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A novel method for tracing coastal water masses using Sr/Ca ratios and salinity in Nanwan Bay, southern Taiwan

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VVV1is formed by the correspondingQ + 1 rows of [ddd0; . . . ; dddQ], and DD

D(!0) is a diagonal matrix with those selected Q + 1 entries from DD

D(!0) on its main diagonal. Since rank(DD(uuu)DD DD(!0)) = Q + 1 and rank(VVV1) = Q + 1, we deduce that rank(888e) = Q + 1, 8 eee 6= 0.

Next, we prove the “only if” part by contradiction. Suppose that for someeee, uuu = 222eee has only Q + 1 < Q + 1 nonzero corresponding entries, that we collect inuuu = [un ; . . . ; un ]T. Then, similarly to the “if” part, we can group the nonzero rows in a matrixDDD(uuu)DDD(!0)VVV1. Now thisVVV1is a(Q + 1) 2 (Q + 1) matrix, while DDD(!0) is a (Q + 1) 2 (Q + 1) matrix. It follows immediately that

rank(VVV1) = Q + 1 < Q + 1

and, hence, rank(888e) < Q + 1, which implies that the maximum diversity cannot be achieved.

REFERENCES

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and reception for communication over multipath fading channels,” IEEE Trans. Commun., vol. 48, pp. 83–94, Jan. 2000.

[3] E. Biglieri, J. Proakis, and S. Shamai (Shitz), “Fading channels: In-formation-theoretic and communications aspects,” IEEE Trans. Inform. Theory, vol. 44, pp. 2619–2692, Oct. 1998.

[4] G. B. Giannakis and C. Tepedelenlio˘glu, “Basis expansion models and diversity techniques for blind identification and equalization of time-varying channels,” Proc. IEEE, vol. 86, pp. 1969–1986, Nov. 1998. [5] W. C. Jakes, Microwave Mobile Communications. New York: Wiley,

1974.

[6] T. Kailath, “Measurements on time-variant communication channels,” IEEE Trans. Inform. Theory, vol. IT-8, pp. S229–S236, Sept. 1962. [7] G. Leus, S. Zhou, and G. B. Giannakis, “Multi-user spreading codes

retaining orthogonality through unknown time- and frequency-selective fading,” in Proc. GLOBECOM Conf., vol. 1, San Antonio, TX, Nov. 25–29, 2001, pp. 259–263.

[8] X. Ma and G. B. Giannakis, “Complex field coded MIMO systems: Per-formance, rate, and tradeoffs,” Wireless Commun. Mobile Comput., pp. 693–717, Nov. 2002.

[9] J. G. Proakis, Digital Communications, 4th ed. New York: McGraw-Hill, 2001.

[10] M. Reinhardt, J. Egle, and J. Lindner, “Transformation methods, coding and equalization for time- and frequency-selective channels,” Europ. Trans. Telecommun., vol. 11, no. 6, pp. 555–565, Nov./Dec. 2000. [11] A. Stamoulis, G. B. Giannakis, and A. Scaglione, “Block FIR

decision-feedback equalizers for filterbank precoded transmissions with blind channel estimation capabilities,” IEEE Trans. Commun., vol. 49, pp. 69–83, Jan. 2001.

[12] A. M. Sayeed and B. Aazhang, “Joint multipath-doppler diversity in mobile wireless communications,” IEEE Trans. Commun., vol. 47, pp. 123–132, Jan. 1999.

[13] M. K. Tsatsanis and G. B. Giannakis, “Equalization of rapidly fading channels: Self-recovering methods,” IEEE Trans. Commun., vol. 44, pp. 619–630, May 1996.

[14] E. Viterbo and J. Boutros, “A universal lattice code decoder for fading channels,” IEEE Trans. Inform. Theory, vol. 45, pp. 1639–1642, July 1999.

[15] Z. Wang and G. B. Giannakis, “Linearly precoded or coded OFDM for wireless channels?,” in Proc. 3rd Workshop Signal Processing Ad-vances in Wireless Communications, Taiwan, ROC, Mar. 20–23, 2001, pp. 267–269.

[16] Y. Xin, Z. Wang, and G. B. Giannakis, “Space-time constellation-ro-tating codes maximizing diversity and coding gains,” IEEE Trans. Wire-less Commun., vol. 2, pp. 294–309, Mar. 25–27, 2003.

On Continuous-Time Optimal Deterministic Traffic Regulation

Chia-Sheng Chang, Member, IEEE, and Kwang-Cheng Chen, Senior Member, IEEE

Abstract—In this correspondence, we study the continuous-time deter-ministic traffic regulation problem. We propose a regulation form shown to be the optimal deterministic traffic regulator in the sense that it outputs the most packets while satisfying the constraint on the output process. We further investigate the subtle relation between continuous-time and discrete-time optimal deterministic regulators, and reduce our general regulation form to the known discrete-time optimal deterministic regulator when restricting arrival (departure) instants to integers and packet size to unity. Therefore, by extending traffic-regulation theory to continuous time, our work provides a fundamental framework for future research regarding quality-of-service (QoS)-guaranteed network design/analysis in continuous-time.

Index Terms—Constrained optimization, network flows, quality-of-ser-vice (QoS), traffic management.

I. INTRODUCTION

Traffic regulation has been widely accepted as an indispensable tech-nique to provide quality-of-service (QoS)-guaranteed multimedia ser-vices in packet-switching communication networks (e.g., TCP/IP net-works [1], asynchronous transfer mode (ATM) [2]). According to [3], a traffic source conforms to a nondecreasing, nonnegative functionf ifR[t 0 ; t)  f() for all ; t  0, where R[t 0 ; t) (bits) de-notes the amount of information bits in packets arriving in time interval [t 0 ; t). A deterministic traffic regulator with constraint function f is a filter shaping an arbitrary traffic input such that the output process conforms tof.

For discrete-time systems, Chang [4] studied discrete-time deter-ministic traffic regulators in great detail and developed a general filtering method for traffic regulation. For continuous-time traffic regulation, a number of results also have been proposed in the liter-ature. A special traffic regulator called leaky bucket(; ) regulator was discussed in [5]–[8]. The (~; ~) regulators were investigated in [3], [9], [10]. However, regulators of the (~; ~) type are lim-ited to those with concave constraint functions. In [11], regulators with nonconcave constraint functions can be realized with a cas-cade of leaky buckets with state-dependent token generation rates, but the detailed implementation was left unspecified. In [12] and [13], Le Boudec successfully applied the continuous-time “network calculus” [14], [4], [15], [16] to traffic shapers, and we can further improve Le Boudec’s results in several aspects. First, the regulators Le Boudec considered are bit-processing devices which assume fluid input streams. In practical packet-switching networks, data arrivals are packets or cells, and thus this assumption is generally not true. Second, the continuity property of constraint functions is very impor-tant to the derivation of the optimal regulation formulas, but this issue

Manuscript received July 15, 2001; revised November 11, 2002. This work was supported by the Ministry of Education, Taiwan, ROC, under Contract 89E-FA06-2-4-7.

C.-S. Chang is with Delta Networks, Inc., Taipei, Taiwan, ROC (e-mail: changchias@ieee.org).

K.-C. Chen is with the Institute of Communications Engineering and De-partment of Electrical Engineering, National Taiwan University, Taipei, Taiwan, ROC (e-mail: chenkc@cc.ee.ntu.edu.tw).

Communicated by V. Anantharam, Associate Editor for Communication Net-works.

Digital Object Identifier 10.1109/TIT.2003.813489 0018-9448/03$17.00 © 2003 IEEE

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was not addressed in [12] and [13]. Third, although the regulators (or shapers) in [12] and [13] are defined in continuous-time setting, Le Boudec considered only systems for which there is a minimum time granularity. This restriction implies these systems are still of discrete-time type. Consequently, in packet-switching networks, how to construct continuous-time optimal deterministic regulators with

general constraint functions is a very interesting problem that still

needs further investigation. In this correspondence, we present con-tinuous-time optimal deterministic traffic regulation formulas which specify the earliest possible departure time of each packet arrival. Furthermore, we also show that discrete-time optimal deterministic regulators discussed in [4] can be regarded as special cases of con-tinuous-time optimal deterministic regulators. This fact demonstrates the subtle relation between continuous-time and discrete-time optimal deterministic regulators.

Throughout this correspondence, we denote the arrival time, the de-parture time, and the length of thenth packet by an,bn, andLn, re-spectively. In addition, we assume 1)Lmax(bits) is the maximal size of the packets in an input source. 2) For anyt  0, there are only a finite number of packet arrivals in[0; t]. Thus, we have limn!1an= 1. 3)f  g means that the inequality holds pointwisely; sup f and inf f are taken in a per-point manner, e.g.,(sup f )(x) = sup f (x). We also denote byT [t 0 ; t) (bits) the amount of information bits in packets departing from a regulator in interval[t0; t). The entire con-tentLnof a packet being released at timebnis assumed to be released at timebn.

The rest of this correspondence is organized as follows. In Sec-tion II, we introduce maximal embedded subadditive funcSec-tions and give some properties regarding these functions. The continuous-time optimal regulation formulas are given in Section III along with the corresponding mathematical proofs. In Section IV, we investigate the subtle relation between continuous-time optimal regulators and the known discrete-time ones. We present a potential realization structure of the proposed optimal regulators in Section V, and, finally, conclude this correspondence in Section VI.

II. MAXIMALEMBEDDEDSUBADDITIVEFUNCTIONS For an arbitrary traffic source active in [0; '], the pointwisely smallest constraint function ~f which this source conforms to can be constructed according to ~f()= supfT [t 0 ; t): t 2 [0; ']g. It is1 not difficult to see that ~f is always subadditive, i.e.,

~

f(x + y)  ~f(x) + ~f(y); forx; y  0:

This observation shows that subadditive functions play an important role in deterministic traffic regulation theory. To begin with, two im-portant definitions are given as follows.

Definition 1: The collection of nonnegative, nondecreasing, and

left-continuous functionsf: +! +withf(0)=0 is denoted by G.

Definition 2: Givenf 2 G, we define f= supfg: g 2 G; g  f; g1

is subadditiveg.

Givenf 2 G, one can easily show by definition that the corre-spondingf is subadditive and f 2 G. Consequently, we call the f corresponding tof 2 G the maximal subadditive function embedded

inf. The next theorem is of great importance for continuous-time

de-terministic regulators, and is actually a continuous-time extension of [17, Lemma 2.1].

Theorem 1: A traffic source conforms tof 2 G if and only if it

conforms tof.

Proof: The necessary condition is clear. We only need to show

the sufficient condition.

Suppose there is a traffic source that does not conform tof. Then, for somet  0 and  > 0, we have T [t 0 ; t) > f(). Hence we have ~f() > f(), where

~

f()= supfT[t 0 ; t): t 2 [0; ']g1

where[0; '] is the time interval over which the source is active. But by its definition, ~f is subadditive and is pointwise upper-bounded by f. This contradicts the definition off as the pointwise largest subadditive function that is pointwise upper-bounded byf.

If we wish a regulator’s output to conform tof 2 G, then f(0+) (where the “+” means the limit from the right) must be larger than or equal toLmax. Otherwise, a packet of lengthLmax would never be allowed to pass the regulator. Therefore,f must satisfy f(0+)  Lmax. The next lemma shows the relation betweenf(0+) and f(0+) for a givenf 2 G.

Lemma 1: Givenf 2 G. Then f(0+) = f(0+).

Proof: Sincef  f, we have f(0+)  f(0+). Conversely,

define^g(x) = 0 for x = 0 and ^g(x) = f(0+) for x > 0. It is not difficult to see that^g  f and ^g is subadditive. Hence, by definition we knowf  ^g and

lim

x!0f(x)  limx!0^g(x) = limx!0f(0+) = f(0+): Thus, we knowf(0+)  f(0+) and the lemma is proved.

Consequently, we know thatf(0+)  Lmax providedf(0+)  Lmax. In summary, Theorem 1 and Lemma 1 tell us that constructing an optimal deterministic regulator with constraint functionf 2 G with f(0+)  Lmaxis equivalent to constructing one with a pointwisely smaller, subadditive constraint functionf 2 G.

In addition, continuous-time inverse functions are also defined as follows.

Definition 3: For a nonnegative, nondecreasing functionf: + ! +, we definef01: + ! +by

f01(x)= inffs  0: x  f(s)g:1

III. OPTIMALDETERMINISTICREGULATIONFORMULAS To produce the output process of a regulator with a subadditive con-straint functionf 2 G with f(0+)  Lmax, we assert that the packet departure times be determined by the following rules:

b1= a1; and bn= maxfan; bn0g; 8 n  2; (1) where b0 n= max bi+ f01 n k=i Lk : 1  i  n 0 1 : (2)

In the following, we show the output process determined by (1) con-forms tof.

Theorem 2: For a given subadditive functionf 2 G with f(0+) 

Lmax, the following three conditions are equivalent: 1) T [t 0 ; t)  f(), for all ; t  0; 2) for a fixed > 0 and 8 n 2

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3) For a fixed > 0 and 8 n 2

Tn[t 0 ; t)  f(); for0  ; t  b n+ 

where Tn[t 0 ; t) is defined using only the first n packet departures.

Proof: SinceT [t0; t) = T [0; t) and Tn[t0; t) = Tn[0; t)

for > t, without loss of generality we may assume   t.

Condition 1) obviously implies condition 2). On the other hand, sup-pose condition 2) is true. Sincelimn!1bn= 1, for an arbitrary t it follows that

T [t 0 ; t)  f(); for all0    t:

To prove that condition 2) and condition 3) are equivalent, note that Tn[t 0 ; t)  T [t 0 ; t); for all0    t

since Tn[t 0 ; t) results from only the first n packet departures. Hence, condition 2) implies condition 3). Conversely, suppose condi-tion 3) is true. Fixingn 2 , we can find m 2 large enough such thatbm bn+ . Then condition 3) implies that

Tm[t 0 ; t)  f(); 8 0    t  b m+ : Sincebk bn+  for k > m, it follows that

T [t 0 ; t) = Tm[t 0 ; t); 8 0    t  b n+  and, consequently, we have

T [t 0 ; t)  f(); 8 0    t  bn+ :

Sincen is arbitrary, it follows condition 3) implies condition 2). Thus, we proved conditions 2) and 3) are equivalent.

Theorem 3: Given a subadditive functionf 2 G with f(0+) 

Lmax. The output process determined by (1) satisfies condition 3) of Theorem 2. In particular, the output process determined by (1) con-forms tof.

Proof: By (1), we knowb1= a1. Clearly, for0  ; t  b1+  we haveT1[t 0 ; t)  L1  f(). Thus, the conclusion holds for n = 1. Suppose

Tn01[t 0 ; t)  f(); for0  ; t  bn01+ : (3) Now we considerTn[t 0 ; t) for 0  ; t  bn+ .

Given departure timesb1; . . . ; bn, consider those 2 [0; bn+ ] such that = bj 0 bifor somei; j 2 f1; . . . ; ng and i < j, it can be seen there are only a finite number of such points. Arrange those  in an increasing order, and denote them by 1; . . . ; l and also let 0 = 0, l+1 = bn+ .

Note that for0  t  bn+ , we can write Tn[t 0 ; t) = n

i=1

Li1 I; i(t)

whereI; i(t) = 1 for t 2 (bi; bi+ ] and I; i(t) = 0 otherwise. With the above decomposition, it can be seen that

hn()= supfT1 n[t 0 ; t): 0  t  bn+ g

is a nondecreasing, left-continuous step function in[0; bn+ ]. That is, [0; bn + ] can be divided into a finite number of subintervals I1; . . . ; Im such thathn() = ci fort interior to Ii. In addition, the only possible discontinuous points of hn() are those k, k = 1; . . . ; l. Consequently, to check whether Tn[t 0 ; t)  f() for 0  ; t  bn+ , we only need to check hn(k+)  f(k+) for k = 1; . . . ; l. By (1), we know bn bi+ f01 n k=i Lk ; 8 i = 1; . . . ; n 0 1

which implies that for alli = 1; . . . ; n 0 1 n k=i Lk f f01 n k=i Lk +  f((bn0 bi)+): (4) Now, with (3) and (4), one can use mathematical induction to show that

hn(k+)  f(k+); fork = 1; . . . ; l and, consequently,Tn[t 0 ; t)  f(), for 0  ; t  bn+ .

By mathematical induction, we conclude that the output process de-termined by (1) satisfies the third condition of Theorem 2. In particular, by Theorem 2, the output process determined by (1) conforms tof.

Having developed continuous-time deterministic regulators with

subadditive constraint functions, now we are ready to define

contin-uous-time deterministic regulators with general constraint functions f 2 G with f(0+)  Lmax.

Definition 4: Suppose the departure time of thenth packet from a

continuous-time deterministic regulatorf 2 G with f(0+)  Lmaxis denoted bybn. Then we setb1= a1andbn= maxfan; bn0g 8 n  2, where b0 n= max bi+ f01 n k=i Lk : 1  i  n 0 1 : (5)

The next theorem shows the optimality of the proposed contin-uous-time deterministic regulators.

Theorem 4: Given an input processf(an; Ln)g. Then for any reg-ulator output processf(cn; Ln)g conforming to f 2 G with f(0+)  Lmaxandcn anfor alln 2 , we must have cn bn, wherebnis the departure time calculated from Definition 4.

Proof: From Definition 4 we knowc1  a1 = b1. Suppose cj  bjholds for all1  j  n 0 1. If

cn< ci+ f01 n k=i

Lk

for somei 2 f1; . . . ; n 0 1g, for 0 <  < ci+ f01 n k=i Lk 0 cn we would have f(cn0 ci+ ) < f f01 n k=i Lk by definition off01  n k=i Lk by the left-continuity off  T [ci; cn+ )

which impliesf(cn; Ln)g did not conform to f (by Theorem 1). How-ever, sincef(cn; Ln)g conforms to f, we must have

cn max ci+ f01 n j=i

Lj : 1  i  n 0 1 :

Hence, the induction hypothesis implies cn max ci+ f01 n j=i Lj : 1  i  n 0 1  max bi+ f01 n j=i Lj : 1  i  n 0 1 1 = bn:

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Note that the conditioncn anis referred to as the causal condition in [4] since the departure time cannot be less than the arrival time.

IV. SPECIALFORM IN THEDISCRETE-TIMECASE

In this section, we consider a special case where packet arrivals and departures occur only att 2 +, and all packets are with unit length. Note that the constraint sequencef must satisfy f(1)  1.

Similar to the continuous-time case, we denote byF the collection of nonnegative, nondecreasing sequencesf: + ! +withf(0) = 0. The maximal subadditive sequence f embedded in f 2 F can be defined similarly to Definition 2.

Definition 5: Givenf 2 F, we define f= supfg: g 2 F; g  f; g1

is sub-additiveg.

According to [4, Lemma 2.2],f (the maximal subadditive sequence embedded inf 2 F) is pointwisely identical to f3(the subadditive closure off). Consequently, given f 2 F, f has the following prop-erties: 1)f is subadditive, 2) f 2 F, 3) f is subadditive if and only if f = f, and 4) a discrete-time traffic source conforms to f if and only if it conforms tof.

In addition, discrete-time inverse sequences are also defined as follows.

Definition 6: For a nonnegative, nondecreasing functionf: +! +. We definef01: +! by f01(x)1= minfs0: xf(s)g01.

The constant “01” in the definition of f01(1) is used to make dis-crete-time regulation formulas identical to their continuous-time coun-terparts. Otherwise, the right-hand side of (6) in Definition 7 would need to append a constant “01.” Also, by definition, one can show the following inequalities

f(f01(x)) < x  f(f01(x) + 1) and

1 + f01(f(x))  x  f01(f(x) + 1):

Analogous to the continuous-time case, we define discrete-time op-timal traffic regulators in a parallel form as follows.

Definition 7: Suppose the departure time of thenth packet from a

discrete-time deterministic regulatorf 2 F with f(1)  1 is denoted bybn. Then, we setb1= a1andbn= maxfan; bn0g 8 n  2, where

b0

n= max bi+ f01(n 0 i + 1): 1  i  n 0 1 : (6)

The following theorems justify Definition 7, and their proofs can be obtained by slightly modifying those of Theorems 2, 3, and 4, respec-tively.

Theorem 5: For a given subadditive sequencef 2F with f(1)1,

the following three conditions are equivalent: 1) T [t 0 ; t)  f(), for all ; t 2 +;

2) for a fixed 2 and 8 n 2 , T [t 0 ; t)  f(), for ; t 2 [0; . . . ; bn+ ];

3) for a fixed 2 and8 n 2 ,Tn[t 0 ; t)  f(), for ; t 2 [0; . . . ; bn+ ], where Tn[t 0 ; t) results from only the firstn packet departures.

Theorem 6: Given a subadditive sequencef 2 F with f(1)  1,

the output process determined by Definition 7 satisfies condition 3) of Theorem 5. In particular, the output process determined by Definition 7 conforms tof.

Theorem 7: Given an input processf(an; Ln)g, then for any regu-lator output processf(cn; Ln)g conforming to f 2 F with f(1)  1 andcn an, for alln 2 , we must have cn  bn, wherebnis the departure time calculated from Definition 7.

We will show later that the discrete-time optimal deterministic reg-ulators in Definition 7 are identical to those discussed in [4].

Lemma 2: Given subadditivef 2 F with f(1)  1. For x, y  1

we havef01(x) + f01(y)  f01(x + y 0 1).

Proof: Letq, r be f01(x), f01(y), respectively (q, r  0 since

x, y  1). By definition, it follows that

f(q + r)  f(q) + f(r)  x + y 0 2 and

f(f01(x + y 0 1) + 1)  x + y 0 1:

Hence, we knowf(q + r) < f(f01(x + y 0 1) + 1), which implies q + r < f01(x + y 0 1) + 1

and

q + r  f01(x + y 0 1): Hence, the result is proved.

With Lemma 2, we can show the following theorem.

Theorem 8: Forn  2, (6) is identical to

b00

n= max ai+ f01(n 0 i + 1): 1  i  n 0 1 : Proof: We prove this by induction. Ifm = 2, it follows that

b20 = a1+ f01(2). By hypothesis, we know that b1 = a1. Hence, the conclusion holds form = 2. Suppose the conclusion also hold for m 2 f2; . . . ; n 0 1g. Now let m = n. It is easy to see that bn0 bn00 sinceai  bi. For thosei 2 f1; . . . ; n 0 1g such that ai = bi, we have

bi+ f01(n 0 i + 1)  ai+ f01(n 0 i + 1):

For thosei 2 f1; . . . ; n 0 1g such that ai < bi, there must exist a j 2 f1; . . . ; i 0 1g such that bi= aj+ f01(i 0 j + 1). However, by Lemma 2, it follows that

bi+ f01(n 0 i + 1) = aj+ f01(i 0 j + 1) + f01(n 0 i + 1)  aj+ f01(n 0 j + 1):

Therefore, we must havebn0  bn00, and have proved the case for m = n.

According to Definition 7,bn= maxfan; bn0g, and note that an= an+ f01(n 0 n + 1). By Theorem 8, we then have

bn= max ai+ f01(n 0 i + 1): 1  i  n : (7) After determiningbn, the departure time of thenth packet, we know B[0; k), the amount of departure in [0; k), can be expressed as

B[0; k) = minfn 0 1: bn k; n 2 g: (8) Discrete-time optimal deterministic regulators have been discussed in [4], and we quote the definition here.

Definition 8: Suppose that each packet has unit length. LetB(i),

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deter-ministic regulator in[0; i). If f 2 F with f(1)  1, then B(i) is constructed by

B(i) = min

0jifR[0; j) + f

3(i 0 j)g (9)

wheref3is the subadditive closure off.

Our primary goal here is to show that the discrete-time regulators defined by (7) are precisely the discrete-time optimal deterministic regulators in Definition 8.

Theorem 9: For eachk 2 +, we have

B[0; k) = minfR[0; s) + f(k 0 s) : 0  s  kg:

Proof: By definition, we knowB[0; k)  R[0; k). Also, for

s 2 f0; . . . ; k 0 1g, it can be seen that

B[0; k) = B[0; s) + B[s; k)  R[0; s) + f(k 0 s): Consequently, we have B[0; k)  minfR[0; s) + f(k 0 s): 0  s  kg: Conversely, ifR[0; k) = 0, then minfR[0; s) + f(k 0 s): 0  s  kg = R[0; k) + f(0) = 0  B[0; k): So we have proved B[0; k) = minfR[0; s) + f(k 0 s): 0  s  kg:

Therefore, we may assumeR[0; k) > 0 and thus a1 < k. Suppose ^s is the argument achieving the minimum of

minfR[0; s) + f(k 0 s): 0  s  kg: By definition, we know that

bR[0; ^s)+f(k0^s)= maxfa1 i+ f01(R[0; ^s) + f(k 0 ^s) 0 i + 1): 1  i  R[0; ^s) + f(k 0 ^s)g: Consideringa1= l, 0  l < k, we partition [1; R[0; ^s)+f(k0^s)] into subintervals [1; R[0; l + 1)]; [R[0; l + 1) + 1; R[0; l + 2)]; . . . ; [R[0; k 0 1) + 1; R[0; ^s) + f(k 0 ^s)]: Then bR[0; ^s)+f(k0^s) = max ai+ f01 R[0; ^s) + f(k 0 ^s) 0 i + 1 :

i is the left boundary point of a nonempty sub-interval :

Fori = 1 = R[0; l)+1, since R[0; ^s)+f(k0^s)  R[0; l)+f(k0l), it can be seen that

a1+ f01(R[0; ^s) + f(k 0 ^s) 0 R[0; l))  l + f01(f(k 0 l))  l + k 0 l 0 1 = k 0 1:

Fori = R[0; j) + 1, it follows that

aR[0; j)+1+ f01(R[0; ^s) + f(k 0 ^s) 0 R[0; j))  j + f01(f(k 0 j))

 j + k 0 j 0 1 = k 0 1 which impliesbR[0; ^s)+f(k0^s) k 0 1.

However, according to (8), we know that

R[0; ^s) + f(k 0 ^s) < B[0; k) + 1 which implies

R[0; ^s) + f(k 0 ^s)  B[0; k) and

B[0; k)  minfR[0; s) + f(k 0 s): 0  s  kg:

From the inequality proved in the first paragraph and the fact thatk 2 +is arbitrary, the theorem is proved.

V. A REALIZATION OF THEOPTIMALREGULATIONFORMULAS The optimal deterministic regulator defined in Definition 4 is not directly realizable since determiningfbng1n=1needs infinite compu-tation steps. In this section, we present a potential realization of the proposed optimal regulators. According to Definition 4, forn  2, we can rewrite (5) as

bn0= max inf bi+ t: t  0; f(t)  n j=i

Lj :1  i  n 0 1 :

Since the setfbi+t: f(t)  nj=iLjg must be of the form of interval [a; 1) or (a; 1), it can be seen that for n  2 (this can be shown by mathematical induction) bn0= inf n01 i=1 bi+ t: t  0; f(t)  n j=1 Lj0 i01 j=1 Lj = inf t  0: min i01 j=1 Lj+ f(t 0 bi): 1  i  n 0 1  n j=1 Lj which implies bn0= Bn01 n j=1 Lj where Bn(t)= min1 i01 j=1 Lj+ f(t 0 bi): 1  i  n 0 1 : (10)

With a subroutineminf1; 1g to find the minimum of two functions, we can setB2(t) = f(t 0 b1) = f(t 0 a1) and recursively update Bn(t) by

Bn+1(t) = min Bn(t); n01 j=1

Lj+ f(t 0 bn) : (11)

In summary, we sketch the complete realization structure in Fig. 1. Note the dashed line implies the update ofBn(t) occurs only after bn

(6)

Fig. 1. The complete block diagram of the proposed realization.

has been determined. When thenth packet with length Lnarrives at t = an,bnis determined andBn(t) is updated according to (11).

VI. CONCLUSION

Discrete-time traffic regulation problem has been systematically solved in [4]. However, to the best of the authors’ knowledge, how to optimally regulate a traffic source in continuous-time setting has remained open till now. In this correspondence, we successfully determined the regulation formulas of continuous-time optimal deterministic regulators. Theorem 4 shows that for all causal output processes conforming to a given constraint function f, the nth departure time of the continuous-time optimal deterministic regulator is earliest for alln 2 .

When comparing the continuous-time regulation formula (5) and its discrete-time counterpart (6), one may find they are actually identical (packet sizes are all unity in the discrete-time case). However, the con-tinuous-time output accumulation functionT [0; t) cannot be written in a form similar to (9). The critical point for this subtle distinction is Lemma 2, whose continuous-time counterpart is not true. Conse-quently, discrete-time optimal deterministic regulators can be regarded as special cases of general continuous-time optimal deterministic reg-ulators.

One important issue we did not discuss in this correspondence is the implementation complexity of the realization structure mentioned in Section V. Without some more carefully designed algorithms, the current structure may be too complex to be realized. For example, how to efficiently represent and recursively updateBn(t) in Fig. 1 is very critical to the feasibility of these optimal regulators. In addition, a fast inverse function computation (B01

n (1)) is also an important component. These implementation issues will be studied in our future work.

Traffic regulation has been widely accepted as an indispensable part of QoS-guaranteed multimedia networks. Therefore, by extending traffic-regulation theory to continuous time, our work provides a fundamental framework for future research regarding QoS-guaranteed network design/analysis in continuous time.

ACKNOWLEDGMENT

The authors wish to thank Prof. Cheng-Shang Chang, who is with the National Tsing Hua University, Hsinchu, Taiwan, ROC, and the Associate Editor, Prof. V. Anantharam, for their helpful comments and suggestions that have improved this correspondence.

REFERENCES

[1] S. Shenker, C. Partridge, and R. Guérin, “Specification of guaranteed quality of service. Request for Comments 2212,” Network Working Group, IETF, 1997.

[2] ATM Forum, Traffic Management Specification, Version 4.1, 1999. [3] R. L. Cruz, “A calculus for network delay—Part I: Network elements in

isolation,” IEEE Trans. Inform. Theory, vol. 37, pp. 114–131, 1991. [4] C.-S. Chang, “On deterministic traffic regulation and service guarantees:

A systematic approach by filtering,” IEEE Trans. Inform. Theory, vol. 44, pp. 1097–1110, May 1998.

[5] A. K. Parekh and R. G. Gallager, “A generalized processor sharing ap-proach to flow control in integrated services networks: The single-node case,” IEEE/ACM Trans. Networking, vol. 1, pp. 344–357, June 1993. [6] V. Anantharam and T. Konstantopoulos, “Optimality and

interchange-ability of leaky buckets in tandem,” in Proc. 32nd Allerton Conf., Mon-ticello, IL, Oct. 1994, pp. 235–244.

[7] T. Konstantopoulos and V. Anantharam, “Optimal flow control schemes that regulate the burstiness of traffic,” IEEE/ACM Trans. Networking, vol. 3, pp. 423–432, Aug. 1995.

[8] F. Lo Presti, Z. L. Zhang, J. Kurose, and D. Towsley, “Source time scale and optimal buffer/bandwidth tradeoff for heterogeneous regulated traffic in a network node,” IEEE/ACM Trans. Networking, vol. 7, pp. 490–501, Aug. 1999.

[9] L. Georgiadis, R. Guérin, V. Peris, and K. Sivarajan, “Efficient network QoS provisioning based on per node traffic shaping,” IEEE/ACM Trans. Networking, vol. 4, pp. 482–501, Aug. 1996.

[10] D. E. Wrege, E. W. Knightly, H. Zhang, and J. Liebeherr, “Determin-istic delay bounds for VBR video in packet-switching networks: Funda-mental limits and practical trade-offs,” IEEE/ACM Trans. Networking, vol. 4, pp. 352–362, June 1996.

[11] E. W. Knightly and H. Zhang, “D-BIND: An accurate traffic model for providing QoS guarantees to VBR traffic,” IEEE/ACM Trans. Net-working, vol. 5, pp. 219–231, Apr. 1997.

[12] J.-Y. Le Boudec, “Network calculus made easy,” Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland, Tech. Rep. EPFL-DI 96/21, 1996.

[13] J.-Y. Le Boudec, “Application of network calculus to guaranteed service networks,” IEEE Trans. Inform. Theory, vol. 44, pp. 1087–1096, May 1998.

[14] H. Sariowan, “A service curve approach to performance guarantees in integrated-service networks,” Ph.D. dissertation, Univ. California, San Diego, La Jolla, 1996.

[15] R. Agrawal and R. Rajan, “Performance bounds for guaranteed and adaptive services,” IBM, Tech. Rep. RC 20649, Dec. 1996.

[16] R. L. Cruz and C. M. Okino, “Service guarantees for window flow con-trol,” in Proc. 34th Allerton Conf. Communication, Control, and Com-puting, Monticello, IL, Oct. 1996.

[17] C.-S. Chang, “Stability, queue length, and delay of deterministic and stochastic queueing networks,” IEEE Trans. Automat. Contr., vol. 39, pp. 913–931, May 1994.

數據

Fig. 1. The complete block diagram of the proposed realization.

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