21 Using this ast inequaity, we check by induction on n 1, 2,... that φ i... i1 φi in 1 φi in p φ i... i1 φi in 1 φi in q K n i p q 22 for arbitrary p and q in [0, 1] L. Furthermore, for each m 1, 2,..., we get φ i i1 φi... i2 φi in+m p φ i i1 φi... i2 φi in p Ki n φ i in+1 φi... in+2 φi in+m p p K n i L uniformy in m. 23 By the fact K i < 1, it foows for each p that the sequence { φ i i1 φi... i1 φi in p, n 1, 2,...} forms a Cauchy sequence, hence is convergent. Eqn. 22 shows that the imit of this convergent sequence is independent of p. This estabishes 10 and the proof is now compete. A carefu examination of the arguments given in the proof of Theorem 1 points to the possibiity of estabishing the convergence of uniform ants for more genera network topoogies. Indeed, this is due to the fact that in uniform ant routing, the evoution of the routing tabes are decouped. It is aso cear that weaker assumptions can be imposed on the ink seection rvs, e.g., the proof given above goes through if each of the sequences { i n, n 1, 2,...} i 1, 0 is assumed stationary and reversibe. VII. A PROOF OF THEOREM 2 We need to show that each of the sequences {p 0 n, n 0, 1,...} and {p 1 n, n 0, 1,...} converges a.s. to the vector 1, 0,, 0. For each i 0, 1, this is equivaent to showing that im n pi 1n 1 a.s. 24 since L 1 pi n 1 for a n 0, 1,.... To estabish 24, we introduce the rvs {Zn, n 0, 1,...} given by Zn p 0 1n + p 1 1n, n 0, 1, and note that the convergence 24 for both i 0 and i 1 wi foow if we can show that im Zn 2 a.s. 26 n To do so, we first note that fc 1 > fc for each 2,, L under the enforced assumptions on f and on the ink costs. Fix i 0, 1 and n 0, 1,.... Combining 9 and 14, we find E [ p i kn + 1 F n ] L 1 nφ i,kp i n a.s. 27 for each k 1,..., L. Next, reporting 27 into the definition 25, and using the constants 12 in 8, we readiy concude that E [Zn + 1 F n ] Zn 28 L fc 1 fc p fc fc 1 np 1 n + p 1 1np 0 n 2 Consequenty, E [Zn + 1 F n ] Zn a.s. 29 and the rvs {Zn, n 0, 1, 2,...} form a bounded F n - submartingae with 0 E [Zn] 2, for a n 0, 1,.... By the Martingae Convergence Theorem [3, Thm. 9.4., p. 334], the submartingae {Zn, n 0, 1, 2,...} converges a.s. to some rv Z, and we must have 0 Z 2. We note from 28 that the a.s. inequaity in 29 becomes an equaity if and ony if p 0 1np 1 n p 1 1np 0 n 0, or equivaenty, if and ony if 2,..., L p 0 1n1 p 1 1n p 1 1n1 p 0 1n 0. In other words, the equaities hod if and ony if either p 0 1 n p 1 1n 0 or p 0 1n p 1 1n 1. It is now a simpe matter to check the foowing say by direct inspection of the dynamics 15 for the routing tabes: If p 0 10 p , then Zn 0 for a n 0, 1,... and Z 0, whie if p p , then Zn 2 for a n 0, 1,... and Z 2. On the other hand, if the initia conditions are seected so that 0 < Z0 < 2, then it is easy to see by induction that 0 < Zn < 2 a.s. for a n 1, 2,....

6 To concude the proof we need to show that Z 2 a.s. here as we. Proceeding by contradiction, we assume P [Z 2] < 1, in which case there exists some ε > 0 such that δ : P [ε < Z < 2 ε] > 0 30 since Z > 0 a.s. By the a.s. convergence of {Zn, n 0, 1,...} to Z we concude that [ ε P 2 2] < Zn < 2 ε δ 2, n n 31 with n determined by ε and δ. Write K : fc 1 fc fc 1 2. Taking advantage of 28, for each n 0, 1,..., we get E [Zn + 1] E [Zn] E [E [Zn + 1 F n ] Zn] L fc 1 fc E [ ] 1 + fc fc 2 [ L K E p 0 1 np 1 n + p 1 1np 0 n ] 2 K E [ p 0 1n1 p 1 1n + p 1 1n1 p 0 1n ] [ [ ε K E 1 2 2]] < Zn < 2 ε Next, with K ε : {x, y [0, 1] 2 : ε 2 < x + y < 2 ε 2 }, we note that Iε : inf{x + y 2xy : x, y K ε } > 0. Therefore, whenever n n, we find that E [Zn + 1] E [Zn] [ ε KIε P 2 2] < Zn < 2 ε The anaysis above aready reveas some of the difficuties inherent in estabishing convergence resuts for reguar ants. Our abiity to estabish extensions of Theorem 2 to more compex topoogies is ikey to depend on a carefu timing of updating events. In that vein, we note that a version of Theorem 2 can aso be obtained through identica arguments when the updates are triggered in an asynchronous manner according to mutuay independent Poisson processes; we omit the detais in the interest of brevity. REFERENCES [1] E. Bonabeau, M. Dorigo and G. Therauaz, Swarm Inteigence: From Natura to Artificia Systems, Santa Fe Institute Studies in the Sciences of Compexity, Oxford University Press, New York NY, [2] P. Biingsey, Convergence of Probabiity Measures, John Wiey & Sons, New York NY, [3] K.L. Chung, A Course in Probabiity Theory, Second. Edition, Academic Press, New York NY, [4] P. Diaconis and D. Freedman, Iterated random functions, SIAM Review , pp [5] G. DiCaro and M. Dorigo, Two ant coony agorithms for besteffort routing in datagram networks, in Proceedings of PDCS 1998: Internationa Conference on Parae and Distributed Computing and Systems, Anaheim CA. [6] M. Günes, U. Sorges and I. Bouazizi, ARA - The ant-coony based routing agorithm for MANETs, in Proceedings of ICPPW 2002: Internationa Conference on Parae Processing Workshops, August 2002, Vancouver, B.C. Canada. [7] R. Schoonderwoerd, O. Hoand, J. Bruten and L. Rothkrantz, Antbased oad baancing in teecommunications networks, Adaptive Behavior , pp [8] D. Subramanian, P. Drusche and J. Chen, Ants and reinforcement earning: A case study in routing in dynamic networks, in Proceedings of IJCAI 1997: The Internationa Joint Conference on Artificia Inteigence, Nagoya Japan, August [9] D. Subramanian, P. Drusche and J. Chen, Ants and reinforcement earning: A case study in routing in dynamic networks, Technica Report TR96-259, Department of Computer Science, Rice University, Houston TX, Updated Juy [10] J.-H. Yoo, R. J. La and A. M. Makowski, Convergence of ant routing agorithms Resuts for a simpe parae network and perspectives, Technica Report CSHCN , Institute for Systems Research, University of Maryand, Coege Park MD, KIε δ 2, As a resut, im inf n E [Zn + 1] E [Zn] > 0, in contradiction with the convergence im n E [Zn] E [Z] that foows from the a.s. convergence of {Zn, n 0, 1,...} and the Bounded Convergence Theorem. The proof of 26 is now compete. The convergence resut of Theorem 2 cannot be obtained by traditiona methods from the theory of Markov chains on genera state spaces when appied to the time-homogeneous Markov chain {pn, n 0, 1,...} on the non-countabe state space [0, 1] L [0, 1] L. Indeed, the genera theory identifies conditions akin to irreducibiity which ensure the weak convergence 11. However, such irreducibiity conditions typicay impy a non-degenerate stationary distribution in the imit, at variance with the situation obtained here 4! 4 That is, if we eiminate the initia condition Z0 0.