FLUID QUEUES DRIVEN BY A BIRTH AND DEATH PROCESS WITH ALTERNATING FLOW RATES

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1 FLUID QUEUES DRIVEN BY A BIRTH AND DEATH PROCESS WITH ALTERNATING FLOW RATES P R PARTHASARATHY K V VIJAYASHREE AND R B LENIN Received 18 January 00 and in revised form 15 July 00 Fluid queue driven by a birth and death process BDP with only one negative effective input rate has been considered in the literature As an alternative here we consider a fluid queue in which the input is characterized by a BDP with alternating positive and negative flow rates on a finite state space Also the BDP has two alternating arrival rates and two alternating service rates Explicit expression for the distribution function of the buffer occupancy is obtained The case where the state space is infinite is also discussed Graphs are presented to visualize the buffer content distribution 1 Introduction Recent measurements have revealed that in high-speed telecommunication networks like the ATM-based broadband ISDN traffic conditions exhibit long-range dependence and burstiness over a wide range of time scales Fluid models characterize such a traffic as a continuous stream with a parameterized flow rate Whitt [6] establishes heavy-traffic stochastic process limits for fluid queue models with multiple on-off sources A fluid model that is typically used to model such a traffic isamarkov Modulated Fluid Model wherein the current state of the underlying Markov process determines the flow rate Fluid models driven by finite state Markov processes that modulate the input rate in the fluid buffer have been analyzed by many authors Lenin and Parthasarathy [] provide closed-form expressions for the eigenvalues and eigenvectors for fluid queues driven by an M/M/1/N queue The case where the state space is infinite has been analyzed by van Doorn and Scheinhardt [5] for a birth and death process BDP In most studies dealing with Markov modulated fluid queues a single negative effective flow rate is assumed Here we consider a more general setting of a fluid queue driven by a BDP on a finite state space in which the flow rates are alternatively positive and negative Our aim is to obtain the stationary distribution function of the buffer occupancy for this fluid model which is modulated by a BDP with two alternating arrival rates and two alternating service rates This modulating Markov process can be visualized as a simple case of a two-state Markov Modulated Poisson Process which is characterized Copyright 004 Hindawi Publishing Corporation Mathematical Problems in Engineering 004: Mathematics Subject Classification: 60K5 60J80 68M0 URL:

2 470 Fluid queues [ ] by a Markov process with an infinitesimal generator Q λ1 λ = 1 λ λ and a diagonal matrix [ ] Λ λ1 0 = 0 λ of arrival probabilities where λ1 and λ denote the rates of arrival when the traffic is bursty and slow respectively The case where the state space of the BDP is infinite is also discussed Some interesting identities of tridiagonal determinants are used for the finite state space and continued fraction methodology is employed for the infinite state space Graphs are presented to visualize the buffer content distribution The model under consideration finds a wide range of application in modelling a communication switch as a fluid stochastic petri net in which two streams of traffic arrive Horton et al [1] One stream is bursty with a high flow rate when the server is busy and the other stream is slow with a low flow rate when the workload of the server is less We will designate the fluid commodity accumulating in the infinite capacity buffer as credit It may be helpful to think of credit as the energy which the server gathers during lean traffic period and consumes when the trafficisbursty Model description We consider an infinite capacity buffer which receives and releases fluid flows modulated by a BDP evolving in the background We denote the background birth-death process by :={Xt t 0} taking values in the state space wherext denotes the state of the process at time t Letλ n and µ n denote the mean arrival and service rates respectively when there are n units in the system The flow rates of the fluid into and out of an infinite capacity buffer are determined by the actual state of the background process Let r j denote the flow rate of the fluid when the background process is in state j The rate of change of content of the buffer Ctwhen Xt = j is given by dct dt r j if Ct > 0 = 0 ifct = 0 r j < 0 1 Clearly the two-dimensional process {XtCt t 0} constitutes a Markov process which possesses a unique stationary distribution under a suitable stability condition The stationary state probabilities p i i ofthebdpcanberepresentedas p i = π i j π j i where π i = λ 0 λ i1 / µ µ i i = 13andπ 0 = 1 are called the potential coefficients In order that a limit distribution for Ct exists as t the stationary net input rate should be negative that is π i r i < 0 3 i=0

3 P R Parthasarathy et al 471 Letting F j tu P Xt = j Ct u j tu 0 4 the Kolmogorov forward equations for the Markov process {XtCt} are given by F 0 tu t F j tu t =r 0 F 0 tu u =r j F j tu u λ 0 F 0 tu+ F 1 tu λ j + µ j Fj tu + λ j1 F j1 tu+µ j+1 F j+1 tu j \{0} tu 0 5 See van Doorn and Scheinhardt [5] When the process is in equilibrium F j tu/ t 0 and let lim t F j tu F j u 3 Finite state space This section deals with a fluid queue modulated by a finite BDP with state space = {01N} The system of equations governing the two-dimensional process {Xt Ct t 0} in equilibrium is r j df j u du r 0 df 0 u du =λ 0 F 0 u+ F 1 u = λ j1 F j1 u λ j + µ j Fj u+µ j+1 F j+1 u for j \{0} u 0 31 In matrix notation 31canbewrittenas dfu du = R1 Q T Fu u 0 3 where Fu = [F 0 uf 1 uf N u] T R = diagr 0 r N and λ 0 λ 0 + Q = µ N1 λ N1 + µ N1 λ N1 µ N µ N N+1 N+1 33

4 47 Fluid queues Hence λ 0 r 0 r 0 λ 0 λ1 + R 1 Q T = µ λ N r N1 λn1 + µ N1 r N1 λ N1 r N µ N r N1 µ N r N N+1 N+1 34 Mitra [3] has shown that R 1 Q T has exactly N + negative eigenvalues N 1 positive eigenvalues and one zero-eigenvalue where N + is the cardinality of the set + := { j : r j > 0 } 35 and N is that of := { j : r j < 0 } 36 Let ξ j j = 01N be the eigenvalues of the matrix R 1 Q T such that ξ j < 0 j = 01N + 1 ξ N + = 0 ξ j > 0 j = N + +1N 37 Since the content of the buffer increases when the net input rate of fluid flow into the buffer is positive it follows that F j u must satisfy the boundary condition F j 0 = 0 for j + 38 Also we have lim F ju = p j for j 39 u where the p j s are the stationary state probabilities of the background BDP The solution to the matrix equation 3is given by N +1 F j u = p j + η lj e ξlu j 310 l=0

5 P R Parthasarathy et al 473 where η lj = k l B j ξl c j0 c j0 = µ µ j r 0 r j1 311 The constants k l are obtained by solving N+1 B j ξl p j + k l = 0 for j + 31 c j0 l=0 The polynomials B j s are defined recursively as B 0 s = 1 B 1 s = s + λ 0 r 0 B j s = s + λ j1 + µ j1 B j1 s λ jµ j1 B j s j = 34N r j1 r j r j1 B N+1 s = s + µ N B N s λ N1µ N B N1 s r N r N1 r N 313 Also B N+1 s = detsi R 1 Q T andb j s is the determinant obtained by considering the first j rows and columns of B N+1 s More specifically we consider a fluid queue model with effective input rates j < 0andj+1 > 0forj = 01N 1/ Under this assumption the system of equations involved in the determination of the constant k l is given in matrix form as B 1 ξ0 B 1 ξ1 c 10 c 10 B 3 ξ0 B 3 ξ1 c 30 c 30 B 5 ξ0 B 5 ξ1 c 50 c 50 B 1 ξ[n/] c 10 B 3 ξ[n/] c 30 B 5 ξ[n/] c 50 [N/] k 0 k 1 k [N/] p 1 p 3 = p 5 [N/] 314 Hence the problem of determining the stationary distribution of the content in the fluid buffer is reduced to that of solving the above matrix equation via Cramer s rule We now give three examples to illustrate the above discussion

6 474 Fluid queues Example 31 N = 1 The steady state probabilities for this two-state Markov process are given by p 0 = /λ 0 + andp 1 = λ 0 /λ 0 + It follows from 3 that the condition r 0 + λ 0 < 0 ensures the stability of the Markov process {XtCt t 0} The matrix R 1 Q T takes the form λ 0 R 1 Q T r = 0 r 0 λ 0 µ with eigenvalues ξ 0 =λ 0 /r 0 + / andξ 1 = 0 The system has one negative root provided λ 0 /r 0 + / > 0 which follows from the stability condition Further η 00 = k 0 η 01 =k 0 r 0 / andfromp 1 + k 0 B 1 ξ 0 /c 10 = 0 we obtain k 0 = /r 0 p 1 Therefore the final solution is given by F 0 u = p 0 + r 0 p 1 e λ0/r0+µ1/r1u F 1 u = p 1 p 1 e λ0/r0+µ1/r1u 316 Observe that P Ct <u = F 0 u+f 1 u = 1 1 r 1 λ 0 e λ0/r0+µ1/r1u 317 r 0 λ 0 + Example 3 N = The steady state probabilities of the modulating Markov process are given by µ p 0 = λ 0 + λ 0 µ + µ λ 0 µ p 1 = λ 0 + λ 0 µ + µ λ p 0 = λ 0 + λ 0 µ + µ 318 It follows from 3 that the stability condition for the Markov process {XtCt t 0} is µ r 0 + λ 0 µ + λ 0 < 0 The matrix R 1 Q T takes the form λ 0 r 0 r 0 R 1 Q T = λ 0 + µ µ 319

7 P R Parthasarathy et al 475 with eigenvalues given by ξ 0 = 1 λ µ r 0 1 λ µ λ0 λ µ + λ 0µ r 0 r 0 r 0 ξ 1 = 0 ξ = 1 λ µ r λ µ λ0 λ µ + λ 0µ r 0 r 0 r 0 30 The constant k 0 is determined as k 0 =p 1 /r 0 B 1 ξ 0 where B s = B 0 s = 1 B 1 s = s + λ 0 r 0 s + λ 0 r 0 s + + λ 0 r 0 31 Therefore the stationary distribution of the buffer content is given by F 0 u = p 0 p 1 r 0 1 B 1 ξ0 e ξ0u F 1 u = p 1 p 1 e ξ0u r F u = p p 1 B ξ0 1 e ξ0u µ B 1 ξ0 3 Observe that P Ct <u = F 0 u+f 1 u+f u = 1 p r 1 B ξ0 e ξ0u r 0 B 1 ξ0 µ B 1 ξ0 33

8 476 Fluid queues In the above discussion we considered two examples in the general case The forthcoming example deals with the model in which the birth and death rates alternate between two constant values with even number of states The case with odd number of states does not lead to explicit expression for eigenvalues Example 33 alternating rates Consider a fluid queue driven by a single server queuing model with state space ={01n 1} whose birth and death rates are given by λ i = µ i = µ for i = 1n1 λ i+1 = λ µ i+1 = for i = 01n 34 with λ 0 = µ n1 = and the effective input rates are i = < 0andi+1 = > 0for i = 01n 1 If ρ = λ / µ < 1 the steady state probabilities are given by k λ1 λ p k = p 0 µ k = 13n1 p k+1 = λ k 1 λ1 λ p 0 µ k = 01n1 35 with p 0 = / + 1 ρ/1 ρ n From 3 it is observed that the condition + < 0 ensures the stability of the Markov process {XtCt t 0} For this specific model the matrix R 1 Q T takes the form R 1 Q T = λ + µ = λ + µ λ + µ λ + µ r n r n 36

9 P R Parthasarathy et al 477 Therefore si R 1 Q T s + s + λ + µ = s + + µ r1 1 s + λ + µ λ r = s λ s + + µ s + r n 1 s + λ + µ λ r 1 s + + n1 37 Using the Identities A A1 anda3 given in the appendix in succession with θ = s + / + / φ = s + λ / + µ / andω = θφ λ / µ / 1weobtain θφ µ r1 λ B n s = s r φ = s φ θφ µ r1 µ 1 ω µ 1 λ ω λ ω λ ω µ 1 λ µ 1 ω θφ λ n1 n

10 478 Fluid queues n1[ = sθ θφ µ r1 r=1 λ λ1 λ µ r1 cos rπ ] n 38 Substituting for θ and φweobtain B n s = s s + + µ 1 n1[ s + + µ 1 s + λ + µ µ r1 r=1 λ λ1 λ µ cos rπ n 39 We observe that B n siszerowhens is the eigenvalue of the tridiagonal matrix R 1 Q T Therefore the eigenvalues of R 1 Q T are given by λ1 ξ 0 = + µ 1 ξ j = 1 λ1 + µ + λ + ξ n = 0 ξ j = 1 λ1 + µ + λ + + λ1 + µ + λ + µ 1 4 λ 4 µ λ1 λ µ +8 cos 1/ jπ j = 1n1 n λ1 + µ + λ + µ 1 4 λ 4 µ λ1 λ µ +8 cos 1/ jπ j=n+1n+n1 n 330

11 P R Parthasarathy et al 479 We give below closed-form expressions for the terms B k sandb k+1 sfork = 0n 1 using the well-known identities of continuants given in the appendix Consider s + B k s = s + λ + µ s + λ + λ s + + µ k s + + µ = µ s + λ + µ s + λ + µ λ s + + µ s + λ + λ s + + µ λ s + + µ k s + λ + k1 331

12 480 Fluid queues UsingIdentitiesA anda4 ω 1 + λ µ λ µ ω 1 B k s = ω 1 λ µ ω 1 k ω 1 + λ µ λ µ ω 1 µ s + + µ ω 1 λ µ ω 1 + λ µ k 33 where ω 1 = s + + µ / s +λ + / / λ µ / Now expanding the first determinantand usingidentity A1 for the second we obtain B k s = ω 1 λ µ ω 1 ω 1 λ µ ω 1 k

13 + λ µ ω 1 λ µ ω 1 µ s + λ + ω 1 λ µ ω 1 λ µ P R Parthasarathy et al 481 ω 1 λ µ ω 1 r k1 λ µ ω 1 ω 1 k1 333 Therefore from Identity A3 the quantity B k s can be expressed in closed form as follows: λ1 λ µ k/ [ x B k s = r1 U k µ s + λ + λ1 λ µ r1 λ / ] x + U k1 k1/ U k1 x 334 where x = r1 s + + µ r s + λ + λ1 λ µ 335 λ1 λ µ and U k x is the Chebyshev polynomial of the second kind Similarly B k+1 s can also be expressed as B k+1 s = s + λ 1 + µ λ1 λ µ k/ x r1 U k µ λ1 λ µ k/ [ x λ1 / ] 1 x r1 U k + U k1 λ µ 336

14 48 Fluid queues Further µ1 µ k if j = k r c 1 j0 = µ1 µ k if j = k Having determined B j s and the roots of R 1 Q T explicitly the constants k l and hence the buffer occupancy distributions are obtained using 310and314 Remark 34 From the above discussion we observe that without loss of generality and can be taken as 1 and +1 respectively This is because the solution remains unaltered when µ andλ are replaced by µ andλ respectively 4 Infinite state space In the previous section we discussed the fluid queue model on a finite state space with alternating positive and negative flow rates Similar analysis on an infinite state space does not lead to explicit expression for the buffer content distribution Hence we analyze the fluid queue driven by an infinite state BDP with a single negative effective flow rate say r 0 < 0 We employ the continued fraction methodology to obtain the stationary distribution of the content of the buffer Let F j tu represent the probability that there are j units in the system and the content of the buffer does not exceed u at time tandlet F j ts denote the corresponding Laplace transform If lim t F j tu = F j u then the governing system of differential-difference equations is r 0 F 0u =λ 0 F 0 u+ F 1 u r j F ju = λ j1 F j1 u λ j + µ j Fj u+µ j+1 F j+1 u j = 1 41 Using the initial condition F 0 0 = athelaplacetransformof41yields F 0 s = a s + λ 0 µ 1 F 1 s r 0 r 0 F 0 s F j s F j1 s = λ j1 r j s + λ j + µ j r j µ j+1 r j j = 13 F j+1 s F j s 4

15 P R Parthasarathy et al 483 This leads to the continued fractions F j s F j1 s = F 0 s = a s + λ 0 r 0 λ j1 r j s + λ j + µ j r j λ 0 r 0 s + + λ j µ j+1 r j r j+1 s + λ j+1 + µ j+1 r j+1 µ s + λ + µ λ j+1 µ j+ r j+1 r j+ s + λ j+ + µ j+ r j+ 43 Define φ j s:= λ j1 r j s + λ j + µ j r j λ j µ j+1 r j r j+1 s + λ j+1 + µ j+1 r j+1 λ j+1 µ j+ r j+1 r j+ s + λ j+ + µ j+ r j+ 44 Then F 0 s = After certain algebra we obtain a s + λ 0 µ 1 φ 1 s r 0 r 0 F j s F j1 s = φ js 45 k µ1 φ1 s k F 0 s = a r k=0 0 s + λ k+1 0 r 0 j F j s = φ k s F 0 s j = 13 k=1 46 On inversion we get k µ1 u F 0 u = a k e λ0/r0u φ k 1 u r k=0 0 k! F j u = φ 1 u φ u φ j u F 0 u j = In the following argument we consider a specific nature of birth and death rates

16 484 Fluid queues λ λ µ µ Figure 41 State-transition diagram of the background Markov process Example 41 Consider a fluid queue driven by an M/M/1 queue with two alternating arrivalratesandtwoalternatingserviceratesasshownin Figure 41 We assume the net input rate of fluid to be r 0 < 0 j = > 0forj = 13 and j+1 = > 0forj = 01 The steady state probabilities are given by 35 with p 0 = µ λ /µ + From 3 the condition r < 0 ensures the stability of the process {XtCt t 0} For the model under consideration f s φ j s = gs if j is odd if j is even 48 where f s = s + λ + λ µ s + + µ s + λ + 49 = s + λ + λ µ s + + µ f s Or r s + λ + f s [ s + + µ r s + λ + λ µ + ] f s+λ1 r1 s + + µ = Using Rouches theorem considering the root that lies inside the unit circle we get f s = 1 [ r1 s + + µ r s + λ + + λ1 λ µ r s + λ + + [ s + + µ r s + λ + + λ µ ] r1 s + + µ r s + λ + 1/ ]

17 P R Parthasarathy et al 485 gs isobtainedfrom f s by interchanging and and λ and and µ The expression for f scan be written as where f s = r [ 1 s + a b ] r1 s + a b 4λ1 s + a c 41 s + a c a = 1 λ1 + µ + λ + b = a λ1 λ + µ + c = 1 λ1 + µ λ On inversion we get f u = λ1 e au { f u 1 n n=0 n k=0 n/n/ 1 n/ k +1 k! } f k 3 u 414 where 4λ1 f 3 u = f1 u f u f 1 u = coshbu + c b sinhbu 415 f u = coshbu c b sinhbu and h k uisthek-fold convolution of hu with itself Also from 45 F k+1 s F k s = f s F k s = gs 416 F k1 s which on solving yield F k+1 s = f s k+1 gs k F 0 s F k s = f s k gs k F 0 s 417

18 486 Fluid queues 10 0 = 0 = 01λ = 003µ = = 0 = 0λ = 003µ = 006 log 10 PC >u = 0 = 03λ = 006µ = 003 = 0 = 04λ = 003µ = 003 λ1 = 0 = 05λ = 003µ = 006 λ1 = 0 = 06λ = 006µ = Buffer content u Figure 51 Buffer content distribution on a finite state space with N = 1 =1 and = 1 On inversion F k+1 u = f k+1 u g k u F 0 u k = 01 F k u = f k u g k u F 0 u k = where F 0 uisobtainedfrom47as 5 Numerical illustrations F 0 u = a k=0 µ1 r 0 k u k e λ0/r0u k! f k u 419 In this section we provide numerical examples for the fluid queue driven by the finite and infinite queuing model discussed in the earlier sections In Figure 51 the buffer overflow probability corresponding to the varying values of the parameters λ andµ is plotted against buffer size for finite state space with N = 1 =1 and = 1 see Remark 34 The overflow probabilities are found to decrease rapidly with increase in The graphs are plotted for two sets of parameters by considering the three different cases wherein the ratio λ /µ is equal to one less than one and greater then one respectively Figure 5 depicts an analogous setting on an infinite state space with N truncated at 5 r 0 =1 and r i = 1fori 1 For constant values of and λ theoverflowprobabilities decrease rapidly with increase in The overflow probability is in the range of to and hence becomes a rare event

19 P R Parthasarathy et al log 10 PC >u = = 48λ = µ = 3 = = 48λ = µ = 6 λ1 = = 484λ = µ = 3 λ1 = = 486λ = µ = Buffer content u Figure 5 Buffer content distribution on an infinite state space with N truncated at 5 r 0 =1 and = 1 = Appendix Some identities of tridiagonal determinants Here we give some identities of tridiagonal determinants Muir [4] which are useful in determining the stationary buffer content distribution of the model under consideration Identity A1 A + a 0 a 0 b 1 A + a 1 + b 1 A + a n1 + b n1 b n a n1 g n a n1 n+1 A + b n g 1 b 1 a 1 g b = A a n g n1 b n1 n A1 where g j = A + a j1 + b j j = 1n

20 488 Fluid queues Identity A θ 1 d 1 1 φ d 1 θ d 3 1 φ d n 1 θ n n1 γ 1 d 1 d γ d 3 = 1 d 4 γ 3 d 5 φ d n4 γ n1 d n3 where γ j = d j + θ j φ + d j1 j = 1nwithd 0 = 0andd n1 = 0 Identity A3 A a b A a b A a b A n n = n j=1 Identity A4 θ 1 d 1 1 φ d 1 θ d 3 1 θ n d n1 1 φ n where γ j = d j + θ j φ + d j1 j = 1nwithd 0 = 0 References d n A jπ nun A abcos = ab ab n +1 d n γ n n γ n n A A3 γ 1 d 1 d γ d 3 d 4 γ 3 d 5 = A4 d n4 γ n1 d n3 [1] G Horton V G Kulkarni D M Nicol and K S Trivedi Fluid stochastic Petri nets: theory applications and solution techniques European J Oper Res no [] R B Lenin and P R Parthasarathy Fluid queues driven by an M/M/1/N queue MathProbl Eng6000 no [3] D Mitra Stochastic theory of a fluid model of producers and consumers coupled by a bufferadv Appl Probab no [4] T Muir A Treatise on the Theory of Determinants Dover Publications New York 1960 revised and enlarged by William H Metzler

21 P R Parthasarathy et al 489 [5] E A van Doorn and W R W Scheinhardt A fluid queue driven by an infinite-state birth-death process Proc 15th International Teletraffic Congress V Ramaswami and P E Wirth eds Elsevier Amsterdam 1997 pp [6] W Whitt Stochastic-Process Limits An Introduction to Stochastic-Process Limits and Their Application to Queues Springer Series in Operations Research Springer-Verlag New York 00 P R Parthasarathy: Department of Mathematics Indian Institute of Technology Madras Chennai India address: prp@iitmacin K V Vijayashree: Department of Mathematics Indian Institute of Technology Madras Chennai India address: ma99p01@violetiitmernetin R B Lenin: Department of Mathematics and Computer Science University of Antwerp Universiteitsplein Antwerpen Belgium address: lenin@uiauaacbe

22 Mathematical Problems in Engineering Special Issue on Modeling Experimental Nonlinear Dynamics and Chaotic Scenarios Call for Papers Thinking about nonlinearity in engineering areas up to the 70s was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system Keying saturation hysteretic phenomena and dead zones were added to existing devices increasing their behavior diversity and precision In this context an intrinsic nonlinearity was treated just as a linear approximation around equilibrium points Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena engineers started using analytical tools from Qualitative Theory of Differential Equations allowing more precise analysis and synthesis in order to produce new vital products and services Bifurcation theory dynamical systems and chaos started to be part of the mandatory set of tools for design engineers This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory relating it with nonlinear and chaotic dynamics for natural and engineered systems Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever sophisticated and at the same time simple and unique analytical environment are the subject of this issue allowing new methods to design high-precision devices and equipment Authors should follow the Mathematical Problems in Engineering manuscript format described at hindawicom/journals/mpe/ Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at mtshindawicom/ according to the following timetable: Guest Editors José Roberto Castilho Piqueira Telecommunication and Control Engineering Department Polytechnic School The University of São Paulo São Paulo Brazil; piqueira@lacuspbr Elbert E Neher Macau Laboratório Associado de Matemática Aplicada e Computação LAC Instituto Nacional de Pesquisas Espaciais INPE São Josè dos Campos São Paulo Brazil ; elbert@lacinpebr Celso Grebogi Center for Applied Dynamics Research King s College University of Aberdeen Aberdeen AB4 3UE UK; grebogi@abdnacuk Manuscript Due Decembe 008 First Round of Reviews March Publication Date June Hindawi Publishing Corporation